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diff --git a/dev/assets/examples_parametric_via_three_wave_mixing.md.BYspXClR.js b/dev/assets/examples_parametric_via_three_wave_mixing.md.DmpGbSE8.js
similarity index 99%
rename from dev/assets/examples_parametric_via_three_wave_mixing.md.BYspXClR.js
rename to dev/assets/examples_parametric_via_three_wave_mixing.md.DmpGbSE8.js
index 12e52518a..71a70da7d 100644
--- a/dev/assets/examples_parametric_via_three_wave_mixing.md.BYspXClR.js
+++ b/dev/assets/examples_parametric_via_three_wave_mixing.md.DmpGbSE8.js
@@ -1,4 +1,4 @@
-import{_ as h,c as a,ai as l,j as s,a as t,o as n}from"./chunks/framework.Bnn8558y.js";const e="/HarmonicBalance.jl/dev/assets/cmjbvef.DVQRnJSE.png",p="/HarmonicBalance.jl/dev/assets/doffefx.2MzQm7AU.png",k="/HarmonicBalance.jl/dev/assets/vcubulm.qoR36MJx.png",r="/HarmonicBalance.jl/dev/assets/aqesnsv.B-Cc1T24.png",b=JSON.parse('{"title":"Parametric Pumping via Three-Wave Mixing","description":"","frontmatter":{},"headers":[],"relativePath":"examples/parametric_via_three_wave_mixing.md","filePath":"examples/parametric_via_three_wave_mixing.md"}'),d={name:"examples/parametric_via_three_wave_mixing.md"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},E={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.281ex",height:"2.034ex",role:"img",focusable:"false",viewBox:"0 -705 566 899","aria-hidden":"true"},o={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},T={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.889ex"},xmlns:"http://www.w3.org/2000/svg",width:"11.212ex",height:"3.096ex",role:"img",focusable:"false",viewBox:"0 -975.7 4955.8 1368.6","aria-hidden":"true"},y={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},Q={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.448ex",height:"1.025ex",role:"img",focusable:"false",viewBox:"0 -442 640 453","aria-hidden":"true"},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.357ex"},xmlns:"http://www.w3.org/2000/svg",width:"3.302ex",height:"1.927ex",role:"img",focusable:"false",viewBox:"0 -694 1459.4 851.8","aria-hidden":"true"};function F(u,i,C,v,x,f){return n(),a("div",null,[i[13]||(i[13]=l(`<h1 id="Parametric-Pumping-via-Three-Wave-Mixing" tabindex="-1">Parametric Pumping via Three-Wave Mixing <a class="header-anchor" href="#Parametric-Pumping-via-Three-Wave-Mixing" aria-label="Permalink to "Parametric Pumping via Three-Wave Mixing {#Parametric-Pumping-via-Three-Wave-Mixing}""></a></h1><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">using</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicBalance, Plots</span></span>
+import{_ as h,c as a,ai as l,j as s,a as t,o as n}from"./chunks/framework.Bnn8558y.js";const e="/HarmonicBalance.jl/dev/assets/cmjbvef.DVQRnJSE.png",p="/HarmonicBalance.jl/dev/assets/doffefx.2MzQm7AU.png",k="/HarmonicBalance.jl/dev/assets/vcubulm.CoWwsMSb.png",r="/HarmonicBalance.jl/dev/assets/aqesnsv.B-Cc1T24.png",b=JSON.parse('{"title":"Parametric Pumping via Three-Wave Mixing","description":"","frontmatter":{},"headers":[],"relativePath":"examples/parametric_via_three_wave_mixing.md","filePath":"examples/parametric_via_three_wave_mixing.md"}'),d={name:"examples/parametric_via_three_wave_mixing.md"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},E={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.281ex",height:"2.034ex",role:"img",focusable:"false",viewBox:"0 -705 566 899","aria-hidden":"true"},o={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},T={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.889ex"},xmlns:"http://www.w3.org/2000/svg",width:"11.212ex",height:"3.096ex",role:"img",focusable:"false",viewBox:"0 -975.7 4955.8 1368.6","aria-hidden":"true"},y={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},Q={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.448ex",height:"1.025ex",role:"img",focusable:"false",viewBox:"0 -442 640 453","aria-hidden":"true"},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.357ex"},xmlns:"http://www.w3.org/2000/svg",width:"3.302ex",height:"1.927ex",role:"img",focusable:"false",viewBox:"0 -694 1459.4 851.8","aria-hidden":"true"};function F(u,i,C,v,x,f){return n(),a("div",null,[i[13]||(i[13]=l(`<h1 id="Parametric-Pumping-via-Three-Wave-Mixing" tabindex="-1">Parametric Pumping via Three-Wave Mixing <a class="header-anchor" href="#Parametric-Pumping-via-Three-Wave-Mixing" aria-label="Permalink to "Parametric Pumping via Three-Wave Mixing {#Parametric-Pumping-via-Three-Wave-Mixing}""></a></h1><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">using</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicBalance, Plots</span></span>
<span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">using</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Measures</span></span>
<span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">using</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Random</span></span></code></pre></div><h2 id="system" tabindex="-1">System <a class="header-anchor" href="#system" aria-label="Permalink to "System""></a></h2><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">@variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> β α ω ω0 F γ t </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># declare constant variables and a function x(t)</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">diff_eq </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
@@ -39,4 +39,4 @@ import{_ as h,c as a,ai as l,j as s,a as t,o as n}from"./chunks/framework.Bnn855
<span class="line"></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">method </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> TotalDegree</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">()</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">result </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(harmonic_eq2, method, varied, fixed)</span></span>
-<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_phase_diagram</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result; class</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"stable"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><img src="`+r+'" alt=""></p><hr><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl" target="_blank" rel="noreferrer">Literate.jl</a>.</em></p>',8))])}const A=h(d,[["render",F]]);export{b as __pageData,A as default};
+<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_phase_diagram</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result; class</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"stable"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><img src="`+r+'" alt=""></p><hr><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl" target="_blank" rel="noreferrer">Literate.jl</a>.</em></p>',8))])}const w=h(d,[["render",F]]);export{b as __pageData,w as default};
diff --git a/dev/assets/examples_parametric_via_three_wave_mixing.md.BYspXClR.lean.js b/dev/assets/examples_parametric_via_three_wave_mixing.md.DmpGbSE8.lean.js
similarity index 98%
rename from dev/assets/examples_parametric_via_three_wave_mixing.md.BYspXClR.lean.js
rename to dev/assets/examples_parametric_via_three_wave_mixing.md.DmpGbSE8.lean.js
index 3b63a11ad..3e37922f9 100644
--- a/dev/assets/examples_parametric_via_three_wave_mixing.md.BYspXClR.lean.js
+++ b/dev/assets/examples_parametric_via_three_wave_mixing.md.DmpGbSE8.lean.js
@@ -1 +1 @@
-import{_ as h,c as a,ai as l,j as s,a as t,o as n}from"./chunks/framework.Bnn8558y.js";const e="/HarmonicBalance.jl/dev/assets/cmjbvef.DVQRnJSE.png",p="/HarmonicBalance.jl/dev/assets/doffefx.2MzQm7AU.png",k="/HarmonicBalance.jl/dev/assets/vcubulm.qoR36MJx.png",r="/HarmonicBalance.jl/dev/assets/aqesnsv.B-Cc1T24.png",b=JSON.parse('{"title":"Parametric Pumping via Three-Wave Mixing","description":"","frontmatter":{},"headers":[],"relativePath":"examples/parametric_via_three_wave_mixing.md","filePath":"examples/parametric_via_three_wave_mixing.md"}'),d={name:"examples/parametric_via_three_wave_mixing.md"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},E={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.281ex",height:"2.034ex",role:"img",focusable:"false",viewBox:"0 -705 566 899","aria-hidden":"true"},o={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},T={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.889ex"},xmlns:"http://www.w3.org/2000/svg",width:"11.212ex",height:"3.096ex",role:"img",focusable:"false",viewBox:"0 -975.7 4955.8 1368.6","aria-hidden":"true"},y={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},Q={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.448ex",height:"1.025ex",role:"img",focusable:"false",viewBox:"0 -442 640 453","aria-hidden":"true"},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.357ex"},xmlns:"http://www.w3.org/2000/svg",width:"3.302ex",height:"1.927ex",role:"img",focusable:"false",viewBox:"0 -694 1459.4 851.8","aria-hidden":"true"};function F(u,i,C,v,x,f){return n(),a("div",null,[i[13]||(i[13]=l("",13)),s("p",null,[i[8]||(i[8]=t("The quadratic nonlinearity ")),s("mjx-container",g,[(n(),a("svg",E,i[0]||(i[0]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D6FD",d:"M29 -194Q23 -188 23 -186Q23 -183 102 134T186 465Q208 533 243 584T309 658Q365 705 429 705H431Q493 705 533 667T573 570Q573 465 469 396L482 383Q533 332 533 252Q533 139 448 65T257 -10Q227 -10 203 -2T165 17T143 40T131 59T126 65L62 -188Q60 -194 42 -194H29ZM353 431Q392 431 427 419L432 422Q436 426 439 429T449 439T461 453T472 471T484 495T493 524T501 560Q503 569 503 593Q503 611 502 616Q487 667 426 667Q384 667 347 643T286 582T247 514T224 455Q219 439 186 308T152 168Q151 163 151 147Q151 99 173 68Q204 26 260 26Q302 26 349 51T425 137Q441 171 449 214T457 279Q457 337 422 372Q380 358 347 358H337Q258 358 258 389Q258 396 261 403Q275 431 353 431Z",style:{"stroke-width":"3"}})])])],-1)]))),i[1]||(i[1]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"β")])],-1))]),i[9]||(i[9]=t(" together with the drive at 2ω gives the effective parametric drive ")),s("mjx-container",o,[(n(),a("svg",T,i[2]||(i[2]=[l("",1)]))),i[3]||(i[3]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"λ"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{"data-mjx-auto-op":"false"},"eff")])]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mi",null,"F"),s("mn",null,"1")]),s("mi",null,"β")]),s("mrow",null,[s("mn",null,"3"),s("mi",null,"m"),s("msup",null,[s("mi",null,"ω"),s("mn",null,"2")])])])])],-1))]),i[10]||(i[10]=t(". But the cubic nonlinearity ")),s("mjx-container",y,[(n(),a("svg",Q,i[4]||(i[4]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D6FC",d:"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z",style:{"stroke-width":"3"}})])])],-1)]))),i[5]||(i[5]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"α")])],-1))]),i[11]||(i[11]=t(" is still needed to get the period doubling bifurcation through ")),s("mjx-container",m,[(n(),a("svg",c,i[6]||(i[6]=[l("",1)]))),i[7]||(i[7]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"λ"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{"data-mjx-auto-op":"false"},"eff")])])])],-1))]),i[12]||(i[12]=t("."))]),i[14]||(i[14]=l("",8))])}const A=h(d,[["render",F]]);export{b as __pageData,A as default};
+import{_ as h,c as a,ai as l,j as s,a as t,o as n}from"./chunks/framework.Bnn8558y.js";const e="/HarmonicBalance.jl/dev/assets/cmjbvef.DVQRnJSE.png",p="/HarmonicBalance.jl/dev/assets/doffefx.2MzQm7AU.png",k="/HarmonicBalance.jl/dev/assets/vcubulm.CoWwsMSb.png",r="/HarmonicBalance.jl/dev/assets/aqesnsv.B-Cc1T24.png",b=JSON.parse('{"title":"Parametric Pumping via Three-Wave Mixing","description":"","frontmatter":{},"headers":[],"relativePath":"examples/parametric_via_three_wave_mixing.md","filePath":"examples/parametric_via_three_wave_mixing.md"}'),d={name:"examples/parametric_via_three_wave_mixing.md"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},E={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.281ex",height:"2.034ex",role:"img",focusable:"false",viewBox:"0 -705 566 899","aria-hidden":"true"},o={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},T={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.889ex"},xmlns:"http://www.w3.org/2000/svg",width:"11.212ex",height:"3.096ex",role:"img",focusable:"false",viewBox:"0 -975.7 4955.8 1368.6","aria-hidden":"true"},y={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},Q={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.448ex",height:"1.025ex",role:"img",focusable:"false",viewBox:"0 -442 640 453","aria-hidden":"true"},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.357ex"},xmlns:"http://www.w3.org/2000/svg",width:"3.302ex",height:"1.927ex",role:"img",focusable:"false",viewBox:"0 -694 1459.4 851.8","aria-hidden":"true"};function F(u,i,C,v,x,f){return n(),a("div",null,[i[13]||(i[13]=l("",13)),s("p",null,[i[8]||(i[8]=t("The quadratic nonlinearity ")),s("mjx-container",g,[(n(),a("svg",E,i[0]||(i[0]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D6FD",d:"M29 -194Q23 -188 23 -186Q23 -183 102 134T186 465Q208 533 243 584T309 658Q365 705 429 705H431Q493 705 533 667T573 570Q573 465 469 396L482 383Q533 332 533 252Q533 139 448 65T257 -10Q227 -10 203 -2T165 17T143 40T131 59T126 65L62 -188Q60 -194 42 -194H29ZM353 431Q392 431 427 419L432 422Q436 426 439 429T449 439T461 453T472 471T484 495T493 524T501 560Q503 569 503 593Q503 611 502 616Q487 667 426 667Q384 667 347 643T286 582T247 514T224 455Q219 439 186 308T152 168Q151 163 151 147Q151 99 173 68Q204 26 260 26Q302 26 349 51T425 137Q441 171 449 214T457 279Q457 337 422 372Q380 358 347 358H337Q258 358 258 389Q258 396 261 403Q275 431 353 431Z",style:{"stroke-width":"3"}})])])],-1)]))),i[1]||(i[1]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"β")])],-1))]),i[9]||(i[9]=t(" together with the drive at 2ω gives the effective parametric drive ")),s("mjx-container",o,[(n(),a("svg",T,i[2]||(i[2]=[l("",1)]))),i[3]||(i[3]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"λ"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{"data-mjx-auto-op":"false"},"eff")])]),s("mo",null,"="),s("mfrac",null,[s("mrow",null,[s("mn",null,"2"),s("msub",null,[s("mi",null,"F"),s("mn",null,"1")]),s("mi",null,"β")]),s("mrow",null,[s("mn",null,"3"),s("mi",null,"m"),s("msup",null,[s("mi",null,"ω"),s("mn",null,"2")])])])])],-1))]),i[10]||(i[10]=t(". But the cubic nonlinearity ")),s("mjx-container",y,[(n(),a("svg",Q,i[4]||(i[4]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D6FC",d:"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z",style:{"stroke-width":"3"}})])])],-1)]))),i[5]||(i[5]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"α")])],-1))]),i[11]||(i[11]=t(" is still needed to get the period doubling bifurcation through ")),s("mjx-container",m,[(n(),a("svg",c,i[6]||(i[6]=[l("",1)]))),i[7]||(i[7]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("msub",null,[s("mi",null,"λ"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{"data-mjx-auto-op":"false"},"eff")])])])],-1))]),i[12]||(i[12]=t("."))]),i[14]||(i[14]=l("",8))])}const w=h(d,[["render",F]]);export{b as __pageData,w as default};
diff --git a/dev/assets/examples_parametron.md.B1i_hIwH.js b/dev/assets/examples_parametron.md.BWW8XtXU.js
similarity index 99%
rename from dev/assets/examples_parametron.md.B1i_hIwH.js
rename to dev/assets/examples_parametron.md.BWW8XtXU.js
index dbe083f26..65eab582c 100644
--- a/dev/assets/examples_parametron.md.B1i_hIwH.js
+++ b/dev/assets/examples_parametron.md.BWW8XtXU.js
@@ -37,8 +37,8 @@ import{_ as l,c as e,j as t,a as s,ai as Q,o as i}from"./chunks/framework.Bnn855
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">varied </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> range</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.8</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">50</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">), λ </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> range</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.001</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.6</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">50</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">result_2D </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(harmonic_eq, varied, fixed);</span></span></code></pre></div><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span></span></span>
<span class="line"><span>Solving for 2500 parameters... 80%|███████████████▉ | ETA: 0:00:00\x1B[K</span></span>
-<span class="line"><span> # parameters solved: 1988\x1B[K</span></span>
-<span class="line"><span> # paths tracked: 9940\x1B[K</span></span>
+<span class="line"><span> # parameters solved: 1989\x1B[K</span></span>
+<span class="line"><span> # paths tracked: 9945\x1B[K</span></span>
<span class="line"><span>\x1B[A</span></span>
<span class="line"><span>\x1B[A</span></span>
<span class="line"><span></span></span>
diff --git a/dev/assets/examples_parametron.md.B1i_hIwH.lean.js b/dev/assets/examples_parametron.md.BWW8XtXU.lean.js
similarity index 100%
rename from dev/assets/examples_parametron.md.B1i_hIwH.lean.js
rename to dev/assets/examples_parametron.md.BWW8XtXU.lean.js
diff --git a/dev/assets/examples_wave_mixing.md.C_1sCxK3.js b/dev/assets/examples_wave_mixing.md.CR10gGqe.js
similarity index 99%
rename from dev/assets/examples_wave_mixing.md.C_1sCxK3.js
rename to dev/assets/examples_wave_mixing.md.CR10gGqe.js
index fcd3c229a..234c34512 100644
--- a/dev/assets/examples_wave_mixing.md.C_1sCxK3.js
+++ b/dev/assets/examples_wave_mixing.md.CR10gGqe.js
@@ -22,7 +22,7 @@ import{_ as l,c as t,ai as n,j as s,a,o as e}from"./chunks/framework.Bnn8558y.js
<span class="line"><span></span></span>
<span class="line"><span>(2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ + u1(T)*u2(T)*β - u1(T)*(ω^2) + u1(T)*(ω0^2) + v3(T)*v2(T)*β + v2(T)*v1(T)*β + v1(T)*γ*ω + u3(T)*u2(T)*β + (3//4)*(u1(T)^3)*α + (3//4)*(u1(T)^2)*u3(T)*α + (3//2)*u1(T)*(v3(T)^2)*α + (3//2)*u1(T)*v3(T)*v1(T)*α + (3//2)*u1(T)*(v2(T)^2)*α + (3//4)*u1(T)*(v1(T)^2)*α + (3//2)*u1(T)*(u3(T)^2)*α + (3//2)*u1(T)*(u2(T)^2)*α + (3//2)*v3(T)*v2(T)*u2(T)*α - (3//4)*(v2(T)^2)*u3(T)*α - (3//4)*(v1(T)^2)*u3(T)*α + (3//4)*u3(T)*(u2(T)^2)*α ~ F</span></span>
<span class="line"><span></span></span>
-<span class="line"><span>Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω + u1(T)*v2(T)*β - u1(T)*γ*ω + v3(T)*u2(T)*β - v2(T)*u3(T)*β - v1(T)*u2(T)*β - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v3(T)*α + (3//4)*(u1(T)^2)*v1(T)*α - (3//2)*u1(T)*v1(T)*u3(T)*α + (3//2)*(v3(T)^2)*v1(T)*α + (3//4)*v3(T)*(v2(T)^2)*α - (3//4)*v3(T)*(v1(T)^2)*α - (3//4)*v3(T)*(u2(T)^2)*α + (3//2)*(v2(T)^2)*v1(T)*α + (3//2)*v2(T)*u3(T)*u2(T)*α + (3//4)*(v1(T)^3)*α + (3//2)*v1(T)*(u3(T)^2)*α + (3//2)*v1(T)*(u2(T)^2)*α ~ 0</span></span>
+<span class="line"><span>Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω + u1(T)*v2(T)*β - u1(T)*γ*ω + v3(T)*u2(T)*β - v2(T)*u3(T)*β - v1(T)*u2(T)*β - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v3(T)*α + (3//4)*(u1(T)^2)*v1(T)*α - (3//2)*u1(T)*v1(T)*u3(T)*α + (3//2)*(v3(T)^2)*v1(T)*α + (3//4)*v3(T)*(v2(T)^2)*α - (3//4)*v3(T)*(v1(T)^2)*α - (3//4)*v3(T)*(u2(T)^2)*α + (3//2)*(v2(T)^2)*v1(T)*α + (3//2)*v2(T)*u3(T)*u2(T)*α + (3//4)*(v1(T)^3)*α + (3//2)*v1(T)*(u3(T)^2)*α + (3//2)*v1(T)*(u2(T)^2)*α ~ 0//1</span></span>
<span class="line"><span></span></span>
<span class="line"><span>Differential(T)(u2(T))*γ + (4//1)*Differential(T)(v2(T))*ω + (1//2)*(u1(T)^2)*β + u1(T)*u3(T)*β + v3(T)*v1(T)*β + (2//1)*v2(T)*γ*ω - (1//2)*(v1(T)^2)*β - (4//1)*u2(T)*(ω^2) + u2(T)*(ω0^2) + (3//2)*(u1(T)^2)*u2(T)*α + (3//2)*u1(T)*v3(T)*v2(T)*α + (3//2)*u1(T)*u3(T)*u2(T)*α + (3//2)*(v3(T)^2)*u2(T)*α - (3//2)*v3(T)*v1(T)*u2(T)*α + (3//4)*(v2(T)^2)*u2(T)*α + (3//2)*v2(T)*v1(T)*u3(T)*α + (3//2)*(v1(T)^2)*u2(T)*α + (3//2)*(u3(T)^2)*u2(T)*α + (3//4)*(u2(T)^3)*α ~ 0//1</span></span>
<span class="line"><span></span></span>
diff --git a/dev/assets/examples_wave_mixing.md.C_1sCxK3.lean.js b/dev/assets/examples_wave_mixing.md.CR10gGqe.lean.js
similarity index 100%
rename from dev/assets/examples_wave_mixing.md.C_1sCxK3.lean.js
rename to dev/assets/examples_wave_mixing.md.CR10gGqe.lean.js
diff --git a/dev/assets/manual_API.md.CC4NrY1l.js b/dev/assets/manual_API.md.DBgY29iG.js
similarity index 96%
rename from dev/assets/manual_API.md.CC4NrY1l.js
rename to dev/assets/manual_API.md.DBgY29iG.js
index 8dee7376e..4f3c553bc 100644
--- a/dev/assets/manual_API.md.CC4NrY1l.js
+++ b/dev/assets/manual_API.md.DBgY29iG.js
@@ -1,4 +1,4 @@
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custom-block",open:""},v={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},A={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.687ex"},xmlns:"http://www.w3.org/2000/svg",width:"27.124ex",height:"2.573ex",role:"img",focusable:"false",viewBox:"0 -833.9 11988.7 1137.4","aria-hidden":"true"},D={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},B={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.407ex",height:"1.027ex",role:"img",focusable:"false",viewBox:"0 -443 622 454","aria-hidden":"true"},j={class:"jldocstring custom-block",open:""},H={class:"jldocstring custom-block",open:""},x={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},w={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"30.769ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 13600.1 1000","aria-hidden":"true"},S={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},L={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"17.717ex",height:"2.587ex",role:"img",focusable:"false",viewBox:"0 -893.3 7831 1143.3","aria-hidden":"true"},P={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},q={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.452ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.534ex",height:"2.149ex",role:"img",focusable:"false",viewBox:"0 -750 1120 950","aria-hidden":"true"},V={class:"jldocstring 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custom-block",open:""},ps={class:"jldocstring custom-block",open:""},hs={class:"jldocstring custom-block",open:""},rs={class:"jldocstring custom-block",open:""};function ks(os,s,ds,gs,Es,cs){const n=r("Badge");return p(),l("div",null,[s[153]||(s[153]=t('<h1 id="doc-API" tabindex="-1">API <a class="header-anchor" href="#doc-API" aria-label="Permalink to "API {#doc-API}""></a></h1><p><strong>Table of contents</strong></p><nav class="table-of-contents"><ul><li><a href="#System-objects-and-types">System objects and types</a></li><li><a href="#Solving-and-transforming-solutions">Solving and transforming solutions</a><ul><li><a href="#methods">Methods</a></li><li><a href="#Access-solutions">Access solutions</a></li><li><a href="#classify">Classify</a></li><li><a href="#plotting">Plotting</a></li></ul></li><li><a href="#Limit-cycles">Limit cycles</a></li><li><a href="#Linear-Response">Linear Response</a></li><li><a href="#extensions">Extensions</a><ul><li><a href="#ordinarydiffeq">OrdinaryDiffEq</a></li><li><a href="#steadystatesweep">SteadyStateSweep</a></li><li><a href="#modelingtoolkit">ModelingToolkit</a></li></ul></li></ul></nav><h2 id="System-objects-and-types" tabindex="-1">System objects and types <a class="header-anchor" href="#System-objects-and-types" aria-label="Permalink to "System objects and types {#System-objects-and-types}""></a></h2>',4)),i("details",o,[i("summary",null,[s[0]||(s[0]=i("a",{id:"HarmonicBalance.d",href:"#HarmonicBalance.d"},[i("span",{class:"jlbinding"},"HarmonicBalance.d")],-1)),s[1]||(s[1]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[2]||(s[2]=i("p",null,"The derivative of f w.r.t. x of degree deg",-1)),s[3]||(s[3]=i("p",null,[i("a",{href:"https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicVariable.jl#L108",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",d,[i("summary",null,[s[4]||(s[4]=i("a",{id:"HarmonicBalance.DifferentialEquation",href:"#HarmonicBalance.DifferentialEquation"},[i("span",{class:"jlbinding"},"HarmonicBalance.DifferentialEquation")],-1)),s[5]||(s[5]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[6]||(s[6]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> DifferentialEquation</span></span></code></pre></div><p>Holds differential equation(s) of motion and a set of harmonics to expand each variable. This is the primary input for <code>HarmonicBalance.jl</code>. After inputting the equations, the harmonics ansatz needs to be specified using <code>add_harmonic!</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>equations::OrderedCollections.OrderedDict{Num, Equation}</code>: Assigns to each variable an equation of motion.</p></li><li><p><code>harmonics::OrderedCollections.OrderedDict{Num, OrderedCollections.OrderedSet{Num}}</code>: Assigns to each variable a set of harmonics.</p></li></ul><p><strong>Example</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> t, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">y</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), ω0, ω, F, k;</span></span>
+import{_ as h,c as l,ai as t,j as i,a,G as e,B as r,o as p}from"./chunks/framework.Bnn8558y.js";const us=JSON.parse('{"title":"API","description":"","frontmatter":{},"headers":[],"relativePath":"manual/API.md","filePath":"manual/API.md"}'),k={name:"manual/API.md"},o={class:"jldocstring custom-block",open:""},d={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""},m={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},b={class:"jldocstring custom-block",open:""},F={class:"jldocstring custom-block",open:""},f={class:"jldocstring custom-block",open:""},Q={class:"jldocstring custom-block",open:""},C={class:"jldocstring custom-block",open:""},_={class:"jldocstring custom-block",open:""},v={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},A={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.687ex"},xmlns:"http://www.w3.org/2000/svg",width:"27.124ex",height:"2.573ex",role:"img",focusable:"false",viewBox:"0 -833.9 11988.7 1137.4","aria-hidden":"true"},D={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},B={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.407ex",height:"1.027ex",role:"img",focusable:"false",viewBox:"0 -443 622 454","aria-hidden":"true"},j={class:"jldocstring custom-block",open:""},H={class:"jldocstring custom-block",open:""},x={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},w={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"30.769ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 13600.1 1000","aria-hidden":"true"},S={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},L={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"17.717ex",height:"2.587ex",role:"img",focusable:"false",viewBox:"0 -893.3 7831 1143.3","aria-hidden":"true"},P={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},q={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.452ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.534ex",height:"2.149ex",role:"img",focusable:"false",viewBox:"0 -750 1120 950","aria-hidden":"true"},V={class:"jldocstring custom-block",open:""},O={class:"jldocstring custom-block",open:""},I={class:"jldocstring custom-block",open:""},N={class:"jldocstring custom-block",open:""},M={class:"jldocstring custom-block",open:""},R={class:"jldocstring custom-block",open:""},Z={class:"jldocstring custom-block",open:""},z={class:"jldocstring custom-block",open:""},W={class:"jldocstring custom-block",open:""},U={class:"jldocstring custom-block",open:""},J={class:"jldocstring custom-block",open:""},G={class:"jldocstring custom-block",open:""},K={class:"jldocstring custom-block",open:""},$={class:"jldocstring custom-block",open:""},X={class:"jldocstring custom-block",open:""},Y={class:"jldocstring custom-block",open:""},ss={class:"jldocstring custom-block",open:""},is={class:"jldocstring custom-block",open:""},as={class:"jldocstring custom-block",open:""},ts={class:"jldocstring custom-block",open:""},ns={class:"jldocstring custom-block",open:""},es={class:"jldocstring custom-block",open:""},ls={class:"jldocstring custom-block",open:""},ps={class:"jldocstring custom-block",open:""},hs={class:"jldocstring custom-block",open:""},rs={class:"jldocstring custom-block",open:""};function ks(os,s,ds,gs,cs,Es){const n=r("Badge");return p(),l("div",null,[s[153]||(s[153]=t('<h1 id="doc-API" tabindex="-1">API <a class="header-anchor" href="#doc-API" aria-label="Permalink to "API {#doc-API}""></a></h1><p><strong>Table of contents</strong></p><nav class="table-of-contents"><ul><li><a href="#System-objects-and-types">System objects and types</a></li><li><a href="#Solving-and-transforming-solutions">Solving and transforming solutions</a><ul><li><a href="#methods">Methods</a></li><li><a href="#Access-solutions">Access solutions</a></li><li><a href="#classify">Classify</a></li><li><a href="#plotting">Plotting</a></li></ul></li><li><a href="#Limit-cycles">Limit cycles</a></li><li><a href="#Linear-Response">Linear Response</a></li><li><a href="#extensions">Extensions</a><ul><li><a href="#ordinarydiffeq">OrdinaryDiffEq</a></li><li><a href="#steadystatesweep">SteadyStateSweep</a></li><li><a href="#modelingtoolkit">ModelingToolkit</a></li></ul></li></ul></nav><h2 id="System-objects-and-types" tabindex="-1">System objects and types <a class="header-anchor" href="#System-objects-and-types" aria-label="Permalink to "System objects and types {#System-objects-and-types}""></a></h2>',4)),i("details",o,[i("summary",null,[s[0]||(s[0]=i("a",{id:"HarmonicBalance.d",href:"#HarmonicBalance.d"},[i("span",{class:"jlbinding"},"HarmonicBalance.d")],-1)),s[1]||(s[1]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[2]||(s[2]=i("p",null,"The derivative of f w.r.t. x of degree deg",-1)),s[3]||(s[3]=i("p",null,[i("a",{href:"https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicVariable.jl#L108",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",d,[i("summary",null,[s[4]||(s[4]=i("a",{id:"HarmonicBalance.DifferentialEquation",href:"#HarmonicBalance.DifferentialEquation"},[i("span",{class:"jlbinding"},"HarmonicBalance.DifferentialEquation")],-1)),s[5]||(s[5]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[6]||(s[6]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> DifferentialEquation</span></span></code></pre></div><p>Holds differential equation(s) of motion and a set of harmonics to expand each variable. This is the primary input for <code>HarmonicBalance.jl</code>. After inputting the equations, the harmonics ansatz needs to be specified using <code>add_harmonic!</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>equations::OrderedCollections.OrderedDict{Num, Equation}</code>: Assigns to each variable an equation of motion.</p></li><li><p><code>harmonics::OrderedCollections.OrderedDict{Num, OrderedCollections.OrderedSet{Num}}</code>: Assigns to each variable a set of harmonics.</p></li></ul><p><strong>Example</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> t, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">y</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), ω0, ω, F, k;</span></span>
<span class="line"></span>
<span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># equivalent ways to enter the simple harmonic oscillator</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">d</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(x,t,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> *</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> x </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">t), x);</span></span>
@@ -7,16 +7,16 @@ import{_ as h,c as l,ai as t,j as i,a,G as e,B as r,o as p}from"./chunks/framewo
<span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># two coupled oscillators, one of them driven</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">d</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(x,t,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> *</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> x </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> k</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">d</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(y,t,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> *</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> y </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> k</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">x] </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], [x,y]</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">);</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L1" target="_blank" rel="noreferrer">source</a></p>`,7))]),i("details",g,[i("summary",null,[s[7]||(s[7]=i("a",{id:"HarmonicBalance.HarmonicVariable",href:"#HarmonicBalance.HarmonicVariable"},[i("span",{class:"jlbinding"},"HarmonicBalance.HarmonicVariable")],-1)),s[8]||(s[8]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[9]||(s[9]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicVariable</span></span></code></pre></div><p>Holds a variable stored under <code>symbol</code> describing the harmonic <code>ω</code> of <code>natural_variable</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>symbol::Num</code>: Symbol of the variable in the HarmonicBalance namespace.</p></li><li><p><code>name::String</code>: Human-readable labels of the variable, used for plotting.</p></li><li><p><code>type::String</code>: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)</p></li><li><p><code>ω::Num</code>: The harmonic being described.</p></li><li><p><code>natural_variable::Num</code>: The natural variable whose harmonic is being described.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicVariable.jl#L1" target="_blank" rel="noreferrer">source</a></p>',5))]),i("details",E,[i("summary",null,[s[10]||(s[10]=i("a",{id:"HarmonicBalance.HarmonicEquation",href:"#HarmonicBalance.HarmonicEquation"},[i("span",{class:"jlbinding"},"HarmonicBalance.HarmonicEquation")],-1)),s[11]||(s[11]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[12]||(s[12]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Holds a set of algebraic equations governing the harmonics of a <code>DifferentialEquation</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>equations::Vector{Equation}</code>: A set of equations governing the harmonics.</p></li><li><p><code>variables::Vector{HarmonicVariable}</code>: A set of variables describing the harmonics.</p></li><li><p><code>parameters::Vector{Num}</code>: The parameters of the equation set.</p></li><li><p><code>natural_equation::DifferentialEquation</code>: The natural equation (before the harmonic ansatz was used).</p></li><li><p><code>jacobian::Matrix{Num}</code>: The Jacobian of the natural equation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L1" target="_blank" rel="noreferrer">source</a></p>',5))]),i("details",c,[i("summary",null,[s[13]||(s[13]=i("a",{id:"HarmonicBalance.rearrange_standard",href:"#HarmonicBalance.rearrange_standard"},[i("span",{class:"jlbinding"},"HarmonicBalance.rearrange_standard")],-1)),s[14]||(s[14]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[15]||(s[15]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">rearrange_standard</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">);</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L1" target="_blank" rel="noreferrer">source</a></p>`,7))]),i("details",g,[i("summary",null,[s[7]||(s[7]=i("a",{id:"HarmonicBalance.HarmonicVariable",href:"#HarmonicBalance.HarmonicVariable"},[i("span",{class:"jlbinding"},"HarmonicBalance.HarmonicVariable")],-1)),s[8]||(s[8]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[9]||(s[9]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicVariable</span></span></code></pre></div><p>Holds a variable stored under <code>symbol</code> describing the harmonic <code>ω</code> of <code>natural_variable</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>symbol::Num</code>: Symbol of the variable in the HarmonicBalance namespace.</p></li><li><p><code>name::String</code>: Human-readable labels of the variable, used for plotting.</p></li><li><p><code>type::String</code>: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)</p></li><li><p><code>ω::Num</code>: The harmonic being described.</p></li><li><p><code>natural_variable::Num</code>: The natural variable whose harmonic is being described.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicVariable.jl#L1" target="_blank" rel="noreferrer">source</a></p>',5))]),i("details",c,[i("summary",null,[s[10]||(s[10]=i("a",{id:"HarmonicBalance.HarmonicEquation",href:"#HarmonicBalance.HarmonicEquation"},[i("span",{class:"jlbinding"},"HarmonicBalance.HarmonicEquation")],-1)),s[11]||(s[11]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[12]||(s[12]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Holds a set of algebraic equations governing the harmonics of a <code>DifferentialEquation</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>equations::Vector{Equation}</code>: A set of equations governing the harmonics.</p></li><li><p><code>variables::Vector{HarmonicVariable}</code>: A set of variables describing the harmonics.</p></li><li><p><code>parameters::Vector{Num}</code>: The parameters of the equation set.</p></li><li><p><code>natural_equation::DifferentialEquation</code>: The natural equation (before the harmonic ansatz was used).</p></li><li><p><code>jacobian::Matrix{Num}</code>: The Jacobian of the natural equation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L1" target="_blank" rel="noreferrer">source</a></p>',5))]),i("details",E,[i("summary",null,[s[13]||(s[13]=i("a",{id:"HarmonicBalance.rearrange_standard",href:"#HarmonicBalance.rearrange_standard"},[i("span",{class:"jlbinding"},"HarmonicBalance.rearrange_standard")],-1)),s[14]||(s[14]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[15]||(s[15]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">rearrange_standard</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Rearrange <code>eom</code> to the standard form, such that the derivatives of the variables are on one side.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L194" target="_blank" rel="noreferrer">source</a></p>`,3))]),i("details",y,[i("summary",null,[s[16]||(s[16]=i("a",{id:"HarmonicBalance.rearrange_standard!",href:"#HarmonicBalance.rearrange_standard!"},[i("span",{class:"jlbinding"},"HarmonicBalance.rearrange_standard!")],-1)),s[17]||(s[17]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[18]||(s[18]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">rearrange_standard!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
-<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">rearrange_standard!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, degree)</span></span></code></pre></div><p>Rearranges the differential equations in <code>eom</code> to standard form, where the highest derivative of each variable (specified by <code>degree</code>, default 2) appears isolated on the left-hand side. Modifies the equations in place.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L160" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">rearrange_standard!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Rearrange <code>eom</code> to the standard form, such that the derivatives of the variables are on one side.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L194" target="_blank" rel="noreferrer">source</a></p>`,3))]),i("details",y,[i("summary",null,[s[16]||(s[16]=i("a",{id:"HarmonicBalance.rearrange_standard!",href:"#HarmonicBalance.rearrange_standard!"},[i("span",{class:"jlbinding"},"HarmonicBalance.rearrange_standard!")],-1)),s[17]||(s[17]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[18]||(s[18]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">rearrange_standard!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
+<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">rearrange_standard!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, degree)</span></span></code></pre></div><p>Rearranges the differential equations in <code>eom</code> to standard form, where the highest derivative of each variable (specified by <code>degree</code>, default 2) appears isolated on the left-hand side. Modifies the equations in place.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L160" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">rearrange_standard!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Rearrange <code>eom</code> to the standard form, such that the derivatives of the variables are on one side.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L204" target="_blank" rel="noreferrer">source</a></p>`,6))]),i("details",u,[i("summary",null,[s[19]||(s[19]=i("a",{id:"HarmonicBalance.first_order_transform!",href:"#HarmonicBalance.first_order_transform!"},[i("span",{class:"jlbinding"},"HarmonicBalance.first_order_transform!")],-1)),s[20]||(s[20]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[21]||(s[21]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">first_order_transform!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, time)</span></span></code></pre></div><p>Transforms a higher-order differential equation system into an equivalent first-order system by introducing additional variables. Modifies the system in place. The <code>time</code> parameter specifies the independent variable used for differentiation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L202" target="_blank" rel="noreferrer">source</a></p>',3))]),i("details",m,[i("summary",null,[s[22]||(s[22]=i("a",{id:"HarmonicBalance.is_rearranged_standard",href:"#HarmonicBalance.is_rearranged_standard"},[i("span",{class:"jlbinding"},"HarmonicBalance.is_rearranged_standard")],-1)),s[23]||(s[23]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[24]||(s[24]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">is_rearranged_standard</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Rearrange <code>eom</code> to the standard form, such that the derivatives of the variables are on one side.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L204" target="_blank" rel="noreferrer">source</a></p>`,6))]),i("details",u,[i("summary",null,[s[19]||(s[19]=i("a",{id:"HarmonicBalance.first_order_transform!",href:"#HarmonicBalance.first_order_transform!"},[i("span",{class:"jlbinding"},"HarmonicBalance.first_order_transform!")],-1)),s[20]||(s[20]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[21]||(s[21]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">first_order_transform!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, time)</span></span></code></pre></div><p>Transforms a higher-order differential equation system into an equivalent first-order system by introducing additional variables. Modifies the system in place. The <code>time</code> parameter specifies the independent variable used for differentiation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L202" target="_blank" rel="noreferrer">source</a></p>',3))]),i("details",m,[i("summary",null,[s[22]||(s[22]=i("a",{id:"HarmonicBalance.is_rearranged_standard",href:"#HarmonicBalance.is_rearranged_standard"},[i("span",{class:"jlbinding"},"HarmonicBalance.is_rearranged_standard")],-1)),s[23]||(s[23]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[24]||(s[24]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">is_rearranged_standard</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span>
<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">is_rearranged_standard</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> degree</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Checks if the differential equations in <code>eom</code> are arranged in standard form, where the highest derivative of each variable appears isolated on the left-hand side. The default degree is 2, corresponding to second-order differential equations.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L147" target="_blank" rel="noreferrer">source</a></p>`,3))]),i("details",b,[i("summary",null,[s[25]||(s[25]=i("a",{id:"HarmonicBalance.get_equations",href:"#HarmonicBalance.get_equations"},[i("span",{class:"jlbinding"},"HarmonicBalance.get_equations")],-1)),s[26]||(s[26]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[27]||(s[27]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_equations</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Equation}</span></span></code></pre></div><p>Return the equations of <code>eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L140" target="_blank" rel="noreferrer">source</a></p>',3))]),i("details",T,[i("summary",null,[s[28]||(s[28]=i("a",{id:"HarmonicBalance.get_harmonic_equations",href:"#HarmonicBalance.get_harmonic_equations"},[i("span",{class:"jlbinding"},"HarmonicBalance.get_harmonic_equations")],-1)),s[29]||(s[29]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[30]||(s[30]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_harmonic_equations</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Checks if the differential equations in <code>eom</code> are arranged in standard form, where the highest derivative of each variable appears isolated on the left-hand side. The default degree is 2, corresponding to second-order differential equations.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L147" target="_blank" rel="noreferrer">source</a></p>`,3))]),i("details",T,[i("summary",null,[s[25]||(s[25]=i("a",{id:"HarmonicBalance.get_equations",href:"#HarmonicBalance.get_equations"},[i("span",{class:"jlbinding"},"HarmonicBalance.get_equations")],-1)),s[26]||(s[26]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[27]||(s[27]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_equations</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Equation}</span></span></code></pre></div><p>Return the equations of <code>eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L140" target="_blank" rel="noreferrer">source</a></p>',3))]),i("details",b,[i("summary",null,[s[28]||(s[28]=i("a",{id:"HarmonicBalance.get_harmonic_equations",href:"#HarmonicBalance.get_harmonic_equations"},[i("span",{class:"jlbinding"},"HarmonicBalance.get_harmonic_equations")],-1)),s[29]||(s[29]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[30]||(s[30]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_harmonic_equations</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> fast_time,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> slow_time,</span></span>
@@ -44,7 +44,7 @@ import{_ as h,c as l,ai as t,j as i,a,G as e,B as r,o as p}from"./chunks/framewo
<span class="line"></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F</span></span>
<span class="line"></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L295-L336" target="_blank" rel="noreferrer">source</a></p>`,7))]),i("details",f,[i("summary",null,[s[31]||(s[31]=i("a",{id:"HarmonicBalance.KrylovBogoliubov.get_krylov_equations",href:"#HarmonicBalance.KrylovBogoliubov.get_krylov_equations"},[i("span",{class:"jlbinding"},"HarmonicBalance.KrylovBogoliubov.get_krylov_equations")],-1)),s[32]||(s[32]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[33]||(s[33]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_krylov_equations</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L295-L336" target="_blank" rel="noreferrer">source</a></p>`,7))]),i("details",F,[i("summary",null,[s[31]||(s[31]=i("a",{id:"HarmonicBalance.KrylovBogoliubov.get_krylov_equations",href:"#HarmonicBalance.KrylovBogoliubov.get_krylov_equations"},[i("span",{class:"jlbinding"},"HarmonicBalance.KrylovBogoliubov.get_krylov_equations")],-1)),s[32]||(s[32]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[33]||(s[33]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_krylov_equations</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> order,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> fast_time,</span></span>
@@ -72,7 +72,7 @@ import{_ as h,c as l,ai as t,j as i,a,G as e,B as r,o as p}from"./chunks/framewo
<span class="line"></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">/</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T))</span></span>
<span class="line"></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">/</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T))</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/KrylovBogoliubov.jl#L25" target="_blank" rel="noreferrer">source</a></p>`,7))]),i("details",F,[i("summary",null,[s[34]||(s[34]=i("a",{id:"HarmonicBalance.add_harmonic!",href:"#HarmonicBalance.add_harmonic!"},[i("span",{class:"jlbinding"},"HarmonicBalance.add_harmonic!")],-1)),s[35]||(s[35]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[36]||(s[36]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">add_harmonic!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, var</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ω)</span></span></code></pre></div><p>Add the harmonic <code>ω</code> to the harmonic ansatz used to expand the variable <code>var</code> in <code>diff_eom</code>.</p><p><strong>Example</strong></p><p><strong>define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> t, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">y</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), ω0, ω, F, k;</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">/</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T))</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/KrylovBogoliubov.jl#L25" target="_blank" rel="noreferrer">source</a></p>`,7))]),i("details",f,[i("summary",null,[s[34]||(s[34]=i("a",{id:"HarmonicBalance.add_harmonic!",href:"#HarmonicBalance.add_harmonic!"},[i("span",{class:"jlbinding"},"HarmonicBalance.add_harmonic!")],-1)),s[35]||(s[35]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[36]||(s[36]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">add_harmonic!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, var</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ω)</span></span></code></pre></div><p>Add the harmonic <code>ω</code> to the harmonic ansatz used to expand the variable <code>var</code> in <code>diff_eom</code>.</p><p><strong>Example</strong></p><p><strong>define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> t, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">y</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), ω0, ω, F, k;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> diff_eq </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">d</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(x,t,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> *</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> x </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">t), x);</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> add_harmonic!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eq, x, ��) </span><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># expand x using ω</span></span>
<span class="line"></span>
@@ -80,11 +80,11 @@ import{_ as h,c as l,ai as t,j as i,a,G as e,B as r,o as p}from"./chunks/framewo
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Variables</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Harmonic ansatz</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω;</span></span>
<span class="line"></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t))) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L88" target="_blank" rel="noreferrer">source</a></p>`,6))]),i("details",Q,[i("summary",null,[s[37]||(s[37]=i("a",{id:"HarmonicBalance.get_independent_variables",href:"#HarmonicBalance.get_independent_variables"},[i("span",{class:"jlbinding"},"HarmonicBalance.get_independent_variables")],-1)),s[38]||(s[38]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[39]||(s[39]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_independent_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t))) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L88" target="_blank" rel="noreferrer">source</a></p>`,6))]),i("details",Q,[i("summary",null,[s[37]||(s[37]=i("a",{id:"HarmonicBalance.get_independent_variables",href:"#HarmonicBalance.get_independent_variables"},[i("span",{class:"jlbinding"},"HarmonicBalance.get_independent_variables")],-1)),s[38]||(s[38]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[39]||(s[39]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_independent_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Return the independent dependent variables of <code>diff_eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L131" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_independent_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Return the independent dependent variables of <code>diff_eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L131" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_independent_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Num}</span></span></code></pre></div><p>Return the independent variables (typically time) of <code>eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L247" target="_blank" rel="noreferrer">source</a></p>`,6))]),i("details",C,[i("summary",null,[s[40]||(s[40]=i("a",{id:"Symbolics.get_variables",href:"#Symbolics.get_variables"},[i("span",{class:"jlbinding"},"Symbolics.get_variables")],-1)),s[41]||(s[41]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[42]||(s[42]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Num}</span></span></code></pre></div><p>Return the dependent variables of <code>diff_eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L113" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Num}</span></span></code></pre></div><p>Get the internal symbols of the independent variables of <code>eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L214" target="_blank" rel="noreferrer">source</a></p>',6))]),s[154]||(s[154]=i("h2",{id:"Solving-and-transforming-solutions",tabindex:"-1"},[a("Solving and transforming solutions "),i("a",{class:"header-anchor",href:"#Solving-and-transforming-solutions","aria-label":'Permalink to "Solving and transforming solutions {#Solving-and-transforming-solutions}"'},"")],-1)),i("details",_,[i("summary",null,[s[43]||(s[43]=i("a",{id:"HarmonicBalance.get_steady_states",href:"#HarmonicBalance.get_steady_states"},[i("span",{class:"jlbinding"},"HarmonicBalance.get_steady_states")],-1)),s[44]||(s[44]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[51]||(s[51]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(problem</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Num}</span></span></code></pre></div><p>Return the independent variables (typically time) of <code>eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L247" target="_blank" rel="noreferrer">source</a></p>`,6))]),i("details",C,[i("summary",null,[s[40]||(s[40]=i("a",{id:"Symbolics.get_variables",href:"#Symbolics.get_variables"},[i("span",{class:"jlbinding"},"Symbolics.get_variables")],-1)),s[41]||(s[41]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[42]||(s[42]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Num}</span></span></code></pre></div><p>Return the dependent variables of <code>diff_eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L113" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Num}</span></span></code></pre></div><p>Get the internal symbols of the independent variables of <code>eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L214" target="_blank" rel="noreferrer">source</a></p>',6))]),s[154]||(s[154]=i("h2",{id:"Solving-and-transforming-solutions",tabindex:"-1"},[a("Solving and transforming solutions "),i("a",{class:"header-anchor",href:"#Solving-and-transforming-solutions","aria-label":'Permalink to "Solving and transforming solutions {#Solving-and-transforming-solutions}"'},"")],-1)),i("details",_,[i("summary",null,[s[43]||(s[43]=i("a",{id:"HarmonicBalance.get_steady_states",href:"#HarmonicBalance.get_steady_states"},[i("span",{class:"jlbinding"},"HarmonicBalance.get_steady_states")],-1)),s[44]||(s[44]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[51]||(s[51]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(problem</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> method</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalanceMethod</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> swept_parameters,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> fixed_parameters;</span></span>
@@ -113,16 +113,16 @@ import{_ as h,c as l,ai as t,j as i,a,G as e,B as r,o as p}from"./chunks/framewo
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> of which real</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> of which stable</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1</span></span>
<span class="line"></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Classes</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> stable, physical, Hopf, binary_labels</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/solve_homotopy.jl#L1-L60" target="_blank" rel="noreferrer">source</a></p>`,4))]),s[155]||(s[155]=i("h3",{id:"methods",tabindex:"-1"},[a("Methods "),i("a",{class:"header-anchor",href:"#methods","aria-label":'Permalink to "Methods"'},"")],-1)),i("details",j,[i("summary",null,[s[53]||(s[53]=i("a",{id:"HarmonicBalance.WarmUp",href:"#HarmonicBalance.WarmUp"},[i("span",{class:"jlbinding"},"HarmonicBalance.WarmUp")],-1)),s[54]||(s[54]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[55]||(s[55]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">WarmUp</span></span></code></pre></div><p>The Warm Up method prepares a warmup system with the Total Degree method using the parameter at <code>index</code> perturbed by <code>perturbation_size</code>. The warmup system is used to perform a homotopy using all other systems in the parameter sweep. It is very efficient for systems with minimal bifurcation in the parameter sweep. The Warm Up method should in theory guarantee to find all solutions, however, if the <code>start_parameters</code> is not proper (to close to the real line) it could miss some solutions.</p><p>See<a href="https://www.juliahomotopycontinuation.org/guides/many-systems/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>warm_up_method::Union{Polyhedral{T}, TotalDegree{T}} where T</code>: Method used for the warmup system.</p></li><li><p><code>start_parameters::Vector</code>: Start parameters.</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/methods.jl#L110-L125" target="_blank" rel="noreferrer">source</a></p>',6))]),i("details",H,[i("summary",null,[s[56]||(s[56]=i("a",{id:"HarmonicBalance.TotalDegree",href:"#HarmonicBalance.TotalDegree"},[i("span",{class:"jlbinding"},"HarmonicBalance.TotalDegree")],-1)),s[57]||(s[57]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[72]||(s[72]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">TotalDegree</span></span></code></pre></div>',1)),i("p",null,[s[64]||(s[64]=a("The Total Degree homotopy method performs a homotopy 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See ")),s[70]||(s[70]=i("a",{href:"https://www.juliahomotopycontinuation.org/guides/totaldegree/",target:"_blank",rel:"noreferrer"},"HomotopyContinuation.jl",-1)),s[71]||(s[71]=a(" for more information."))]),s[73]||(s[73]=t('<p><strong>Fields</strong></p><ul><li><p><code>gamma::Complex</code>: Complex multiplying factor of the start system G(x) for the homotopy</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/methods.jl#L8-L20" target="_blank" rel="noreferrer">source</a></p>',3))]),i("details",V,[i("summary",null,[s[74]||(s[74]=i("a",{id:"HarmonicBalance.Polyhedral",href:"#HarmonicBalance.Polyhedral"},[i("span",{class:"jlbinding"},"HarmonicBalance.Polyhedral")],-1)),s[75]||(s[75]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[76]||(s[76]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Polyhedral</span></span></code></pre></div><p>The Polyhedral homotopy method constructs a homotopy based on the polyhedral structure of the polynomial system. It is more efficient than the Total Degree method for sparse systems, meaning most of the coefficients are zero. It can be especially useful if you don't need to find the zero solutions (<code>only_non_zero = true</code>), resulting in a speed up. See <a href="https://www.juliahomotopycontinuation.org/guides/polyhedral/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>only_non_zero::Bool</code>: Boolean indicating if only non-zero solutions are considered.</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/methods.jl#L57-L69" target="_blank" rel="noreferrer">source</a></p>',5))]),s[156]||(s[156]=i("h3",{id:"Access-solutions",tabindex:"-1"},[a("Access solutions "),i("a",{class:"header-anchor",href:"#Access-solutions","aria-label":'Permalink to "Access solutions {#Access-solutions}"'},"")],-1)),i("details",O,[i("summary",null,[s[77]||(s[77]=i("a",{id:"HarmonicBalance.get_single_solution",href:"#HarmonicBalance.get_single_solution"},[i("span",{class:"jlbinding"},"HarmonicBalance.get_single_solution")],-1)),s[78]||(s[78]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[79]||(s[79]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_single_solution</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Classes</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> stable, physical, Hopf, binary_labels</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/solve_homotopy.jl#L1-L60" target="_blank" rel="noreferrer">source</a></p>`,4))]),s[155]||(s[155]=i("h3",{id:"methods",tabindex:"-1"},[a("Methods "),i("a",{class:"header-anchor",href:"#methods","aria-label":'Permalink to "Methods"'},"")],-1)),i("details",j,[i("summary",null,[s[53]||(s[53]=i("a",{id:"HarmonicBalance.WarmUp",href:"#HarmonicBalance.WarmUp"},[i("span",{class:"jlbinding"},"HarmonicBalance.WarmUp")],-1)),s[54]||(s[54]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[55]||(s[55]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">WarmUp</span></span></code></pre></div><p>The Warm Up method prepares a warmup system with the Total Degree method using the parameter at <code>index</code> perturbed by <code>perturbation_size</code>. The warmup system is used to perform a homotopy using all other systems in the parameter sweep. It is very efficient for systems with minimal bifurcation in the parameter sweep. The Warm Up method should in theory guarantee to find all solutions, however, if the <code>start_parameters</code> is not proper (to close to the real line) it could miss some solutions.</p><p>See<a href="https://www.juliahomotopycontinuation.org/guides/many-systems/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>warm_up_method::Union{Polyhedral{T}, TotalDegree{T}} where T</code>: Method used for the warmup system.</p></li><li><p><code>start_parameters::Vector</code>: Start parameters.</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/methods.jl#L110-L125" target="_blank" rel="noreferrer">source</a></p>',6))]),i("details",H,[i("summary",null,[s[56]||(s[56]=i("a",{id:"HarmonicBalance.TotalDegree",href:"#HarmonicBalance.TotalDegree"},[i("span",{class:"jlbinding"},"HarmonicBalance.TotalDegree")],-1)),s[57]||(s[57]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[72]||(s[72]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">TotalDegree</span></span></code></pre></div>',1)),i("p",null,[s[64]||(s[64]=a("The Total Degree homotopy method performs a homotopy 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30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(520,0)"><text data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">ᵢ</text></g></g></g>',1)]))),s[63]||(s[63]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"d"),i("mi",null,"ᵢ")])],-1))]),s[67]||(s[67]=a(" determined by the ")),s[68]||(s[68]=i("a",{href:"https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem",target:"_blank",rel:"noreferrer"},"Bezout bound",-1)),s[69]||(s[69]=a(". The method guarantees to find all solutions, however, it comes with a high computational cost. See ")),s[70]||(s[70]=i("a",{href:"https://www.juliahomotopycontinuation.org/guides/totaldegree/",target:"_blank",rel:"noreferrer"},"HomotopyContinuation.jl",-1)),s[71]||(s[71]=a(" for more information."))]),s[73]||(s[73]=t('<p><strong>Fields</strong></p><ul><li><p><code>gamma::Complex</code>: Complex multiplying factor of the start system G(x) for the homotopy</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/methods.jl#L8-L20" target="_blank" rel="noreferrer">source</a></p>',3))]),i("details",V,[i("summary",null,[s[74]||(s[74]=i("a",{id:"HarmonicBalance.Polyhedral",href:"#HarmonicBalance.Polyhedral"},[i("span",{class:"jlbinding"},"HarmonicBalance.Polyhedral")],-1)),s[75]||(s[75]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[76]||(s[76]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Polyhedral</span></span></code></pre></div><p>The Polyhedral homotopy method constructs a homotopy based on the polyhedral structure of the polynomial system. It is more efficient than the Total Degree method for sparse systems, meaning most of the coefficients are zero. It can be especially useful if you don't need to find the zero solutions (<code>only_non_zero = true</code>), resulting in a speed up. See <a href="https://www.juliahomotopycontinuation.org/guides/polyhedral/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>only_non_zero::Bool</code>: Boolean indicating if only non-zero solutions are considered.</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/methods.jl#L57-L69" target="_blank" rel="noreferrer">source</a></p>',5))]),s[156]||(s[156]=i("h3",{id:"Access-solutions",tabindex:"-1"},[a("Access solutions "),i("a",{class:"header-anchor",href:"#Access-solutions","aria-label":'Permalink to "Access solutions {#Access-solutions}"'},"")],-1)),i("details",O,[i("summary",null,[s[77]||(s[77]=i("a",{id:"HarmonicBalance.get_single_solution",href:"#HarmonicBalance.get_single_solution"},[i("span",{class:"jlbinding"},"HarmonicBalance.get_single_solution")],-1)),s[78]||(s[78]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[79]||(s[79]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_single_solution</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result{S, P, D, F}</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> where</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> {D, F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">FunctionWrappers.FunctionWrapper{Array{S, 2}, Tuple{Array{S, 1}}}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">};</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> branch,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> index</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Return an ordered dictionary specifying all variables and parameters of the solution in <code>result</code> on <code>branch</code> at the position <code>index</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/transform_solutions.jl#L1" target="_blank" rel="noreferrer">source</a></p>`,3))]),i("details",I,[i("summary",null,[s[80]||(s[80]=i("a",{id:"HarmonicBalance.transform_solutions",href:"#HarmonicBalance.transform_solutions"},[i("span",{class:"jlbinding"},"HarmonicBalance.transform_solutions")],-1)),s[81]||(s[81]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[82]||(s[82]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">transform_solutions</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Return an ordered dictionary specifying all variables and parameters of the solution in <code>result</code> on <code>branch</code> at the position <code>index</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/transform_solutions.jl#L1" target="_blank" rel="noreferrer">source</a></p>`,3))]),i("details",I,[i("summary",null,[s[80]||(s[80]=i("a",{id:"HarmonicBalance.transform_solutions",href:"#HarmonicBalance.transform_solutions"},[i("span",{class:"jlbinding"},"HarmonicBalance.transform_solutions")],-1)),s[81]||(s[81]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[82]||(s[82]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">transform_solutions</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result{S, ParType, D, F}</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> where</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> {ParType</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Number</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, D, F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">FunctionWrappers.FunctionWrapper{Array{S, 2}, Tuple{Array{S, 1}}}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">},</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> func;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> branches,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> realify</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector</span></span></code></pre></div><p>Takes a <code>Result</code> object and a string <code>f</code> representing a Symbolics.jl expression. Returns an array with the values of <code>f</code> evaluated for the respective solutions. Additional substitution rules can be specified in <code>rules</code> in the format <code>("a" => val)</code> or <code>(a => val)</code></p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/transform_solutions.jl#L64" target="_blank" rel="noreferrer">source</a></p>`,3))]),s[157]||(s[157]=i("h3",{id:"classify",tabindex:"-1"},[a("Classify "),i("a",{class:"header-anchor",href:"#classify","aria-label":'Permalink to "Classify"'},"")],-1)),i("details",N,[i("summary",null,[s[83]||(s[83]=i("a",{id:"HarmonicBalance.classify_solutions!",href:"#HarmonicBalance.classify_solutions!"},[i("span",{class:"jlbinding"},"HarmonicBalance.classify_solutions!")],-1)),s[84]||(s[84]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[85]||(s[85]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector</span></span></code></pre></div><p>Takes a <code>Result</code> object and a string <code>f</code> representing a Symbolics.jl expression. Returns an array with the values of <code>f</code> evaluated for the respective solutions. Additional substitution rules can be specified in <code>rules</code> in the format <code>("a" => val)</code> or <code>(a => val)</code></p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/transform_solutions.jl#L64" target="_blank" rel="noreferrer">source</a></p>`,3))]),s[157]||(s[157]=i("h3",{id:"classify",tabindex:"-1"},[a("Classify "),i("a",{class:"header-anchor",href:"#classify","aria-label":'Permalink to "Classify"'},"")],-1)),i("details",N,[i("summary",null,[s[83]||(s[83]=i("a",{id:"HarmonicBalance.classify_solutions!",href:"#HarmonicBalance.classify_solutions!"},[i("span",{class:"jlbinding"},"HarmonicBalance.classify_solutions!")],-1)),s[84]||(s[84]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[85]||(s[85]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> func</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Union{Function, String}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> name</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
@@ -131,34 +131,34 @@ import{_ as h,c as l,ai as t,j as i,a,G as e,B as r,o as p}from"./chunks/framewo
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">res </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(problem, swept_parameters, fixed_parameters)</span></span>
<span class="line"></span>
<span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># classify, store in result.classes["large_amplitude"]</span></span>
-<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u1^2 + v1^2) > 1.0"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> , </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"large_amplitude"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/classification.jl#L1" target="_blank" rel="noreferrer">source</a></p>`,5))]),i("details",M,[i("summary",null,[s[86]||(s[86]=i("a",{id:"HarmonicBalance.get_class",href:"#HarmonicBalance.get_class"},[i("span",{class:"jlbinding"},"HarmonicBalance.get_class")],-1)),s[87]||(s[87]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[88]||(s[88]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_class</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u1^2 + v1^2) > 1.0"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> , </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"large_amplitude"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/classification.jl#L1" target="_blank" rel="noreferrer">source</a></p>`,5))]),i("details",M,[i("summary",null,[s[86]||(s[86]=i("a",{id:"HarmonicBalance.get_class",href:"#HarmonicBalance.get_class"},[i("span",{class:"jlbinding"},"HarmonicBalance.get_class")],-1)),s[87]||(s[87]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[88]||(s[88]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_class</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int64</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> class</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Returns an array of booleans classifying <code>branch</code> in the solutions in <code>res</code> according to <code>class</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/classification.jl#L43" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_class</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Returns an array of booleans classifying <code>branch</code> in the solutions in <code>res</code> according to <code>class</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/classification.jl#L43" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_class</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> class</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector</span></span></code></pre></div><p>Returns an array of booleans classifying each branch in the solutions in <code>res</code> according to <code>class</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/classification.jl#L53" target="_blank" rel="noreferrer">source</a></p>`,6))]),s[158]||(s[158]=i("h3",{id:"plotting",tabindex:"-1"},[a("Plotting "),i("a",{class:"header-anchor",href:"#plotting","aria-label":'Permalink to "Plotting"'},"")],-1)),i("details",R,[i("summary",null,[s[89]||(s[89]=i("a",{id:"RecipesBase.plot",href:"#RecipesBase.plot"},[i("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[90]||(s[90]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[91]||(s[91]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESolution</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, f</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, harm_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a function <code>f</code> of a time-dependent solution <code>soln</code> of <code>harm_eq</code>.</p><p><strong>As a function of time</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p><code>f</code> is parsed by Symbolics.jl</p><p><strong>parametric plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p>Parametric plot of f[1] against f[2]</p><p>Also callable as plot!</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/plotting.jl#L4-L22" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector</span></span></code></pre></div><p>Returns an array of booleans classifying each branch in the solutions in <code>res</code> according to <code>class</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/classification.jl#L53" target="_blank" rel="noreferrer">source</a></p>`,6))]),s[158]||(s[158]=i("h3",{id:"plotting",tabindex:"-1"},[a("Plotting "),i("a",{class:"header-anchor",href:"#plotting","aria-label":'Permalink to "Plotting"'},"")],-1)),i("details",R,[i("summary",null,[s[89]||(s[89]=i("a",{id:"RecipesBase.plot",href:"#RecipesBase.plot"},[i("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[90]||(s[90]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[91]||(s[91]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESolution</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, f</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, harm_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a function <code>f</code> of a time-dependent solution <code>soln</code> of <code>harm_eq</code>.</p><p><strong>As a function of time</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p><code>f</code> is parsed by Symbolics.jl</p><p><strong>parametric plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p>Parametric plot of f[1] against f[2]</p><p>Also callable as plot!</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/plotting.jl#L4-L22" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result{S, P, D, F}</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> where</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">FunctionWrappers.FunctionWrapper{Array{S, 2}, Tuple{Array{S, 1}}}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> varargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> cut,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Plot</span></span></code></pre></div><p><strong>Plot a <code>Result</code> object.</strong></p><p>Class selection done by passing <code>String</code> or <code>Vector{String}</code> as kwarg:</p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>class : only plot solutions in this class(es) ("all" --> plot everything)</span></span>
<span class="line"><span>not_class : do not plot solutions in this class(es)</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr().</p><p>See also <code>plot!</code></p><p>The x,y,z arguments are Strings compatible with Symbolics.jl, e.g., <code>y=2*sqrt(u1^2+v1^2)</code> plots the amplitude of the first quadratures multiplied by 2.</p><p><strong>1D plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(res::Result; x::String, y::String, class="default", not_class=[], kwargs...)</span></span>
-<span class="line"><span>plot(res::Result, y::String; kwargs...) # take x automatically from Result</span></span></code></pre></div><p>Default behaviour is to plot stable solutions as full lines, unstable as dashed.</p><p>If a sweep in two parameters were done, i.e., <code>dim(res)==2</code>, a one dimensional cut can be plotted by using the keyword <code>cut</code> were it takes a <code>Pair{Num, Float}</code> type entry. For example, <code>plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2))</code> plots a cut at <code>λ = 0.2</code>.</p><hr><p><strong>2D plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)</span></span></code></pre></div><p>To make the 2d plot less chaotic it is required to specify the specific <code>branch</code> to plot, labeled by a <code>Int64</code>.</p><p>The x and y axes are taken automatically from <code>res</code></p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/plotting_Plots.jl#L11" target="_blank" rel="noreferrer">source</a></p>`,27))]),i("details",Z,[i("summary",null,[s[92]||(s[92]=i("a",{id:"RecipesBase.plot!",href:"#RecipesBase.plot!"},[i("span",{class:"jlbinding"},"RecipesBase.plot!")],-1)),s[93]||(s[93]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[94]||(s[94]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span>plot(res::Result, y::String; kwargs...) # take x automatically from Result</span></span></code></pre></div><p>Default behaviour is to plot stable solutions as full lines, unstable as dashed.</p><p>If a sweep in two parameters were done, i.e., <code>dim(res)==2</code>, a one dimensional cut can be plotted by using the keyword <code>cut</code> were it takes a <code>Pair{Num, Float}</code> type entry. For example, <code>plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2))</code> plots a cut at <code>λ = 0.2</code>.</p><hr><p><strong>2D plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)</span></span></code></pre></div><p>To make the 2d plot less chaotic it is required to specify the specific <code>branch</code> to plot, labeled by a <code>Int64</code>.</p><p>The x and y axes are taken automatically from <code>res</code></p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/plotting_Plots.jl#L11" target="_blank" rel="noreferrer">source</a></p>`,27))]),i("details",Z,[i("summary",null,[s[92]||(s[92]=i("a",{id:"RecipesBase.plot!",href:"#RecipesBase.plot!"},[i("span",{class:"jlbinding"},"RecipesBase.plot!")],-1)),s[93]||(s[93]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[94]||(s[94]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> varargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Plot</span></span></code></pre></div><p>Similar to <code>plot</code> but adds a plot onto an existing plot.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/plotting_Plots.jl#L62" target="_blank" rel="noreferrer">source</a></p>`,3))]),i("details",z,[i("summary",null,[s[95]||(s[95]=i("a",{id:"HarmonicBalance.plot_phase_diagram",href:"#HarmonicBalance.plot_phase_diagram"},[i("span",{class:"jlbinding"},"HarmonicBalance.plot_phase_diagram")],-1)),s[96]||(s[96]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[97]||(s[97]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_phase_diagram</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Plot</span></span></code></pre></div><p>Similar to <code>plot</code> but adds a plot onto an existing plot.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/plotting_Plots.jl#L62" target="_blank" rel="noreferrer">source</a></p>`,3))]),i("details",z,[i("summary",null,[s[95]||(s[95]=i("a",{id:"HarmonicBalance.plot_phase_diagram",href:"#HarmonicBalance.plot_phase_diagram"},[i("span",{class:"jlbinding"},"HarmonicBalance.plot_phase_diagram")],-1)),s[96]||(s[96]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[97]||(s[97]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_phase_diagram</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Plot</span></span></code></pre></div><p>Plot the number of solutions in a <code>Result</code> object as a function of the parameters. Works with 1D and 2D datasets.</p><p>Class selection done by passing <code>String</code> or <code>Vector{String}</code> as kwarg:</p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>class::String : only count solutions in this class ("all" --> plot everything)</span></span>
-<span class="line"><span>not_class::String : do not count solutions in this class</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr()</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/plotting_Plots.jl#L275" target="_blank" rel="noreferrer">source</a></p>`,6))]),i("details",W,[i("summary",null,[s[98]||(s[98]=i("a",{id:"HarmonicBalance.plot_spaghetti",href:"#HarmonicBalance.plot_spaghetti"},[i("span",{class:"jlbinding"},"HarmonicBalance.plot_spaghetti")],-1)),s[99]||(s[99]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[100]||(s[100]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_spaghetti</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; x, y, z, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a three dimension line plot of a <code>Result</code> object as a function of the parameters. Works with 1D and 2D datasets.</p><p>Class selection done by passing <code>String</code> or <code>Vector{String}</code> as kwarg:</p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>class::String : only count solutions in this class ("all" --> plot everything)</span></span>
-<span class="line"><span>not_class::String : do not count solutions in this class</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr()</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/plotting_Plots.jl#L342-L354" target="_blank" rel="noreferrer">source</a></p>`,6))]),s[159]||(s[159]=i("h2",{id:"Limit-cycles",tabindex:"-1"},[a("Limit cycles "),i("a",{class:"header-anchor",href:"#Limit-cycles","aria-label":'Permalink to "Limit cycles {#Limit-cycles}"'},"")],-1)),i("details",U,[i("summary",null,[s[101]||(s[101]=i("a",{id:"HarmonicBalance.LimitCycles.get_limit_cycles",href:"#HarmonicBalance.LimitCycles.get_limit_cycles"},[i("span",{class:"jlbinding"},"HarmonicBalance.LimitCycles.get_limit_cycles")],-1)),s[102]||(s[102]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[103]||(s[103]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_limit_cycles</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, method</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalanceMethod</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, swept, fixed, ω_lc; kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Variant of <code>get_steady_states</code> for a limit cycle problem characterised by a Hopf frequency (usually called ω_lc)</p><p>Solutions with ω_lc = 0 are labelled unphysical since this contradicts the assumption of distinct harmonic variables corresponding to distinct harmonics.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LimitCycles/gauge_fixing.jl#L106-L113" target="_blank" rel="noreferrer">source</a></p>`,4))]),i("details",J,[i("summary",null,[s[104]||(s[104]=i("a",{id:"HarmonicBalance.LimitCycles.get_cycle_variables",href:"#HarmonicBalance.LimitCycles.get_cycle_variables"},[i("span",{class:"jlbinding"},"HarmonicBalance.LimitCycles.get_cycle_variables")],-1)),s[105]||(s[105]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[106]||(s[106]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_cycle_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span>not_class::String : do not count solutions in this class</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr()</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/plotting_Plots.jl#L275" target="_blank" rel="noreferrer">source</a></p>`,6))]),i("details",W,[i("summary",null,[s[98]||(s[98]=i("a",{id:"HarmonicBalance.plot_spaghetti",href:"#HarmonicBalance.plot_spaghetti"},[i("span",{class:"jlbinding"},"HarmonicBalance.plot_spaghetti")],-1)),s[99]||(s[99]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[100]||(s[100]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_spaghetti</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; x, y, z, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a three dimension line plot of a <code>Result</code> object as a function of the parameters. Works with 1D and 2D datasets.</p><p>Class selection done by passing <code>String</code> or <code>Vector{String}</code> as kwarg:</p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>class::String : only count solutions in this class ("all" --> plot everything)</span></span>
+<span class="line"><span>not_class::String : do not count solutions in this class</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr()</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/plotting_Plots.jl#L342-L354" target="_blank" rel="noreferrer">source</a></p>`,6))]),s[159]||(s[159]=i("h2",{id:"Limit-cycles",tabindex:"-1"},[a("Limit cycles "),i("a",{class:"header-anchor",href:"#Limit-cycles","aria-label":'Permalink to "Limit cycles {#Limit-cycles}"'},"")],-1)),i("details",U,[i("summary",null,[s[101]||(s[101]=i("a",{id:"HarmonicBalance.LimitCycles.get_limit_cycles",href:"#HarmonicBalance.LimitCycles.get_limit_cycles"},[i("span",{class:"jlbinding"},"HarmonicBalance.LimitCycles.get_limit_cycles")],-1)),s[102]||(s[102]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[103]||(s[103]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_limit_cycles</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, method</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalanceMethod</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, swept, fixed, ω_lc; kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Variant of <code>get_steady_states</code> for a limit cycle problem characterised by a Hopf frequency (usually called ω_lc)</p><p>Solutions with ω_lc = 0 are labelled unphysical since this contradicts the assumption of distinct harmonic variables corresponding to distinct harmonics.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LimitCycles/gauge_fixing.jl#L106-L113" target="_blank" rel="noreferrer">source</a></p>`,4))]),i("details",J,[i("summary",null,[s[104]||(s[104]=i("a",{id:"HarmonicBalance.LimitCycles.get_cycle_variables",href:"#HarmonicBalance.LimitCycles.get_cycle_variables"},[i("span",{class:"jlbinding"},"HarmonicBalance.LimitCycles.get_cycle_variables")],-1)),s[105]||(s[105]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[106]||(s[106]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_cycle_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω_lc</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{HarmonicVariable}</span></span></code></pre></div><p>Return the harmonic variables which participate in the limit cycle labelled by <code>ω_lc</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LimitCycles/gauge_fixing.jl#L22" target="_blank" rel="noreferrer">source</a></p>`,3))]),i("details",G,[i("summary",null,[s[107]||(s[107]=i("a",{id:"HarmonicBalance.LimitCycles.add_pairs!",href:"#HarmonicBalance.LimitCycles.add_pairs!"},[i("span",{class:"jlbinding"},"HarmonicBalance.LimitCycles.add_pairs!")],-1)),s[108]||(s[108]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[109]||(s[109]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">add_pairs!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; ω_lc</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, n</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Add a limit cycle harmonic <code>ω_lc</code> to the system Equivalent to adding <code>n</code> pairs of harmonics ω +- ω_lc for each existing ω.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LimitCycles/gauge_fixing.jl#L9-L14" target="_blank" rel="noreferrer">source</a></p>',3))]),s[160]||(s[160]=i("h2",{id:"Linear-Response",tabindex:"-1"},[a("Linear Response "),i("a",{class:"header-anchor",href:"#Linear-Response","aria-label":'Permalink to "Linear Response {#Linear-Response}"'},"")],-1)),i("details",K,[i("summary",null,[s[110]||(s[110]=i("a",{id:"HarmonicBalance.LinearResponse.plot_eigenvalues-Union{Tuple{HarmonicBalance.Result{S, P, D, F} where {D, F<:FunctionWrappers.FunctionWrapper{Matrix{S}, Tuple{Vector{S}}}}}, Tuple{P}, Tuple{S}} where {S, P}",href:"#HarmonicBalance.LinearResponse.plot_eigenvalues-Union{Tuple{HarmonicBalance.Result{S, P, D, F} where {D, F<:FunctionWrappers.FunctionWrapper{Matrix{S}, Tuple{Vector{S}}}}}, Tuple{P}, Tuple{S}} where {S, P}"},[i("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.plot_eigenvalues")],-1)),s[111]||(s[111]=a()),e(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[112]||(s[112]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_eigenvalues</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, class</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">[</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"physical"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], type</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">:imag</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, projection</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">v </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, cscheme</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">:default</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot the eigenvalues of the jacobian in the rotating frame for Result <code>res</code> on <code>branch</code>. Either the real (<code>type=:real``) or imaginary part (</code>type=:imag``) can be plotted. The<code>projection</code> function ℜᵈ → ℜ is applied to the eigenvectors and defines the color of the eigenvalues. The color scheme can be set to a custom one or to the default one.</p><p>Any kwargs are fed to Plots' gr().</p><p>Solutions not belonging to the <code>physical</code> class are ignored.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/plotting.jl#L259-L267" target="_blank" rel="noreferrer">source</a></p>',5))]),i("details",$,[i("summary",null,[s[113]||(s[113]=i("a",{id:"HarmonicBalance.LinearResponse.plot_linear_response-Tuple{HarmonicBalance.Result, Num}",href:"#HarmonicBalance.LinearResponse.plot_linear_response-Tuple{HarmonicBalance.Result, Num}"},[i("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.plot_linear_response")],-1)),s[114]||(s[114]=a()),e(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[115]||(s[115]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_linear_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, nat_var</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; Ω_range, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, order</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, logscale</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">false</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, show_progress</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot the linear response to white noise of the variable <code>nat_var</code> for Result <code>res</code> on <code>branch</code> for input frequencies <code>Ω_range</code>. Slow-time derivatives up to <code>order</code> are kept in the process.</p><p>Any kwargs are fed to Plots' gr().</p><p>Solutions not belonging to the <code>physical</code> class are ignored.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/plotting.jl#L120-L129" target="_blank" rel="noreferrer">source</a></p>',5))]),i("details",X,[i("summary",null,[s[116]||(s[116]=i("a",{id:"HarmonicBalance.LinearResponse.plot_rotframe_jacobian_response-Union{Tuple{HarmonicBalance.Result{S, P, D, F} where {D, F<:FunctionWrappers.FunctionWrapper{Matrix{S}, Tuple{Vector{S}}}}}, Tuple{P}, Tuple{S}} where {S, P}",href:"#HarmonicBalance.LinearResponse.plot_rotframe_jacobian_response-Union{Tuple{HarmonicBalance.Result{S, P, D, F} where {D, F<:FunctionWrappers.FunctionWrapper{Matrix{S}, Tuple{Vector{S}}}}}, Tuple{P}, Tuple{S}} where {S, P}"},[i("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.plot_rotframe_jacobian_response")],-1)),s[117]||(s[117]=a()),e(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[118]||(s[118]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_rotframe_jacobian_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; Ω_range, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, logscale</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, damping_mod </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, show_progress</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot the linear response to white noise in the rotating frame for Result <code>res</code> on <code>branch</code> for input frequencies <code>Ω_range</code>. 'damping_mod' gets multiplied by the real part of the eigenvalues of the Jacobian in order to be able to make peaks with similar frequency separately identifiable.</p><p>Any kwargs are fed to Plots' gr().</p><p>Solutions not belonging to the <code>physical</code> class are ignored.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/plotting.jl#L214-L222" target="_blank" rel="noreferrer">source</a></p>',5))]),i("details",Y,[i("summary",null,[s[119]||(s[119]=i("a",{id:"HarmonicBalance.get_Jacobian",href:"#HarmonicBalance.get_Jacobian"},[i("span",{class:"jlbinding"},"HarmonicBalance.get_Jacobian")],-1)),s[120]||(s[120]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[121]||(s[121]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_Jacobian</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom)</span></span></code></pre></div><p>Obtain the symbolic Jacobian matrix of <code>eom</code>. This is the linearised left-hand side of F(u) = du/dT.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/Jacobian.jl#L59" target="_blank" rel="noreferrer">source</a></p><p>Obtain a Jacobian from a <code>DifferentialEquation</code> by first converting it into a <code>HarmonicEquation</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/Jacobian.jl#L77" target="_blank" rel="noreferrer">source</a></p><p>Get the Jacobian of a set of equations <code>eqs</code> with respect to the variables <code>vars</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/Jacobian.jl#L86" target="_blank" rel="noreferrer">source</a></p>',7))]),s[161]||(s[161]=i("h2",{id:"extensions",tabindex:"-1"},[a("Extensions "),i("a",{class:"header-anchor",href:"#extensions","aria-label":'Permalink to "Extensions"'},"")],-1)),s[162]||(s[162]=i("h3",{id:"ordinarydiffeq",tabindex:"-1"},[a("OrdinaryDiffEq "),i("a",{class:"header-anchor",href:"#ordinarydiffeq","aria-label":'Permalink to "OrdinaryDiffEq"'},"")],-1)),i("details",ss,[i("summary",null,[s[122]||(s[122]=i("a",{id:"HarmonicBalance.AdiabaticSweep",href:"#HarmonicBalance.AdiabaticSweep"},[i("span",{class:"jlbinding"},"HarmonicBalance.AdiabaticSweep")],-1)),s[123]||(s[123]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[124]||(s[124]=t(`<p>Represents a sweep of one or more parameters of a <code>HarmonicEquation</code>. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.</p><p>Sweeps of different variables can be combined using <code>+</code>.</p><p><strong>Fields</strong></p><ul><li><code>functions::Dict{Num, Function}</code>: Maps each swept parameter to a function.</li></ul><p><strong>Examples</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># create a sweep of parameter a from 0 to 1 over time 0 -> 100</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{HarmonicVariable}</span></span></code></pre></div><p>Return the harmonic variables which participate in the limit cycle labelled by <code>ω_lc</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LimitCycles/gauge_fixing.jl#L22" target="_blank" rel="noreferrer">source</a></p>`,3))]),i("details",G,[i("summary",null,[s[107]||(s[107]=i("a",{id:"HarmonicBalance.LimitCycles.add_pairs!",href:"#HarmonicBalance.LimitCycles.add_pairs!"},[i("span",{class:"jlbinding"},"HarmonicBalance.LimitCycles.add_pairs!")],-1)),s[108]||(s[108]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[109]||(s[109]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">add_pairs!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; ω_lc</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, n</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Add a limit cycle harmonic <code>ω_lc</code> to the system Equivalent to adding <code>n</code> pairs of harmonics ω +- ω_lc for each existing ω.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LimitCycles/gauge_fixing.jl#L9-L14" target="_blank" rel="noreferrer">source</a></p>',3))]),s[160]||(s[160]=i("h2",{id:"Linear-Response",tabindex:"-1"},[a("Linear Response "),i("a",{class:"header-anchor",href:"#Linear-Response","aria-label":'Permalink to "Linear Response {#Linear-Response}"'},"")],-1)),i("details",K,[i("summary",null,[s[110]||(s[110]=i("a",{id:"HarmonicBalance.LinearResponse.plot_eigenvalues-Union{Tuple{HarmonicBalance.Result{S, P, D, F} where {D, F<:FunctionWrappers.FunctionWrapper{Matrix{S}, Tuple{Vector{S}}}}}, Tuple{P}, Tuple{S}} where {S, P}",href:"#HarmonicBalance.LinearResponse.plot_eigenvalues-Union{Tuple{HarmonicBalance.Result{S, P, D, F} where {D, F<:FunctionWrappers.FunctionWrapper{Matrix{S}, Tuple{Vector{S}}}}}, Tuple{P}, Tuple{S}} where {S, P}"},[i("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.plot_eigenvalues")],-1)),s[111]||(s[111]=a()),e(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[112]||(s[112]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_eigenvalues</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, class</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">[</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"physical"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], type</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">:imag</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, projection</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">v </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, cscheme</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">:default</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot the eigenvalues of the jacobian in the rotating frame for Result <code>res</code> on <code>branch</code>. Either the real (<code>type=:real``) or imaginary part (</code>type=:imag``) can be plotted. The<code>projection</code> function ℜᵈ → ℜ is applied to the eigenvectors and defines the color of the eigenvalues. The color scheme can be set to a custom one or to the default one.</p><p>Any kwargs are fed to Plots' gr().</p><p>Solutions not belonging to the <code>physical</code> class are ignored.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/plotting.jl#L259-L267" target="_blank" rel="noreferrer">source</a></p>',5))]),i("details",$,[i("summary",null,[s[113]||(s[113]=i("a",{id:"HarmonicBalance.LinearResponse.plot_linear_response-Tuple{HarmonicBalance.Result, Num}",href:"#HarmonicBalance.LinearResponse.plot_linear_response-Tuple{HarmonicBalance.Result, Num}"},[i("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.plot_linear_response")],-1)),s[114]||(s[114]=a()),e(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[115]||(s[115]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_linear_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, nat_var</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; Ω_range, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, order</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, logscale</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">false</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, show_progress</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot the linear response to white noise of the variable <code>nat_var</code> for Result <code>res</code> on <code>branch</code> for input frequencies <code>Ω_range</code>. Slow-time derivatives up to <code>order</code> are kept in the process.</p><p>Any kwargs are fed to Plots' gr().</p><p>Solutions not belonging to the <code>physical</code> class are ignored.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/plotting.jl#L120-L129" target="_blank" rel="noreferrer">source</a></p>',5))]),i("details",X,[i("summary",null,[s[116]||(s[116]=i("a",{id:"HarmonicBalance.LinearResponse.plot_rotframe_jacobian_response-Union{Tuple{HarmonicBalance.Result{S, P, D, F} where {D, F<:FunctionWrappers.FunctionWrapper{Matrix{S}, Tuple{Vector{S}}}}}, Tuple{P}, Tuple{S}} where {S, P}",href:"#HarmonicBalance.LinearResponse.plot_rotframe_jacobian_response-Union{Tuple{HarmonicBalance.Result{S, P, D, F} where {D, F<:FunctionWrappers.FunctionWrapper{Matrix{S}, Tuple{Vector{S}}}}}, Tuple{P}, Tuple{S}} where {S, P}"},[i("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.plot_rotframe_jacobian_response")],-1)),s[117]||(s[117]=a()),e(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[118]||(s[118]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_rotframe_jacobian_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; Ω_range, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, logscale</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, damping_mod </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, show_progress</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot the linear response to white noise in the rotating frame for Result <code>res</code> on <code>branch</code> for input frequencies <code>Ω_range</code>. 'damping_mod' gets multiplied by the real part of the eigenvalues of the Jacobian in order to be able to make peaks with similar frequency separately identifiable.</p><p>Any kwargs are fed to Plots' gr().</p><p>Solutions not belonging to the <code>physical</code> class are ignored.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/plotting.jl#L214-L222" target="_blank" rel="noreferrer">source</a></p>',5))]),i("details",Y,[i("summary",null,[s[119]||(s[119]=i("a",{id:"HarmonicBalance.get_Jacobian",href:"#HarmonicBalance.get_Jacobian"},[i("span",{class:"jlbinding"},"HarmonicBalance.get_Jacobian")],-1)),s[120]||(s[120]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[121]||(s[121]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_Jacobian</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom)</span></span></code></pre></div><p>Obtain the symbolic Jacobian matrix of <code>eom</code>. This is the linearised left-hand side of F(u) = du/dT.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/Jacobian.jl#L59" target="_blank" rel="noreferrer">source</a></p><p>Obtain a Jacobian from a <code>DifferentialEquation</code> by first converting it into a <code>HarmonicEquation</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/Jacobian.jl#L77" target="_blank" rel="noreferrer">source</a></p><p>Get the Jacobian of a set of equations <code>eqs</code> with respect to the variables <code>vars</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/Jacobian.jl#L86" target="_blank" rel="noreferrer">source</a></p>',7))]),s[161]||(s[161]=i("h2",{id:"extensions",tabindex:"-1"},[a("Extensions "),i("a",{class:"header-anchor",href:"#extensions","aria-label":'Permalink to "Extensions"'},"")],-1)),s[162]||(s[162]=i("h3",{id:"ordinarydiffeq",tabindex:"-1"},[a("OrdinaryDiffEq "),i("a",{class:"header-anchor",href:"#ordinarydiffeq","aria-label":'Permalink to "OrdinaryDiffEq"'},"")],-1)),i("details",ss,[i("summary",null,[s[122]||(s[122]=i("a",{id:"HarmonicBalance.AdiabaticSweep",href:"#HarmonicBalance.AdiabaticSweep"},[i("span",{class:"jlbinding"},"HarmonicBalance.AdiabaticSweep")],-1)),s[123]||(s[123]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[124]||(s[124]=t(`<p>Represents a sweep of one or more parameters of a <code>HarmonicEquation</code>. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.</p><p>Sweeps of different variables can be combined using <code>+</code>.</p><p><strong>Fields</strong></p><ul><li><code>functions::Dict{Num, Function}</code>: Maps each swept parameter to a function.</li></ul><p><strong>Examples</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># create a sweep of parameter a from 0 to 1 over time 0 -> 100</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> a,b;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(a </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">100</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">));</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep[a](</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">50</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
@@ -170,22 +170,22 @@ import{_ as h,c as l,ai as t,j as i,a,G as e,B as r,o as p}from"./chunks/framewo
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">([a </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], b </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">]], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">100</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span></code></pre></div><p>Successive sweeps can be combined,</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep1 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.95</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep2 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(λ </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.05</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.01</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">4e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep1 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep2</span></span></code></pre></div><p>multiple parameters can be swept simultaneously,</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">([ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.95</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], λ </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">5e-2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1e-2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">]], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span></code></pre></div><p>and custom sweep functions may be used.</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#6F42C1;--shiki-dark:#B392F0;">ωfunc</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ωfunc)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/types.jl#L9-L48" target="_blank" rel="noreferrer">source</a></p>`,13))]),i("details",is,[i("summary",null,[s[125]||(s[125]=i("a",{id:"HarmonicBalance.follow_branch",href:"#HarmonicBalance.follow_branch"},[i("span",{class:"jlbinding"},"HarmonicBalance.follow_branch")],-1)),s[126]||(s[126]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[127]||(s[127]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">follow_branch</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ωfunc)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/types.jl#L9-L48" target="_blank" rel="noreferrer">source</a></p>`,13))]),i("details",is,[i("summary",null,[s[125]||(s[125]=i("a",{id:"HarmonicBalance.follow_branch",href:"#HarmonicBalance.follow_branch"},[i("span",{class:"jlbinding"},"HarmonicBalance.follow_branch")],-1)),s[126]||(s[126]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[127]||(s[127]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">follow_branch</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> starting_branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int64</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> y,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> tf,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ϵ</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Tuple{Any, Any}</span></span></code></pre></div><p>Return the indexes and values following stable branches along a 1D sweep. When a no stable solutions are found (e.g. in a bifurcation), the next stable solution is calculated by time evolving the previous solution (quench).</p><p><strong>Keyword arguments</strong></p><ul><li><p><code>y</code>: Dependent variable expression (parsed into Symbolics.jl) to evaluate the followed solution branches on .</p></li><li><p><code>sweep</code>: Direction for the sweeping of solutions. A <code>right</code> (<code>left</code>) sweep proceeds from the first (last) solution, ordered as the sweeping parameter.</p></li><li><p><code>tf</code>: time to reach steady</p></li><li><p><code>ϵ</code>: small random perturbation applied to quenched solution, in a bifurcation in order to favour convergence in cases where multiple solutions are identically accessible (e.g. symmetry breaking into two equal amplitude states)</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/hysteresis_sweep.jl#L15" target="_blank" rel="noreferrer">source</a></p>`,5))]),i("details",as,[i("summary",null,[s[128]||(s[128]=i("a",{id:"HarmonicBalance.plot_1D_solutions_branch",href:"#HarmonicBalance.plot_1D_solutions_branch"},[i("span",{class:"jlbinding"},"HarmonicBalance.plot_1D_solutions_branch")],-1)),s[129]||(s[129]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[130]||(s[130]=i("p",null,"1D plot with the followed branch highlighted",-1)),s[131]||(s[131]=i("p",null,[i("a",{href:"https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/plotting.jl#L62-L64",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",ts,[i("summary",null,[s[132]||(s[132]=i("a",{id:"HarmonicBalance.follow_branch-Tuple{Int64, HarmonicBalance.Result}-manual-API",href:"#HarmonicBalance.follow_branch-Tuple{Int64, HarmonicBalance.Result}-manual-API"},[i("span",{class:"jlbinding"},"HarmonicBalance.follow_branch")],-1)),s[133]||(s[133]=a()),e(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[134]||(s[134]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">follow_branch</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Tuple{Any, Any}</span></span></code></pre></div><p>Return the indexes and values following stable branches along a 1D sweep. When a no stable solutions are found (e.g. in a bifurcation), the next stable solution is calculated by time evolving the previous solution (quench).</p><p><strong>Keyword arguments</strong></p><ul><li><p><code>y</code>: Dependent variable expression (parsed into Symbolics.jl) to evaluate the followed solution branches on .</p></li><li><p><code>sweep</code>: Direction for the sweeping of solutions. A <code>right</code> (<code>left</code>) sweep proceeds from the first (last) solution, ordered as the sweeping parameter.</p></li><li><p><code>tf</code>: time to reach steady</p></li><li><p><code>ϵ</code>: small random perturbation applied to quenched solution, in a bifurcation in order to favour convergence in cases where multiple solutions are identically accessible (e.g. symmetry breaking into two equal amplitude states)</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/hysteresis_sweep.jl#L15" target="_blank" rel="noreferrer">source</a></p>`,5))]),i("details",as,[i("summary",null,[s[128]||(s[128]=i("a",{id:"HarmonicBalance.plot_1D_solutions_branch",href:"#HarmonicBalance.plot_1D_solutions_branch"},[i("span",{class:"jlbinding"},"HarmonicBalance.plot_1D_solutions_branch")],-1)),s[129]||(s[129]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[130]||(s[130]=i("p",null,"1D plot with the followed branch highlighted",-1)),s[131]||(s[131]=i("p",null,[i("a",{href:"https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/plotting.jl#L62-L64",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",ts,[i("summary",null,[s[132]||(s[132]=i("a",{id:"HarmonicBalance.follow_branch-Tuple{Int64, HarmonicBalance.Result}-manual-API",href:"#HarmonicBalance.follow_branch-Tuple{Int64, HarmonicBalance.Result}-manual-API"},[i("span",{class:"jlbinding"},"HarmonicBalance.follow_branch")],-1)),s[133]||(s[133]=a()),e(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[134]||(s[134]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">follow_branch</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> starting_branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int64</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> y,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> tf,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ϵ</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Tuple{Any, Any}</span></span></code></pre></div><p>Return the indexes and values following stable branches along a 1D sweep. When a no stable solutions are found (e.g. in a bifurcation), the next stable solution is calculated by time evolving the previous solution (quench).</p><p><strong>Keyword arguments</strong></p><ul><li><p><code>y</code>: Dependent variable expression (parsed into Symbolics.jl) to evaluate the followed solution branches on .</p></li><li><p><code>sweep</code>: Direction for the sweeping of solutions. A <code>right</code> (<code>left</code>) sweep proceeds from the first (last) solution, ordered as the sweeping parameter.</p></li><li><p><code>tf</code>: time to reach steady</p></li><li><p><code>ϵ</code>: small random perturbation applied to quenched solution, in a bifurcation in order to favour convergence in cases where multiple solutions are identically accessible (e.g. symmetry breaking into two equal amplitude states)</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/hysteresis_sweep.jl#L15" target="_blank" rel="noreferrer">source</a></p>`,5))]),i("details",ns,[i("summary",null,[s[135]||(s[135]=i("a",{id:"RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-API",href:"#RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-API"},[i("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[136]||(s[136]=a()),e(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[137]||(s[137]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESolution</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, f</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, harm_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a function <code>f</code> of a time-dependent solution <code>soln</code> of <code>harm_eq</code>.</p><p><strong>As a function of time</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p><code>f</code> is parsed by Symbolics.jl</p><p><strong>parametric plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p>Parametric plot of f[1] against f[2]</p><p>Also callable as plot!</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/plotting.jl#L4-L22" target="_blank" rel="noreferrer">source</a></p>',10))]),s[163]||(s[163]=i("h3",{id:"steadystatesweep",tabindex:"-1"},[a("SteadyStateSweep "),i("a",{class:"header-anchor",href:"#steadystatesweep","aria-label":'Permalink to "SteadyStateSweep"'},"")],-1)),i("details",es,[i("summary",null,[s[138]||(s[138]=i("a",{id:"HarmonicBalance.steady_state_sweep",href:"#HarmonicBalance.steady_state_sweep"},[i("span",{class:"jlbinding"},"HarmonicBalance.steady_state_sweep")],-1)),s[139]||(s[139]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[140]||(s[140]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">steady_state_sweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(prob</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">SteadyStateProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, alg</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DynamicSS</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; varied</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Pair</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Sweeps through a range of parameter values using a dynamic steady state solver <code>DynamicSS</code> of the <code>SteadyStateDiffEq.jl</code> package. Given a steady state problem and a parameter to vary, computes the steady state solution for each value in the sweep range. The solutions are returned as a vector where each element corresponds to the steady state found at that parameter value.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/SteadyStateDiffEqExt.jl#L10-L18" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">steady_state_sweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(prob_np</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">NonlinearProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, prob_ss</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">SteadyStateProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> alg_np, alg_ss</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DynamicSS</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; varied</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Pair</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Performs a parameter sweep by combining nonlinear root <code>alg_np</code> and steady state solvers <code>alg_ss</code>. For each parameter value, it first attempts a direct nonlinear root solver and checks its stability. If the solution is unstable or not found, it switches to a dynamic steady state solver. This hybrid approach is much faster then only using a steady state solver.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/SteadyStateDiffEqExt.jl#L37-L45" target="_blank" rel="noreferrer">source</a></p>`,6))]),s[164]||(s[164]=i("h3",{id:"modelingtoolkit",tabindex:"-1"},[a("ModelingToolkit "),i("a",{class:"header-anchor",href:"#modelingtoolkit","aria-label":'Permalink to "ModelingToolkit"'},"")],-1)),i("details",ls,[i("summary",null,[s[141]||(s[141]=i("a",{id:"SciMLBase.ODEProblem",href:"#SciMLBase.ODEProblem"},[i("span",{class:"jlbinding"},"SciMLBase.ODEProblem")],-1)),s[142]||(s[142]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[143]||(s[143]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODEProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Tuple{Any, Any}</span></span></code></pre></div><p>Return the indexes and values following stable branches along a 1D sweep. When a no stable solutions are found (e.g. in a bifurcation), the next stable solution is calculated by time evolving the previous solution (quench).</p><p><strong>Keyword arguments</strong></p><ul><li><p><code>y</code>: Dependent variable expression (parsed into Symbolics.jl) to evaluate the followed solution branches on .</p></li><li><p><code>sweep</code>: Direction for the sweeping of solutions. A <code>right</code> (<code>left</code>) sweep proceeds from the first (last) solution, ordered as the sweeping parameter.</p></li><li><p><code>tf</code>: time to reach steady</p></li><li><p><code>ϵ</code>: small random perturbation applied to quenched solution, in a bifurcation in order to favour convergence in cases where multiple solutions are identically accessible (e.g. symmetry breaking into two equal amplitude states)</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/hysteresis_sweep.jl#L15" target="_blank" rel="noreferrer">source</a></p>`,5))]),i("details",ns,[i("summary",null,[s[135]||(s[135]=i("a",{id:"RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-API",href:"#RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-API"},[i("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[136]||(s[136]=a()),e(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[137]||(s[137]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESolution</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, f</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, harm_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a function <code>f</code> of a time-dependent solution <code>soln</code> of <code>harm_eq</code>.</p><p><strong>As a function of time</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p><code>f</code> is parsed by Symbolics.jl</p><p><strong>parametric plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p>Parametric plot of f[1] against f[2]</p><p>Also callable as plot!</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/plotting.jl#L4-L22" target="_blank" rel="noreferrer">source</a></p>',10))]),s[163]||(s[163]=i("h3",{id:"steadystatesweep",tabindex:"-1"},[a("SteadyStateSweep "),i("a",{class:"header-anchor",href:"#steadystatesweep","aria-label":'Permalink to "SteadyStateSweep"'},"")],-1)),i("details",es,[i("summary",null,[s[138]||(s[138]=i("a",{id:"HarmonicBalance.steady_state_sweep",href:"#HarmonicBalance.steady_state_sweep"},[i("span",{class:"jlbinding"},"HarmonicBalance.steady_state_sweep")],-1)),s[139]||(s[139]=a()),e(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[140]||(s[140]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">steady_state_sweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(prob</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">SteadyStateProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, alg</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DynamicSS</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; varied</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Pair</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Sweeps through a range of parameter values using a dynamic steady state solver <code>DynamicSS</code> of the <code>SteadyStateDiffEq.jl</code> package. Given a steady state problem and a parameter to vary, computes the steady state solution for each value in the sweep range. The solutions are returned as a vector where each element corresponds to the steady state found at that parameter value.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/SteadyStateDiffEqExt.jl#L10-L18" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">steady_state_sweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(prob_np</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">NonlinearProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, prob_ss</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">SteadyStateProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> alg_np, alg_ss</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DynamicSS</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; varied</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Pair</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Performs a parameter sweep by combining nonlinear root <code>alg_np</code> and steady state solvers <code>alg_ss</code>. For each parameter value, it first attempts a direct nonlinear root solver and checks its stability. If the solution is unstable or not found, it switches to a dynamic steady state solver. This hybrid approach is much faster then only using a steady state solver.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/SteadyStateDiffEqExt.jl#L37-L45" target="_blank" rel="noreferrer">source</a></p>`,6))]),s[164]||(s[164]=i("h3",{id:"modelingtoolkit",tabindex:"-1"},[a("ModelingToolkit "),i("a",{class:"header-anchor",href:"#modelingtoolkit","aria-label":'Permalink to "ModelingToolkit"'},"")],-1)),i("details",ls,[i("summary",null,[s[141]||(s[141]=i("a",{id:"SciMLBase.ODEProblem",href:"#SciMLBase.ODEProblem"},[i("span",{class:"jlbinding"},"SciMLBase.ODEProblem")],-1)),s[142]||(s[142]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[143]||(s[143]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODEProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> fixed_parameters;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep,</span></span>
@@ -193,23 +193,23 @@ import{_ as h,c as l,ai as t,j as i,a,G as e,B as r,o as p}from"./chunks/framewo
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> timespan,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> perturb_initial,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Creates an ODEProblem object used by OrdinaryDiffEqTsit5.jl from the equations in <code>eom</code> to simulate time-evolution within <code>timespan</code>. <code>fixed_parameters</code> must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If <code>u0</code> is specified, it is used as an initial condition; otherwise the values from <code>fixed_parameters</code> are used.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/ODEProblem.jl#L3-L9" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODEProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Creates an ODEProblem object used by OrdinaryDiffEqTsit5.jl from the equations in <code>eom</code> to simulate time-evolution within <code>timespan</code>. <code>fixed_parameters</code> must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If <code>u0</code> is specified, it is used as an initial condition; otherwise the values from <code>fixed_parameters</code> are used.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/ODEProblem.jl#L3-L9" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODEProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Union{DifferentialEquation, HarmonicEquation}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> u0,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> tspan</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Tuple</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> p</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">AbstractDict</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> in_place,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.ODEProblem from a DifferentialEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/ModelingToolkitExt.jl#L100" target="_blank" rel="noreferrer">source</a></p>`,6))]),i("details",ps,[i("summary",null,[s[144]||(s[144]=i("a",{id:"ModelingToolkit.ODESystem",href:"#ModelingToolkit.ODESystem"},[i("span",{class:"jlbinding"},"ModelingToolkit.ODESystem")],-1)),s[145]||(s[145]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[146]||(s[146]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESystem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.ODESystem from a HarmonicEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/ModelingToolkitExt.jl#L39" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESystem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.ODESystem from a DifferentialEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/ModelingToolkitExt.jl#L69" target="_blank" rel="noreferrer">source</a></p>',6))]),i("details",hs,[i("summary",null,[s[147]||(s[147]=i("a",{id:"SciMLBase.SteadyStateProblem",href:"#SciMLBase.SteadyStateProblem"},[i("span",{class:"jlbinding"},"SciMLBase.SteadyStateProblem")],-1)),s[148]||(s[148]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[149]||(s[149]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">SteadyStateProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.ODEProblem from a DifferentialEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/ModelingToolkitExt.jl#L100" target="_blank" rel="noreferrer">source</a></p>`,6))]),i("details",ps,[i("summary",null,[s[144]||(s[144]=i("a",{id:"ModelingToolkit.ODESystem",href:"#ModelingToolkit.ODESystem"},[i("span",{class:"jlbinding"},"ModelingToolkit.ODESystem")],-1)),s[145]||(s[145]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[146]||(s[146]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESystem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.ODESystem from a HarmonicEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/ModelingToolkitExt.jl#L39" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESystem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.ODESystem from a DifferentialEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/ModelingToolkitExt.jl#L69" target="_blank" rel="noreferrer">source</a></p>',6))]),i("details",hs,[i("summary",null,[s[147]||(s[147]=i("a",{id:"SciMLBase.SteadyStateProblem",href:"#SciMLBase.SteadyStateProblem"},[i("span",{class:"jlbinding"},"SciMLBase.SteadyStateProblem")],-1)),s[148]||(s[148]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[149]||(s[149]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">SteadyStateProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> u0,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> p</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">AbstractDict</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> in_place,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.SteadyStateProblem from a DifferentialEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/ModelingToolkitExt.jl#L135" target="_blank" rel="noreferrer">source</a></p>`,3))]),i("details",rs,[i("summary",null,[s[150]||(s[150]=i("a",{id:"SciMLBase.NonlinearProblem",href:"#SciMLBase.NonlinearProblem"},[i("span",{class:"jlbinding"},"SciMLBase.NonlinearProblem")],-1)),s[151]||(s[151]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[152]||(s[152]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">NonlinearProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.SteadyStateProblem from a DifferentialEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/ModelingToolkitExt.jl#L135" target="_blank" rel="noreferrer">source</a></p>`,3))]),i("details",rs,[i("summary",null,[s[150]||(s[150]=i("a",{id:"SciMLBase.NonlinearProblem",href:"#SciMLBase.NonlinearProblem"},[i("span",{class:"jlbinding"},"SciMLBase.NonlinearProblem")],-1)),s[151]||(s[151]=a()),e(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[152]||(s[152]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">NonlinearProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> u0,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> p</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">AbstractDict</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> in_place,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.NonlinearProblem from a DifferentialEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/ModelingToolkitExt.jl#L123" target="_blank" rel="noreferrer">source</a></p>`,3))])])}const ms=h(k,[["render",ks]]);export{us as __pageData,ms as default};
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.NonlinearProblem from a DifferentialEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/ModelingToolkitExt.jl#L123" target="_blank" rel="noreferrer">source</a></p>`,3))])])}const ms=h(k,[["render",ks]]);export{us as __pageData,ms as default};
diff --git a/dev/assets/manual_API.md.CC4NrY1l.lean.js b/dev/assets/manual_API.md.DBgY29iG.lean.js
similarity index 96%
rename from dev/assets/manual_API.md.CC4NrY1l.lean.js
rename to dev/assets/manual_API.md.DBgY29iG.lean.js
index 8ea3956c5..d7ef57722 100644
--- a/dev/assets/manual_API.md.CC4NrY1l.lean.js
+++ b/dev/assets/manual_API.md.DBgY29iG.lean.js
@@ -1 +1 @@
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diff --git a/dev/assets/manual_Krylov-Bogoliubov_method.md.DUYIP91V.js b/dev/assets/manual_Krylov-Bogoliubov_method.md.CqAi1HuT.js
similarity index 99%
rename from dev/assets/manual_Krylov-Bogoliubov_method.md.DUYIP91V.js
rename to dev/assets/manual_Krylov-Bogoliubov_method.md.CqAi1HuT.js
index 4995d76ac..f24764656 100644
--- a/dev/assets/manual_Krylov-Bogoliubov_method.md.DUYIP91V.js
+++ b/dev/assets/manual_Krylov-Bogoliubov_method.md.CqAi1HuT.js
@@ -26,4 +26,4 @@ import{_ as h,c as t,j as i,a,ai as n,G as k,B as p,o as e}from"./chunks/framewo
<span class="line"></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">/</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T))</span></span>
<span class="line"></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">/</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T))</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/KrylovBogoliubov.jl#L25" target="_blank" rel="noreferrer">source</a></p>`,7))]),s[13]||(s[13]=i("p",null,[a("For further information and a detailed understanding of this method, refer to "),i("a",{href:"https://en.wikipedia.org/wiki/Krylov%E2%80%93Bogoliubov_averaging_method",target:"_blank",rel:"noreferrer"},"Krylov-Bogoliubov averaging method on Wikipedia"),a(".")],-1))])}const f=h(r,[["render",E]]);export{C as __pageData,f as default};
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">/</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T))</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/KrylovBogoliubov.jl#L25" target="_blank" rel="noreferrer">source</a></p>`,7))]),s[13]||(s[13]=i("p",null,[a("For further information and a detailed understanding of this method, refer to "),i("a",{href:"https://en.wikipedia.org/wiki/Krylov%E2%80%93Bogoliubov_averaging_method",target:"_blank",rel:"noreferrer"},"Krylov-Bogoliubov averaging method on Wikipedia"),a(".")],-1))])}const f=h(r,[["render",E]]);export{C as __pageData,f as default};
diff --git a/dev/assets/manual_Krylov-Bogoliubov_method.md.DUYIP91V.lean.js b/dev/assets/manual_Krylov-Bogoliubov_method.md.CqAi1HuT.lean.js
similarity index 100%
rename from dev/assets/manual_Krylov-Bogoliubov_method.md.DUYIP91V.lean.js
rename to dev/assets/manual_Krylov-Bogoliubov_method.md.CqAi1HuT.lean.js
diff --git a/dev/assets/manual_entering_eom.md.BPBBr3gB.js b/dev/assets/manual_entering_eom.md.DPEo_ILC.js
similarity index 97%
rename from dev/assets/manual_entering_eom.md.BPBBr3gB.js
rename to dev/assets/manual_entering_eom.md.DPEo_ILC.js
index 64604a5a7..5d374909d 100644
--- a/dev/assets/manual_entering_eom.md.BPBBr3gB.js
+++ b/dev/assets/manual_entering_eom.md.DPEo_ILC.js
@@ -1,4 +1,4 @@
-import{_ as l,c as h,j as s,a,G as t,ai as e,B as k,o as p}from"./chunks/framework.Bnn8558y.js";const D=JSON.parse('{"title":"Entering equations of motion","description":"","frontmatter":{},"headers":[],"relativePath":"manual/entering_eom.md","filePath":"manual/entering_eom.md"}'),r={name:"manual/entering_eom.md"},d={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""},o={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""};function c(u,i,F,f,m,b){const n=k("Badge");return p(),h("div",null,[i[16]||(i[16]=s("h1",{id:"Entering-equations-of-motion",tabindex:"-1"},[a("Entering equations of motion "),s("a",{class:"header-anchor",href:"#Entering-equations-of-motion","aria-label":'Permalink to "Entering equations of motion {#Entering-equations-of-motion}"'},"")],-1)),i[17]||(i[17]=s("p",null,[a("The struct "),s("code",null,"DifferentialEquation"),a(" is the primary input method; it holds an ODE or a coupled system of ODEs composed of terms with harmonic time-dependence The dependent variables are specified during input, any other symbols are identified as parameters. Information on which variable is to be expanded in which harmonic is specified using "),s("code",null,"add_harmonic!"),a(".")],-1)),i[18]||(i[18]=s("p",null,[s("code",null,"DifferentialEquation.equations"),a(" stores a dictionary assigning variables to equations. This information is necessary because the harmonics belonging to a variable are later used to Fourier-transform its corresponding ODE.")],-1)),s("details",d,[s("summary",null,[i[0]||(i[0]=s("a",{id:"HarmonicBalance.d-manual-entering_eom",href:"#HarmonicBalance.d-manual-entering_eom"},[s("span",{class:"jlbinding"},"HarmonicBalance.d")],-1)),i[1]||(i[1]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[2]||(i[2]=s("p",null,"The derivative of f w.r.t. x of degree deg",-1)),i[3]||(i[3]=s("p",null,[s("a",{href:"https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicVariable.jl#L108",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",E,[s("summary",null,[i[4]||(i[4]=s("a",{id:"HarmonicBalance.DifferentialEquation-manual-entering_eom",href:"#HarmonicBalance.DifferentialEquation-manual-entering_eom"},[s("span",{class:"jlbinding"},"HarmonicBalance.DifferentialEquation")],-1)),i[5]||(i[5]=a()),t(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[6]||(i[6]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> DifferentialEquation</span></span></code></pre></div><p>Holds differential equation(s) of motion and a set of harmonics to expand each variable. This is the primary input for <code>HarmonicBalance.jl</code>. After inputting the equations, the harmonics ansatz needs to be specified using <code>add_harmonic!</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>equations::OrderedCollections.OrderedDict{Num, Equation}</code>: Assigns to each variable an equation of motion.</p></li><li><p><code>harmonics::OrderedCollections.OrderedDict{Num, OrderedCollections.OrderedSet{Num}}</code>: Assigns to each variable a set of harmonics.</p></li></ul><p><strong>Example</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> t, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">y</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), ω0, ω, F, k;</span></span>
+import{_ as l,c as h,j as s,a,G as t,ai as e,B as k,o as p}from"./chunks/framework.Bnn8558y.js";const D=JSON.parse('{"title":"Entering equations of motion","description":"","frontmatter":{},"headers":[],"relativePath":"manual/entering_eom.md","filePath":"manual/entering_eom.md"}'),r={name:"manual/entering_eom.md"},d={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""},o={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""};function c(u,i,F,m,f,C){const n=k("Badge");return p(),h("div",null,[i[16]||(i[16]=s("h1",{id:"Entering-equations-of-motion",tabindex:"-1"},[a("Entering equations of motion "),s("a",{class:"header-anchor",href:"#Entering-equations-of-motion","aria-label":'Permalink to "Entering equations of motion {#Entering-equations-of-motion}"'},"")],-1)),i[17]||(i[17]=s("p",null,[a("The struct "),s("code",null,"DifferentialEquation"),a(" is the primary input method; it holds an ODE or a coupled system of ODEs composed of terms with harmonic time-dependence The dependent variables are specified during input, any other symbols are identified as parameters. Information on which variable is to be expanded in which harmonic is specified using "),s("code",null,"add_harmonic!"),a(".")],-1)),i[18]||(i[18]=s("p",null,[s("code",null,"DifferentialEquation.equations"),a(" stores a dictionary assigning variables to equations. This information is necessary because the harmonics belonging to a variable are later used to Fourier-transform its corresponding ODE.")],-1)),s("details",d,[s("summary",null,[i[0]||(i[0]=s("a",{id:"HarmonicBalance.d-manual-entering_eom",href:"#HarmonicBalance.d-manual-entering_eom"},[s("span",{class:"jlbinding"},"HarmonicBalance.d")],-1)),i[1]||(i[1]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[2]||(i[2]=s("p",null,"The derivative of f w.r.t. x of degree deg",-1)),i[3]||(i[3]=s("p",null,[s("a",{href:"https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicVariable.jl#L108",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",E,[s("summary",null,[i[4]||(i[4]=s("a",{id:"HarmonicBalance.DifferentialEquation-manual-entering_eom",href:"#HarmonicBalance.DifferentialEquation-manual-entering_eom"},[s("span",{class:"jlbinding"},"HarmonicBalance.DifferentialEquation")],-1)),i[5]||(i[5]=a()),t(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[6]||(i[6]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> DifferentialEquation</span></span></code></pre></div><p>Holds differential equation(s) of motion and a set of harmonics to expand each variable. This is the primary input for <code>HarmonicBalance.jl</code>. After inputting the equations, the harmonics ansatz needs to be specified using <code>add_harmonic!</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>equations::OrderedCollections.OrderedDict{Num, Equation}</code>: Assigns to each variable an equation of motion.</p></li><li><p><code>harmonics::OrderedCollections.OrderedDict{Num, OrderedCollections.OrderedSet{Num}}</code>: Assigns to each variable a set of harmonics.</p></li></ul><p><strong>Example</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> t, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">y</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), ω0, ω, F, k;</span></span>
<span class="line"></span>
<span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># equivalent ways to enter the simple harmonic oscillator</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">d</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(x,t,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> *</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> x </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">t), x);</span></span>
@@ -7,7 +7,7 @@ import{_ as l,c as h,j as s,a,G as t,ai as e,B as k,o as p}from"./chunks/framewo
<span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># two coupled oscillators, one of them driven</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">d</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(x,t,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> *</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> x </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> k</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">d</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(y,t,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> *</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> y </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> k</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">x] </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], [x,y]</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">);</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L1" target="_blank" rel="noreferrer">source</a></p>`,7))]),s("details",o,[s("summary",null,[i[7]||(i[7]=s("a",{id:"HarmonicBalance.add_harmonic!-manual-entering_eom",href:"#HarmonicBalance.add_harmonic!-manual-entering_eom"},[s("span",{class:"jlbinding"},"HarmonicBalance.add_harmonic!")],-1)),i[8]||(i[8]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[9]||(i[9]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">add_harmonic!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, var</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ω)</span></span></code></pre></div><p>Add the harmonic <code>ω</code> to the harmonic ansatz used to expand the variable <code>var</code> in <code>diff_eom</code>.</p><p><strong>Example</strong></p><p><strong>define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> t, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">y</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), ω0, ω, F, k;</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">);</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L1" target="_blank" rel="noreferrer">source</a></p>`,7))]),s("details",o,[s("summary",null,[i[7]||(i[7]=s("a",{id:"HarmonicBalance.add_harmonic!-manual-entering_eom",href:"#HarmonicBalance.add_harmonic!-manual-entering_eom"},[s("span",{class:"jlbinding"},"HarmonicBalance.add_harmonic!")],-1)),i[8]||(i[8]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[9]||(i[9]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">add_harmonic!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, var</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ω)</span></span></code></pre></div><p>Add the harmonic <code>ω</code> to the harmonic ansatz used to expand the variable <code>var</code> in <code>diff_eom</code>.</p><p><strong>Example</strong></p><p><strong>define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> t, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">y</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), ω0, ω, F, k;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> diff_eq </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">d</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(x,t,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> *</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> x </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">t), x);</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> add_harmonic!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eq, x, ��) </span><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># expand x using ω</span></span>
<span class="line"></span>
@@ -15,6 +15,6 @@ import{_ as l,c as h,j as s,a,G as t,ai as e,B as k,o as p}from"./chunks/framewo
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Variables</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Harmonic ansatz</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω;</span></span>
<span class="line"></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t))) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L88" target="_blank" rel="noreferrer">source</a></p>`,6))]),s("details",g,[s("summary",null,[i[10]||(i[10]=s("a",{id:"Symbolics.get_variables-Tuple{DifferentialEquation}-manual-entering_eom",href:"#Symbolics.get_variables-Tuple{DifferentialEquation}-manual-entering_eom"},[s("span",{class:"jlbinding"},"Symbolics.get_variables")],-1)),i[11]||(i[11]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[12]||(i[12]=e('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Num}</span></span></code></pre></div><p>Return the dependent variables of <code>diff_eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L113" target="_blank" rel="noreferrer">source</a></p>',3))]),s("details",y,[s("summary",null,[i[13]||(i[13]=s("a",{id:"HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}-manual-entering_eom",href:"#HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}-manual-entering_eom"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_independent_variables")],-1)),i[14]||(i[14]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[15]||(i[15]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_independent_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t))) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L88" target="_blank" rel="noreferrer">source</a></p>`,6))]),s("details",g,[s("summary",null,[i[10]||(i[10]=s("a",{id:"Symbolics.get_variables-Tuple{DifferentialEquation}-manual-entering_eom",href:"#Symbolics.get_variables-Tuple{DifferentialEquation}-manual-entering_eom"},[s("span",{class:"jlbinding"},"Symbolics.get_variables")],-1)),i[11]||(i[11]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[12]||(i[12]=e('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Num}</span></span></code></pre></div><p>Return the dependent variables of <code>diff_eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L113" target="_blank" rel="noreferrer">source</a></p>',3))]),s("details",y,[s("summary",null,[i[13]||(i[13]=s("a",{id:"HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}-manual-entering_eom",href:"#HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}-manual-entering_eom"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_independent_variables")],-1)),i[14]||(i[14]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[15]||(i[15]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_independent_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Return the independent dependent variables of <code>diff_eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L131" target="_blank" rel="noreferrer">source</a></p>`,3))])])}const _=l(r,[["render",c]]);export{D as __pageData,_ as default};
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Return the independent dependent variables of <code>diff_eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L131" target="_blank" rel="noreferrer">source</a></p>`,3))])])}const _=l(r,[["render",c]]);export{D as __pageData,_ as default};
diff --git a/dev/assets/manual_entering_eom.md.BPBBr3gB.lean.js b/dev/assets/manual_entering_eom.md.DPEo_ILC.lean.js
similarity index 96%
rename from dev/assets/manual_entering_eom.md.BPBBr3gB.lean.js
rename to dev/assets/manual_entering_eom.md.DPEo_ILC.lean.js
index 5bc4286ec..b1ec02ef5 100644
--- a/dev/assets/manual_entering_eom.md.BPBBr3gB.lean.js
+++ b/dev/assets/manual_entering_eom.md.DPEo_ILC.lean.js
@@ -1 +1 @@
-import{_ as l,c as h,j as s,a,G as t,ai as e,B as k,o as p}from"./chunks/framework.Bnn8558y.js";const D=JSON.parse('{"title":"Entering equations of motion","description":"","frontmatter":{},"headers":[],"relativePath":"manual/entering_eom.md","filePath":"manual/entering_eom.md"}'),r={name:"manual/entering_eom.md"},d={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""},o={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""};function c(u,i,F,f,m,b){const n=k("Badge");return p(),h("div",null,[i[16]||(i[16]=s("h1",{id:"Entering-equations-of-motion",tabindex:"-1"},[a("Entering equations of motion "),s("a",{class:"header-anchor",href:"#Entering-equations-of-motion","aria-label":'Permalink to "Entering equations of motion {#Entering-equations-of-motion}"'},"")],-1)),i[17]||(i[17]=s("p",null,[a("The struct "),s("code",null,"DifferentialEquation"),a(" is the primary input method; it holds an ODE or a coupled system of ODEs composed of terms with harmonic time-dependence The dependent variables are specified during input, any other symbols are identified as parameters. Information on which variable is to be expanded in which harmonic is specified using "),s("code",null,"add_harmonic!"),a(".")],-1)),i[18]||(i[18]=s("p",null,[s("code",null,"DifferentialEquation.equations"),a(" stores a dictionary assigning variables to equations. This information is necessary because the harmonics belonging to a variable are later used to Fourier-transform its corresponding ODE.")],-1)),s("details",d,[s("summary",null,[i[0]||(i[0]=s("a",{id:"HarmonicBalance.d-manual-entering_eom",href:"#HarmonicBalance.d-manual-entering_eom"},[s("span",{class:"jlbinding"},"HarmonicBalance.d")],-1)),i[1]||(i[1]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[2]||(i[2]=s("p",null,"The derivative of f w.r.t. x of degree deg",-1)),i[3]||(i[3]=s("p",null,[s("a",{href:"https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicVariable.jl#L108",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",E,[s("summary",null,[i[4]||(i[4]=s("a",{id:"HarmonicBalance.DifferentialEquation-manual-entering_eom",href:"#HarmonicBalance.DifferentialEquation-manual-entering_eom"},[s("span",{class:"jlbinding"},"HarmonicBalance.DifferentialEquation")],-1)),i[5]||(i[5]=a()),t(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[6]||(i[6]=e("",7))]),s("details",o,[s("summary",null,[i[7]||(i[7]=s("a",{id:"HarmonicBalance.add_harmonic!-manual-entering_eom",href:"#HarmonicBalance.add_harmonic!-manual-entering_eom"},[s("span",{class:"jlbinding"},"HarmonicBalance.add_harmonic!")],-1)),i[8]||(i[8]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[9]||(i[9]=e("",6))]),s("details",g,[s("summary",null,[i[10]||(i[10]=s("a",{id:"Symbolics.get_variables-Tuple{DifferentialEquation}-manual-entering_eom",href:"#Symbolics.get_variables-Tuple{DifferentialEquation}-manual-entering_eom"},[s("span",{class:"jlbinding"},"Symbolics.get_variables")],-1)),i[11]||(i[11]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[12]||(i[12]=e("",3))]),s("details",y,[s("summary",null,[i[13]||(i[13]=s("a",{id:"HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}-manual-entering_eom",href:"#HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}-manual-entering_eom"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_independent_variables")],-1)),i[14]||(i[14]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[15]||(i[15]=e("",3))])])}const _=l(r,[["render",c]]);export{D as __pageData,_ as default};
+import{_ as l,c as h,j as s,a,G as t,ai as e,B as k,o as p}from"./chunks/framework.Bnn8558y.js";const D=JSON.parse('{"title":"Entering equations of motion","description":"","frontmatter":{},"headers":[],"relativePath":"manual/entering_eom.md","filePath":"manual/entering_eom.md"}'),r={name:"manual/entering_eom.md"},d={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""},o={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""};function c(u,i,F,m,f,C){const n=k("Badge");return p(),h("div",null,[i[16]||(i[16]=s("h1",{id:"Entering-equations-of-motion",tabindex:"-1"},[a("Entering equations of motion "),s("a",{class:"header-anchor",href:"#Entering-equations-of-motion","aria-label":'Permalink to "Entering equations of motion {#Entering-equations-of-motion}"'},"")],-1)),i[17]||(i[17]=s("p",null,[a("The struct "),s("code",null,"DifferentialEquation"),a(" is the primary input method; it holds an ODE or a coupled system of ODEs composed of terms with harmonic time-dependence The dependent variables are specified during input, any other symbols are identified as parameters. Information on which variable is to be expanded in which harmonic is specified using "),s("code",null,"add_harmonic!"),a(".")],-1)),i[18]||(i[18]=s("p",null,[s("code",null,"DifferentialEquation.equations"),a(" stores a dictionary assigning variables to equations. This information is necessary because the harmonics belonging to a variable are later used to Fourier-transform its corresponding ODE.")],-1)),s("details",d,[s("summary",null,[i[0]||(i[0]=s("a",{id:"HarmonicBalance.d-manual-entering_eom",href:"#HarmonicBalance.d-manual-entering_eom"},[s("span",{class:"jlbinding"},"HarmonicBalance.d")],-1)),i[1]||(i[1]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[2]||(i[2]=s("p",null,"The derivative of f w.r.t. x of degree deg",-1)),i[3]||(i[3]=s("p",null,[s("a",{href:"https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicVariable.jl#L108",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",E,[s("summary",null,[i[4]||(i[4]=s("a",{id:"HarmonicBalance.DifferentialEquation-manual-entering_eom",href:"#HarmonicBalance.DifferentialEquation-manual-entering_eom"},[s("span",{class:"jlbinding"},"HarmonicBalance.DifferentialEquation")],-1)),i[5]||(i[5]=a()),t(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[6]||(i[6]=e("",7))]),s("details",o,[s("summary",null,[i[7]||(i[7]=s("a",{id:"HarmonicBalance.add_harmonic!-manual-entering_eom",href:"#HarmonicBalance.add_harmonic!-manual-entering_eom"},[s("span",{class:"jlbinding"},"HarmonicBalance.add_harmonic!")],-1)),i[8]||(i[8]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[9]||(i[9]=e("",6))]),s("details",g,[s("summary",null,[i[10]||(i[10]=s("a",{id:"Symbolics.get_variables-Tuple{DifferentialEquation}-manual-entering_eom",href:"#Symbolics.get_variables-Tuple{DifferentialEquation}-manual-entering_eom"},[s("span",{class:"jlbinding"},"Symbolics.get_variables")],-1)),i[11]||(i[11]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[12]||(i[12]=e("",3))]),s("details",y,[s("summary",null,[i[13]||(i[13]=s("a",{id:"HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}-manual-entering_eom",href:"#HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}-manual-entering_eom"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_independent_variables")],-1)),i[14]||(i[14]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[15]||(i[15]=e("",3))])])}const _=l(r,[["render",c]]);export{D as __pageData,_ as default};
diff --git a/dev/assets/manual_extracting_harmonics.md.2TPGZJbG.js b/dev/assets/manual_extracting_harmonics.md.DPolJlR9.js
similarity index 98%
rename from dev/assets/manual_extracting_harmonics.md.2TPGZJbG.js
rename to dev/assets/manual_extracting_harmonics.md.DPolJlR9.js
index 165a70ce5..fa2150bc9 100644
--- a/dev/assets/manual_extracting_harmonics.md.2TPGZJbG.js
+++ b/dev/assets/manual_extracting_harmonics.md.DPolJlR9.js
@@ -1,4 +1,4 @@
-import{_ as o,c as e,ai as t,j as a,a as i,G as T,B as r,o as n}from"./chunks/framework.Bnn8558y.js";const I=JSON.parse('{"title":"Extracting harmonic equations","description":"","frontmatter":{},"headers":[],"relativePath":"manual/extracting_harmonics.md","filePath":"manual/extracting_harmonics.md"}'),Q={name:"manual/extracting_harmonics.md"},h={class:"jldocstring custom-block",open:""},d={class:"jldocstring custom-block",open:""},p={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},m={class:"jldocstring custom-block",open:""},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.972ex"},xmlns:"http://www.w3.org/2000/svg",width:"47.051ex",height:"3.144ex",role:"img",focusable:"false",viewBox:"0 -960 20796.4 1389.6","aria-hidden":"true"},u={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},E={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.666ex"},xmlns:"http://www.w3.org/2000/svg",width:"8.845ex",height:"2.363ex",role:"img",focusable:"false",viewBox:"0 -750 3909.4 1044.2","aria-hidden":"true"},y={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},f={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.666ex"},xmlns:"http://www.w3.org/2000/svg",width:"3.251ex",height:"1.668ex",role:"img",focusable:"false",viewBox:"0 -443 1436.9 737.2","aria-hidden":"true"},x={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},w={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.611ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2038 1000","aria-hidden":"true"},b={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},H={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.611ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2038 1000","aria-hidden":"true"},F={class:"jldocstring custom-block",open:""},_={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},C={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"2.378ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 1051 683","aria-hidden":"true"},L={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},v={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"2.009ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 888 683","aria-hidden":"true"},D={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},M={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"5.518ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 2439 683","aria-hidden":"true"},A={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},j={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"5.518ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 2439 683","aria-hidden":"true"},V={class:"jldocstring custom-block",open:""};function B(Z,s,q,S,N,P){const l=r("Badge");return n(),e("div",null,[s[56]||(s[56]=t('<h1 id="Extracting-harmonic-equations" tabindex="-1">Extracting harmonic equations <a class="header-anchor" href="#Extracting-harmonic-equations" aria-label="Permalink to "Extracting harmonic equations {#Extracting-harmonic-equations}""></a></h1><h2 id="Harmonic-Balance-method" tabindex="-1">Harmonic Balance method <a class="header-anchor" href="#Harmonic-Balance-method" aria-label="Permalink to "Harmonic Balance method {#Harmonic-Balance-method}""></a></h2><p>Once a <code>DifferentialEquation</code> is defined and its harmonics specified, one can extract the harmonic equations using <code>get_harmonic_equations</code>, which itself is composed of the subroutines <code>harmonic_ansatz</code>, <code>slow_flow</code>, <code>fourier_transform!</code> and <code>drop_powers</code>.</p><p>The harmonic equations use an additional time variable specified as <code>slow_time</code> in <code>get_harmonic_equations</code>. This is essentially a label distinguishing the time dependence of the harmonic variables (expected to be slow) from that of the oscillating terms (expected to be fast). When the equations are Fourier-transformed to remove oscillating terms, <code>slow_time</code> is treated as a constant. Such an approach is exact when looking for steady states.</p>',4)),a("details",h,[a("summary",null,[s[0]||(s[0]=a("a",{id:"HarmonicBalance.get_harmonic_equations-manual-extracting_harmonics",href:"#HarmonicBalance.get_harmonic_equations-manual-extracting_harmonics"},[a("span",{class:"jlbinding"},"HarmonicBalance.get_harmonic_equations")],-1)),s[1]||(s[1]=i()),T(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[2]||(s[2]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_harmonic_equations</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+import{_ as o,c as e,ai as t,j as a,a as i,G as T,B as r,o as n}from"./chunks/framework.Bnn8558y.js";const I=JSON.parse('{"title":"Extracting harmonic equations","description":"","frontmatter":{},"headers":[],"relativePath":"manual/extracting_harmonics.md","filePath":"manual/extracting_harmonics.md"}'),Q={name:"manual/extracting_harmonics.md"},h={class:"jldocstring custom-block",open:""},d={class:"jldocstring custom-block",open:""},p={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},m={class:"jldocstring custom-block",open:""},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.972ex"},xmlns:"http://www.w3.org/2000/svg",width:"47.051ex",height:"3.144ex",role:"img",focusable:"false",viewBox:"0 -960 20796.4 1389.6","aria-hidden":"true"},u={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},E={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.666ex"},xmlns:"http://www.w3.org/2000/svg",width:"8.845ex",height:"2.363ex",role:"img",focusable:"false",viewBox:"0 -750 3909.4 1044.2","aria-hidden":"true"},y={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},x={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.666ex"},xmlns:"http://www.w3.org/2000/svg",width:"3.251ex",height:"1.668ex",role:"img",focusable:"false",viewBox:"0 -443 1436.9 737.2","aria-hidden":"true"},f={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},w={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.611ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2038 1000","aria-hidden":"true"},H={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},b={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.611ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2038 1000","aria-hidden":"true"},F={class:"jldocstring custom-block",open:""},_={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},C={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"2.378ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 1051 683","aria-hidden":"true"},L={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},v={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"2.009ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 888 683","aria-hidden":"true"},D={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},M={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"5.518ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 2439 683","aria-hidden":"true"},A={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},j={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"5.518ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 2439 683","aria-hidden":"true"},V={class:"jldocstring custom-block",open:""};function B(Z,s,q,S,N,P){const l=r("Badge");return n(),e("div",null,[s[56]||(s[56]=t('<h1 id="Extracting-harmonic-equations" tabindex="-1">Extracting harmonic equations <a class="header-anchor" href="#Extracting-harmonic-equations" aria-label="Permalink to "Extracting harmonic equations {#Extracting-harmonic-equations}""></a></h1><h2 id="Harmonic-Balance-method" tabindex="-1">Harmonic Balance method <a class="header-anchor" href="#Harmonic-Balance-method" aria-label="Permalink to "Harmonic Balance method {#Harmonic-Balance-method}""></a></h2><p>Once a <code>DifferentialEquation</code> is defined and its harmonics specified, one can extract the harmonic equations using <code>get_harmonic_equations</code>, which itself is composed of the subroutines <code>harmonic_ansatz</code>, <code>slow_flow</code>, <code>fourier_transform!</code> and <code>drop_powers</code>.</p><p>The harmonic equations use an additional time variable specified as <code>slow_time</code> in <code>get_harmonic_equations</code>. This is essentially a label distinguishing the time dependence of the harmonic variables (expected to be slow) from that of the oscillating terms (expected to be fast). When the equations are Fourier-transformed to remove oscillating terms, <code>slow_time</code> is treated as a constant. Such an approach is exact when looking for steady states.</p>',4)),a("details",h,[a("summary",null,[s[0]||(s[0]=a("a",{id:"HarmonicBalance.get_harmonic_equations-manual-extracting_harmonics",href:"#HarmonicBalance.get_harmonic_equations-manual-extracting_harmonics"},[a("span",{class:"jlbinding"},"HarmonicBalance.get_harmonic_equations")],-1)),s[1]||(s[1]=i()),T(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[2]||(s[2]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_harmonic_equations</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> fast_time,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> slow_time,</span></span>
@@ -26,13 +26,13 @@ import{_ as o,c as e,ai as t,j as a,a as i,G as T,B as r,o as n}from"./chunks/fr
<span class="line"></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F</span></span>
<span class="line"></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L295-L336" target="_blank" rel="noreferrer">source</a></p>`,7))]),a("details",d,[a("summary",null,[s[3]||(s[3]=a("a",{id:"HarmonicBalance.harmonic_ansatz-manual-extracting_harmonics",href:"#HarmonicBalance.harmonic_ansatz-manual-extracting_harmonics"},[a("span",{class:"jlbinding"},"HarmonicBalance.harmonic_ansatz")],-1)),s[4]||(s[4]=i()),T(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[5]||(s[5]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">harmonic_ansatz</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, time</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; coordinates</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"Cartesian"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Expand each variable of <code>diff_eom</code> using the harmonics assigned to it with <code>time</code> as the time variable. For each harmonic of each variable, instance(s) of <code>HarmonicVariable</code> are automatically created and named.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L72-L78" target="_blank" rel="noreferrer">source</a></p>',3))]),a("details",p,[a("summary",null,[s[6]||(s[6]=a("a",{id:"HarmonicBalance.slow_flow-manual-extracting_harmonics",href:"#HarmonicBalance.slow_flow-manual-extracting_harmonics"},[a("span",{class:"jlbinding"},"HarmonicBalance.slow_flow")],-1)),s[7]||(s[7]=i()),T(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[8]||(s[8]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">slow_flow</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; fast_time</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, slow_time</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, degree</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Removes all derivatives w.r.t <code>fast_time</code> (and their products) in <code>eom</code> of power <code>degree</code>. In the remaining derivatives, <code>fast_time</code> is replaced by <code>slow_time</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L139-L144" target="_blank" rel="noreferrer">source</a></p>',3))]),a("details",k,[a("summary",null,[s[9]||(s[9]=a("a",{id:"HarmonicBalance.fourier_transform-manual-extracting_harmonics",href:"#HarmonicBalance.fourier_transform-manual-extracting_harmonics"},[a("span",{class:"jlbinding"},"HarmonicBalance.fourier_transform")],-1)),s[10]||(s[10]=i()),T(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">fourier_transform</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L295-L336" target="_blank" rel="noreferrer">source</a></p>`,7))]),a("details",d,[a("summary",null,[s[3]||(s[3]=a("a",{id:"HarmonicBalance.harmonic_ansatz-manual-extracting_harmonics",href:"#HarmonicBalance.harmonic_ansatz-manual-extracting_harmonics"},[a("span",{class:"jlbinding"},"HarmonicBalance.harmonic_ansatz")],-1)),s[4]||(s[4]=i()),T(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[5]||(s[5]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">harmonic_ansatz</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, time</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; coordinates</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"Cartesian"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Expand each variable of <code>diff_eom</code> using the harmonics assigned to it with <code>time</code> as the time variable. For each harmonic of each variable, instance(s) of <code>HarmonicVariable</code> are automatically created and named.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L72-L78" target="_blank" rel="noreferrer">source</a></p>',3))]),a("details",p,[a("summary",null,[s[6]||(s[6]=a("a",{id:"HarmonicBalance.slow_flow-manual-extracting_harmonics",href:"#HarmonicBalance.slow_flow-manual-extracting_harmonics"},[a("span",{class:"jlbinding"},"HarmonicBalance.slow_flow")],-1)),s[7]||(s[7]=i()),T(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[8]||(s[8]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">slow_flow</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; fast_time</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, slow_time</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, degree</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Removes all derivatives w.r.t <code>fast_time</code> (and their products) in <code>eom</code> of power <code>degree</code>. In the remaining derivatives, <code>fast_time</code> is replaced by <code>slow_time</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L139-L144" target="_blank" rel="noreferrer">source</a></p>',3))]),a("details",k,[a("summary",null,[s[9]||(s[9]=a("a",{id:"HarmonicBalance.fourier_transform-manual-extracting_harmonics",href:"#HarmonicBalance.fourier_transform-manual-extracting_harmonics"},[a("span",{class:"jlbinding"},"HarmonicBalance.fourier_transform")],-1)),s[10]||(s[10]=i()),T(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">fourier_transform</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> time</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Extract the Fourier components of <code>eom</code> corresponding to the harmonics specified in <code>eom.variables</code>. For each non-zero harmonic of each variable, 2 equations are generated (cos and sin Fourier coefficients). For each zero (constant) harmonic, 1 equation is generated <code>time</code> does not appear in the resulting equations anymore.</p><p>Underlying assumption: all time-dependences are harmonic.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L256" target="_blank" rel="noreferrer">source</a></p>`,4))]),a("details",m,[a("summary",null,[s[12]||(s[12]=a("a",{id:"HarmonicBalance.ExprUtils.drop_powers-manual-extracting_harmonics",href:"#HarmonicBalance.ExprUtils.drop_powers-manual-extracting_harmonics"},[a("span",{class:"jlbinding"},"HarmonicBalance.ExprUtils.drop_powers")],-1)),s[13]||(s[13]=i()),T(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[14]||(s[14]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">drop_powers</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(expr, vars, deg)</span></span></code></pre></div><p>Remove parts of <code>expr</code> where the combined power of <code>vars</code> is => <code>deg</code>.</p><p><strong>Example</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> x,y;</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Extract the Fourier components of <code>eom</code> corresponding to the harmonics specified in <code>eom.variables</code>. For each non-zero harmonic of each variable, 2 equations are generated (cos and sin Fourier coefficients). For each zero (constant) harmonic, 1 equation is generated <code>time</code> does not appear in the resulting equations anymore.</p><p>Underlying assumption: all time-dependences are harmonic.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L256" target="_blank" rel="noreferrer">source</a></p>`,4))]),a("details",m,[a("summary",null,[s[12]||(s[12]=a("a",{id:"HarmonicBalance.ExprUtils.drop_powers-manual-extracting_harmonics",href:"#HarmonicBalance.ExprUtils.drop_powers-manual-extracting_harmonics"},[a("span",{class:"jlbinding"},"HarmonicBalance.ExprUtils.drop_powers")],-1)),s[13]||(s[13]=i()),T(l,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[14]||(s[14]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">drop_powers</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(expr, vars, deg)</span></span></code></pre></div><p>Remove parts of <code>expr</code> where the combined power of <code>vars</code> is => <code>deg</code>.</p><p><strong>Example</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> x,y;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">drop_powers</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, x, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> +</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">drop_powers</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, [x,y], </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">drop_powers</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> +</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">3</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, [x,y], </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">3</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> +</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> y</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> +</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/ExprUtils/drop_powers.jl#L1" target="_blank" rel="noreferrer">source</a></p>`,5))]),s[57]||(s[57]=a("h2",{id:"HarmonicVariable-and-HarmonicEquation-types",tabindex:"-1"},[i("HarmonicVariable and HarmonicEquation types "),a("a",{class:"header-anchor",href:"#HarmonicVariable-and-HarmonicEquation-types","aria-label":'Permalink to "HarmonicVariable and HarmonicEquation types {#HarmonicVariable-and-HarmonicEquation-types}"'},"")],-1)),a("p",null,[s[25]||(s[25]=i("The equations governing the harmonics are stored using the two following structs. 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shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicVariable</span></span></code></pre></div><p>Holds a variable stored under <code>symbol</code> describing the harmonic <code>ω</code> of <code>natural_variable</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>symbol::Num</code>: Symbol of the variable in the HarmonicBalance namespace.</p></li><li><p><code>name::String</code>: Human-readable labels of the variable, used for plotting.</p></li><li><p><code>type::String</code>: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)</p></li><li><p><code>ω::Num</code>: The harmonic being described.</p></li><li><p><code>natural_variable::Num</code>: The natural variable whose harmonic is being described.</p></li></ul><p><a 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+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> +</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> y</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> +</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/ExprUtils/drop_powers.jl#L1" target="_blank" rel="noreferrer">source</a></p>`,5))]),s[57]||(s[57]=a("h2",{id:"HarmonicVariable-and-HarmonicEquation-types",tabindex:"-1"},[i("HarmonicVariable and HarmonicEquation types "),a("a",{class:"header-anchor",href:"#HarmonicVariable-and-HarmonicEquation-types","aria-label":'Permalink to "HarmonicVariable and HarmonicEquation types {#HarmonicVariable-and-HarmonicEquation-types}"'},"")],-1)),a("p",null,[s[25]||(s[25]=i("The equations governing the harmonics are stored using the two following structs. 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shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicVariable</span></span></code></pre></div><p>Holds a variable stored under <code>symbol</code> describing the harmonic <code>ω</code> of <code>natural_variable</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>symbol::Num</code>: Symbol of the variable in the HarmonicBalance namespace.</p></li><li><p><code>name::String</code>: Human-readable labels of the variable, used for plotting.</p></li><li><p><code>type::String</code>: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)</p></li><li><p><code>ω::Num</code>: The harmonic being described.</p></li><li><p><code>natural_variable::Num</code>: The natural variable whose harmonic is being described.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicVariable.jl#L1" target="_blank" rel="noreferrer">source</a></p>',5))]),a("p",null,[s[44]||(s[44]=i("When the full set of equations of motion is expanded using the harmonic ansatz, the result is stored as a ")),s[45]||(s[45]=a("code",null,"HarmonicEquation",-1)),s[46]||(s[46]=i(". 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Each symbol not corresponding to a variable is identified as a parameter."))]),s[58]||(s[58]=a("p",null,[i("A "),a("code",null,"HarmonicEquation"),i(" can be either parsed into a steady-state "),a("code",null,"Problem"),i(" or solved using a dynamical ODE solver.")],-1)),a("details",V,[a("summary",null,[s[53]||(s[53]=a("a",{id:"HarmonicBalance.HarmonicEquation-manual-extracting_harmonics",href:"#HarmonicBalance.HarmonicEquation-manual-extracting_harmonics"},[a("span",{class:"jlbinding"},"HarmonicBalance.HarmonicEquation")],-1)),s[54]||(s[54]=i()),T(l,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[55]||(s[55]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Holds a set of algebraic equations governing the harmonics of a <code>DifferentialEquation</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>equations::Vector{Equation}</code>: A set of equations governing the harmonics.</p></li><li><p><code>variables::Vector{HarmonicVariable}</code>: A set of variables describing the harmonics.</p></li><li><p><code>parameters::Vector{Num}</code>: The parameters of the equation set.</p></li><li><p><code>natural_equation::DifferentialEquation</code>: The natural equation (before the harmonic ansatz was used).</p></li><li><p><code>jacobian::Matrix{Num}</code>: The Jacobian of the natural equation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L1" target="_blank" rel="noreferrer">source</a></p>',5))]),s[59]||(s[59]=a("p",null,"MarkdownAST.LineBreak()",-1))])}const R=o(Q,[["render",B]]);export{I as __pageData,R as default};
diff --git a/dev/assets/manual_extracting_harmonics.md.2TPGZJbG.lean.js b/dev/assets/manual_extracting_harmonics.md.DPolJlR9.lean.js
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diff --git a/dev/assets/manual_linear_response.md.DwIcdztY.js b/dev/assets/manual_linear_response.md.Bchqv-cb.js
similarity index 94%
rename from dev/assets/manual_linear_response.md.DwIcdztY.js
rename to dev/assets/manual_linear_response.md.Bchqv-cb.js
index 7f39f0d4e..07ea1ccb7 100644
--- a/dev/assets/manual_linear_response.md.DwIcdztY.js
+++ b/dev/assets/manual_linear_response.md.Bchqv-cb.js
@@ -1,5 +1,5 @@
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One is calculating the first-order Jacobian, used to obtain stability and approximate (but inexpensive) the linear response of steady states. The other is calculating the full response matrix as a function of frequency; this is more accurate but more expensive.</p><p>The methodology used is explained in <a href="https://www.doi.org/10.3929/ethz-b-000589190" target="_blank" rel="noreferrer">Jan Kosata phd thesis</a>.</p><h2 id="stability" tabindex="-1">Stability <a class="header-anchor" href="#stability" aria-label="Permalink to "Stability""></a></h2><p>The Jacobian is used to evaluate stability of the solutions. It can be shown explicitly,</p>',5)),s("details",h,[s("summary",null,[e[0]||(e[0]=s("a",{id:"HarmonicBalance.get_Jacobian-manual-linear_response",href:"#HarmonicBalance.get_Jacobian-manual-linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_Jacobian")],-1)),e[1]||(e[1]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[2]||(e[2]=i('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_Jacobian</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom)</span></span></code></pre></div><p>Obtain the symbolic Jacobian matrix of <code>eom</code>. This is the linearised left-hand side of F(u) = du/dT.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/Jacobian.jl#L59" target="_blank" rel="noreferrer">source</a></p><p>Obtain a Jacobian from a <code>DifferentialEquation</code> by first converting it into a <code>HarmonicEquation</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/Jacobian.jl#L77" target="_blank" rel="noreferrer">source</a></p><p>Get the Jacobian of a set of equations <code>eqs</code> with respect to the variables <code>vars</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/Jacobian.jl#L86" target="_blank" rel="noreferrer">source</a></p>',7))]),e[32]||(e[32]=s("h2",{id:"Linear-response",tabindex:"-1"},[a("Linear response "),s("a",{class:"header-anchor",href:"#Linear-response","aria-label":'Permalink to "Linear response {#Linear-response}"'},"")],-1)),e[33]||(e[33]=s("p",null,[a("The response to white noise can be shown with "),s("code",null,"plot_linear_response"),a(". Depending on the "),s("code",null,"order"),a(" argument, different methods are used.")],-1)),s("details",c,[s("summary",null,[e[3]||(e[3]=s("a",{id:"HarmonicBalance.LinearResponse.plot_linear_response-manual-linear_response",href:"#HarmonicBalance.LinearResponse.plot_linear_response-manual-linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.plot_linear_response")],-1)),e[4]||(e[4]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[5]||(e[5]=i('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_linear_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, nat_var</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; Ω_range, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, order</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, logscale</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">false</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, show_progress</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot the linear response to white noise of the variable <code>nat_var</code> for Result <code>res</code> on <code>branch</code> for input frequencies <code>Ω_range</code>. Slow-time derivatives up to <code>order</code> are kept in the process.</p><p>Any kwargs are fed to Plots' gr().</p><p>Solutions not belonging to the <code>physical</code> class are ignored.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/plotting.jl#L120-L129" target="_blank" rel="noreferrer">source</a></p>',5))]),e[34]||(e[34]=s("h3",{id:"First-order",tabindex:"-1"},[a("First order "),s("a",{class:"header-anchor",href:"#First-order","aria-label":'Permalink to "First order {#First-order}"'},"")],-1)),s("p",null,[e[12]||(e[12]=a("The simplest way to extract the linear response of a steady state is to evaluate the Jacobian of the harmonic equations. Each of its eigenvalues ")),s("mjx-container",k,[(r(),l("svg",m,e[6]||(e[6]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),e[7]||(e[7]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2041,0)"><path data-c="5D" d="M22 710V750H159V-250H22V-210H119V710H22Z" style="stroke-width:3;"></path></g></g></g>',1)]))),e[9]||(e[9]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mtext",null,"Re"),s("mo",{stretchy:"false"},"["),s("mi",null,"λ"),s("mo",{stretchy:"false"},"]")])],-1))]),e[14]||(e[14]=a(" gives its center and 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262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2055,0)"><path data-c="5D" d="M22 710V750H159V-250H22V-210H119V710H22Z" style="stroke-width:3;"></path></g></g></g>',1)]))),e[11]||(e[11]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mtext",null,"Im"),s("mo",{stretchy:"false"},"["),s("mi",null,"λ"),s("mo",{stretchy:"false"},"]")])],-1))]),e[15]||(e[15]=a(" its width. Transforming the harmonic variables into the non-rotating frame (that is, inverting the harmonic ansatz) then gives the response as it would be observed in an experiment."))]),e[35]||(e[35]=s("p",null,"The advantage of this method is that for a given parameter set, only one matrix diagonalization is needed to fully describe the response spectrum. However, the method is inaccurate for response frequencies far from the frequencies used in the harmonic ansatz (it relies on the response oscillating slowly in the rotating frame).",-1)),e[36]||(e[36]=s("p",null,[a("Behind the scenes, the spectra are stored using the dedicated structs "),s("code",null,"Lorentzian"),a(" and "),s("code",null,"JacobianSpectrum"),a(".")],-1)),s("details",f,[s("summary",null,[e[16]||(e[16]=s("a",{id:"HarmonicBalance.LinearResponse.JacobianSpectrum-manual-linear_response",href:"#HarmonicBalance.LinearResponse.JacobianSpectrum-manual-linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.JacobianSpectrum")],-1)),e[17]||(e[17]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[18]||(e[18]=i('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> JacobianSpectrum{T</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Real</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}</span></span></code></pre></div><p>Holds a set of <code>Lorentzian</code> objects belonging to a variable.</p><p><strong>Fields</strong></p><ul><li><code>peaks::Array{HarmonicBalance.LinearResponse.Lorentzian{T}, 1} where T<:Real</code></li></ul><p><strong>Constructor</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">JacobianSpectrum</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; index</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/types.jl#L21" target="_blank" rel="noreferrer">source</a></p>',7))]),s("details",y,[s("summary",null,[e[19]||(e[19]=s("a",{id:"HarmonicBalance.LinearResponse.Lorentzian-manual-linear_response",href:"#HarmonicBalance.LinearResponse.Lorentzian-manual-linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.Lorentzian")],-1)),e[20]||(e[20]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[21]||(e[21]=i('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Lorentzian{T</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Real</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}</span></span></code></pre></div><p>Holds the three parameters of a Lorentzian peak, defined as A / sqrt((ω-ω0)² + Γ²).</p><p><strong>Fields</strong></p><ul><li><p><code>ω0::Real</code></p></li><li><p><code>Γ::Real</code></p></li><li><p><code>A::Real</code></p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/types.jl#L1" target="_blank" rel="noreferrer">source</a></p>',5))]),e[37]||(e[37]=s("h3",{id:"Higher-orders",tabindex:"-1"},[a("Higher orders "),s("a",{class:"header-anchor",href:"#Higher-orders","aria-label":'Permalink to "Higher orders {#Higher-orders}"'},"")],-1)),e[38]||(e[38]=s("p",null,[a("Setting "),s("code",null,"order > 1"),a(" increases the accuracy of the response spectra. However, unlike for the Jacobian, here we must perform a matrix inversion for each response frequency.")],-1)),s("details",_,[s("summary",null,[e[22]||(e[22]=s("a",{id:"HarmonicBalance.LinearResponse.ResponseMatrix-manual-linear_response",href:"#HarmonicBalance.LinearResponse.ResponseMatrix-manual-linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.ResponseMatrix")],-1)),e[23]||(e[23]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[24]||(e[24]=i('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ResponseMatrix</span></span></code></pre></div><p>Holds the compiled response matrix of a system.</p><p><strong>Fields</strong></p><ul><li><p><code>matrix::Matrix{Function}</code>: The response matrix (compiled).</p></li><li><p><code>symbols::Vector{Num}</code>: Any symbolic variables in <code>matrix</code> to be substituted at evaluation.</p></li><li><p><code>variables::Vector{HarmonicVariable}</code>: The frequencies of the harmonic variables underlying <code>matrix</code>. These are needed to transform the harmonic variables to the non-rotating frame.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/types.jl#L40" target="_blank" rel="noreferrer">source</a></p>',5))]),s("details",E,[s("summary",null,[e[25]||(e[25]=s("a",{id:"HarmonicBalance.LinearResponse.get_response-manual-linear_response",href:"#HarmonicBalance.LinearResponse.get_response-manual-linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.get_response")],-1)),e[26]||(e[26]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[27]||(e[27]=i(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+import{_ as o,c as l,ai as i,j as s,a,G as n,B as p,o as r}from"./chunks/framework.Bnn8558y.js";const L=JSON.parse('{"title":"Linear response (WIP)","description":"","frontmatter":{},"headers":[],"relativePath":"manual/linear_response.md","filePath":"manual/linear_response.md"}'),d={name:"manual/linear_response.md"},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},k={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},m={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.027ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.319ex",height:"1.597ex",role:"img",focusable:"false",viewBox:"0 -694 583 706","aria-hidden":"true"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.247ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2319 1000","aria-hidden":"true"},T={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},b={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.278ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2333 1000","aria-hidden":"true"},y={class:"jldocstring custom-block",open:""},f={class:"jldocstring custom-block",open:""},_={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""},Q={class:"jldocstring custom-block",open:""};function x(H,e,v,C,w,F){const t=p("Badge");return r(),l("div",null,[e[31]||(e[31]=i('<h1 id="linresp_man" tabindex="-1">Linear response (WIP) <a class="header-anchor" href="#linresp_man" aria-label="Permalink to "Linear response (WIP) {#linresp_man}""></a></h1><p>This module currently has two goals. One is calculating the first-order Jacobian, used to obtain stability and approximate (but inexpensive) the linear response of steady states. The other is calculating the full response matrix as a function of frequency; this is more accurate but more expensive.</p><p>The methodology used is explained in <a href="https://www.doi.org/10.3929/ethz-b-000589190" target="_blank" rel="noreferrer">Jan Kosata phd thesis</a>.</p><h2 id="stability" tabindex="-1">Stability <a class="header-anchor" href="#stability" aria-label="Permalink to "Stability""></a></h2><p>The Jacobian is used to evaluate stability of the solutions. It can be shown explicitly,</p>',5)),s("details",h,[s("summary",null,[e[0]||(e[0]=s("a",{id:"HarmonicBalance.get_Jacobian-manual-linear_response",href:"#HarmonicBalance.get_Jacobian-manual-linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_Jacobian")],-1)),e[1]||(e[1]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[2]||(e[2]=i('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_Jacobian</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom)</span></span></code></pre></div><p>Obtain the symbolic Jacobian matrix of <code>eom</code>. This is the linearised left-hand side of F(u) = du/dT.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/Jacobian.jl#L59" target="_blank" rel="noreferrer">source</a></p><p>Obtain a Jacobian from a <code>DifferentialEquation</code> by first converting it into a <code>HarmonicEquation</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/Jacobian.jl#L77" target="_blank" rel="noreferrer">source</a></p><p>Get the Jacobian of a set of equations <code>eqs</code> with respect to the variables <code>vars</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/Jacobian.jl#L86" target="_blank" rel="noreferrer">source</a></p>',7))]),e[32]||(e[32]=s("h2",{id:"Linear-response",tabindex:"-1"},[a("Linear response "),s("a",{class:"header-anchor",href:"#Linear-response","aria-label":'Permalink to "Linear response {#Linear-response}"'},"")],-1)),e[33]||(e[33]=s("p",null,[a("The response to white noise can be shown with "),s("code",null,"plot_linear_response"),a(". Depending on the "),s("code",null,"order"),a(" argument, different methods are used.")],-1)),s("details",c,[s("summary",null,[e[3]||(e[3]=s("a",{id:"HarmonicBalance.LinearResponse.plot_linear_response-manual-linear_response",href:"#HarmonicBalance.LinearResponse.plot_linear_response-manual-linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.plot_linear_response")],-1)),e[4]||(e[4]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[5]||(e[5]=i('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_linear_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, nat_var</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; Ω_range, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, order</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, logscale</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">false</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, show_progress</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot the linear response to white noise of the variable <code>nat_var</code> for Result <code>res</code> on <code>branch</code> for input frequencies <code>Ω_range</code>. Slow-time derivatives up to <code>order</code> are kept in the process.</p><p>Any kwargs are fed to Plots' gr().</p><p>Solutions not belonging to the <code>physical</code> class are ignored.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/plotting.jl#L120-L129" target="_blank" rel="noreferrer">source</a></p>',5))]),e[34]||(e[34]=s("h3",{id:"First-order",tabindex:"-1"},[a("First order "),s("a",{class:"header-anchor",href:"#First-order","aria-label":'Permalink to "First order {#First-order}"'},"")],-1)),s("p",null,[e[12]||(e[12]=a("The simplest way to extract the linear response of a steady state is to evaluate the Jacobian of the harmonic equations. Each of its eigenvalues ")),s("mjx-container",k,[(r(),l("svg",m,e[6]||(e[6]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),e[7]||(e[7]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2055,0)"><path data-c="5D" d="M22 710V750H159V-250H22V-210H119V710H22Z" style="stroke-width:3;"></path></g></g></g>',1)]))),e[11]||(e[11]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mtext",null,"Im"),s("mo",{stretchy:"false"},"["),s("mi",null,"λ"),s("mo",{stretchy:"false"},"]")])],-1))]),e[15]||(e[15]=a(" its width. Transforming the harmonic variables into the non-rotating frame (that is, inverting the harmonic ansatz) then gives the response as it would be observed in an experiment."))]),e[35]||(e[35]=s("p",null,"The advantage of this method is that for a given parameter set, only one matrix diagonalization is needed to fully describe the response spectrum. However, the method is inaccurate for response frequencies far from the frequencies used in the harmonic ansatz (it relies on the response oscillating slowly in the rotating frame).",-1)),e[36]||(e[36]=s("p",null,[a("Behind the scenes, the spectra are stored using the dedicated structs "),s("code",null,"Lorentzian"),a(" and "),s("code",null,"JacobianSpectrum"),a(".")],-1)),s("details",y,[s("summary",null,[e[16]||(e[16]=s("a",{id:"HarmonicBalance.LinearResponse.JacobianSpectrum-manual-linear_response",href:"#HarmonicBalance.LinearResponse.JacobianSpectrum-manual-linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.JacobianSpectrum")],-1)),e[17]||(e[17]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[18]||(e[18]=i('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> JacobianSpectrum{T</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Real</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}</span></span></code></pre></div><p>Holds a set of <code>Lorentzian</code> objects belonging to a variable.</p><p><strong>Fields</strong></p><ul><li><code>peaks::Array{HarmonicBalance.LinearResponse.Lorentzian{T}, 1} where T<:Real</code></li></ul><p><strong>Constructor</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">JacobianSpectrum</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; index</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/types.jl#L21" target="_blank" rel="noreferrer">source</a></p>',7))]),s("details",f,[s("summary",null,[e[19]||(e[19]=s("a",{id:"HarmonicBalance.LinearResponse.Lorentzian-manual-linear_response",href:"#HarmonicBalance.LinearResponse.Lorentzian-manual-linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.Lorentzian")],-1)),e[20]||(e[20]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[21]||(e[21]=i('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Lorentzian{T</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Real</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}</span></span></code></pre></div><p>Holds the three parameters of a Lorentzian peak, defined as A / sqrt((ω-ω0)² + Γ²).</p><p><strong>Fields</strong></p><ul><li><p><code>ω0::Real</code></p></li><li><p><code>Γ::Real</code></p></li><li><p><code>A::Real</code></p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/types.jl#L1" target="_blank" rel="noreferrer">source</a></p>',5))]),e[37]||(e[37]=s("h3",{id:"Higher-orders",tabindex:"-1"},[a("Higher orders "),s("a",{class:"header-anchor",href:"#Higher-orders","aria-label":'Permalink to "Higher orders {#Higher-orders}"'},"")],-1)),e[38]||(e[38]=s("p",null,[a("Setting "),s("code",null,"order > 1"),a(" increases the accuracy of the response spectra. However, unlike for the Jacobian, here we must perform a matrix inversion for each response frequency.")],-1)),s("details",_,[s("summary",null,[e[22]||(e[22]=s("a",{id:"HarmonicBalance.LinearResponse.ResponseMatrix-manual-linear_response",href:"#HarmonicBalance.LinearResponse.ResponseMatrix-manual-linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.ResponseMatrix")],-1)),e[23]||(e[23]=a()),n(t,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[24]||(e[24]=i('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ResponseMatrix</span></span></code></pre></div><p>Holds the compiled response matrix of a system.</p><p><strong>Fields</strong></p><ul><li><p><code>matrix::Matrix{Function}</code>: The response matrix (compiled).</p></li><li><p><code>symbols::Vector{Num}</code>: Any symbolic variables in <code>matrix</code> to be substituted at evaluation.</p></li><li><p><code>variables::Vector{HarmonicVariable}</code>: The frequencies of the harmonic variables underlying <code>matrix</code>. These are needed to transform the harmonic variables to the non-rotating frame.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/types.jl#L40" target="_blank" rel="noreferrer">source</a></p>',5))]),s("details",E,[s("summary",null,[e[25]||(e[25]=s("a",{id:"HarmonicBalance.LinearResponse.get_response-manual-linear_response",href:"#HarmonicBalance.LinearResponse.get_response-manual-linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.get_response")],-1)),e[26]||(e[26]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[27]||(e[27]=i(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> rmat</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.LinearResponse.ResponseMatrix</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> s</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">OrderedCollections.OrderedDict</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Ω</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>For <code>rmat</code> and a solution dictionary <code>s</code>, calculate the total response to a perturbative force at frequency <code>Ω</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/response.jl#L63" target="_blank" rel="noreferrer">source</a></p>`,3))]),s("details",Q,[s("summary",null,[e[28]||(e[28]=s("a",{id:"HarmonicBalance.LinearResponse.get_response_matrix-manual-linear_response",href:"#HarmonicBalance.LinearResponse.get_response_matrix-manual-linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.get_response_matrix")],-1)),e[29]||(e[29]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[30]||(e[30]=i('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_response_matrix</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, freq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; order</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Obtain the symbolic linear response matrix of a <code>diff_eq</code> corresponding to a perturbation frequency <code>freq</code>. This routine cannot accept a <code>HarmonicEquation</code> since there, some time-derivatives are already dropped. <code>order</code> denotes the highest differential order to be considered.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/response.jl#L1-L8" target="_blank" rel="noreferrer">source</a></p>',3))])])}const V=o(d,[["render",x]]);export{L as __pageData,V as default};
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>For <code>rmat</code> and a solution dictionary <code>s</code>, calculate the total response to a perturbative force at frequency <code>Ω</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/response.jl#L63" target="_blank" rel="noreferrer">source</a></p>`,3))]),s("details",Q,[s("summary",null,[e[28]||(e[28]=s("a",{id:"HarmonicBalance.LinearResponse.get_response_matrix-manual-linear_response",href:"#HarmonicBalance.LinearResponse.get_response_matrix-manual-linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.get_response_matrix")],-1)),e[29]||(e[29]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[30]||(e[30]=i('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_response_matrix</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, freq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; order</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Obtain the symbolic linear response matrix of a <code>diff_eq</code> corresponding to a perturbation frequency <code>freq</code>. This routine cannot accept a <code>HarmonicEquation</code> since there, some time-derivatives are already dropped. <code>order</code> denotes the highest differential order to be considered.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/response.jl#L1-L8" target="_blank" rel="noreferrer">source</a></p>',3))])])}const V=o(d,[["render",x]]);export{L as __pageData,V as default};
diff --git a/dev/assets/manual_linear_response.md.DwIcdztY.lean.js b/dev/assets/manual_linear_response.md.Bchqv-cb.lean.js
similarity index 95%
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-import{_ as o,c as l,ai as i,j as s,a,G as n,B as p,o as r}from"./chunks/framework.Bnn8558y.js";const L=JSON.parse('{"title":"Linear response (WIP)","description":"","frontmatter":{},"headers":[],"relativePath":"manual/linear_response.md","filePath":"manual/linear_response.md"}'),d={name:"manual/linear_response.md"},h={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},k={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},m={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.027ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.319ex",height:"1.597ex",role:"img",focusable:"false",viewBox:"0 -694 583 706","aria-hidden":"true"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.247ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2319 1000","aria-hidden":"true"},b={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},T={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.278ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2333 1000","aria-hidden":"true"},f={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""},_={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""},Q={class:"jldocstring custom-block",open:""};function x(H,e,v,C,w,F){const t=p("Badge");return r(),l("div",null,[e[31]||(e[31]=i("",5)),s("details",h,[s("summary",null,[e[0]||(e[0]=s("a",{id:"HarmonicBalance.get_Jacobian-manual-linear_response",href:"#HarmonicBalance.get_Jacobian-manual-linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_Jacobian")],-1)),e[1]||(e[1]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[2]||(e[2]=i("",7))]),e[32]||(e[32]=s("h2",{id:"Linear-response",tabindex:"-1"},[a("Linear response "),s("a",{class:"header-anchor",href:"#Linear-response","aria-label":'Permalink to "Linear response {#Linear-response}"'},"")],-1)),e[33]||(e[33]=s("p",null,[a("The response to white noise can be shown with "),s("code",null,"plot_linear_response"),a(". Depending on the "),s("code",null,"order"),a(" argument, different methods are used.")],-1)),s("details",c,[s("summary",null,[e[3]||(e[3]=s("a",{id:"HarmonicBalance.LinearResponse.plot_linear_response-manual-linear_response",href:"#HarmonicBalance.LinearResponse.plot_linear_response-manual-linear_response"},[s("span",{class:"jlbinding"},"HarmonicBalance.LinearResponse.plot_linear_response")],-1)),e[4]||(e[4]=a()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[5]||(e[5]=i("",5))]),e[34]||(e[34]=s("h3",{id:"First-order",tabindex:"-1"},[a("First order "),s("a",{class:"header-anchor",href:"#First-order","aria-label":'Permalink to "First order {#First-order}"'},"")],-1)),s("p",null,[e[12]||(e[12]=a("The simplest way to extract the linear response of a steady state is to evaluate the Jacobian of the harmonic equations. Each of its eigenvalues ")),s("mjx-container",k,[(r(),l("svg",m,e[6]||(e[6]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D706",d:"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z",style:{"stroke-width":"3"}})])])],-1)]))),e[7]||(e[7]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"λ")])],-1))]),e[13]||(e[13]=a(" describes a Lorentzian peak in the response; ")),s("mjx-container",g,[(r(),l("svg",u,e[8]||(e[8]=[i("",1)]))),e[9]||(e[9]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mtext",null,"Re"),s("mo",{stretchy:"false"},"["),s("mi",null,"λ"),s("mo",{stretchy:"false"},"]")])],-1))]),e[14]||(e[14]=a(" gives its center and ")),s("mjx-container",b,[(r(),l("svg",T,e[10]||(e[10]=[i("",1)]))),e[11]||(e[11]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mtext",null,"Im"),s("mo",{stretchy:"false"},"["),s("mi",null,"λ"),s("mo",{stretchy:"false"},"]")])],-1))]),e[15]||(e[15]=a(" its width. 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diff --git a/dev/assets/manual_methods.md.DHpPWVkR.js b/dev/assets/manual_methods.md.ClFJSA9j.js
similarity index 97%
rename from dev/assets/manual_methods.md.DHpPWVkR.js
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index 295b2c0ce..b4ba79f40 100644
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+++ b/dev/assets/manual_methods.md.ClFJSA9j.js
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The method guarantees to find all solutions, however, it comes with a high computational cost. See ")),e[14]||(e[14]=t("a",{href:"https://www.juliahomotopycontinuation.org/guides/totaldegree/",target:"_blank",rel:"noreferrer"},"HomotopyContinuation.jl",-1)),e[15]||(e[15]=a(" for more information."))]),e[17]||(e[17]=o('<p><strong>Fields</strong></p><ul><li><p><code>gamma::Complex</code>: Complex multiplying factor of the start system G(x) for the homotopy</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/methods.jl#L8-L20" target="_blank" rel="noreferrer">source</a></p>',3))]),e[27]||(e[27]=t("h2",{id:"Polyhedral-Method",tabindex:"-1"},[a("Polyhedral Method "),t("a",{class:"header-anchor",href:"#Polyhedral-Method","aria-label":'Permalink to "Polyhedral Method {#Polyhedral-Method}"'},"")],-1)),t("details",u,[t("summary",null,[e[18]||(e[18]=t("a",{id:"HarmonicBalance.Polyhedral-manual-methods",href:"#HarmonicBalance.Polyhedral-manual-methods"},[t("span",{class:"jlbinding"},"HarmonicBalance.Polyhedral")],-1)),e[19]||(e[19]=a()),Q(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[20]||(e[20]=o('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Polyhedral</span></span></code></pre></div><p>The Polyhedral homotopy method constructs a homotopy based on the polyhedral structure of the polynomial system. It is more efficient than the Total Degree method for sparse systems, meaning most of the coefficients are zero. It can be especially useful if you don't need to find the zero solutions (<code>only_non_zero = true</code>), resulting in a speed up. See <a href="https://www.juliahomotopycontinuation.org/guides/polyhedral/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>only_non_zero::Bool</code>: Boolean indicating if only non-zero solutions are considered.</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/methods.jl#L57-L69" target="_blank" rel="noreferrer">source</a></p>',5))]),e[28]||(e[28]=t("h2",{id:"Warm-Up-Method",tabindex:"-1"},[a("Warm Up Method "),t("a",{class:"header-anchor",href:"#Warm-Up-Method","aria-label":'Permalink to "Warm Up Method {#Warm-Up-Method}"'},"")],-1)),t("details",H,[t("summary",null,[e[21]||(e[21]=t("a",{id:"HarmonicBalance.WarmUp-manual-methods",href:"#HarmonicBalance.WarmUp-manual-methods"},[t("span",{class:"jlbinding"},"HarmonicBalance.WarmUp")],-1)),e[22]||(e[22]=a()),Q(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[23]||(e[23]=o('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">WarmUp</span></span></code></pre></div><p>The Warm Up method prepares a warmup system with the Total Degree method using the parameter at <code>index</code> perturbed by <code>perturbation_size</code>. The warmup system is used to perform a homotopy using all other systems in the parameter sweep. It is very efficient for systems with minimal bifurcation in the parameter sweep. The Warm Up method should in theory guarantee to find all solutions, however, if the <code>start_parameters</code> is not proper (to close to the real line) it could miss some solutions.</p><p>See<a href="https://www.juliahomotopycontinuation.org/guides/many-systems/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>warm_up_method::Union{Polyhedral{T}, TotalDegree{T}} where T</code>: Method used for the warmup system.</p></li><li><p><code>start_parameters::Vector</code>: Start parameters.</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/methods.jl#L110-L125" target="_blank" rel="noreferrer">source</a></p>',6))])])}const v=r(T,[["render",y]]);export{L as __pageData,v as default};
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Each method has different tradeoffs between speed, robustness, and completeness.",-1)),e[26]||(e[26]=t("h2",{id:"Total-Degree-Method",tabindex:"-1"},[a("Total Degree Method "),t("a",{class:"header-anchor",href:"#Total-Degree-Method","aria-label":'Permalink to "Total Degree Method {#Total-Degree-Method}"'},"")],-1)),t("details",d,[t("summary",null,[e[0]||(e[0]=t("a",{id:"HarmonicBalance.TotalDegree-manual-methods",href:"#HarmonicBalance.TotalDegree-manual-methods"},[t("span",{class:"jlbinding"},"HarmonicBalance.TotalDegree")],-1)),e[1]||(e[1]=a()),Q(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[16]||(e[16]=o('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">TotalDegree</span></span></code></pre></div>',1)),t("p",null,[e[8]||(e[8]=a("The 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The method guarantees to find all solutions, however, it comes with a high computational cost. See ")),e[14]||(e[14]=t("a",{href:"https://www.juliahomotopycontinuation.org/guides/totaldegree/",target:"_blank",rel:"noreferrer"},"HomotopyContinuation.jl",-1)),e[15]||(e[15]=a(" for more information."))]),e[17]||(e[17]=o('<p><strong>Fields</strong></p><ul><li><p><code>gamma::Complex</code>: Complex multiplying factor of the start system G(x) for the homotopy</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/methods.jl#L8-L20" target="_blank" rel="noreferrer">source</a></p>',3))]),e[27]||(e[27]=t("h2",{id:"Polyhedral-Method",tabindex:"-1"},[a("Polyhedral Method "),t("a",{class:"header-anchor",href:"#Polyhedral-Method","aria-label":'Permalink to "Polyhedral Method {#Polyhedral-Method}"'},"")],-1)),t("details",f,[t("summary",null,[e[18]||(e[18]=t("a",{id:"HarmonicBalance.Polyhedral-manual-methods",href:"#HarmonicBalance.Polyhedral-manual-methods"},[t("span",{class:"jlbinding"},"HarmonicBalance.Polyhedral")],-1)),e[19]||(e[19]=a()),Q(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[20]||(e[20]=o('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Polyhedral</span></span></code></pre></div><p>The Polyhedral homotopy method constructs a homotopy based on the polyhedral structure of the polynomial system. It is more efficient than the Total Degree method for sparse systems, meaning most of the coefficients are zero. It can be especially useful if you don't need to find the zero solutions (<code>only_non_zero = true</code>), resulting in a speed up. See <a href="https://www.juliahomotopycontinuation.org/guides/polyhedral/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>only_non_zero::Bool</code>: Boolean indicating if only non-zero solutions are considered.</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/methods.jl#L57-L69" target="_blank" rel="noreferrer">source</a></p>',5))]),e[28]||(e[28]=t("h2",{id:"Warm-Up-Method",tabindex:"-1"},[a("Warm Up Method "),t("a",{class:"header-anchor",href:"#Warm-Up-Method","aria-label":'Permalink to "Warm Up Method {#Warm-Up-Method}"'},"")],-1)),t("details",H,[t("summary",null,[e[21]||(e[21]=t("a",{id:"HarmonicBalance.WarmUp-manual-methods",href:"#HarmonicBalance.WarmUp-manual-methods"},[t("span",{class:"jlbinding"},"HarmonicBalance.WarmUp")],-1)),e[22]||(e[22]=a()),Q(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),e[23]||(e[23]=o('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">WarmUp</span></span></code></pre></div><p>The Warm Up method prepares a warmup system with the Total Degree method using the parameter at <code>index</code> perturbed by <code>perturbation_size</code>. The warmup system is used to perform a homotopy using all other systems in the parameter sweep. It is very efficient for systems with minimal bifurcation in the parameter sweep. The Warm Up method should in theory guarantee to find all solutions, however, if the <code>start_parameters</code> is not proper (to close to the real line) it could miss some solutions.</p><p>See<a href="https://www.juliahomotopycontinuation.org/guides/many-systems/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>warm_up_method::Union{Polyhedral{T}, TotalDegree{T}} where T</code>: Method used for the warmup system.</p></li><li><p><code>start_parameters::Vector</code>: Start parameters.</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/methods.jl#L110-L125" target="_blank" rel="noreferrer">source</a></p>',6))])])}const v=r(T,[["render",y]]);export{L as __pageData,v as default};
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+import{_ as r,c as l,j as t,a,G as Q,ai as o,B as i,o as s}from"./chunks/framework.Bnn8558y.js";const L=JSON.parse('{"title":"Methods","description":"","frontmatter":{},"headers":[],"relativePath":"manual/methods.md","filePath":"manual/methods.md"}'),T={name:"manual/methods.md"},d={class:"jldocstring custom-block",open:""},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},p={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"30.769ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 13600.1 1000","aria-hidden":"true"},h={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"17.717ex",height:"2.587ex",role:"img",focusable:"false",viewBox:"0 -893.3 7831 1143.3","aria-hidden":"true"},g={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.452ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.534ex",height:"2.149ex",role:"img",focusable:"false",viewBox:"0 -750 1120 950","aria-hidden":"true"},f={class:"jldocstring custom-block",open:""},H={class:"jldocstring custom-block",open:""};function y(w,e,x,b,k,_){const n=i("Badge");return s(),l("div",null,[e[24]||(e[24]=t("h1",{id:"methods",tabindex:"-1"},[a("Methods "),t("a",{class:"header-anchor",href:"#methods","aria-label":'Permalink to "Methods"'},"")],-1)),e[25]||(e[25]=t("p",null,"We offer several methods for solving the nonlinear algebraic equations that arise from the harmonic balance procedure. 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diff --git a/dev/assets/manual_plotting.md.BEcwTRGc.js b/dev/assets/manual_plotting.md.ClQxtj0e.js
similarity index 96%
rename from dev/assets/manual_plotting.md.BEcwTRGc.js
rename to dev/assets/manual_plotting.md.ClQxtj0e.js
index bff4e6bed..e26fc566d 100644
--- a/dev/assets/manual_plotting.md.BEcwTRGc.js
+++ b/dev/assets/manual_plotting.md.ClQxtj0e.js
@@ -1,22 +1,22 @@
-import{_ as l,c as o,j as a,a as i,G as n,ai as e,B as p,o as r}from"./chunks/framework.Bnn8558y.js";const F=JSON.parse('{"title":"Analysis and plotting","description":"","frontmatter":{},"headers":[],"relativePath":"manual/plotting.md","filePath":"manual/plotting.md"}'),h={name:"manual/plotting.md"},d={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""};function E(y,s,b,m,f,_){const t=p("Badge");return r(),o("div",null,[s[15]||(s[15]=a("h1",{id:"Analysis-and-plotting",tabindex:"-1"},[i("Analysis and plotting "),a("a",{class:"header-anchor",href:"#Analysis-and-plotting","aria-label":'Permalink to "Analysis and plotting {#Analysis-and-plotting}"'},"")],-1)),s[16]||(s[16]=a("p",null,[i("The key method for visualization is "),a("code",null,"transform_solutions"),i(", which parses a string into a symbolic expression and evaluates it for every steady state solution.")],-1)),a("details",d,[a("summary",null,[s[0]||(s[0]=a("a",{id:"HarmonicBalance.transform_solutions-manual-plotting",href:"#HarmonicBalance.transform_solutions-manual-plotting"},[a("span",{class:"jlbinding"},"HarmonicBalance.transform_solutions")],-1)),s[1]||(s[1]=i()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[2]||(s[2]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">transform_solutions</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
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<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result{S, ParType, D, F}</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> where</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> {ParType</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Number</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, D, F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">FunctionWrappers.FunctionWrapper{Array{S, 2}, Tuple{Array{S, 1}}}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">},</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> func;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> branches,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> realify</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector</span></span></code></pre></div><p>Takes a <code>Result</code> object and a string <code>f</code> representing a Symbolics.jl expression. Returns an array with the values of <code>f</code> evaluated for the respective solutions. Additional substitution rules can be specified in <code>rules</code> in the format <code>("a" => val)</code> or <code>(a => val)</code></p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/transform_solutions.jl#L64" target="_blank" rel="noreferrer">source</a></p>`,3))]),s[17]||(s[17]=a("h2",{id:"Plotting-solutions",tabindex:"-1"},[i("Plotting solutions "),a("a",{class:"header-anchor",href:"#Plotting-solutions","aria-label":'Permalink to "Plotting solutions {#Plotting-solutions}"'},"")],-1)),s[18]||(s[18]=a("p",null,[i("The function "),a("code",null,"plot"),i(" is multiple-dispatched to plot 1D and 2D datasets. In 1D, the solutions are colour-coded according to the branches obtained by "),a("code",null,"sort_solutions"),i(".")],-1)),a("details",c,[a("summary",null,[s[3]||(s[3]=a("a",{id:"RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}-manual-plotting",href:"#RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}-manual-plotting"},[a("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[4]||(s[4]=i()),n(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[5]||(s[5]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector</span></span></code></pre></div><p>Takes a <code>Result</code> object and a string <code>f</code> representing a Symbolics.jl expression. Returns an array with the values of <code>f</code> evaluated for the respective solutions. Additional substitution rules can be specified in <code>rules</code> in the format <code>("a" => val)</code> or <code>(a => val)</code></p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/transform_solutions.jl#L64" target="_blank" rel="noreferrer">source</a></p>`,3))]),s[17]||(s[17]=a("h2",{id:"Plotting-solutions",tabindex:"-1"},[i("Plotting solutions "),a("a",{class:"header-anchor",href:"#Plotting-solutions","aria-label":'Permalink to "Plotting solutions {#Plotting-solutions}"'},"")],-1)),s[18]||(s[18]=a("p",null,[i("The function "),a("code",null,"plot"),i(" is multiple-dispatched to plot 1D and 2D datasets. In 1D, the solutions are colour-coded according to the branches obtained by "),a("code",null,"sort_solutions"),i(".")],-1)),a("details",c,[a("summary",null,[s[3]||(s[3]=a("a",{id:"RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}-manual-plotting",href:"#RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}-manual-plotting"},[a("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[4]||(s[4]=i()),n(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[5]||(s[5]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result{S, P, D, F}</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> where</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">FunctionWrappers.FunctionWrapper{Array{S, 2}, Tuple{Array{S, 1}}}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> varargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> cut,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Plot</span></span></code></pre></div><p><strong>Plot a <code>Result</code> object.</strong></p><p>Class selection done by passing <code>String</code> or <code>Vector{String}</code> as kwarg:</p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>class : only plot solutions in this class(es) ("all" --> plot everything)</span></span>
<span class="line"><span>not_class : do not plot solutions in this class(es)</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr().</p><p>See also <code>plot!</code></p><p>The x,y,z arguments are Strings compatible with Symbolics.jl, e.g., <code>y=2*sqrt(u1^2+v1^2)</code> plots the amplitude of the first quadratures multiplied by 2.</p><p><strong>1D plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(res::Result; x::String, y::String, class="default", not_class=[], kwargs...)</span></span>
-<span class="line"><span>plot(res::Result, y::String; kwargs...) # take x automatically from Result</span></span></code></pre></div><p>Default behaviour is to plot stable solutions as full lines, unstable as dashed.</p><p>If a sweep in two parameters were done, i.e., <code>dim(res)==2</code>, a one dimensional cut can be plotted by using the keyword <code>cut</code> were it takes a <code>Pair{Num, Float}</code> type entry. For example, <code>plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2))</code> plots a cut at <code>λ = 0.2</code>.</p><hr><p><strong>2D plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)</span></span></code></pre></div><p>To make the 2d plot less chaotic it is required to specify the specific <code>branch</code> to plot, labeled by a <code>Int64</code>.</p><p>The x and y axes are taken automatically from <code>res</code></p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/plotting_Plots.jl#L11" target="_blank" rel="noreferrer">source</a></p>`,17))]),a("details",k,[a("summary",null,[s[6]||(s[6]=a("a",{id:"RecipesBase.plot!-manual-plotting",href:"#RecipesBase.plot!-manual-plotting"},[a("span",{class:"jlbinding"},"RecipesBase.plot!")],-1)),s[7]||(s[7]=i()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[8]||(s[8]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span>plot(res::Result, y::String; kwargs...) # take x automatically from Result</span></span></code></pre></div><p>Default behaviour is to plot stable solutions as full lines, unstable as dashed.</p><p>If a sweep in two parameters were done, i.e., <code>dim(res)==2</code>, a one dimensional cut can be plotted by using the keyword <code>cut</code> were it takes a <code>Pair{Num, Float}</code> type entry. For example, <code>plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2))</code> plots a cut at <code>λ = 0.2</code>.</p><hr><p><strong>2D plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)</span></span></code></pre></div><p>To make the 2d plot less chaotic it is required to specify the specific <code>branch</code> to plot, labeled by a <code>Int64</code>.</p><p>The x and y axes are taken automatically from <code>res</code></p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/plotting_Plots.jl#L11" target="_blank" rel="noreferrer">source</a></p>`,17))]),a("details",k,[a("summary",null,[s[6]||(s[6]=a("a",{id:"RecipesBase.plot!-manual-plotting",href:"#RecipesBase.plot!-manual-plotting"},[a("span",{class:"jlbinding"},"RecipesBase.plot!")],-1)),s[7]||(s[7]=i()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[8]||(s[8]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> varargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Plot</span></span></code></pre></div><p>Similar to <code>plot</code> but adds a plot onto an existing plot.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/plotting_Plots.jl#L62" target="_blank" rel="noreferrer">source</a></p>`,3))]),s[19]||(s[19]=a("h2",{id:"Plotting-phase-diagrams",tabindex:"-1"},[i("Plotting phase diagrams "),a("a",{class:"header-anchor",href:"#Plotting-phase-diagrams","aria-label":'Permalink to "Plotting phase diagrams {#Plotting-phase-diagrams}"'},"")],-1)),s[20]||(s[20]=a("p",null,[i("In many problems, rather than in any property of the solutions themselves, we are interested in the phase diagrams, encoding the number of (stable) solutions in different regions of the parameter space. "),a("code",null,"plot_phase_diagram"),i(" handles this for 1D and 2D datasets.")],-1)),a("details",g,[a("summary",null,[s[9]||(s[9]=a("a",{id:"HarmonicBalance.plot_phase_diagram-manual-plotting",href:"#HarmonicBalance.plot_phase_diagram-manual-plotting"},[a("span",{class:"jlbinding"},"HarmonicBalance.plot_phase_diagram")],-1)),s[10]||(s[10]=i()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_phase_diagram</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Plot</span></span></code></pre></div><p>Similar to <code>plot</code> but adds a plot onto an existing plot.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/plotting_Plots.jl#L62" target="_blank" rel="noreferrer">source</a></p>`,3))]),s[19]||(s[19]=a("h2",{id:"Plotting-phase-diagrams",tabindex:"-1"},[i("Plotting phase diagrams "),a("a",{class:"header-anchor",href:"#Plotting-phase-diagrams","aria-label":'Permalink to "Plotting phase diagrams {#Plotting-phase-diagrams}"'},"")],-1)),s[20]||(s[20]=a("p",null,[i("In many problems, rather than in any property of the solutions themselves, we are interested in the phase diagrams, encoding the number of (stable) solutions in different regions of the parameter space. "),a("code",null,"plot_phase_diagram"),i(" handles this for 1D and 2D datasets.")],-1)),a("details",g,[a("summary",null,[s[9]||(s[9]=a("a",{id:"HarmonicBalance.plot_phase_diagram-manual-plotting",href:"#HarmonicBalance.plot_phase_diagram-manual-plotting"},[a("span",{class:"jlbinding"},"HarmonicBalance.plot_phase_diagram")],-1)),s[10]||(s[10]=i()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_phase_diagram</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Plot</span></span></code></pre></div><p>Plot the number of solutions in a <code>Result</code> object as a function of the parameters. Works with 1D and 2D datasets.</p><p>Class selection done by passing <code>String</code> or <code>Vector{String}</code> as kwarg:</p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>class::String : only count solutions in this class ("all" --> plot everything)</span></span>
-<span class="line"><span>not_class::String : do not count solutions in this class</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr()</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/plotting_Plots.jl#L275" target="_blank" rel="noreferrer">source</a></p>`,6))]),s[21]||(s[21]=a("h2",{id:"Plot-spaghetti-plot",tabindex:"-1"},[i("Plot spaghetti plot "),a("a",{class:"header-anchor",href:"#Plot-spaghetti-plot","aria-label":'Permalink to "Plot spaghetti plot {#Plot-spaghetti-plot}"'},"")],-1)),s[22]||(s[22]=a("p",null,[i("Sometimes, it is useful to plot the quadratures of the steady states (u, v) in function of a swept parameter. This is done with "),a("code",null,"plot_spaghetti"),i(".")],-1)),a("details",u,[a("summary",null,[s[12]||(s[12]=a("a",{id:"HarmonicBalance.plot_spaghetti-manual-plotting",href:"#HarmonicBalance.plot_spaghetti-manual-plotting"},[a("span",{class:"jlbinding"},"HarmonicBalance.plot_spaghetti")],-1)),s[13]||(s[13]=i()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[14]||(s[14]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_spaghetti</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; x, y, z, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a three dimension line plot of a <code>Result</code> object as a function of the parameters. Works with 1D and 2D datasets.</p><p>Class selection done by passing <code>String</code> or <code>Vector{String}</code> as kwarg:</p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>class::String : only count solutions in this class ("all" --> plot everything)</span></span>
-<span class="line"><span>not_class::String : do not count solutions in this class</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr()</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/plotting_Plots.jl#L342-L354" target="_blank" rel="noreferrer">source</a></p>`,6))])])}const C=l(h,[["render",E]]);export{F as __pageData,C as default};
+<span class="line"><span>not_class::String : do not count solutions in this class</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr()</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/plotting_Plots.jl#L275" target="_blank" rel="noreferrer">source</a></p>`,6))]),s[21]||(s[21]=a("h2",{id:"Plot-spaghetti-plot",tabindex:"-1"},[i("Plot spaghetti plot "),a("a",{class:"header-anchor",href:"#Plot-spaghetti-plot","aria-label":'Permalink to "Plot spaghetti plot {#Plot-spaghetti-plot}"'},"")],-1)),s[22]||(s[22]=a("p",null,[i("Sometimes, it is useful to plot the quadratures of the steady states (u, v) in function of a swept parameter. This is done with "),a("code",null,"plot_spaghetti"),i(".")],-1)),a("details",u,[a("summary",null,[s[12]||(s[12]=a("a",{id:"HarmonicBalance.plot_spaghetti-manual-plotting",href:"#HarmonicBalance.plot_spaghetti-manual-plotting"},[a("span",{class:"jlbinding"},"HarmonicBalance.plot_spaghetti")],-1)),s[13]||(s[13]=i()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[14]||(s[14]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_spaghetti</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; x, y, z, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a three dimension line plot of a <code>Result</code> object as a function of the parameters. Works with 1D and 2D datasets.</p><p>Class selection done by passing <code>String</code> or <code>Vector{String}</code> as kwarg:</p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>class::String : only count solutions in this class ("all" --> plot everything)</span></span>
+<span class="line"><span>not_class::String : do not count solutions in this class</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr()</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/plotting_Plots.jl#L342-L354" target="_blank" rel="noreferrer">source</a></p>`,6))])])}const C=l(h,[["render",E]]);export{F as __pageData,C as default};
diff --git a/dev/assets/manual_plotting.md.BEcwTRGc.lean.js b/dev/assets/manual_plotting.md.ClQxtj0e.lean.js
similarity index 98%
rename from dev/assets/manual_plotting.md.BEcwTRGc.lean.js
rename to dev/assets/manual_plotting.md.ClQxtj0e.lean.js
index 3aaaa3e4a..14d27bea9 100644
--- a/dev/assets/manual_plotting.md.BEcwTRGc.lean.js
+++ b/dev/assets/manual_plotting.md.ClQxtj0e.lean.js
@@ -1 +1 @@
-import{_ as l,c as o,j as a,a as i,G as n,ai as e,B as p,o as r}from"./chunks/framework.Bnn8558y.js";const F=JSON.parse('{"title":"Analysis and plotting","description":"","frontmatter":{},"headers":[],"relativePath":"manual/plotting.md","filePath":"manual/plotting.md"}'),h={name:"manual/plotting.md"},d={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""};function E(y,s,b,m,f,_){const t=p("Badge");return r(),o("div",null,[s[15]||(s[15]=a("h1",{id:"Analysis-and-plotting",tabindex:"-1"},[i("Analysis and plotting "),a("a",{class:"header-anchor",href:"#Analysis-and-plotting","aria-label":'Permalink to "Analysis and plotting {#Analysis-and-plotting}"'},"")],-1)),s[16]||(s[16]=a("p",null,[i("The key method for visualization is "),a("code",null,"transform_solutions"),i(", which parses a string into a symbolic expression and evaluates it for every steady state solution.")],-1)),a("details",d,[a("summary",null,[s[0]||(s[0]=a("a",{id:"HarmonicBalance.transform_solutions-manual-plotting",href:"#HarmonicBalance.transform_solutions-manual-plotting"},[a("span",{class:"jlbinding"},"HarmonicBalance.transform_solutions")],-1)),s[1]||(s[1]=i()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[2]||(s[2]=e("",3))]),s[17]||(s[17]=a("h2",{id:"Plotting-solutions",tabindex:"-1"},[i("Plotting solutions "),a("a",{class:"header-anchor",href:"#Plotting-solutions","aria-label":'Permalink to "Plotting solutions {#Plotting-solutions}"'},"")],-1)),s[18]||(s[18]=a("p",null,[i("The function "),a("code",null,"plot"),i(" is multiple-dispatched to plot 1D and 2D datasets. In 1D, the solutions are colour-coded according to the branches obtained by "),a("code",null,"sort_solutions"),i(".")],-1)),a("details",c,[a("summary",null,[s[3]||(s[3]=a("a",{id:"RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}-manual-plotting",href:"#RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}-manual-plotting"},[a("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[4]||(s[4]=i()),n(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[5]||(s[5]=e("",17))]),a("details",k,[a("summary",null,[s[6]||(s[6]=a("a",{id:"RecipesBase.plot!-manual-plotting",href:"#RecipesBase.plot!-manual-plotting"},[a("span",{class:"jlbinding"},"RecipesBase.plot!")],-1)),s[7]||(s[7]=i()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[8]||(s[8]=e("",3))]),s[19]||(s[19]=a("h2",{id:"Plotting-phase-diagrams",tabindex:"-1"},[i("Plotting phase diagrams "),a("a",{class:"header-anchor",href:"#Plotting-phase-diagrams","aria-label":'Permalink to "Plotting phase diagrams {#Plotting-phase-diagrams}"'},"")],-1)),s[20]||(s[20]=a("p",null,[i("In many problems, rather than in any property of the solutions themselves, we are interested in the phase diagrams, encoding the number of (stable) solutions in different regions of the parameter space. "),a("code",null,"plot_phase_diagram"),i(" handles this for 1D and 2D datasets.")],-1)),a("details",g,[a("summary",null,[s[9]||(s[9]=a("a",{id:"HarmonicBalance.plot_phase_diagram-manual-plotting",href:"#HarmonicBalance.plot_phase_diagram-manual-plotting"},[a("span",{class:"jlbinding"},"HarmonicBalance.plot_phase_diagram")],-1)),s[10]||(s[10]=i()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=e("",6))]),s[21]||(s[21]=a("h2",{id:"Plot-spaghetti-plot",tabindex:"-1"},[i("Plot spaghetti plot "),a("a",{class:"header-anchor",href:"#Plot-spaghetti-plot","aria-label":'Permalink to "Plot spaghetti plot {#Plot-spaghetti-plot}"'},"")],-1)),s[22]||(s[22]=a("p",null,[i("Sometimes, it is useful to plot the quadratures of the steady states (u, v) in function of a swept parameter. This is done with "),a("code",null,"plot_spaghetti"),i(".")],-1)),a("details",u,[a("summary",null,[s[12]||(s[12]=a("a",{id:"HarmonicBalance.plot_spaghetti-manual-plotting",href:"#HarmonicBalance.plot_spaghetti-manual-plotting"},[a("span",{class:"jlbinding"},"HarmonicBalance.plot_spaghetti")],-1)),s[13]||(s[13]=i()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[14]||(s[14]=e("",6))])])}const C=l(h,[["render",E]]);export{F as __pageData,C as default};
+import{_ as l,c as o,j as a,a as i,G as n,ai as e,B as p,o as r}from"./chunks/framework.Bnn8558y.js";const F=JSON.parse('{"title":"Analysis and plotting","description":"","frontmatter":{},"headers":[],"relativePath":"manual/plotting.md","filePath":"manual/plotting.md"}'),h={name:"manual/plotting.md"},d={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""};function E(y,s,m,b,f,_){const t=p("Badge");return r(),o("div",null,[s[15]||(s[15]=a("h1",{id:"Analysis-and-plotting",tabindex:"-1"},[i("Analysis and plotting "),a("a",{class:"header-anchor",href:"#Analysis-and-plotting","aria-label":'Permalink to "Analysis and plotting {#Analysis-and-plotting}"'},"")],-1)),s[16]||(s[16]=a("p",null,[i("The key method for visualization is "),a("code",null,"transform_solutions"),i(", which parses a string into a symbolic expression and evaluates it for every steady state solution.")],-1)),a("details",d,[a("summary",null,[s[0]||(s[0]=a("a",{id:"HarmonicBalance.transform_solutions-manual-plotting",href:"#HarmonicBalance.transform_solutions-manual-plotting"},[a("span",{class:"jlbinding"},"HarmonicBalance.transform_solutions")],-1)),s[1]||(s[1]=i()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[2]||(s[2]=e("",3))]),s[17]||(s[17]=a("h2",{id:"Plotting-solutions",tabindex:"-1"},[i("Plotting solutions "),a("a",{class:"header-anchor",href:"#Plotting-solutions","aria-label":'Permalink to "Plotting solutions {#Plotting-solutions}"'},"")],-1)),s[18]||(s[18]=a("p",null,[i("The function "),a("code",null,"plot"),i(" is multiple-dispatched to plot 1D and 2D datasets. In 1D, the solutions are colour-coded according to the branches obtained by "),a("code",null,"sort_solutions"),i(".")],-1)),a("details",c,[a("summary",null,[s[3]||(s[3]=a("a",{id:"RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}-manual-plotting",href:"#RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}-manual-plotting"},[a("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[4]||(s[4]=i()),n(t,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[5]||(s[5]=e("",17))]),a("details",k,[a("summary",null,[s[6]||(s[6]=a("a",{id:"RecipesBase.plot!-manual-plotting",href:"#RecipesBase.plot!-manual-plotting"},[a("span",{class:"jlbinding"},"RecipesBase.plot!")],-1)),s[7]||(s[7]=i()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[8]||(s[8]=e("",3))]),s[19]||(s[19]=a("h2",{id:"Plotting-phase-diagrams",tabindex:"-1"},[i("Plotting phase diagrams "),a("a",{class:"header-anchor",href:"#Plotting-phase-diagrams","aria-label":'Permalink to "Plotting phase diagrams {#Plotting-phase-diagrams}"'},"")],-1)),s[20]||(s[20]=a("p",null,[i("In many problems, rather than in any property of the solutions themselves, we are interested in the phase diagrams, encoding the number of (stable) solutions in different regions of the parameter space. "),a("code",null,"plot_phase_diagram"),i(" handles this for 1D and 2D datasets.")],-1)),a("details",g,[a("summary",null,[s[9]||(s[9]=a("a",{id:"HarmonicBalance.plot_phase_diagram-manual-plotting",href:"#HarmonicBalance.plot_phase_diagram-manual-plotting"},[a("span",{class:"jlbinding"},"HarmonicBalance.plot_phase_diagram")],-1)),s[10]||(s[10]=i()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=e("",6))]),s[21]||(s[21]=a("h2",{id:"Plot-spaghetti-plot",tabindex:"-1"},[i("Plot spaghetti plot "),a("a",{class:"header-anchor",href:"#Plot-spaghetti-plot","aria-label":'Permalink to "Plot spaghetti plot {#Plot-spaghetti-plot}"'},"")],-1)),s[22]||(s[22]=a("p",null,[i("Sometimes, it is useful to plot the quadratures of the steady states (u, v) in function of a swept parameter. This is done with "),a("code",null,"plot_spaghetti"),i(".")],-1)),a("details",u,[a("summary",null,[s[12]||(s[12]=a("a",{id:"HarmonicBalance.plot_spaghetti-manual-plotting",href:"#HarmonicBalance.plot_spaghetti-manual-plotting"},[a("span",{class:"jlbinding"},"HarmonicBalance.plot_spaghetti")],-1)),s[13]||(s[13]=i()),n(t,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[14]||(s[14]=e("",6))])])}const C=l(h,[["render",E]]);export{F as __pageData,C as default};
diff --git a/dev/assets/manual_saving.md.DxNepvaB.js b/dev/assets/manual_saving.md.DzEBajjy.js
similarity index 91%
rename from dev/assets/manual_saving.md.DxNepvaB.js
rename to dev/assets/manual_saving.md.DzEBajjy.js
index 739493ed2..449c38a1f 100644
--- a/dev/assets/manual_saving.md.DxNepvaB.js
+++ b/dev/assets/manual_saving.md.DzEBajjy.js
@@ -1 +1 @@
-import{_ as t,c as l,ai as n,j as e,a as i,G as o,B as c,o as d}from"./chunks/framework.Bnn8558y.js";const j=JSON.parse('{"title":"Saving and loading","description":"","frontmatter":{},"headers":[],"relativePath":"manual/saving.md","filePath":"manual/saving.md"}'),r={name:"manual/saving.md"},p={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""};function m(b,a,h,_,f,v){const s=c("Badge");return d(),l("div",null,[a[9]||(a[9]=n('<h1 id="Saving-and-loading" tabindex="-1">Saving and loading <a class="header-anchor" href="#Saving-and-loading" aria-label="Permalink to "Saving and loading {#Saving-and-loading}""></a></h1><p>All of the types native to <code>HarmonicBalance.jl</code> can be saved into a <code>.jld2</code> file using <code>save</code> and loaded using <code>load</code>. Most of the saving/loading is performed using the package <code>JLD2.jl</code>, with the addition of reinstating the symbolic variables in the <code>HarmonicBalance</code> namespace (needed to parse expressions used in the plotting functions) and recompiling stored functions (needed to evaluate Jacobians). As a consequence, composite objects such as <code>Result</code> can be saved and loaded with no loss of information.</p><p>The function <code>export_csv</code> saves a .csv file which can be plot elsewhere.</p>',3)),e("details",p,[e("summary",null,[a[0]||(a[0]=e("a",{id:"HarmonicBalance.save-manual-saving",href:"#HarmonicBalance.save-manual-saving"},[e("span",{class:"jlbinding"},"HarmonicBalance.save")],-1)),a[1]||(a[1]=i()),o(s,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[2]||(a[2]=n('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">save</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(filename, object)</span></span></code></pre></div><p>Saves <code>object</code> into <code>.jld2</code> file <code>filename</code> (the suffix is added automatically if not entered). The resulting file contains a dictionary with a single entry.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/saving.jl#L1-L7" target="_blank" rel="noreferrer">source</a></p>',3))]),e("details",g,[e("summary",null,[a[3]||(a[3]=e("a",{id:"HarmonicBalance.load-manual-saving",href:"#HarmonicBalance.load-manual-saving"},[e("span",{class:"jlbinding"},"HarmonicBalance.load")],-1)),a[4]||(a[4]=i()),o(s,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[5]||(a[5]=n('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">load</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(filename)</span></span></code></pre></div><p>Loads an object from <code>filename</code>. For objects containing symbolic expressions such as <code>HarmonicEquation</code>, the symbolic variables are reinstated in the <code>HarmonicBalance</code> namespace.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/saving.jl#L22-L28" target="_blank" rel="noreferrer">source</a></p>',3))]),e("details",u,[e("summary",null,[a[6]||(a[6]=e("a",{id:"HarmonicBalance.export_csv-manual-saving",href:"#HarmonicBalance.export_csv-manual-saving"},[e("span",{class:"jlbinding"},"HarmonicBalance.export_csv")],-1)),a[7]||(a[7]=i()),o(s,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[8]||(a[8]=n('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">export_csv</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(filename, res, branch)</span></span></code></pre></div><p>Saves into <code>filename</code> a specified solution <code>branch</code> of the Result <code>res</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/saving.jl#L77-L81" target="_blank" rel="noreferrer">source</a></p>',3))]),a[10]||(a[10]=n('<div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span></span></span></code></pre></div>',1))])}const T=t(r,[["render",m]]);export{j as __pageData,T as default};
+import{_ as t,c as l,ai as n,j as e,a as i,G as o,B as c,o as d}from"./chunks/framework.Bnn8558y.js";const j=JSON.parse('{"title":"Saving and loading","description":"","frontmatter":{},"headers":[],"relativePath":"manual/saving.md","filePath":"manual/saving.md"}'),r={name:"manual/saving.md"},p={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""};function m(h,a,b,_,v,f){const s=c("Badge");return d(),l("div",null,[a[9]||(a[9]=n('<h1 id="Saving-and-loading" tabindex="-1">Saving and loading <a class="header-anchor" href="#Saving-and-loading" aria-label="Permalink to "Saving and loading {#Saving-and-loading}""></a></h1><p>All of the types native to <code>HarmonicBalance.jl</code> can be saved into a <code>.jld2</code> file using <code>save</code> and loaded using <code>load</code>. Most of the saving/loading is performed using the package <code>JLD2.jl</code>, with the addition of reinstating the symbolic variables in the <code>HarmonicBalance</code> namespace (needed to parse expressions used in the plotting functions) and recompiling stored functions (needed to evaluate Jacobians). As a consequence, composite objects such as <code>Result</code> can be saved and loaded with no loss of information.</p><p>The function <code>export_csv</code> saves a .csv file which can be plot elsewhere.</p>',3)),e("details",p,[e("summary",null,[a[0]||(a[0]=e("a",{id:"HarmonicBalance.save-manual-saving",href:"#HarmonicBalance.save-manual-saving"},[e("span",{class:"jlbinding"},"HarmonicBalance.save")],-1)),a[1]||(a[1]=i()),o(s,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[2]||(a[2]=n('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">save</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(filename, object)</span></span></code></pre></div><p>Saves <code>object</code> into <code>.jld2</code> file <code>filename</code> (the suffix is added automatically if not entered). The resulting file contains a dictionary with a single entry.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/saving.jl#L1-L7" target="_blank" rel="noreferrer">source</a></p>',3))]),e("details",g,[e("summary",null,[a[3]||(a[3]=e("a",{id:"HarmonicBalance.load-manual-saving",href:"#HarmonicBalance.load-manual-saving"},[e("span",{class:"jlbinding"},"HarmonicBalance.load")],-1)),a[4]||(a[4]=i()),o(s,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[5]||(a[5]=n('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">load</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(filename)</span></span></code></pre></div><p>Loads an object from <code>filename</code>. For objects containing symbolic expressions such as <code>HarmonicEquation</code>, the symbolic variables are reinstated in the <code>HarmonicBalance</code> namespace.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/saving.jl#L22-L28" target="_blank" rel="noreferrer">source</a></p>',3))]),e("details",u,[e("summary",null,[a[6]||(a[6]=e("a",{id:"HarmonicBalance.export_csv-manual-saving",href:"#HarmonicBalance.export_csv-manual-saving"},[e("span",{class:"jlbinding"},"HarmonicBalance.export_csv")],-1)),a[7]||(a[7]=i()),o(s,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[8]||(a[8]=n('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">export_csv</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(filename, res, branch)</span></span></code></pre></div><p>Saves into <code>filename</code> a specified solution <code>branch</code> of the Result <code>res</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/saving.jl#L77-L81" target="_blank" rel="noreferrer">source</a></p>',3))]),a[10]||(a[10]=n('<div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span></span></span></code></pre></div>',1))])}const T=t(r,[["render",m]]);export{j as __pageData,T as default};
diff --git a/dev/assets/manual_saving.md.DxNepvaB.lean.js b/dev/assets/manual_saving.md.DzEBajjy.lean.js
similarity index 91%
rename from dev/assets/manual_saving.md.DxNepvaB.lean.js
rename to dev/assets/manual_saving.md.DzEBajjy.lean.js
index 130391ef4..5658779eb 100644
--- a/dev/assets/manual_saving.md.DxNepvaB.lean.js
+++ b/dev/assets/manual_saving.md.DzEBajjy.lean.js
@@ -1 +1 @@
-import{_ as t,c as l,ai as n,j as e,a as i,G as o,B as c,o as d}from"./chunks/framework.Bnn8558y.js";const j=JSON.parse('{"title":"Saving and loading","description":"","frontmatter":{},"headers":[],"relativePath":"manual/saving.md","filePath":"manual/saving.md"}'),r={name:"manual/saving.md"},p={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""};function m(b,a,h,_,f,v){const s=c("Badge");return d(),l("div",null,[a[9]||(a[9]=n("",3)),e("details",p,[e("summary",null,[a[0]||(a[0]=e("a",{id:"HarmonicBalance.save-manual-saving",href:"#HarmonicBalance.save-manual-saving"},[e("span",{class:"jlbinding"},"HarmonicBalance.save")],-1)),a[1]||(a[1]=i()),o(s,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[2]||(a[2]=n("",3))]),e("details",g,[e("summary",null,[a[3]||(a[3]=e("a",{id:"HarmonicBalance.load-manual-saving",href:"#HarmonicBalance.load-manual-saving"},[e("span",{class:"jlbinding"},"HarmonicBalance.load")],-1)),a[4]||(a[4]=i()),o(s,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[5]||(a[5]=n("",3))]),e("details",u,[e("summary",null,[a[6]||(a[6]=e("a",{id:"HarmonicBalance.export_csv-manual-saving",href:"#HarmonicBalance.export_csv-manual-saving"},[e("span",{class:"jlbinding"},"HarmonicBalance.export_csv")],-1)),a[7]||(a[7]=i()),o(s,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[8]||(a[8]=n("",3))]),a[10]||(a[10]=n("",1))])}const T=t(r,[["render",m]]);export{j as __pageData,T as default};
+import{_ as t,c as l,ai as n,j as e,a as i,G as o,B as c,o as d}from"./chunks/framework.Bnn8558y.js";const j=JSON.parse('{"title":"Saving and loading","description":"","frontmatter":{},"headers":[],"relativePath":"manual/saving.md","filePath":"manual/saving.md"}'),r={name:"manual/saving.md"},p={class:"jldocstring custom-block",open:""},g={class:"jldocstring custom-block",open:""},u={class:"jldocstring custom-block",open:""};function m(h,a,b,_,v,f){const s=c("Badge");return d(),l("div",null,[a[9]||(a[9]=n("",3)),e("details",p,[e("summary",null,[a[0]||(a[0]=e("a",{id:"HarmonicBalance.save-manual-saving",href:"#HarmonicBalance.save-manual-saving"},[e("span",{class:"jlbinding"},"HarmonicBalance.save")],-1)),a[1]||(a[1]=i()),o(s,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[2]||(a[2]=n("",3))]),e("details",g,[e("summary",null,[a[3]||(a[3]=e("a",{id:"HarmonicBalance.load-manual-saving",href:"#HarmonicBalance.load-manual-saving"},[e("span",{class:"jlbinding"},"HarmonicBalance.load")],-1)),a[4]||(a[4]=i()),o(s,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[5]||(a[5]=n("",3))]),e("details",u,[e("summary",null,[a[6]||(a[6]=e("a",{id:"HarmonicBalance.export_csv-manual-saving",href:"#HarmonicBalance.export_csv-manual-saving"},[e("span",{class:"jlbinding"},"HarmonicBalance.export_csv")],-1)),a[7]||(a[7]=i()),o(s,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),a[8]||(a[8]=n("",3))]),a[10]||(a[10]=n("",1))])}const T=t(r,[["render",m]]);export{j as __pageData,T as default};
diff --git a/dev/assets/manual_solving_harmonics.md.BVaL1Npd.js b/dev/assets/manual_solving_harmonics.md.DwC-AGfx.js
similarity index 98%
rename from dev/assets/manual_solving_harmonics.md.BVaL1Npd.js
rename to dev/assets/manual_solving_harmonics.md.DwC-AGfx.js
index 13179ab43..0bdf07ea5 100644
--- a/dev/assets/manual_solving_harmonics.md.BVaL1Npd.js
+++ b/dev/assets/manual_solving_harmonics.md.DwC-AGfx.js
@@ -1,9 +1,9 @@
-import{_ as p,c as l,ai as t,j as s,a,G as n,B as r,o}from"./chunks/framework.Bnn8558y.js";const w=JSON.parse('{"title":"Solving harmonic equations","description":"","frontmatter":{},"headers":[],"relativePath":"manual/solving_harmonics.md","filePath":"manual/solving_harmonics.md"}'),h={name:"manual/solving_harmonics.md"},d={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},c={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},g={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.687ex"},xmlns:"http://www.w3.org/2000/svg",width:"27.124ex",height:"2.573ex",role:"img",focusable:"false",viewBox:"0 -833.9 11988.7 1137.4","aria-hidden":"true"},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.407ex",height:"1.027ex",role:"img",focusable:"false",viewBox:"0 -443 622 454","aria-hidden":"true"},E={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""};function Q(b,i,f,F,C,_){const e=r("Badge");return o(),l("div",null,[i[22]||(i[22]=t('<h1 id="Solving-harmonic-equations" tabindex="-1">Solving harmonic equations <a class="header-anchor" href="#Solving-harmonic-equations" aria-label="Permalink to "Solving harmonic equations {#Solving-harmonic-equations}""></a></h1><p>Once a differential equation of motion has been defined in <code>DifferentialEquation</code> and converted to a <code>HarmonicEquation</code>, we may use the homotopy continuation method (as implemented in HomotopyContinuation.jl) to find steady states. This means that, having called <code>get_harmonic_equations</code>, we need to set all time-derivatives to zero and parse the resulting algebraic equations into a <code>Problem</code>.</p><p><code>Problem</code> holds the steady-state equations, and (optionally) the symbolic Jacobian which is needed for stability / linear response calculations.</p><p>Once defined, a <code>Problem</code> can be solved for a set of input parameters using <code>get_steady_states</code> to obtain <code>Result</code>.</p>',4)),s("details",d,[s("summary",null,[i[0]||(i[0]=s("a",{id:"HarmonicBalance.Problem-manual-solving_harmonics",href:"#HarmonicBalance.Problem-manual-solving_harmonics"},[s("span",{class:"jlbinding"},"HarmonicBalance.Problem")],-1)),i[1]||(i[1]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[2]||(i[2]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Problem{ParType</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Number</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, Jac</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">FunctionWrappers.FunctionWrapper{Matrix{ComplexF64}, Tuple{Vector{ComplexF64}}}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}</span></span></code></pre></div><p>Holds a set of algebraic equations describing the steady state of a system.</p><p><strong>Fields</strong></p><ul><li><p><code>variables::Vector{Num}</code>: The harmonic variables to be solved for.</p></li><li><p><code>parameters::Vector{Num}</code>: All symbols which are not the harmonic variables.</p></li><li><p><code>swept_parameters::OrderedCollections.OrderedDict{Num, Vector{ParType}} where ParType<:Number</code>: The swept parameters in the homotopy.</p></li><li><p><code>fixed_parameters::OrderedCollections.OrderedDict{Num, ParType} where ParType<:Number</code>: The fixed parameters in the homotopy.</p></li><li><p><code>system::HomotopyContinuation.ModelKit.System</code>: The input object for HomotopyContinuation.jl solver methods.</p></li><li><p><code>jacobian::FunctionWrappers.FunctionWrapper{Matrix{ComplexF64}, Tuple{Vector{ComplexF64}}}</code>: The Jacobian matrix (possibly symbolic or compiled function). If <code>Matrix{Nan}</code> and implicit function is compiled when a <code>Result</code> is created.</p></li><li><p><code>eom::HarmonicEquation</code>: The HarmonicEquation object used to generate this <code>Problem</code>.</p></li></ul><p><strong>Constructors</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Problem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+import{_ as p,c as l,ai as t,j as s,a,G as n,B as r,o}from"./chunks/framework.Bnn8558y.js";const w=JSON.parse('{"title":"Solving harmonic equations","description":"","frontmatter":{},"headers":[],"relativePath":"manual/solving_harmonics.md","filePath":"manual/solving_harmonics.md"}'),h={name:"manual/solving_harmonics.md"},d={class:"jldocstring custom-block",open:""},k={class:"jldocstring custom-block",open:""},c={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},g={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.687ex"},xmlns:"http://www.w3.org/2000/svg",width:"27.124ex",height:"2.573ex",role:"img",focusable:"false",viewBox:"0 -833.9 11988.7 1137.4","aria-hidden":"true"},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.407ex",height:"1.027ex",role:"img",focusable:"false",viewBox:"0 -443 622 454","aria-hidden":"true"},E={class:"jldocstring custom-block",open:""},T={class:"jldocstring custom-block",open:""},y={class:"jldocstring custom-block",open:""};function Q(b,i,F,f,C,_){const e=r("Badge");return o(),l("div",null,[i[22]||(i[22]=t('<h1 id="Solving-harmonic-equations" tabindex="-1">Solving harmonic equations <a class="header-anchor" href="#Solving-harmonic-equations" aria-label="Permalink to "Solving harmonic equations {#Solving-harmonic-equations}""></a></h1><p>Once a differential equation of motion has been defined in <code>DifferentialEquation</code> and converted to a <code>HarmonicEquation</code>, we may use the homotopy continuation method (as implemented in HomotopyContinuation.jl) to find steady states. This means that, having called <code>get_harmonic_equations</code>, we need to set all time-derivatives to zero and parse the resulting algebraic equations into a <code>Problem</code>.</p><p><code>Problem</code> holds the steady-state equations, and (optionally) the symbolic Jacobian which is needed for stability / linear response calculations.</p><p>Once defined, a <code>Problem</code> can be solved for a set of input parameters using <code>get_steady_states</code> to obtain <code>Result</code>.</p>',4)),s("details",d,[s("summary",null,[i[0]||(i[0]=s("a",{id:"HarmonicBalance.Problem-manual-solving_harmonics",href:"#HarmonicBalance.Problem-manual-solving_harmonics"},[s("span",{class:"jlbinding"},"HarmonicBalance.Problem")],-1)),i[1]||(i[1]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[2]||(i[2]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Problem{ParType</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Number</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, Jac</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">FunctionWrappers.FunctionWrapper{Matrix{ComplexF64}, Tuple{Vector{ComplexF64}}}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}</span></span></code></pre></div><p>Holds a set of algebraic equations describing the steady state of a system.</p><p><strong>Fields</strong></p><ul><li><p><code>variables::Vector{Num}</code>: The harmonic variables to be solved for.</p></li><li><p><code>parameters::Vector{Num}</code>: All symbols which are not the harmonic variables.</p></li><li><p><code>swept_parameters::OrderedCollections.OrderedDict{Num, Vector{ParType}} where ParType<:Number</code>: The swept parameters in the homotopy.</p></li><li><p><code>fixed_parameters::OrderedCollections.OrderedDict{Num, ParType} where ParType<:Number</code>: The fixed parameters in the homotopy.</p></li><li><p><code>system::HomotopyContinuation.ModelKit.System</code>: The input object for HomotopyContinuation.jl solver methods.</p></li><li><p><code>jacobian::FunctionWrappers.FunctionWrapper{Matrix{ComplexF64}, Tuple{Vector{ComplexF64}}}</code>: The Jacobian matrix (possibly symbolic or compiled function). If <code>Matrix{Nan}</code> and implicit function is compiled when a <code>Result</code> is created.</p></li><li><p><code>eom::HarmonicEquation</code>: The HarmonicEquation object used to generate this <code>Problem</code>.</p></li></ul><p><strong>Constructors</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Problem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> swept</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">AbstractDict</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> fixed</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">AbstractDict</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> compute_Jacobian</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Bool</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/Problem.jl#L3" target="_blank" rel="noreferrer">source</a></p>`,7))]),s("details",k,[s("summary",null,[i[3]||(i[3]=s("a",{id:"HarmonicBalance.get_steady_states-manual-solving_harmonics",href:"#HarmonicBalance.get_steady_states-manual-solving_harmonics"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_steady_states")],-1)),i[4]||(i[4]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[11]||(i[11]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(problem</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/Problem.jl#L3" target="_blank" rel="noreferrer">source</a></p>`,7))]),s("details",k,[s("summary",null,[i[3]||(i[3]=s("a",{id:"HarmonicBalance.get_steady_states-manual-solving_harmonics",href:"#HarmonicBalance.get_steady_states-manual-solving_harmonics"},[s("span",{class:"jlbinding"},"HarmonicBalance.get_steady_states")],-1)),i[4]||(i[4]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[11]||(i[11]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(problem</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> method</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalanceMethod</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> swept_parameters,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> fixed_parameters;</span></span>
@@ -32,7 +32,7 @@ import{_ as p,c as l,ai as t,j as s,a,G as n,B as r,o}from"./chunks/framework.Bn
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> of which real</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> of which stable</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1</span></span>
<span class="line"></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Classes</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> stable, physical, Hopf, binary_labels</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/solve_homotopy.jl#L1-L60" target="_blank" rel="noreferrer">source</a></p>`,4))]),s("details",E,[s("summary",null,[i[13]||(i[13]=s("a",{id:"HarmonicBalance.Result-manual-solving_harmonics",href:"#HarmonicBalance.Result-manual-solving_harmonics"},[s("span",{class:"jlbinding"},"HarmonicBalance.Result")],-1)),i[14]||(i[14]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[15]||(i[15]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Result{SolType</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Number</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ParType</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Number</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, D, F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">FunctionWrappers.FunctionWrapper{Array{SolType<:Number, 2}, Tuple{Array{SolType<:Number, 1}}}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}</span></span></code></pre></div><p>Stores the steady states of a HarmonicEquation.</p><p><strong>Fields</strong></p><ul><li><p><code>solutions::Array{Array{Vector{SolType}, 1}} where SolType<:Number</code>: The variable values of steady-state solutions.</p></li><li><p><code>swept_parameters::OrderedCollections.OrderedDict{Num, Vector{ParType}} where ParType<:Number</code>: Values of all parameters for all solutions.</p></li><li><p><code>fixed_parameters::OrderedCollections.OrderedDict{Num, ParType} where ParType<:Number</code>: The parameters fixed throughout the solutions.</p></li><li><p><code>problem::HarmonicBalance.Problem{ParType, F} where {SolType<:Number, ParType<:Number, F<:FunctionWrappers.FunctionWrapper{Matrix{SolType}, Tuple{Vector{SolType}}}}</code>: The <code>Problem</code> used to generate this.</p></li><li><p><code>classes::Dict{String, Array{BitVector, D}} where D</code>: Maps strings such as "stable", "physical" etc to arrays of values, classifying the solutions (see method <code>classify_solutions!</code>).</p></li><li><p><code>binary_labels::Array{Int64}</code>: Create binary classification of the solutions, such that each solution point receives an identifier based on its permutation of stable branches (allows to distinguish between different phases, which may have the same number of stable solutions). It works by converting each bitstring <code>[is_stable(solution_1), is_stable(solution_2), ...,]</code> into unique labels.</p></li><li><p><code>jacobian::FunctionWrappers.FunctionWrapper{Matrix{SolType}, Tuple{Vector{SolType}}} where SolType<:Number</code>: The Jacobian function with <code>fixed_parameters</code> already substituted. Accepts a vector specifying the solution. If problem.jacobian is a symbolic matrix, this holds a compiled function.</p></li><li><p><code>seed::UInt32</code>: Seed used for the solver</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/Result.jl#L1" target="_blank" rel="noreferrer">source</a></p>',5))]),i[23]||(i[23]=s("h2",{id:"Classifying-solutions",tabindex:"-1"},[a("Classifying solutions "),s("a",{class:"header-anchor",href:"#Classifying-solutions","aria-label":'Permalink to "Classifying solutions {#Classifying-solutions}"'},"")],-1)),i[24]||(i[24]=s("p",null,[a("The solutions in "),s("code",null,"Result"),a(" are accompanied by similarly-sized boolean arrays stored in the dictionary "),s("code",null,"Result.classes"),a(". The classes can be used by the plotting functions to show/hide/label certain solutions.")],-1)),i[25]||(i[25]=s("p",null,[a('By default, classes "physical", "stable" and "binary_labels" are created. User-defined classification is possible with '),s("code",null,"classify_solutions!"),a(".")],-1)),s("details",T,[s("summary",null,[i[16]||(i[16]=s("a",{id:"HarmonicBalance.classify_solutions!-manual-solving_harmonics",href:"#HarmonicBalance.classify_solutions!-manual-solving_harmonics"},[s("span",{class:"jlbinding"},"HarmonicBalance.classify_solutions!")],-1)),i[17]||(i[17]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[18]||(i[18]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Classes</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> stable, physical, Hopf, binary_labels</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/solve_homotopy.jl#L1-L60" target="_blank" rel="noreferrer">source</a></p>`,4))]),s("details",E,[s("summary",null,[i[13]||(i[13]=s("a",{id:"HarmonicBalance.Result-manual-solving_harmonics",href:"#HarmonicBalance.Result-manual-solving_harmonics"},[s("span",{class:"jlbinding"},"HarmonicBalance.Result")],-1)),i[14]||(i[14]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[15]||(i[15]=t('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Result{SolType</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Number</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ParType</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Number</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, D, F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">FunctionWrappers.FunctionWrapper{Array{SolType<:Number, 2}, Tuple{Array{SolType<:Number, 1}}}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}</span></span></code></pre></div><p>Stores the steady states of a HarmonicEquation.</p><p><strong>Fields</strong></p><ul><li><p><code>solutions::Array{Array{Vector{SolType}, 1}} where SolType<:Number</code>: The variable values of steady-state solutions.</p></li><li><p><code>swept_parameters::OrderedCollections.OrderedDict{Num, Vector{ParType}} where ParType<:Number</code>: Values of all parameters for all solutions.</p></li><li><p><code>fixed_parameters::OrderedCollections.OrderedDict{Num, ParType} where ParType<:Number</code>: The parameters fixed throughout the solutions.</p></li><li><p><code>problem::HarmonicBalance.Problem{ParType, F} where {SolType<:Number, ParType<:Number, F<:FunctionWrappers.FunctionWrapper{Matrix{SolType}, Tuple{Vector{SolType}}}}</code>: The <code>Problem</code> used to generate this.</p></li><li><p><code>classes::Dict{String, Array{BitVector, D}} where D</code>: Maps strings such as "stable", "physical" etc to arrays of values, classifying the solutions (see method <code>classify_solutions!</code>).</p></li><li><p><code>binary_labels::Array{Int64}</code>: Create binary classification of the solutions, such that each solution point receives an identifier based on its permutation of stable branches (allows to distinguish between different phases, which may have the same number of stable solutions). It works by converting each bitstring <code>[is_stable(solution_1), is_stable(solution_2), ...,]</code> into unique labels.</p></li><li><p><code>jacobian::FunctionWrappers.FunctionWrapper{Matrix{SolType}, Tuple{Vector{SolType}}} where SolType<:Number</code>: The Jacobian function with <code>fixed_parameters</code> already substituted. Accepts a vector specifying the solution. If problem.jacobian is a symbolic matrix, this holds a compiled function.</p></li><li><p><code>seed::UInt32</code>: Seed used for the solver</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/Result.jl#L1" target="_blank" rel="noreferrer">source</a></p>',5))]),i[23]||(i[23]=s("h2",{id:"Classifying-solutions",tabindex:"-1"},[a("Classifying solutions "),s("a",{class:"header-anchor",href:"#Classifying-solutions","aria-label":'Permalink to "Classifying solutions {#Classifying-solutions}"'},"")],-1)),i[24]||(i[24]=s("p",null,[a("The solutions in "),s("code",null,"Result"),a(" are accompanied by similarly-sized boolean arrays stored in the dictionary "),s("code",null,"Result.classes"),a(". The classes can be used by the plotting functions to show/hide/label certain solutions.")],-1)),i[25]||(i[25]=s("p",null,[a('By default, classes "physical", "stable" and "binary_labels" are created. User-defined classification is possible with '),s("code",null,"classify_solutions!"),a(".")],-1)),s("details",T,[s("summary",null,[i[16]||(i[16]=s("a",{id:"HarmonicBalance.classify_solutions!-manual-solving_harmonics",href:"#HarmonicBalance.classify_solutions!-manual-solving_harmonics"},[s("span",{class:"jlbinding"},"HarmonicBalance.classify_solutions!")],-1)),i[17]||(i[17]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[18]||(i[18]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> func</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Union{Function, String}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> name</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
@@ -41,8 +41,8 @@ import{_ as p,c as l,ai as t,j as s,a,G as n,B as r,o}from"./chunks/framework.Bn
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">res </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(problem, swept_parameters, fixed_parameters)</span></span>
<span class="line"></span>
<span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># classify, store in result.classes["large_amplitude"]</span></span>
-<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u1^2 + v1^2) > 1.0"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> , </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"large_amplitude"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/classification.jl#L1" target="_blank" rel="noreferrer">source</a></p>`,5))]),i[26]||(i[26]=t('<h2 id="Sorting-solutions" tabindex="-1">Sorting solutions <a class="header-anchor" href="#Sorting-solutions" aria-label="Permalink to "Sorting solutions {#Sorting-solutions}""></a></h2><p>Solving a steady-state problem over a range of parameters returns a solution set for each parameter. For a continuous change of parameters, each solution in a set usually also changes continuously; it is said to form a ''solution branch''. For an example, see the three colour-coded branches for the Duffing oscillator in Example 1.</p><p>For stable states, the branches describe a system's behaviour under adiabatic parameter changes.</p><p>Therefore, after solving for a parameter range, we want to order each solution set such that the solutions' order reflects the branches.</p><p>The function <code>sort_solutions</code> goes over the the raw output of <code>get_steady_states</code> and sorts each entry such that neighboring solution sets minimize Euclidean distance.</p><p>Currently, <code>sort_solutions</code> is compatible with 1D and 2D arrays of solution sets.</p>',6)),s("details",y,[s("summary",null,[i[19]||(i[19]=s("a",{id:"HarmonicBalance.sort_solutions-manual-solving_harmonics",href:"#HarmonicBalance.sort_solutions-manual-solving_harmonics"},[s("span",{class:"jlbinding"},"HarmonicBalance.sort_solutions")],-1)),i[20]||(i[20]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[21]||(i[21]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">sort_solutions</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u1^2 + v1^2) > 1.0"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> , </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"large_amplitude"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/classification.jl#L1" target="_blank" rel="noreferrer">source</a></p>`,5))]),i[26]||(i[26]=t('<h2 id="Sorting-solutions" tabindex="-1">Sorting solutions <a class="header-anchor" href="#Sorting-solutions" aria-label="Permalink to "Sorting solutions {#Sorting-solutions}""></a></h2><p>Solving a steady-state problem over a range of parameters returns a solution set for each parameter. For a continuous change of parameters, each solution in a set usually also changes continuously; it is said to form a ''solution branch''. For an example, see the three colour-coded branches for the Duffing oscillator in Example 1.</p><p>For stable states, the branches describe a system's behaviour under adiabatic parameter changes.</p><p>Therefore, after solving for a parameter range, we want to order each solution set such that the solutions' order reflects the branches.</p><p>The function <code>sort_solutions</code> goes over the the raw output of <code>get_steady_states</code> and sorts each entry such that neighboring solution sets minimize Euclidean distance.</p><p>Currently, <code>sort_solutions</code> is compatible with 1D and 2D arrays of solution sets.</p>',6)),s("details",y,[s("summary",null,[i[19]||(i[19]=s("a",{id:"HarmonicBalance.sort_solutions-manual-solving_harmonics",href:"#HarmonicBalance.sort_solutions-manual-solving_harmonics"},[s("span",{class:"jlbinding"},"HarmonicBalance.sort_solutions")],-1)),i[20]||(i[20]=a()),n(e,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),i[21]||(i[21]=t(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">sort_solutions</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> solutions</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Union</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">{Array{Array{Array{T, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}, Array{Array{Array{T, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}};</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sorting,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> show_progress</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Sorts <code>solutions</code> into branches according to the specified <code>sorting</code> method.</p><p><code>solutions</code> is an n-dimensional array of <code>Vector{Vector}</code>. Each element describes a set of solutions for a given parameter set. The output is a similar array, with each solution set rearranged such that neighboring solution sets have the smallest Euclidean distance.</p><p>The <code>sorting</code> keyword argument specifies the method used to get continuous solution branches. Options are <code>"hilbert"</code> (1D sorting along a Hilbert curve), <code>"nearest"</code> (nearest-neighbor sorting), and <code>"none"</code>. The <code>show_progress</code> keyword argument indicates whether a progress bar should be displayed.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/sorting.jl#L1-L13" target="_blank" rel="noreferrer">source</a></p>`,5))])])}const x=p(h,[["render",Q]]);export{w as __pageData,x as default};
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Sorts <code>solutions</code> into branches according to the specified <code>sorting</code> method.</p><p><code>solutions</code> is an n-dimensional array of <code>Vector{Vector}</code>. Each element describes a set of solutions for a given parameter set. The output is a similar array, with each solution set rearranged such that neighboring solution sets have the smallest Euclidean distance.</p><p>The <code>sorting</code> keyword argument specifies the method used to get continuous solution branches. Options are <code>"hilbert"</code> (1D sorting along a Hilbert curve), <code>"nearest"</code> (nearest-neighbor sorting), and <code>"none"</code>. The <code>show_progress</code> keyword argument indicates whether a progress bar should be displayed.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/sorting.jl#L1-L13" target="_blank" rel="noreferrer">source</a></p>`,5))])])}const x=p(h,[["render",Q]]);export{w as __pageData,x as default};
diff --git a/dev/assets/manual_solving_harmonics.md.BVaL1Npd.lean.js b/dev/assets/manual_solving_harmonics.md.DwC-AGfx.lean.js
similarity index 98%
rename from dev/assets/manual_solving_harmonics.md.BVaL1Npd.lean.js
rename to dev/assets/manual_solving_harmonics.md.DwC-AGfx.lean.js
index a070dfdc2..f42e2c742 100644
--- a/dev/assets/manual_solving_harmonics.md.BVaL1Npd.lean.js
+++ b/dev/assets/manual_solving_harmonics.md.DwC-AGfx.lean.js
@@ -1 +1 @@
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0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"ω")])],-1))])]),i[12]||(i[12]=t("",4))]),s("details",E,[s("summary",null,[i[13]||(i[13]=s("a",{id:"HarmonicBalance.Result-manual-solving_harmonics",href:"#HarmonicBalance.Result-manual-solving_harmonics"},[s("span",{class:"jlbinding"},"HarmonicBalance.Result")],-1)),i[14]||(i[14]=a()),n(e,{type:"info",class:"jlObjectType jlType",text:"Type"})]),i[15]||(i[15]=t("",5))]),i[23]||(i[23]=s("h2",{id:"Classifying-solutions",tabindex:"-1"},[a("Classifying solutions "),s("a",{class:"header-anchor",href:"#Classifying-solutions","aria-label":'Permalink to "Classifying solutions {#Classifying-solutions}"'},"")],-1)),i[24]||(i[24]=s("p",null,[a("The solutions in "),s("code",null,"Result"),a(" are accompanied by similarly-sized boolean arrays stored in the dictionary "),s("code",null,"Result.classes"),a(". 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diff --git a/dev/assets/manual_time_dependent.md.CnOG8uBM.js b/dev/assets/manual_time_dependent.md.D8z0BdFi.js
similarity index 96%
rename from dev/assets/manual_time_dependent.md.CnOG8uBM.js
rename to dev/assets/manual_time_dependent.md.D8z0BdFi.js
index e3b58a879..0d9d7fa4f 100644
--- a/dev/assets/manual_time_dependent.md.CnOG8uBM.js
+++ b/dev/assets/manual_time_dependent.md.D8z0BdFi.js
@@ -1,4 +1,4 @@
-import{_ as l,c as p,ai as e,j as i,a,G as t,B as h,o as k}from"./chunks/framework.Bnn8558y.js";const f=JSON.parse('{"title":"Time evolution","description":"","frontmatter":{},"headers":[],"relativePath":"manual/time_dependent.md","filePath":"manual/time_dependent.md"}'),o={name:"manual/time_dependent.md"},r={class:"jldocstring custom-block",open:""},d={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""};function g(y,s,u,m,b,F){const n=h("Badge");return k(),p("div",null,[s[12]||(s[12]=e('<h1 id="Time-evolution" tabindex="-1">Time evolution <a class="header-anchor" href="#Time-evolution" aria-label="Permalink to "Time evolution {#Time-evolution}""></a></h1><p>Generally, solving the ODE of oscillatory systems in time requires numerically tracking the oscillations. This is a computationally expensive process; however, using the harmonic ansatz removes the oscillatory time-dependence. Simulating instead the harmonic variables of a <code>HarmonicEquation</code> is vastly more efficient - a steady state of the system appears as a fixed point in multidimensional space rather than an oscillatory function.</p><p>The extension <code>TimeEvolution</code> is used to interface <code>HarmonicEquation</code> with the solvers contained in <code>OrdinaryDiffEq.jl</code>. Time-dependent parameter sweeps are defined using the object <code>AdiabaticSweep</code>. To use the <code>TimeEvolution</code> extension, one must first load the <code>OrdinaryDiffEq.jl</code> package.</p>',3)),i("details",r,[i("summary",null,[s[0]||(s[0]=i("a",{id:"SciMLBase.ODEProblem-Tuple{HarmonicEquation, Any}-manual-time_dependent",href:"#SciMLBase.ODEProblem-Tuple{HarmonicEquation, Any}-manual-time_dependent"},[i("span",{class:"jlbinding"},"SciMLBase.ODEProblem")],-1)),s[1]||(s[1]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[2]||(s[2]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODEProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+import{_ as l,c as p,ai as e,j as i,a,G as t,B as h,o as k}from"./chunks/framework.Bnn8558y.js";const f=JSON.parse('{"title":"Time evolution","description":"","frontmatter":{},"headers":[],"relativePath":"manual/time_dependent.md","filePath":"manual/time_dependent.md"}'),o={name:"manual/time_dependent.md"},d={class:"jldocstring custom-block",open:""},r={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""};function g(y,s,u,m,b,F){const n=h("Badge");return k(),p("div",null,[s[12]||(s[12]=e('<h1 id="Time-evolution" tabindex="-1">Time evolution <a class="header-anchor" href="#Time-evolution" aria-label="Permalink to "Time evolution {#Time-evolution}""></a></h1><p>Generally, solving the ODE of oscillatory systems in time requires numerically tracking the oscillations. This is a computationally expensive process; however, using the harmonic ansatz removes the oscillatory time-dependence. Simulating instead the harmonic variables of a <code>HarmonicEquation</code> is vastly more efficient - a steady state of the system appears as a fixed point in multidimensional space rather than an oscillatory function.</p><p>The extension <code>TimeEvolution</code> is used to interface <code>HarmonicEquation</code> with the solvers contained in <code>OrdinaryDiffEq.jl</code>. Time-dependent parameter sweeps are defined using the object <code>AdiabaticSweep</code>. To use the <code>TimeEvolution</code> extension, one must first load the <code>OrdinaryDiffEq.jl</code> package.</p>',3)),i("details",d,[i("summary",null,[s[0]||(s[0]=i("a",{id:"SciMLBase.ODEProblem-Tuple{HarmonicEquation, Any}-manual-time_dependent",href:"#SciMLBase.ODEProblem-Tuple{HarmonicEquation, Any}-manual-time_dependent"},[i("span",{class:"jlbinding"},"SciMLBase.ODEProblem")],-1)),s[1]||(s[1]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[2]||(s[2]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODEProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> fixed_parameters;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep,</span></span>
@@ -6,7 +6,7 @@ import{_ as l,c as p,ai as e,j as i,a,G as t,B as h,o as k}from"./chunks/framewo
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> timespan,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> perturb_initial,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Creates an ODEProblem object used by OrdinaryDiffEqTsit5.jl from the equations in <code>eom</code> to simulate time-evolution within <code>timespan</code>. <code>fixed_parameters</code> must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If <code>u0</code> is specified, it is used as an initial condition; otherwise the values from <code>fixed_parameters</code> are used.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/ODEProblem.jl#L3-L9" target="_blank" rel="noreferrer">source</a></p>`,3))]),i("details",d,[i("summary",null,[s[3]||(s[3]=i("a",{id:"HarmonicBalance.AdiabaticSweep-manual-time_dependent",href:"#HarmonicBalance.AdiabaticSweep-manual-time_dependent"},[i("span",{class:"jlbinding"},"HarmonicBalance.AdiabaticSweep")],-1)),s[4]||(s[4]=a()),t(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[5]||(s[5]=e(`<p>Represents a sweep of one or more parameters of a <code>HarmonicEquation</code>. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.</p><p>Sweeps of different variables can be combined using <code>+</code>.</p><p><strong>Fields</strong></p><ul><li><code>functions::Dict{Num, Function}</code>: Maps each swept parameter to a function.</li></ul><p><strong>Examples</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># create a sweep of parameter a from 0 to 1 over time 0 -> 100</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Creates an ODEProblem object used by OrdinaryDiffEqTsit5.jl from the equations in <code>eom</code> to simulate time-evolution within <code>timespan</code>. <code>fixed_parameters</code> must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If <code>u0</code> is specified, it is used as an initial condition; otherwise the values from <code>fixed_parameters</code> are used.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/ODEProblem.jl#L3-L9" target="_blank" rel="noreferrer">source</a></p>`,3))]),i("details",r,[i("summary",null,[s[3]||(s[3]=i("a",{id:"HarmonicBalance.AdiabaticSweep-manual-time_dependent",href:"#HarmonicBalance.AdiabaticSweep-manual-time_dependent"},[i("span",{class:"jlbinding"},"HarmonicBalance.AdiabaticSweep")],-1)),s[4]||(s[4]=a()),t(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[5]||(s[5]=e(`<p>Represents a sweep of one or more parameters of a <code>HarmonicEquation</code>. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.</p><p>Sweeps of different variables can be combined using <code>+</code>.</p><p><strong>Fields</strong></p><ul><li><code>functions::Dict{Num, Function}</code>: Maps each swept parameter to a function.</li></ul><p><strong>Examples</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># create a sweep of parameter a from 0 to 1 over time 0 -> 100</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> a,b;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(a </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">100</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">));</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep[a](</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">50</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
@@ -18,14 +18,14 @@ import{_ as l,c as p,ai as e,j as i,a,G as t,B as h,o as k}from"./chunks/framewo
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">([a </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], b </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">]], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">100</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span></code></pre></div><p>Successive sweeps can be combined,</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep1 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.95</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep2 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(λ </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.05</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.01</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">4e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep1 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep2</span></span></code></pre></div><p>multiple parameters can be swept simultaneously,</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">([ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.95</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], λ </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">5e-2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1e-2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">]], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span></code></pre></div><p>and custom sweep functions may be used.</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#6F42C1;--shiki-dark:#B392F0;">ωfunc</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ωfunc)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/types.jl#L9-L48" target="_blank" rel="noreferrer">source</a></p>`,13))]),s[13]||(s[13]=i("h2",{id:"plotting",tabindex:"-1"},[a("Plotting "),i("a",{class:"header-anchor",href:"#plotting","aria-label":'Permalink to "Plotting"'},"")],-1)),i("details",E,[i("summary",null,[s[6]||(s[6]=i("a",{id:"RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-time_dependent",href:"#RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-time_dependent"},[i("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[7]||(s[7]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[8]||(s[8]=e('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESolution</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, f</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, harm_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a function <code>f</code> of a time-dependent solution <code>soln</code> of <code>harm_eq</code>.</p><p><strong>As a function of time</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p><code>f</code> is parsed by Symbolics.jl</p><p><strong>parametric plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p>Parametric plot of f[1] against f[2]</p><p>Also callable as plot!</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/plotting.jl#L4-L22" target="_blank" rel="noreferrer">source</a></p>',10))]),s[14]||(s[14]=i("h2",{id:"miscellaneous",tabindex:"-1"},[a("Miscellaneous "),i("a",{class:"header-anchor",href:"#miscellaneous","aria-label":'Permalink to "Miscellaneous"'},"")],-1)),s[15]||(s[15]=i("p",null,"Using a time-dependent simulation can verify solution stability in cases where the Jacobian is too expensive to compute.",-1)),i("details",c,[i("summary",null,[s[9]||(s[9]=i("a",{id:"HarmonicBalance.is_stable-manual-time_dependent",href:"#HarmonicBalance.is_stable-manual-time_dependent"},[i("span",{class:"jlbinding"},"HarmonicBalance.is_stable")],-1)),s[10]||(s[10]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">is_stable</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ωfunc)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/types.jl#L9-L48" target="_blank" rel="noreferrer">source</a></p>`,13))]),s[13]||(s[13]=i("h2",{id:"plotting",tabindex:"-1"},[a("Plotting "),i("a",{class:"header-anchor",href:"#plotting","aria-label":'Permalink to "Plotting"'},"")],-1)),i("details",E,[i("summary",null,[s[6]||(s[6]=i("a",{id:"RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-time_dependent",href:"#RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-time_dependent"},[i("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[7]||(s[7]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[8]||(s[8]=e('<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESolution</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, f</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, harm_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a function <code>f</code> of a time-dependent solution <code>soln</code> of <code>harm_eq</code>.</p><p><strong>As a function of time</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p><code>f</code> is parsed by Symbolics.jl</p><p><strong>parametric plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p>Parametric plot of f[1] against f[2]</p><p>Also callable as plot!</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/plotting.jl#L4-L22" target="_blank" rel="noreferrer">source</a></p>',10))]),s[14]||(s[14]=i("h2",{id:"miscellaneous",tabindex:"-1"},[a("Miscellaneous "),i("a",{class:"header-anchor",href:"#miscellaneous","aria-label":'Permalink to "Miscellaneous"'},"")],-1)),s[15]||(s[15]=i("p",null,"Using a time-dependent simulation can verify solution stability in cases where the Jacobian is too expensive to compute.",-1)),i("details",c,[i("summary",null,[s[9]||(s[9]=i("a",{id:"HarmonicBalance.is_stable-manual-time_dependent",href:"#HarmonicBalance.is_stable-manual-time_dependent"},[i("span",{class:"jlbinding"},"HarmonicBalance.is_stable")],-1)),s[10]||(s[10]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=e(`<div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">is_stable</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">OrderedCollections.OrderedDict</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> timespan,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> tol,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> perturb_initial</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Numerically investigate the stability of a solution <code>soln</code> of <code>eom</code> within <code>timespan</code>. The initial condition is displaced by <code>perturb_initial</code>.</p><p>Return <code>true</code> the solution evolves within <code>tol</code> of the initial value (interpreted as stable).</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/ODEProblem.jl#L61" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">is_stable</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Numerically investigate the stability of a solution <code>soln</code> of <code>eom</code> within <code>timespan</code>. The initial condition is displaced by <code>perturb_initial</code>.</p><p>Return <code>true</code> the solution evolves within <code>tol</code> of the initial value (interpreted as stable).</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/ODEProblem.jl#L61" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">is_stable</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">OrderedCollections.OrderedDict</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Returns true if the solution <code>soln</code> of the Result <code>res</code> is stable. Stable solutions are real and have all Jacobian eigenvalues Re(λ) <= 0. <code>im_tol</code> : an absolute threshold to distinguish real/complex numbers. <code>rel_tol</code>: Re(λ) considered <=0 if real.(λ) < rel_tol*abs(λmax)</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/classification.jl#L77" target="_blank" rel="noreferrer">source</a></p>`,7))])])}const v=l(o,[["render",g]]);export{f as __pageData,v as default};
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Returns true if the solution <code>soln</code> of the Result <code>res</code> is stable. Stable solutions are real and have all Jacobian eigenvalues Re(λ) <= 0. <code>im_tol</code> : an absolute threshold to distinguish real/complex numbers. <code>rel_tol</code>: Re(λ) considered <=0 if real.(λ) < rel_tol*abs(λmax)</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/classification.jl#L77" target="_blank" rel="noreferrer">source</a></p>`,7))])])}const v=l(o,[["render",g]]);export{f as __pageData,v as default};
diff --git a/dev/assets/manual_time_dependent.md.CnOG8uBM.lean.js b/dev/assets/manual_time_dependent.md.D8z0BdFi.lean.js
similarity index 90%
rename from dev/assets/manual_time_dependent.md.CnOG8uBM.lean.js
rename to dev/assets/manual_time_dependent.md.D8z0BdFi.lean.js
index fb171ef97..66669fe23 100644
--- a/dev/assets/manual_time_dependent.md.CnOG8uBM.lean.js
+++ b/dev/assets/manual_time_dependent.md.D8z0BdFi.lean.js
@@ -1 +1 @@
-import{_ as l,c as p,ai as e,j as i,a,G as t,B as h,o as k}from"./chunks/framework.Bnn8558y.js";const f=JSON.parse('{"title":"Time evolution","description":"","frontmatter":{},"headers":[],"relativePath":"manual/time_dependent.md","filePath":"manual/time_dependent.md"}'),o={name:"manual/time_dependent.md"},r={class:"jldocstring custom-block",open:""},d={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""};function g(y,s,u,m,b,F){const n=h("Badge");return k(),p("div",null,[s[12]||(s[12]=e("",3)),i("details",r,[i("summary",null,[s[0]||(s[0]=i("a",{id:"SciMLBase.ODEProblem-Tuple{HarmonicEquation, Any}-manual-time_dependent",href:"#SciMLBase.ODEProblem-Tuple{HarmonicEquation, Any}-manual-time_dependent"},[i("span",{class:"jlbinding"},"SciMLBase.ODEProblem")],-1)),s[1]||(s[1]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[2]||(s[2]=e("",3))]),i("details",d,[i("summary",null,[s[3]||(s[3]=i("a",{id:"HarmonicBalance.AdiabaticSweep-manual-time_dependent",href:"#HarmonicBalance.AdiabaticSweep-manual-time_dependent"},[i("span",{class:"jlbinding"},"HarmonicBalance.AdiabaticSweep")],-1)),s[4]||(s[4]=a()),t(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[5]||(s[5]=e("",13))]),s[13]||(s[13]=i("h2",{id:"plotting",tabindex:"-1"},[a("Plotting "),i("a",{class:"header-anchor",href:"#plotting","aria-label":'Permalink to "Plotting"'},"")],-1)),i("details",E,[i("summary",null,[s[6]||(s[6]=i("a",{id:"RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-time_dependent",href:"#RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-time_dependent"},[i("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[7]||(s[7]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[8]||(s[8]=e("",10))]),s[14]||(s[14]=i("h2",{id:"miscellaneous",tabindex:"-1"},[a("Miscellaneous "),i("a",{class:"header-anchor",href:"#miscellaneous","aria-label":'Permalink to "Miscellaneous"'},"")],-1)),s[15]||(s[15]=i("p",null,"Using a time-dependent simulation can verify solution stability in cases where the Jacobian is too expensive to compute.",-1)),i("details",c,[i("summary",null,[s[9]||(s[9]=i("a",{id:"HarmonicBalance.is_stable-manual-time_dependent",href:"#HarmonicBalance.is_stable-manual-time_dependent"},[i("span",{class:"jlbinding"},"HarmonicBalance.is_stable")],-1)),s[10]||(s[10]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=e("",7))])])}const v=l(o,[["render",g]]);export{f as __pageData,v as default};
+import{_ as l,c as p,ai as e,j as i,a,G as t,B as h,o as k}from"./chunks/framework.Bnn8558y.js";const f=JSON.parse('{"title":"Time evolution","description":"","frontmatter":{},"headers":[],"relativePath":"manual/time_dependent.md","filePath":"manual/time_dependent.md"}'),o={name:"manual/time_dependent.md"},d={class:"jldocstring custom-block",open:""},r={class:"jldocstring custom-block",open:""},E={class:"jldocstring custom-block",open:""},c={class:"jldocstring custom-block",open:""};function g(y,s,u,m,b,F){const n=h("Badge");return k(),p("div",null,[s[12]||(s[12]=e("",3)),i("details",d,[i("summary",null,[s[0]||(s[0]=i("a",{id:"SciMLBase.ODEProblem-Tuple{HarmonicEquation, Any}-manual-time_dependent",href:"#SciMLBase.ODEProblem-Tuple{HarmonicEquation, Any}-manual-time_dependent"},[i("span",{class:"jlbinding"},"SciMLBase.ODEProblem")],-1)),s[1]||(s[1]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[2]||(s[2]=e("",3))]),i("details",r,[i("summary",null,[s[3]||(s[3]=i("a",{id:"HarmonicBalance.AdiabaticSweep-manual-time_dependent",href:"#HarmonicBalance.AdiabaticSweep-manual-time_dependent"},[i("span",{class:"jlbinding"},"HarmonicBalance.AdiabaticSweep")],-1)),s[4]||(s[4]=a()),t(n,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[5]||(s[5]=e("",13))]),s[13]||(s[13]=i("h2",{id:"plotting",tabindex:"-1"},[a("Plotting "),i("a",{class:"header-anchor",href:"#plotting","aria-label":'Permalink to "Plotting"'},"")],-1)),i("details",E,[i("summary",null,[s[6]||(s[6]=i("a",{id:"RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-time_dependent",href:"#RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-time_dependent"},[i("span",{class:"jlbinding"},"RecipesBase.plot")],-1)),s[7]||(s[7]=a()),t(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[8]||(s[8]=e("",10))]),s[14]||(s[14]=i("h2",{id:"miscellaneous",tabindex:"-1"},[a("Miscellaneous "),i("a",{class:"header-anchor",href:"#miscellaneous","aria-label":'Permalink to "Miscellaneous"'},"")],-1)),s[15]||(s[15]=i("p",null,"Using a time-dependent simulation can verify solution stability in cases where the Jacobian is too expensive to compute.",-1)),i("details",c,[i("summary",null,[s[9]||(s[9]=i("a",{id:"HarmonicBalance.is_stable-manual-time_dependent",href:"#HarmonicBalance.is_stable-manual-time_dependent"},[i("span",{class:"jlbinding"},"HarmonicBalance.is_stable")],-1)),s[10]||(s[10]=a()),t(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[11]||(s[11]=e("",7))])])}const v=l(o,[["render",g]]);export{f as __pageData,v as default};
diff --git a/dev/assets/tutorials_steady_states.md.CagED6_V.js b/dev/assets/tutorials_steady_states.md.ERR8LiBX.js
similarity index 98%
rename from dev/assets/tutorials_steady_states.md.CagED6_V.js
rename to dev/assets/tutorials_steady_states.md.ERR8LiBX.js
index 4748025fc..c13dd841e 100644
--- a/dev/assets/tutorials_steady_states.md.CagED6_V.js
+++ b/dev/assets/tutorials_steady_states.md.ERR8LiBX.js
@@ -14,7 +14,7 @@ import{_ as l,c as T,j as t,a,ai as e,o as s}from"./chunks/framework.Bnn8558y.js
<span class="line"><span></span></span>
<span class="line"><span>(2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω0^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*u1(T)*(v1(T)^2)*α ~ F</span></span>
<span class="line"><span></span></span>
-<span class="line"><span>Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v1(T)^3)*α ~ 0</span></span></code></pre></div><p>The variables <code>u1</code> and <code>v1</code> were declared automatically to construct the harmonic ansatz. The <em>slow time</em> variable <code>T</code> describes variation of the quadratures on timescales much slower than <code>ω</code>. For a steady state, all derivatives w.r.t <code>T</code> vanish, leaving only algebraic equations to be solved.</p><p>We are ready to start plugging in numbers! Let us find steady states by solving <code>harmonic_eq</code> for numerical parameters. Homotopy continuation is especially suited to <a href="https://www.juliahomotopycontinuation.org/guides/parameter-homotopies/" target="_blank" rel="noreferrer">solving over a range of parameter values</a>. Here we will solve over a range of driving frequencies <code>ω</code> – these are stored as <code>Pairs{Sym, Vector{Float}}</code>:</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">varied </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> range</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.9</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">100</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">); </span><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># range of parameter values</span></span></code></pre></div><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>ω => 0.9:0.0030303030303030303:1.2</span></span></code></pre></div><p>The other parameters we be fixed – these are declared as <code>Pairs{Sym, Float}</code> pairs:</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">fixed </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (α </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ω0 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0.01</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, γ </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0.01</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">); </span><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># fixed parameters</span></span></code></pre></div><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>(α => 1.0, ω0 => 1.0, F => 0.01, γ => 0.01)</span></span></code></pre></div><p>Now everything is ready to crank the handle. <code>get_steady_states</code> solves our <code>harmonic_eq</code> using the varied and fixed parameters:</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">result </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(harmonic_eq, varied, fixed)</span></span></code></pre></div><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>A steady state result for 100 parameter points</span></span>
+<span class="line"><span>Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v1(T)^3)*α ~ 0//1</span></span></code></pre></div><p>The variables <code>u1</code> and <code>v1</code> were declared automatically to construct the harmonic ansatz. The <em>slow time</em> variable <code>T</code> describes variation of the quadratures on timescales much slower than <code>ω</code>. For a steady state, all derivatives w.r.t <code>T</code> vanish, leaving only algebraic equations to be solved.</p><p>We are ready to start plugging in numbers! Let us find steady states by solving <code>harmonic_eq</code> for numerical parameters. Homotopy continuation is especially suited to <a href="https://www.juliahomotopycontinuation.org/guides/parameter-homotopies/" target="_blank" rel="noreferrer">solving over a range of parameter values</a>. Here we will solve over a range of driving frequencies <code>ω</code> – these are stored as <code>Pairs{Sym, Vector{Float}}</code>:</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">varied </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> range</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.9</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">100</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">); </span><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># range of parameter values</span></span></code></pre></div><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>ω => 0.9:0.0030303030303030303:1.2</span></span></code></pre></div><p>The other parameters we be fixed – these are declared as <code>Pairs{Sym, Float}</code> pairs:</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">fixed </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (α </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ω0 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0.01</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, γ </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0.01</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">); </span><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># fixed parameters</span></span></code></pre></div><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>(α => 1.0, ω0 => 1.0, F => 0.01, γ => 0.01)</span></span></code></pre></div><p>Now everything is ready to crank the handle. <code>get_steady_states</code> solves our <code>harmonic_eq</code> using the varied and fixed parameters:</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">result </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(harmonic_eq, varied, fixed)</span></span></code></pre></div><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>A steady state result for 100 parameter points</span></span>
<span class="line"><span></span></span>
<span class="line"><span>Solution branches: 3</span></span>
<span class="line"><span> of which real: 3</span></span>
diff --git a/dev/assets/tutorials_steady_states.md.CagED6_V.lean.js b/dev/assets/tutorials_steady_states.md.ERR8LiBX.lean.js
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class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Transient Dynamics</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Examples</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Three Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Oscillator</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>State Dependent Perturbation</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Resources</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/API" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>API</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Bibliography</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Manual</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Entering Equations of Motion</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Effective System</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/methods" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Krylov-Bogoliubov</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Time Evolution</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Plotting</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/saving" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Saving and Loading</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-a9a9e638 data-v-91765379><div class="VPDoc has-sidebar has-aside" data-v-91765379 data-v-83890dd9><!--[--><!--]--><div class="container" data-v-83890dd9><div class="aside" data-v-83890dd9><div class="aside-curtain" data-v-83890dd9></div><div class="aside-container" data-v-83890dd9><div class="aside-content" data-v-83890dd9><div class="VPDocAside" data-v-83890dd9 data-v-6d7b3c46><!--[--><!--]--><!--[--><!--]--><nav aria-labelledby="doc-outline-aria-label" class="VPDocAsideOutline" data-v-6d7b3c46 data-v-b38bf2ff><div class="content" data-v-b38bf2ff><div class="outline-marker" data-v-b38bf2ff></div><div aria-level="2" class="outline-title" id="doc-outline-aria-label" role="heading" data-v-b38bf2ff>On this page</div><ul class="VPDocOutlineItem root" data-v-b38bf2ff data-v-3f927ebe><!--[--><!--]--></ul></div></nav><!--[--><!--]--><div class="spacer" data-v-6d7b3c46></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--]--></div></div></div></div><div class="content" data-v-83890dd9><div class="content-container" data-v-83890dd9><!--[--><!--]--><main class="main" data-v-83890dd9><div style="position:relative;" class="vp-doc _HarmonicBalance_jl_dev_background_harmonic_balance" data-v-83890dd9><div><h1 id="intro_hb" tabindex="-1">The method of harmonic balance <a class="header-anchor" href="#intro_hb" aria-label="Permalink to "The method of harmonic balance {#intro_hb}""></a></h1><p>HarmonicBalance.jl focuses on systems with equations of motion that are subject to time-dependent terms of harmonic type.</p><h2 id="prelude" tabindex="-1">Harmonic generation in oscillating nonlinear systems <a class="header-anchor" href="#prelude" aria-label="Permalink to "Harmonic generation in oscillating nonlinear systems {#prelude}""></a></h2><p>Let us take a general nonlinear system of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="2.009ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 888 683" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></mjx-assistive-mml></mjx-container> as the independent variable,</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="21.167ex" height="2.294ex" role="img" focusable="false" viewBox="0 -764 9355.7 1014" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D431" d="M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 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stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>x</mi><mi>N</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mtext>T</mtext></mrow></msup></math></mjx-assistive-mml></mjx-container> fully describes the state of the system. 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display="block" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo>¨</mo></mover></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>γ</mi><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo>˙</mo></mover></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>ω</mi><mn>0</mn><mn>2</mn></msubsup><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>d</mi></msub><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo></math></mjx-assistive-mml></mjx-container><p>where <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.489ex;" xmlns="http://www.w3.org/2000/svg" width="1.229ex" height="1.486ex" role="img" focusable="false" viewBox="0 -441 543 657" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D6FE" d="M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 -190T303 -216Q286 -216 286 -188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math></mjx-assistive-mml></mjx-container> is the damping coefficient and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.395ex" height="1.377ex" role="img" focusable="false" viewBox="0 -443 1058.6 608.6" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path 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style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mn>0</mn></msub></math></mjx-assistive-mml></mjx-container> the natural frequency. 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transform="translate(9778.7,0)"><path data-c="5D" d="M22 710V750H159V-250H22V-210H119V710H22Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mstyle" transform="translate(21401.3,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mo" transform="translate(21568.3,0)"><path data-c="2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">(</mo><msubsup><mi>ω</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>i</mi><mi>ω</mi><mi>γ</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>F</mi><mn>2</mn></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>ω</mi><mo>+</mo><msub><mi>ω</mi><mi>d</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>ω</mi><mo>−</mo><msub><mi>ω</mi><mi>d</mi></msub><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">]</mo></mrow><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mo>.</mo></math></mjx-assistive-mml></mjx-container><p>Evidently, <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.462ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1972 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 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1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> is only non-vanishing for <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.355ex;" xmlns="http://www.w3.org/2000/svg" width="8.611ex" height="1.862ex" role="img" focusable="false" viewBox="0 -666 3806.3 823.1" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(899.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1955.6,0)"><path data-c="B1" d="M56 320T56 333T70 353H369V502Q369 651 371 655Q376 666 388 666Q402 666 405 654T409 596V500V353H707Q722 345 722 333Q722 320 707 313H409V40H707Q722 32 722 20T707 0H70Q56 7 56 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-10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mo>±</mo><msub><mi>ω</mi><mi>d</mi></msub></math></mjx-assistive-mml></mjx-container>. The system thus responds solely at the driving frequency, i.e., the time evolution can be captured by a single harmonic. This illustrates the general point that <em>linear systems are exactly solvable</em> by transforming to Fourier space, where the equations are diagonal.</p><p>The situation becomes more complex if nonlinear terms are present, as these cause <em>frequency conversion</em>. 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76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(655,363) scale(0.707)"><g data-c="2033"><path data-c="2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z" style="stroke-width:3;"></path><path data-c="2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z" transform="translate(275,0)" style="stroke-width:3;"></path></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>,</mo><msup><mi>ω</mi><mo data-mjx-alternate="1">′</mo></msup><mo>,</mo><msup><mi>ω</mi><mo data-mjx-alternate="1">″</mo></msup></math></mjx-assistive-mml></mjx-container> such that <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.186ex;" xmlns="http://www.w3.org/2000/svg" width="15.597ex" height="1.903ex" role="img" focusable="false" viewBox="0 -759 6893.8 841" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(844.2,0)"><path data-c="2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z" style="stroke-width:3;"></path></g><g data-mml-node="msup" transform="translate(1844.4,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(655,363) scale(0.707)"><path data-c="2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(2966.1,0)"><path data-c="2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z" style="stroke-width:3;"></path></g><g data-mml-node="msup" transform="translate(3966.3,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(655,363) scale(0.707)"><g data-c="2033"><path data-c="2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z" style="stroke-width:3;"></path><path data-c="2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z" transform="translate(275,0)" style="stroke-width:3;"></path></g></g></g><g data-mml-node="mo" transform="translate(5338,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(6393.8,0)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>+</mo><msup><mi>ω</mi><mo data-mjx-alternate="1">′</mo></msup><mo>+</mo><msup><mi>ω</mi><mo data-mjx-alternate="1">″</mo></msup><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container>. To the lowest order, this means that the induced motion at the drive frequency generates a higher harmonic, <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.355ex;" xmlns="http://www.w3.org/2000/svg" width="9.504ex" height="1.862ex" role="img" focusable="false" viewBox="0 -666 4200.9 823.1" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1350.5,0)"><path data-c="2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(2628.3,0)"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(3128.3,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>d</mi></msub><mo stretchy="false">→</mo><mn>2</mn><msub><mi>ω</mi><mi>d</mi></msub></math></mjx-assistive-mml></mjx-container>. The frequency conversion couples the response at different frequencies and propagates through the spectrum, thus, <em>coupling an infinite number of harmonics</em>. Hence, the system is not easily solvable in Fourier space anymore!</p><h2 id="Harmonic-ansatz-and-harmonic-equations" tabindex="-1">Harmonic ansatz & harmonic equations <a class="header-anchor" href="#Harmonic-ansatz-and-harmonic-equations" aria-label="Permalink to "Harmonic ansatz &amp; harmonic equations {#Harmonic-ansatz-and-harmonic-equations}""></a></h2><p>Even though we need an infinity of Fourier components to describe our system exactly, some components are more important than others. The strategy of harmonic balance is to describe the motion of any variable <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.611ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2038 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 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182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> in a truncated Fourier space</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-3.014ex;" xmlns="http://www.w3.org/2000/svg" width="46.044ex" height="6.986ex" role="img" focusable="false" viewBox="0 -1755.5 20351.4 3087.7" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 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stretchy="false">)</mo><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mo>.</mo></math></mjx-assistive-mml></mjx-container><p>Under the assumption that <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.799ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2121 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 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0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.666ex;" xmlns="http://www.w3.org/2000/svg" width="8.606ex" height="2.363ex" role="img" focusable="false" viewBox="0 -750 3803.9 1044.2" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 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Collecting these, we obtain a set of coupled 1st order nonlinear ODEs,</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-1.575ex;" xmlns="http://www.w3.org/2000/svg" width="15.838ex" height="4.878ex" role="img" focusable="false" viewBox="0 -1460 7000.6 2156" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(220,710)"><g data-mml-node="mi"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 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0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mrow><mi>d</mi><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mi>T</mi></mrow></mfrac><mo>=</mo><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">)</mo><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mo>,</mo></math></mjx-assistive-mml></mjx-container><p>which we call the <em>harmonic equations</em>. The main purpose of HarmonicBalance.jl is to obtain and solve these Harmonic equations. We are primarily interested in <em>steady states</em> <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.433ex" height="1.393ex" role="img" focusable="false" viewBox="0 -450 1075.6 615.6" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub></math></mjx-assistive-mml></mjx-container> defined by <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="9.98ex" height="2.672ex" role="img" focusable="false" viewBox="0 -931 4411.1 1181" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D405" d="M425 0L228 3Q63 3 51 0H39V62H147V618H39V680H644V676Q647 670 659 552T675 428V424H613Q613 433 605 477Q599 511 589 535T562 574T530 599T488 612T441 617T387 618H368H304V371H333Q389 373 411 390T437 468V488H499V192H437V212Q436 244 430 263T408 292T378 305T333 309H304V62H439V0H425Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(362,241) translate(-250 0)"><path data-c="AF" d="M69 544V590H430V544H69Z" style="stroke-width:3;"></path></g></g></g><g data-mml-node="mo" transform="translate(724,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(1113,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(2188.6,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2855.3,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(3911.1,0)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container>.</p><p>The process of obtaining the harmonic equations is best shown using the example below:</p><h2 id="Duffing_harmeq" tabindex="-1">Example: the Duffing oscillator <a class="header-anchor" href="#Duffing_harmeq" aria-label="Permalink to "Example: the Duffing oscillator {#Duffing_harmeq}""></a></h2><p>Here, we derive the harmonic equations for a single Duffing resonator, governed by the equation</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 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d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(16814.4,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g><g data-mml-node="mstyle" transform="translate(17203.4,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mo" transform="translate(17370.4,0)"><path data-c="2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo>¨</mo></mover></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>ω</mi><mn>0</mn><mn>2</mn></msubsup><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>α</mi><msup><mi>x</mi><mn>3</mn></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>d</mi></msub><mi>t</mi><mo>+</mo><mi>θ</mi><mo stretchy="false">)</mo><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mo>.</mo></math></mjx-assistive-mml></mjx-container><p>As explained <a href="/HarmonicBalance.jl/dev/background/harmonic_balance#prelude">above</a>, for a periodic driving at frequency <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.355ex;" xmlns="http://www.w3.org/2000/svg" width="2.427ex" height="1.358ex" role="img" focusable="false" viewBox="0 -443 1072.7 600.1" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>d</mi></msub></math></mjx-assistive-mml></mjx-container> and a weak nonlinearity <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.448ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 640 453" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D6FC" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math></mjx-assistive-mml></mjx-container>, we expect the response at frequency <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.355ex;" xmlns="http://www.w3.org/2000/svg" width="2.427ex" height="1.358ex" role="img" focusable="false" viewBox="0 -443 1072.7 600.1" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>d</mi></msub></math></mjx-assistive-mml></mjx-container> to dominate, followed by a response at <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.355ex;" xmlns="http://www.w3.org/2000/svg" width="3.558ex" height="1.86ex" role="img" focusable="false" viewBox="0 -665 1572.7 822.1" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><path data-c="33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(500,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>ω</mi><mi>d</mi></msub></math></mjx-assistive-mml></mjx-container> due to frequency conversion.</p><h3 id="Single-frequency-ansatz" tabindex="-1">Single-frequency ansatz <a class="header-anchor" href="#Single-frequency-ansatz" aria-label="Permalink to "Single-frequency ansatz {#Single-frequency-ansatz}""></a></h3><p>We first attempt to describe the steady states of the Duffing equations of motion using only a single harmonic, <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg 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0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>d</mi></msub></math></mjx-assistive-mml></mjx-container>, in the ansatz, i.e., our starting point is the following harmonic ansatz for <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 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style="stroke-width:3;"></path></g><g data-mml-node="mstyle" transform="translate(15610.7,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mo" transform="translate(15832.7,0)"><path data-c="2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>u</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>d</mi></msub><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>v</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>d</mi></msub><mi>t</mi><mo stretchy="false">)</mo><mstyle scriptlevel="0"><mspace width="0.222em"></mspace></mstyle><mo>,</mo></math></mjx-assistive-mml></mjx-container><p>with the harmonic variables <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math></mjx-assistive-mml></mjx-container> and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.097ex" height="1.027ex" role="img" focusable="false" viewBox="0 -443 485 454" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math></mjx-assistive-mml></mjx-container>. The <em>slow time</em> <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="1.593ex" height="1.532ex" role="img" focusable="false" viewBox="0 -677 704 677" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></mjx-assistive-mml></mjx-container> is, for now, equivalent to <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.025ex;" xmlns="http://www.w3.org/2000/svg" width="0.817ex" height="1.441ex" role="img" focusable="false" viewBox="0 -626 361 637" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></mjx-assistive-mml></mjx-container>. Substituting this ansatz into the Duffing equation of motion results in</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-9.551ex;" xmlns="http://www.w3.org/2000/svg" width="66.02ex" height="20.233ex" role="img" focusable="false" viewBox="0 -4721.5 29180.7 8943" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtable"><g data-mml-node="mtr" transform="translate(0,2972)"><g data-mml-node="mtd" transform="translate(660.7,0)"><g data-mml-node="mrow"><g data-mml-node="mo" transform="translate(0 -0.5)"><path data-c="5B" d="M269 -1249V1750H577V1677H342V-1176H577V-1249H269Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(583,0)"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(313.8,-7) translate(-250 0)"><path data-c="A8" d="M95 612Q95 633 112 651T153 669T193 652T210 612Q210 588 194 571T152 554L127 560Q95 577 95 612ZM289 611Q289 634 304 649T335 668Q336 668 340 668T346 669Q369 669 386 652T404 612T387 572T346 554Q323 554 306 570T289 611Z" style="stroke-width:3;"></path></g></g></g><g data-mml-node="mo" transform="translate(1377.2,0)"><path data-c="2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(2377.4,0)"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(2877.4,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 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400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(3950.1,0)"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(270.3,-1) 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columnalign="right left right" columnspacing="0em 2em" rowspacing="3pt"><mtr><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo>¨</mo></mover></mrow><mo>+</mo><mn>2</mn><msub><mi>ω</mi><mi>d</mi></msub><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo>˙</mo></mover></mrow><mo>+</mo><mi>u</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msubsup><mi>ω</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>ω</mi><mi>d</mi><mn>2</mn></msubsup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mfrac><mrow><mn>3</mn><mi>α</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>u</mi><mn>3</mn></msup><mo>+</mo><mi>u</mi><msup><mi>v</mi><mn>2</mn></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow><mn>4</mn></mfrac><mo>+</mo><mi>F</mi><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mi>θ</mi></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd><mtd><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>d</mi></msub><mi>t</mi><mo stretchy="false">)</mo></mtd><mtd></mtd></mtr><mtr><mtd><mo>+</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo>¨</mo></mover></mrow><mo>−</mo><mn>2</mn><msub><mi>ω</mi><mi>d</mi></msub><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo>˙</mo></mover></mrow><mo>+</mo><mi>v</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msubsup><mi>ω</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>ω</mi><mi>d</mi><mn>2</mn></msubsup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mfrac><mrow><mn>3</mn><mi>α</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>v</mi><mn>3</mn></msup><mo>+</mo><msup><mi>u</mi><mn>2</mn></msup><mi>v</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow><mn>4</mn></mfrac><mo>−</mo><mi>F</mi><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mi>θ</mi></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd><mtd><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>d</mi></msub><mi>t</mi><mo stretchy="false">)</mo></mtd><mtd></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mrow><mi>α</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>u</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><mi>u</mi><msup><mi>v</mi><mn>2</mn></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow><mn>4</mn></mfrac><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><mn>3</mn><msub><mi>ω</mi><mi>d</mi></msub><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mi>α</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>3</mn><msup><mi>u</mi><mn>2</mn></msup><mi>v</mi><mo>−</mo><msup><mi>v</mi><mn>3</mn></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow><mn>4</mn></mfrac><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><mn>3</mn><msub><mi>ω</mi><mi>d</mi></msub><mi>t</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mn>0.</mn></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>We see that the <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.91ex" role="img" focusable="false" viewBox="0 -833.2 1008.6 844.2" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 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130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup></math></mjx-assistive-mml></mjx-container> term has generated terms that oscillate at <mjx-container class="MathJax" jax="SVG" 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473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>ω</mi><mi>d</mi></msub></math></mjx-assistive-mml></mjx-container>, describing the process of frequency upconversion. We now Fourier-transform both sides of the above equations with respect to <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.355ex;" xmlns="http://www.w3.org/2000/svg" width="2.427ex" height="1.358ex" role="img" focusable="false" viewBox="0 -443 1072.7 600.1" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>d</mi></msub></math></mjx-assistive-mml></mjx-container> to obtain the harmonic equations. This process is equivalent to extracting the respective coefficients of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="8.031ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3549.7 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z" style="stroke-width:3;"></path><path data-c="6F" d="M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z" transform="translate(444,0)" style="stroke-width:3;"></path><path data-c="73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z" transform="translate(944,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1338,0)"><path data-c="2061" d="" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1338,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(1727,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mi" transform="translate(2799.7,0)"><path data-c="1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(3160.7,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>d</mi></msub><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="7.782ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3439.7 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z" style="stroke-width:3;"></path><path data-c="69" d="M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z" transform="translate(394,0)" style="stroke-width:3;"></path><path data-c="6E" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q450 438 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z" transform="translate(672,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1228,0)"><path data-c="2061" d="" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1228,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(1617,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mi" transform="translate(2689.7,0)"><path data-c="1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(3050.7,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>d</mi></msub><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>. Here, the distinction between <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.025ex;" xmlns="http://www.w3.org/2000/svg" width="0.817ex" height="1.441ex" role="img" focusable="false" viewBox="0 -626 361 637" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></mjx-assistive-mml></mjx-container> and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="1.593ex" height="1.532ex" role="img" focusable="false" viewBox="0 -677 704 677" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></mjx-assistive-mml></mjx-container> becomes important: since the evolution of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.647ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2054 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(572,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(961,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 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0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.45ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1967 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 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data-c="2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>T</mi></mrow></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>u</mi></mtd></mtr><mtr><mtd><mi>v</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mn>8</mn><msub><mi>ω</mi><mi>d</mi></msub></mrow></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>4</mn><mi>v</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msubsup><mi>ω</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>ω</mi><mi>d</mi><mn>2</mn></msubsup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mn>3</mn><mi>α</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>v</mi><mn>3</mn></msup><mo>+</mo><msup><mi>u</mi><mn>2</mn></msup><mi>v</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>−</mo><mn>4</mn><mi>F</mi><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mi>θ</mi></mrow></mtd></mtr><mtr><mtd><mn>4</mn><mi>u</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msubsup><mi>ω</mi><mi>d</mi><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>ω</mi><mn>0</mn><mn>2</mn></msubsup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>−</mo><mn>3</mn><mi>α</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>u</mi><mn>3</mn></msup><mo>+</mo><mi>u</mi><msup><mi>v</mi><mn>2</mn></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>−</mo><mn>4</mn><mi>F</mi><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mi>θ</mi></mrow></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mo>.</mo></math></mjx-assistive-mml></mjx-container><p>Steady states can now be found by setting the l.h.s. to zero, i.e., assuming <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.647ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2054 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(572,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(961,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1665,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.45ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1967 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(485,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(874,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1578,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> constant and neglecting any transient behaviour. This results in a set of 2 nonlinear polynomial equations of order 3, for which the maximum number of solutions set by <a href="https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem" target="_blank" rel="noreferrer">Bézout's theorem</a> is <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.186ex;" xmlns="http://www.w3.org/2000/svg" width="6.267ex" height="2.072ex" role="img" focusable="false" viewBox="0 -833.9 2770.1 915.9" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mn"><path data-c="33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(533,363) scale(0.707)"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1214.3,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(2270.1,0)"><path data-c="39" d="M352 287Q304 211 232 211Q154 211 104 270T44 396Q42 412 42 436V444Q42 537 111 606Q171 666 243 666Q245 666 249 666T257 665H261Q273 665 286 663T323 651T370 619T413 560Q456 472 456 334Q456 194 396 97Q361 41 312 10T208 -22Q147 -22 108 7T68 93T121 149Q143 149 158 135T173 96Q173 78 164 65T148 49T135 44L131 43Q131 41 138 37T164 27T206 22H212Q272 22 313 86Q352 142 352 280V287ZM244 248Q292 248 321 297T351 430Q351 508 343 542Q341 552 337 562T323 588T293 615T246 625Q208 625 181 598Q160 576 154 546T147 441Q147 358 152 329T172 282Q197 248 244 248Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>3</mn><mn>2</mn></msup><mo>=</mo><mn>9</mn></math></mjx-assistive-mml></mjx-container>. Depending on the parameters, the number of real solutions is known to be between 1 and 3 [<a href="https://www.sciencedirect.com/science/article/pii/S0021782423001563?via%3Dihub" target="_blank" rel="noreferrer">see proof</a>].</p><h3 id="Sidenote:-perturbative-approach" tabindex="-1">Sidenote: perturbative approach <a class="header-anchor" href="#Sidenote:-perturbative-approach" aria-label="Permalink to "Sidenote: perturbative approach {#Sidenote:-perturbative-approach}""></a></h3><p>The steady states describe a response that may be recast as <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="23.259ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 10280.5 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g 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0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>X</mi><mn>0</mn></msub><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>d</mi></msub><mi>t</mi><mo>+</mo><mi>ϕ</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>, where <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.375ex;" xmlns="http://www.w3.org/2000/svg" width="14.94ex" height="2.561ex" role="img" focusable="false" viewBox="0 -966.5 6603.7 1132" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D44B" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 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width="3152.6" height="60" x="853" y="846.5"></rect></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>0</mn></msub><mo>=</mo><msqrt><msup><mi>u</mi><mn>2</mn></msup><mo>+</mo><msup><mi>v</mi><mn>2</mn></msup></msqrt></math></mjx-assistive-mml></mjx-container> and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="15.809ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 6987.6 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D719" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(873.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 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data-c="74" d="M27 422Q80 426 109 478T141 600V615H181V431H316V385H181V241Q182 116 182 100T189 68Q203 29 238 29Q282 29 292 100Q293 108 293 146V181H333V146V134Q333 57 291 17Q264 -10 221 -10Q187 -10 162 2T124 33T105 68T98 100Q97 107 97 248V385H18V422H27Z" transform="translate(500,0)" style="stroke-width:3;"></path><path data-c="61" d="M137 305T115 305T78 320T63 359Q63 394 97 421T218 448Q291 448 336 416T396 340Q401 326 401 309T402 194V124Q402 76 407 58T428 40Q443 40 448 56T453 109V145H493V106Q492 66 490 59Q481 29 455 12T400 -6T353 12T329 54V58L327 55Q325 52 322 49T314 40T302 29T287 17T269 6T247 -2T221 -8T190 -11Q130 -11 82 20T34 107Q34 128 41 147T68 188T116 225T194 253T304 268H318V290Q318 324 312 340Q290 411 215 411Q197 411 181 410T156 406T148 403Q170 388 170 359Q170 334 154 320ZM126 106Q126 75 150 51T209 26Q247 26 276 49T315 109Q317 116 318 175Q318 233 317 233Q309 233 296 232T251 223T193 203T147 166T126 106Z" transform="translate(889,0)" style="stroke-width:3;"></path><path data-c="6E" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q450 438 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z" transform="translate(1389,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(4652.6,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 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transform="translate(6026.6,0)"><path data-c="1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(6598.6,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi><mo>=</mo><mo>−</mo><mtext>atan</mtext><mo stretchy="false">(</mo><mi>v</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>u</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>. Frequency conversion from <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.355ex;" xmlns="http://www.w3.org/2000/svg" width="2.427ex" height="1.358ex" role="img" focusable="false" viewBox="0 -443 1072.7 600.1" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>d</mi></msub></math></mjx-assistive-mml></mjx-container> to <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.355ex;" xmlns="http://www.w3.org/2000/svg" width="3.558ex" height="1.86ex" role="img" focusable="false" viewBox="0 -665 1572.7 822.1" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><path data-c="33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(500,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>ω</mi><mi>d</mi></msub></math></mjx-assistive-mml></mjx-container> can be found by setting <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="19.388ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 8569.6 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(572,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(961,0)"><path data-c="1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1322,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1988.8,0)"><path data-c="2261" d="M56 444Q56 457 70 464H707Q722 456 722 444Q722 430 706 424H72Q56 429 56 444ZM56 237T56 250T70 270H707Q722 262 722 250T707 230H70Q56 237 56 250ZM56 56Q56 71 72 76H706Q722 70 722 56Q722 44 707 36H70Q56 43 56 56Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(3044.6,0)"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" 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0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≡</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>δ</mi><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> with <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="15.769ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 6970.1 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo" transform="translate(0 -0.5)"><path data-c="7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 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data-mjx-texclass="ORD" stretchy="false">|</mo><mi>δ</mi><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mo>≪</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo data-mjx-texclass="ORD" stretchy="false">|</mo></math></mjx-assistive-mml></mjx-container> and expanding the equations of motion to first-order in <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.876ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2155 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(444,0)"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1016,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1405,0)"><path data-c="1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1766,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>. 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441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(22409.6,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g><g data-mml-node="mstyle" transform="translate(22798.6,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mo" transform="translate(22965.6,0)"><path data-c="2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 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data-mjx-texclass="CLOSE">]</mo></mrow><mi>δ</mi><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mrow><mi>α</mi><msubsup><mi>X</mi><mn>0</mn><mn>3</mn></msubsup></mrow><mn>4</mn></mfrac><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><mn>3</mn><msub><mi>ω</mi><mi>d</mi></msub><mi>t</mi><mo>+</mo><mn>3</mn><mi>ϕ</mi><mo stretchy="false">)</mo><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mo>,</mo></math></mjx-assistive-mml></mjx-container><p>describes a simple harmonic oscillator, which is exactly soluble. Correspondingly, a response of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.876ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2155 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(444,0)"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1016,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 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250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> at frequency <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.355ex;" xmlns="http://www.w3.org/2000/svg" width="3.558ex" height="1.86ex" role="img" focusable="false" viewBox="0 -665 1572.7 822.1" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><path data-c="33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(500,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>ω</mi><mi>d</mi></msub></math></mjx-assistive-mml></mjx-container> is observed. Since this response is obtained 'on top of' each steady state of the equations of motion, no previously-unknown solutions are generated in the process.</p><h3 id="Two-frequency-ansatz" tabindex="-1">Two-frequency ansatz <a class="header-anchor" href="#Two-frequency-ansatz" aria-label="Permalink to "Two-frequency ansatz {#Two-frequency-ansatz}""></a></h3><p>An approach in the spirit of harmonic balance is to use both harmonics <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.355ex;" xmlns="http://www.w3.org/2000/svg" width="2.427ex" height="1.358ex" role="img" focusable="false" viewBox="0 -443 1072.7 600.1" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>d</mi></msub></math></mjx-assistive-mml></mjx-container> and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.355ex;" xmlns="http://www.w3.org/2000/svg" width="3.558ex" height="1.86ex" role="img" focusable="false" viewBox="0 -665 1572.7 822.1" aria-hidden="true"><g 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278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>ω</mi><mi>d</mi></msub></math></mjx-assistive-mml></mjx-container> on the same footing, i.e., to insert the ansatz</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" 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46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>,</mo><msub><mi>u</mi><mn>2</mn></msub><mo>,</mo><msub><mi>v</mi><mn>1</mn></msub><mo>,</mo><msub><mi>v</mi><mn>2</mn></msub></math></mjx-assistive-mml></mjx-container> being the harmonic variables. As before we substitute the ansatz into the equations of motion, drop second derivatives with respect to <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="1.593ex" height="1.532ex" role="img" focusable="false" viewBox="0 -677 704 677" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></mjx-assistive-mml></mjx-container> and Fourier-transform both sides. Now, the respective coefficients correspond to <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="8.031ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3549.7 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z" style="stroke-width:3;"></path><path data-c="6F" d="M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 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0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>d</mi></msub><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>, <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="7.782ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3439.7 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 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transform="translate(3660.7,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><mn>3</mn><msub><mi>ω</mi><mi>d</mi></msub><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="8.913ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3939.7 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z" style="stroke-width:3;"></path><path data-c="69" d="M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z" transform="translate(394,0)" style="stroke-width:3;"></path><path data-c="6E" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q450 438 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z" transform="translate(672,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1228,0)"><path data-c="2061" d="" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1228,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(1617,0)"><path data-c="33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(2117,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g 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615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(3550.7,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><mn>3</mn><msub><mi>ω</mi><mi>d</mi></msub><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>. Rearranging, we obtain</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-10.887ex;" xmlns="http://www.w3.org/2000/svg" width="96.636ex" height="22.905ex" role="img" focusable="false" viewBox="0 -5312.1 42713.2 10124.1" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtable"><g data-mml-node="mtr"><g data-mml-node="mtd"><g data-mml-node="mtable"><g data-mml-node="mtr" transform="translate(0,3862.6)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mi"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(520,0)"><g data-mml-node="mi"><path data-c="1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><path data-c="31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z" style="stroke-width:3;"></path></g></g></g><g data-mml-node="mrow" transform="translate(372.3,-686)"><g data-mml-node="mi"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(520,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 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transform="translate(220,-686)"><g data-mml-node="mn"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(500,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 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data-mjx-texclass="OPEN">(</mo><msubsup><mi>v</mi><mn>1</mn><mn>3</mn></msubsup><mo>+</mo><msubsup><mi>u</mi><mn>1</mn><mn>2</mn></msubsup><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><msubsup><mi>u</mi><mn>1</mn><mn>2</mn></msubsup><msub><mi>v</mi><mn>2</mn></msub><mo>−</mo><msubsup><mi>v</mi><mn>1</mn><mn>2</mn></msubsup><msub><mi>v</mi><mn>2</mn></msub><mo>+</mo><mn>2</mn><msubsup><mi>u</mi><mn>2</mn><mn>2</mn></msubsup><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><msubsup><mi>v</mi><mn>2</mn><mn>2</mn></msubsup><msub><mi>v</mi><mn>1</mn></msub><mo>−</mo><mn>2</mn><msub><mi>u</mi><mn>1</mn></msub><msub><mi>u</mi><mn>2</mn></msub><msub><mi>v</mi><mn>1</mn></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mi>F</mi><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mi>θ</mi></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi>d</mi><msub><mi>v</mi><mn>1</mn></msub></mrow><mrow><mi>d</mi><mi>T</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msub><mi>ω</mi><mi>d</mi></msub></mrow></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mrow data-mjx-texclass="ORD"><msub><mi>ω</mi><mi>d</mi></msub></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>−</mo><msubsup><mi>ω</mi><mn>0</mn><mn>2</mn></msubsup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><msub><mi>u</mi><mn>1</mn></msub></mrow><mo>−</mo><mfrac><mrow><mn>3</mn><mi>α</mi></mrow><mn>4</mn></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msubsup><mi>u</mi><mn>1</mn><mn>3</mn></msubsup><mo>+</mo><msubsup><mi>u</mi><mn>1</mn><mn>2</mn></msubsup><msub><mi>u</mi><mn>2</mn></msub><mo>+</mo><msubsup><mi>v</mi><mn>1</mn><mn>2</mn></msubsup><msub><mi>u</mi><mn>1</mn></msub><mo>−</mo><msubsup><mi>v</mi><mn>1</mn><mn>2</mn></msubsup><msub><mi>u</mi><mn>2</mn></msub><mo>+</mo><mn>2</mn><msubsup><mi>u</mi><mn>2</mn><mn>2</mn></msubsup><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><msubsup><mi>v</mi><mn>2</mn><mn>2</mn></msubsup><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><msub><mi>u</mi><mn>1</mn></msub><msub><mi>v</mi><mn>1</mn></msub><msub><mi>v</mi><mn>2</mn></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>−</mo><mi>F</mi><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mi>θ</mi></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi>d</mi><msub><mi>u</mi><mn>2</mn></msub></mrow><mrow><mi>d</mi><mi>T</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>6</mn><msub><mi>ω</mi><mi>d</mi></msub></mrow></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msubsup><mi>ω</mi><mn>0</mn><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msubsup><mo>−</mo><mn>9</mn><msubsup><mi>ω</mi><mi>d</mi><mn>2</mn></msubsup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><msub><mi>v</mi><mn>2</mn></msub></mrow><mo>+</mo><mfrac><mi>α</mi><mn>4</mn></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><msubsup><mi>v</mi><mn>1</mn><mn>3</mn></msubsup><mo>+</mo><mn>3</mn><msubsup><mi>v</mi><mn>2</mn><mn>3</mn></msubsup><mo>+</mo><mn>3</mn><msubsup><mi>u</mi><mn>1</mn><mn>2</mn></msubsup><msub><mi>v</mi><mn>1</mn></msub><mo>+</mo><mn>6</mn><msubsup><mi>u</mi><mn>1</mn><mn>2</mn></msubsup><msub><mi>v</mi><mn>2</mn></msub><mo>+</mo><mn>3</mn><msubsup><mi>u</mi><mn>2</mn><mn>2</mn></msubsup><msub><mi>v</mi><mn>2</mn></msub><mo>+</mo><mn>6</mn><msubsup><mi>v</mi><mn>1</mn><mn>2</mn></msubsup><msub><mi>v</mi><mn>2</mn></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi>d</mi><msub><mi>v</mi><mn>2</mn></msub></mrow><mrow><mi>d</mi><mi>T</mi></mrow></mfrac></mtd><mtd><mi></mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>6</mn><msub><mi>ω</mi><mi>d</mi></msub></mrow></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>9</mn><msubsup><mi>ω</mi><mi>d</mi><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>ω</mi><mn>0</mn><mn>2</mn></msubsup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><msub><mi>u</mi><mn>2</mn></msub></mrow><mo>−</mo><mfrac><mi>α</mi><mn>4</mn></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msubsup><mi>u</mi><mn>1</mn><mn>3</mn></msubsup><mo>+</mo><mn>3</mn><msubsup><mi>u</mi><mn>2</mn><mn>3</mn></msubsup><mo>+</mo><mn>6</mn><msubsup><mi>u</mi><mn>1</mn><mn>2</mn></msubsup><msub><mi>u</mi><mn>2</mn></msub><mo>−</mo><mn>3</mn><msubsup><mi>v</mi><mn>1</mn><mn>2</mn></msubsup><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mn>3</mn><msubsup><mi>v</mi><mn>2</mn><mn>2</mn></msubsup><msub><mi>u</mi><mn>2</mn></msub><mo>+</mo><mn>6</mn><msubsup><mi>v</mi><mn>1</mn><mn>2</mn></msubsup><msub><mi>u</mi><mn>2</mn></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mstyle scriptlevel="0"><mspace width="0.222em"></mspace></mstyle><mo>.</mo></mtd></mtr></mtable></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>In contrast to the single-frequency ansatz, we now have 4 equations of order 3, allowing up to <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.398ex" height="2.09ex" role="img" focusable="false" viewBox="0 -841.7 3270.1 923.7" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mn"><path data-c="33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(533,363) scale(0.707)"><path data-c="34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1214.3,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(2270.1,0)"><path data-c="38" d="M70 417T70 494T124 618T248 666Q319 666 374 624T429 515Q429 485 418 459T392 417T361 389T335 371T324 363L338 354Q352 344 366 334T382 323Q457 264 457 174Q457 95 399 37T249 -22Q159 -22 101 29T43 155Q43 263 172 335L154 348Q133 361 127 368Q70 417 70 494ZM286 386L292 390Q298 394 301 396T311 403T323 413T334 425T345 438T355 454T364 471T369 491T371 513Q371 556 342 586T275 624Q268 625 242 625Q201 625 165 599T128 534Q128 511 141 492T167 463T217 431Q224 426 228 424L286 386ZM250 21Q308 21 350 55T392 137Q392 154 387 169T375 194T353 216T330 234T301 253T274 270Q260 279 244 289T218 306L210 311Q204 311 181 294T133 239T107 157Q107 98 150 60T250 21Z" style="stroke-width:3;"></path><path data-c="31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z" transform="translate(500,0)" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>3</mn><mn>4</mn></msup><mo>=</mo><mn>81</mn></math></mjx-assistive-mml></mjx-container> solutions (the number of unique real ones is again generally far smaller). The larger number of solutions is explained by higher harmonics which cannot be captured perturbatively by the single-frequency ansatz. In particular, those where the <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.355ex;" xmlns="http://www.w3.org/2000/svg" width="3.558ex" height="1.86ex" role="img" focusable="false" viewBox="0 -665 1572.7 822.1" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><path data-c="33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(500,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>ω</mi><mi>d</mi></msub></math></mjx-assistive-mml></mjx-container> component is significant. Such solutions appear, e.g., for <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="10.101ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4464.8 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1350.5,0)"><path data-c="2248" d="M55 319Q55 360 72 393T114 444T163 472T205 482Q207 482 213 482T223 483Q262 483 296 468T393 413L443 381Q502 346 553 346Q609 346 649 375T694 454Q694 465 698 474T708 483Q722 483 722 452Q722 386 675 338T555 289Q514 289 468 310T388 357T308 404T224 426Q164 426 125 393T83 318Q81 289 69 289Q55 289 55 319ZM55 85Q55 126 72 159T114 210T163 238T205 248Q207 248 213 248T223 249Q262 249 296 234T393 179L443 147Q502 112 553 112Q609 112 649 141T694 220Q694 249 708 249T722 217Q722 153 675 104T555 55Q514 55 468 76T388 123T308 170T224 192Q164 192 125 159T83 84Q80 55 69 55Q55 55 55 85Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(2406.3,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(655,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(3464.8,0)"><g data-mml-node="mo"><path data-c="2F" d="M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(3964.8,0)"><path data-c="33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math 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82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(500,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g 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1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>ω</mi><mi>d</mi></msub></math></mjx-assistive-mml></mjx-container> harmonic is close to the natural resonant frequency. See the <a href="/HarmonicBalance.jl/dev/tutorials/steady_states#Duffing">examples</a> for numerical results.</p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/background/harmonic_balance.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/introduction/index" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Getting Started</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/background/stability_response" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Stability and Linear Response</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. 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Started</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible has-active" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Theoretical Background</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/harmonic_balance" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Floquet expansions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" 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class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/steady_states" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/classification" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Classifying Solutions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Transient Dynamics</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Examples</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Three Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Oscillator</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>State Dependent Perturbation</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Resources</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/API" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>API</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Bibliography</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Manual</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Entering Equations of Motion</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Effective System</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/methods" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Krylov-Bogoliubov</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Time Evolution</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Plotting</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/saving" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Saving and Loading</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-a9a9e638 data-v-91765379><div class="VPDoc has-sidebar has-aside" data-v-91765379 data-v-83890dd9><!--[--><!--]--><div class="container" data-v-83890dd9><div class="aside" data-v-83890dd9><div class="aside-curtain" data-v-83890dd9></div><div class="aside-container" data-v-83890dd9><div class="aside-content" data-v-83890dd9><div class="VPDocAside" data-v-83890dd9 data-v-6d7b3c46><!--[--><!--]--><!--[--><!--]--><nav aria-labelledby="doc-outline-aria-label" class="VPDocAsideOutline" data-v-6d7b3c46 data-v-b38bf2ff><div class="content" data-v-b38bf2ff><div class="outline-marker" data-v-b38bf2ff></div><div aria-level="2" class="outline-title" id="doc-outline-aria-label" role="heading" data-v-b38bf2ff>On this page</div><ul class="VPDocOutlineItem root" data-v-b38bf2ff data-v-3f927ebe><!--[--><!--]--></ul></div></nav><!--[--><!--]--><div class="spacer" data-v-6d7b3c46></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--]--></div></div></div></div><div class="content" data-v-83890dd9><div class="content-container" data-v-83890dd9><!--[--><!--]--><main class="main" data-v-83890dd9><div style="position:relative;" class="vp-doc _HarmonicBalance_jl_dev_background_limit_cycles" data-v-83890dd9><div><h1 id="limit_cycles_bg" tabindex="-1">Limit cycles <a class="header-anchor" href="#limit_cycles_bg" aria-label="Permalink to "Limit cycles {#limit_cycles_bg}""></a></h1><p>We explain how HarmonicBalance.jl uses a new technique to find limit cycles in systems of nonlinear ODEs. For a more in depth overview see Chapter 6 in <a href="https://www.doi.org/10.3929/ethz-b-000589190" target="_blank" rel="noreferrer">Jan Košata's PhD theses</a> or <a href="https://www.doi.org/10.1103/PhysRevResearch.6.03318" target="_blank" rel="noreferrer">del_Pino_2024</a>.</p><h2 id="Limit-cycles-from-a-Hopf-bifurcation" tabindex="-1">Limit cycles from a Hopf bifurcation <a class="header-anchor" href="#Limit-cycles-from-a-Hopf-bifurcation" aria-label="Permalink to "Limit cycles from a Hopf bifurcation {#Limit-cycles-from-a-Hopf-bifurcation}""></a></h2><p>The end product of the <a href="/HarmonicBalance.jl/dev/background/harmonic_balance#intro_hb">harmonic balance technique</a> are what we call the harmonic equations, i.e., first-order ODEs for the harmonic variables <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.355ex" 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style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(5953.6,0)"><g data-mml-node="mi"><path data-c="1D414" d="M570 686Q588 683 703 683T836 686H845V624H737V420Q737 390 737 345T738 284Q738 205 729 164T689 83Q614 -11 465 -11Q321 -11 240 51T148 207Q147 214 147 421V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V405V370V268Q304 181 311 146T346 87Q387 52 466 52Q642 52 667 195Q668 204 669 415V624H561V686H570Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(6838.6,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mrow><mi>d</mi><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">U</mi></mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mi>T</mi></mrow></mfrac><mo>=</mo><mover><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">G</mi></mrow><mo accent="true">―</mo></mover><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">U</mi></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><p>These Odes have no explicit time-dependence - they are autonomous. We have mostly been searching for steady states, which likewise show no time dependence. However, time-dependent solutions to autonomous ODEs can also exist. One mechanism for their creation is a <a href="https://en.wikipedia.org/wiki/Hopf_bifurcation" target="_blank" rel="noreferrer">Hopf bifurcation</a> - a critical point where a stable solution transitions into an unstable one. For a stable solution, the associated eigenvalues <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.027ex;" xmlns="http://www.w3.org/2000/svg" width="1.319ex" height="1.597ex" role="img" focusable="false" viewBox="0 -694 583 706" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></mjx-assistive-mml></mjx-container> of the linearisation all satisfy <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="9.897ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4374.6 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="52" d="M130 622Q123 629 119 631T103 634T60 637H27V683H202H236H300Q376 683 417 677T500 648Q595 600 609 517Q610 512 610 501Q610 468 594 439T556 392T511 361T472 343L456 338Q459 335 467 332Q497 316 516 298T545 254T559 211T568 155T578 94Q588 46 602 31T640 16H645Q660 16 674 32T692 87Q692 98 696 101T712 105T728 103T732 90Q732 59 716 27T672 -16Q656 -22 630 -22Q481 -16 458 90Q456 101 456 163T449 246Q430 304 373 320L363 322L297 323H231V192L232 61Q238 51 249 49T301 46H334V0H323Q302 3 181 3Q59 3 38 0H27V46H60Q102 47 111 49T130 61V622ZM491 499V509Q491 527 490 539T481 570T462 601T424 623T362 636Q360 636 340 636T304 637H283Q238 637 234 628Q231 624 231 492V360H289Q390 360 434 378T489 456Q491 467 491 499Z" style="stroke-width:3;"></path><path data-c="65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z" transform="translate(736,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1180,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1569,0)"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2152,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 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style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Re</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container>. When a Hopf bifurcation takes place, one complex-conjugate pair of eigenvalues crosses the real axis such that <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="9.897ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4374.6 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="52" d="M130 622Q123 629 119 631T103 634T60 637H27V683H202H236H300Q376 683 417 677T500 648Q595 600 609 517Q610 512 610 501Q610 468 594 439T556 392T511 361T472 343L456 338Q459 335 467 332Q497 316 516 298T545 254T559 211T568 155T578 94Q588 46 602 31T640 16H645Q660 16 674 32T692 87Q692 98 696 101T712 105T728 103T732 90Q732 59 716 27T672 -16Q656 -22 630 -22Q481 -16 458 90Q456 101 456 163T449 246Q430 304 373 320L363 322L297 323H231V192L232 61Q238 51 249 49T301 46H334V0H323Q302 3 181 3Q59 3 38 0H27V46H60Q102 47 111 49T130 61V622ZM491 499V509Q491 527 490 539T481 570T462 601T424 623T362 636Q360 636 340 636T304 637H283Q238 637 234 628Q231 624 231 492V360H289Q390 360 434 378T489 456Q491 467 491 499Z" style="stroke-width:3;"></path><path data-c="65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z" transform="translate(736,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1180,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1569,0)"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2152,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2818.8,0)"><path data-c="3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(3874.6,0)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Re</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></math></mjx-assistive-mml></mjx-container>. The state is then, strictly speaking, unstable. However, instead of evolving into another steady state, the system may assume a periodic orbit in phase space, giving a solution of the form</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="31.471ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 13910.4 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D414" d="M570 686Q588 683 703 683T836 686H845V624H737V420Q737 390 737 345T738 284Q738 205 729 164T689 83Q614 -11 465 -11Q321 -11 240 51T148 207Q147 214 147 421V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V405V370V268Q304 181 311 146T346 87Q387 52 466 52Q642 52 667 195Q668 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380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z" transform="translate(444,0)" style="stroke-width:3;"></path><path data-c="73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z" transform="translate(944,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(9227.8,0)"><path data-c="2061" d="" style="stroke-width:3;"></path></g><g data-mml-node="mrow" transform="translate(9394.4,0)"><g data-mml-node="mo"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(389,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="6C" d="M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z" style="stroke-width:3;"></path><path data-c="63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z" transform="translate(278,0)" style="stroke-width:3;"></path></g></g></g></g><g data-mml-node="mi" transform="translate(1604.5,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2530.8,0)"><path data-c="2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(3531,0)"><path data-c="1D719" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(4127,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">U</mi></mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">U</mi></mrow><mn>0</mn></msub><mo>+</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">U</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">lc</mi></mrow></mrow></msub><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">lc</mi></mrow></mrow></msub><mi>T</mi><mo>+</mo><mi>ϕ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></math></mjx-assistive-mml></mjx-container><p>which is an example of a limit cycle. We denote the originating steady state as Hopf-unstable.</p><p>We can continue to use harmonic balance as the solution still describes a harmonic response <a href="https://www.doi.org/10.1017/S0305004100054128" target="_blank" rel="noreferrer">Allwright (1977)</a>. If we translate back to the the lab frame [variable <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="3.871ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1711 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(572,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(961,0)"><path data-c="1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1322,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>], clearly, each frequency <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.666ex;" xmlns="http://www.w3.org/2000/svg" width="2.254ex" height="1.668ex" role="img" focusable="false" viewBox="0 -443 996.3 737.2" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>j</mi></msub></math></mjx-assistive-mml></mjx-container> constituting our harmonic ansatz [<mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.355ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2367 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D414" d="M570 686Q588 683 703 683T836 686H845V624H737V420Q737 390 737 345T738 284Q738 205 729 164T689 83Q614 -11 465 -11Q321 -11 240 51T148 207Q147 214 147 421V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V405V370V268Q304 181 311 146T346 87Q387 52 466 52Q642 52 667 195Q668 204 669 415V624H561V686H570Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(885,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1274,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1978,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">U</mi></mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>], we obtain frequencies <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.666ex;" xmlns="http://www.w3.org/2000/svg" width="2.254ex" height="1.668ex" role="img" focusable="false" viewBox="0 -443 996.3 737.2" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>j</mi></msub></math></mjx-assistive-mml></mjx-container> as well as <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.666ex;" xmlns="http://www.w3.org/2000/svg" width="8.17ex" height="2.173ex" role="img" focusable="false" viewBox="0 -666 3611.1 960.2" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><path data-c="1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1218.6,0)"><path data-c="B1" d="M56 320T56 333T70 353H369V502Q369 651 371 655Q376 666 388 666Q402 666 405 654T409 596V500V353H707Q722 345 722 333Q722 320 707 313H409V40H707Q722 32 722 20T707 0H70Q56 7 56 20T70 40H369V313H70Q56 320 56 333Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(2218.8,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mtext"><path data-c="6C" d="M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z" style="stroke-width:3;"></path><path data-c="63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z" transform="translate(278,0)" style="stroke-width:3;"></path><path data-c="A0" d="" transform="translate(722,0)" style="stroke-width:3;"></path></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>j</mi></msub><mo>±</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mtext>lc </mtext></mrow></msub></math></mjx-assistive-mml></mjx-container> in the lab frame. Furthermore, as multiple harmonics now co-exist in the system, frequency conversion may take place, spawning further pairs <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.666ex;" xmlns="http://www.w3.org/2000/svg" width="9.349ex" height="2.236ex" role="img" focusable="false" viewBox="0 -694 4132.1 988.2" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 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style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(2739.8,0)"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mtext"><path data-c="6C" d="M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z" style="stroke-width:3;"></path><path data-c="63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z" transform="translate(278,0)" style="stroke-width:3;"></path><path data-c="A0" d="" transform="translate(722,0)" style="stroke-width:3;"></path></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>j</mi></msub><mo>±</mo><mi>k</mi><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mtext>lc </mtext></mrow></msub></math></mjx-assistive-mml></mjx-container> with integer <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.179ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 521 705" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></mjx-assistive-mml></mjx-container>. Therefore, to construct a harmonic ansatz capturing limit cycles, we simply add an integer number <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="2.011ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 889 683" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D43E" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi></math></mjx-assistive-mml></mjx-container> of such pairs to our existing set of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="2.378ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 1051 683" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D440" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></mjx-assistive-mml></mjx-container> harmonics,</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="55.96ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 24734.4 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo"><path data-c="7B" d="M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 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data-mjx-texclass="CLOSE">}</mo></mrow><mo stretchy="false">→</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><msub><mi>ω</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ω</mi><mn>1</mn></msub><mo>±</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">lc</mi></mrow></mrow></msub><mo>,</mo><msub><mi>ω</mi><mn>1</mn></msub><mo>±</mo><mn>2</mn><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">lc</mi></mrow></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>ω</mi><mi>M</mi></msub><mo>±</mo><mi>K</mi><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">lc</mi></mrow></mrow></msub><mo data-mjx-texclass="CLOSE">}</mo></mrow></math></mjx-assistive-mml></mjx-container><h2 id="ansatz" tabindex="-1">Ansatz <a class="header-anchor" href="#ansatz" aria-label="Permalink to "Ansatz""></a></h2><h3 id="Original-ansatz" tabindex="-1">Original ansatz <a class="header-anchor" href="#Original-ansatz" aria-label="Permalink to "Original ansatz {#Original-ansatz}""></a></h3><p>Having seen how limit cycles are formed, we now proceed to tackle a key problem: how to find their frequency <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.357ex;" xmlns="http://www.w3.org/2000/svg" width="2.75ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 1215.5 600.8" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="6C" d="M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z" style="stroke-width:3;"></path><path data-c="63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z" transform="translate(278,0)" style="stroke-width:3;"></path></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">lc</mi></mrow></mrow></msub></math></mjx-assistive-mml></mjx-container>. We again demonstrate by considering a single variable <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="3.871ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1711 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(572,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(961,0)"><path data-c="1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1322,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>. We may try the simplest ansatz for a system driven at frequency <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.407ex" height="1.027ex" role="img" focusable="false" viewBox="0 -443 622 454" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math></mjx-assistive-mml></mjx-container>,</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="35.254ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 15582.4 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g 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style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(12593.4,0)"><path data-c="73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z" style="stroke-width:3;"></path><path data-c="69" d="M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 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transform="translate(13821.4,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(14210.4,0)"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(14832.4,0)"><path data-c="1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(15193.4,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>u</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><mi>ω</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><mi>ω</mi><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><p>In this formulation, limit cycles may be obtained by solving the resulting harmonic equations with a Runge-Kutta type solver to obtain the time evolution of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.635ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2490.6 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><path data-c="31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1008.6,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1397.6,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2101.6,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.438ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2403.6 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(518,-150) scale(0.707)"><path data-c="31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(921.6,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1310.6,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2014.6,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>. See the <a href="/HarmonicBalance.jl/dev/tutorials/limit_cycles#limit_cycles">limit cycle tutorial</a> for an example.</p><h3 id="Extended-ansatz" tabindex="-1">Extended ansatz <a class="header-anchor" href="#Extended-ansatz" aria-label="Permalink to "Extended ansatz {#Extended-ansatz}""></a></h3><p>Including newly-emergent pairs of harmonics is in principle straightforward. Suppose a limit cycle has formed in our system with a frequency <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.357ex;" xmlns="http://www.w3.org/2000/svg" width="2.75ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 1215.5 600.8" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="6C" d="M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z" style="stroke-width:3;"></path><path data-c="63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z" transform="translate(278,0)" style="stroke-width:3;"></path></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">lc</mi></mrow></mrow></msub></math></mjx-assistive-mml></mjx-container>, prompting the ansatz</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-3.507ex;" xmlns="http://www.w3.org/2000/svg" width="45.402ex" height="8.145ex" role="img" focusable="false" viewBox="0 -2050 20067.6 3600" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtable"><g data-mml-node="mtr" transform="translate(0,1300)"><g data-mml-node="mtd"></g><g data-mml-node="mtd"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 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d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60ZM525 60Q525 84 542 102T585 120Q609 120 627 104T646 61Q646 36 629 18T586 0T543 17T525 60ZM972 60Q972 84 989 102T1032 120Q1056 120 1074 104T1093 61Q1093 36 1076 18T1033 0T990 17T972 60Z" style="stroke-width:3;"></path></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd></mtd><mtd><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>u</mi><mn>1</mn></msub><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><mi>ω</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>v</mi><mn>1</mn></msub><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><mi>ω</mi><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>+</mo><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><msub><mi>u</mi><mn>2</mn></msub><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ω</mi><mo>+</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">lc</mi></mrow></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>t</mi><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>+</mo><msub><mi>v</mi><mn>2</mn></msub><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ω</mi><mo>+</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">lc</mi></mrow></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>t</mi><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mo>+</mo><msub><mi>u</mi><mn>3</mn></msub><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ω</mi><mo>−</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">lc</mi></mrow></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>t</mi><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>+</mo><msub><mi>v</mi><mn>3</mn></msub><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ω</mi><mo>−</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">lc</mi></mrow></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>t</mi><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>+</mo><mo>…</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>where each of the <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.357ex;" xmlns="http://www.w3.org/2000/svg" width="8.502ex" height="1.927ex" role="img" focusable="false" viewBox="0 -694 3757.8 851.8" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(844.2,0)"><path data-c="B1" d="M56 320T56 333T70 353H369V502Q369 651 371 655Q376 666 388 666Q402 666 405 654T409 596V500V353H707Q722 345 722 333Q722 320 707 313H409V40H707Q722 32 722 20T707 0H70Q56 7 56 20T70 40H369V313H70Q56 320 56 333Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1844.4,0)"><path data-c="1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 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429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mtext"><path data-c="6C" d="M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z" style="stroke-width:3;"></path><path data-c="63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z" transform="translate(278,0)" style="stroke-width:3;"></path><path data-c="A0" d="" transform="translate(722,0)" style="stroke-width:3;"></path></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>±</mo><mi>k</mi><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mtext>lc </mtext></mrow></msub></math></mjx-assistive-mml></mjx-container> pairs contributes 4 harmonic variables. The limit cycle frequency <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.357ex;" xmlns="http://www.w3.org/2000/svg" width="2.75ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 1215.5 600.8" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="6C" d="M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z" style="stroke-width:3;"></path><path data-c="63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z" transform="translate(278,0)" style="stroke-width:3;"></path></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">lc</mi></mrow></mrow></msub></math></mjx-assistive-mml></mjx-container> is also a variable in this formulation, but does not contribute a harmonic equation, since <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="11.975ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 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443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="6C" d="M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z" style="stroke-width:3;"></path><path data-c="63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z" transform="translate(278,0)" style="stroke-width:3;"></path></g></g></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(1735.5,0)"><g data-mml-node="mo"><path data-c="2F" d="M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mi" transform="translate(2235.5,0)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(2755.5,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(3737.3,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(4793.1,0)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">lc</mi></mrow></mrow></msub><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>d</mi><mi>T</mi><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container> by construction. We thus arrive at a total of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.039ex" height="1.731ex" role="img" focusable="false" viewBox="0 -683 3111.4 765" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(722.2,0)"><path data-c="2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(1722.4,0)"><path data-c="34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(2222.4,0)"><path data-c="1D43E" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>+</mo><mn>4</mn><mi>K</mi></math></mjx-assistive-mml></mjx-container> harmonic equations in <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.186ex;" xmlns="http://www.w3.org/2000/svg" width="10.936ex" height="1.731ex" role="img" focusable="false" viewBox="0 -683 4833.9 765" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(722.2,0)"><path data-c="2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(1722.4,0)"><path data-c="34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(2222.4,0)"><path data-c="1D43E" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(3333.7,0)"><path data-c="2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(4333.9,0)"><path data-c="31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>+</mo><mn>4</mn><mi>K</mi><mo>+</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container> variables. To obtain steady states, we must thus solve an underdetermined system, which has an infinite number of solutions. Given that we expect the limit cycles to possess <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.627ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2045 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D448" d="M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 86 183 55T269 24Q336 24 403 69T501 205L552 406Q599 598 599 606Q599 633 535 637Q511 637 511 648Q511 650 513 660Q517 676 519 679T529 683Q532 683 561 682T645 680Q696 680 723 681T752 682Q767 682 767 672Q767 650 759 642Q756 637 737 637Q666 633 648 597Q646 592 598 404Q557 235 548 205Q515 105 433 42T263 -22Q171 -22 116 34T60 167V183Q60 201 115 421Q164 622 164 628Q164 635 107 637Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(767,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(1156,0)"><path data-c="31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1656,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> gauge freedom, this is a sensible observation. We may still use iterative numerical procedures such as the Newton method to find solutions one by one, but homotopy continuation is not applicable. In this formulation, steady staes states are characterised by zero entries for <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.471ex;" xmlns="http://www.w3.org/2000/svg" width="21.714ex" height="1.473ex" role="img" focusable="false" viewBox="0 -443 9597.5 651" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1008.6,0)"><path data-c="2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 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xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub><mo>,</mo><msub><mi>v</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><msub><mi>u</mi><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>K</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>v</mi><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>K</mi><mo>+</mo><mn>1</mn></mrow></msub></math></mjx-assistive-mml></mjx-container>. The variable <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.357ex;" xmlns="http://www.w3.org/2000/svg" width="3.15ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 1392.3 600.8" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mtext"><path data-c="6C" d="M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z" style="stroke-width:3;"></path><path data-c="63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z" transform="translate(278,0)" style="stroke-width:3;"></path><path data-c="A0" d="" transform="translate(722,0)" style="stroke-width:3;"></path></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mtext>lc </mtext></mrow></msub></math></mjx-assistive-mml></mjx-container> is redundant and may take any value - the states therefore also appear infinitely degenerate, which, however, has no physical grounds. 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500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1389,0)"><path data-c="2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 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xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub><mo>,</mo><msub><mi>v</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><msub><mi>u</mi><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>K</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>v</mi><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>K</mi><mo>+</mo><mn>1</mn></mrow></msub></math></mjx-assistive-mml></mjx-container> are nonzero, but <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.357ex;" xmlns="http://www.w3.org/2000/svg" width="7.298ex" height="1.864ex" role="img" focusable="false" viewBox="0 -666 3225.9 823.8" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mtext"><path data-c="6C" d="M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z" style="stroke-width:3;"></path><path data-c="63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z" transform="translate(278,0)" style="stroke-width:3;"></path><path data-c="A0" d="" transform="translate(722,0)" style="stroke-width:3;"></path></g></g></g><g data-mml-node="mo" transform="translate(1670.1,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(2725.9,0)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mtext>lc </mtext></mrow></msub><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container>. These violate our assumption of distinct harmonic variables corresponding to distinct frequencies and are therefore discarded.</p><h3 id="gauge_fixing" tabindex="-1">Gauge fixing <a class="header-anchor" href="#gauge_fixing" aria-label="Permalink to "Gauge fixing {#gauge_fixing}""></a></h3><p>We now constrain the system to remove the <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.627ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2045 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D448" d="M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 86 183 55T269 24Q336 24 403 69T501 205L552 406Q599 598 599 606Q599 633 535 637Q511 637 511 648Q511 650 513 660Q517 676 519 679T529 683Q532 683 561 682T645 680Q696 680 723 681T752 682Q767 682 767 672Q767 650 759 642Q756 637 737 637Q666 633 648 597Q646 592 598 404Q557 235 548 205Q515 105 433 42T263 -22Q171 -22 116 34T60 167V183Q60 201 115 421Q164 622 164 628Q164 635 107 637Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(767,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(1156,0)"><path data-c="31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1656,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> gauge freedom. This is best done by explicitly writing out the free phase. Recall that our solution must be symmetric under a time translation symmetry, that is, taking <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="12.878ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 5692 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(638.8,0)"><path data-c="2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1916.6,0)"><path data-c="1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2499.8,0)"><path data-c="2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(3500,0)"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(4000,0)"><path data-c="1D70B" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(4570,0)"><g data-mml-node="mo"><path data-c="2F" d="M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mi" transform="translate(5070,0)"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo stretchy="false">→</mo><mi>t</mi><mo>+</mo><mn>2</mn><mi>π</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>ω</mi></math></mjx-assistive-mml></mjx-container>. Applying this <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.357ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 600 453" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container> times transforms <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="3.871ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1711 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(572,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 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columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>u</mi><mn>1</mn></msub><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><mi>ω</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo></mtd><mtd><msub><mi>v</mi><mn>1</mn></msub><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mo stretchy="false">(</mo><mi>ω</mi><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mo>+</mo><msub><mi>u</mi><mn>2</mn></msub><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ω</mi><mo>+</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">lc</mi></mrow></mrow></msub><mo 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style="stroke-width:3;"></path></g></g><g data-mml-node="mi" transform="translate(5491.9,0)"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi><mo>=</mo><mn>2</mn><mi>π</mi><mi>n</mi><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mtext>lc </mtext></mrow></msub><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>ω</mi></math></mjx-assistive-mml></mjx-container>. Since <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.348ex" height="2.034ex" role="img" focusable="false" viewBox="0 -694 596 899" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D719" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math></mjx-assistive-mml></mjx-container> is free, we can fix it to, for example,</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="18.67ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 8252 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D719" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z" style="stroke-width:3;"></path></g><g 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data-mml-node="msub" transform="translate(5821.9,0)"><g data-mml-node="mi"><path data-c="1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z" style="stroke-width:3;"></path></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(6830.4,0)"><g data-mml-node="mo"><path data-c="2F" d="M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z" style="stroke-width:3;"></path></g></g><g data-mml-node="msub" transform="translate(7330.4,0)"><g data-mml-node="mi"><path data-c="1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(518,-150) scale(0.707)"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>ϕ</mi><mo>=</mo><mo>−</mo><mi>arctan</mi><mo data-mjx-texclass="NONE"></mo><msub><mi>u</mi><mn>2</mn></msub><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><msub><mi>v</mi><mn>2</mn></msub></math></mjx-assistive-mml></mjx-container><p>which turns into</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-2.036ex;" xmlns="http://www.w3.org/2000/svg" width="73.801ex" height="5.204ex" role="img" focusable="false" viewBox="0 -1400 32619.9 2300" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtable"><g data-mml-node="mtr" transform="translate(0,650)"><g data-mml-node="mtd" transform="translate(2053.8,0)"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(572,0)"><path data-c="28" d="M94 250Q94 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data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ω</mi><mo>−</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">lc</mi></mrow></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>t</mi><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>+</mo><mo>…</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>We see that fixing the free phase has effectively removed one of the variables, since <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="14.562ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 6436.4 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z" style="stroke-width:3;"></path><path data-c="6F" d="M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z" transform="translate(444,0)" style="stroke-width:3;"></path><path data-c="73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z" transform="translate(944,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1338,0)"><path data-c="2061" d="" style="stroke-width:3;"></path></g><g data-mml-node="mrow" transform="translate(1504.7,0)"><g data-mml-node="mo"><path data-c="5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z" style="stroke-width:3;"></path></g><g data-mml-node="mrow" transform="translate(278,0)"><g data-mml-node="mo"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ω</mi><mo>+</mo><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mtext>lc </mtext></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>t</mi><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container> does not appear any more. Discarding <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.339ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.339ex" role="img" focusable="false" viewBox="0 -442 1008.6 592" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub></math></mjx-assistive-mml></mjx-container>, we can therefore use <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.039ex" height="1.731ex" role="img" focusable="false" viewBox="0 -683 3111.4 765" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(722.2,0)"><path data-c="2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(1722.4,0)"><path data-c="34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(2222.4,0)"><path data-c="1D43E" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>+</mo><mn>4</mn><mi>K</mi></math></mjx-assistive-mml></mjx-container> variables as our harmonic ansatz, i.e.,</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-9.106ex;" xmlns="http://www.w3.org/2000/svg" width="14.53ex" height="19.344ex" role="img" focusable="false" viewBox="0 -4525 6422.4 8550" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D414" d="M570 686Q588 683 703 683T836 686H845V624H737V420Q737 390 737 345T738 284Q738 205 729 164T689 83Q614 -11 465 -11Q321 -11 240 51T148 207Q147 214 147 421V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V405V370V268Q304 181 311 146T346 87Q387 52 466 52Q642 52 667 195Q668 204 669 415V624H561V686H570Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1162.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="mrow" transform="translate(2218.6,0)"><g data-mml-node="mo"><path data-c="239B" d="M837 1154Q843 1148 843 1145Q843 1141 818 1106T753 1002T667 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610Q548 610 558 610T573 610T578 609Q582 609 582 592T583 473V127Q583 -9 579 -9Z" transform="scale(1,12.416)" style="stroke-width:3;"></path></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">U</mi></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>u</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>v</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>v</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><mrow data-mjx-texclass="ORD"><mo>⋮</mo></mrow></mtd></mtr><mtr><mtd><msub><mi>v</mi><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>K</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">lc</mi></mrow></mrow></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">)</mo></mrow></math></mjx-assistive-mml></mjx-container><p>to remove the infinite degeneracy. Note that <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.348ex" height="2.034ex" role="img" focusable="false" viewBox="0 -694 596 899" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D719" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math></mjx-assistive-mml></mjx-container> is only defined modulo <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.29ex" height="1ex" role="img" focusable="false" viewBox="0 -431 570 442" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D70B" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></mjx-assistive-mml></mjx-container>, but its effect on the harmonic variables is not. 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style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi><mo>=</mo><mo>−</mo><mi>arctan</mi><mo data-mjx-texclass="NONE"></mo><msub><mi>u</mi><mn>2</mn></msub><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><msub><mi>v</mi><mn>2</mn></msub><mo>+</mo><mi>π</mi></math></mjx-assistive-mml></mjx-container> would invert the signs of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.439ex;" xmlns="http://www.w3.org/2000/svg" width="8.464ex" height="1.441ex" role="img" focusable="false" viewBox="0 -443 3741 637" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(518,-150) scale(0.707)"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(921.6,0)"><path data-c="2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(1366.2,0)"><g data-mml-node="mi"><path data-c="1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><path data-c="33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 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style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(518,-150) scale(0.707)"><path data-c="33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 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As a result, each solution is doubly degenerate. Combined with the sign ambiguity of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.357ex;" xmlns="http://www.w3.org/2000/svg" width="3.15ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 1392.3 600.8" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mtext"><path data-c="6C" d="M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z" style="stroke-width:3;"></path><path data-c="63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z" transform="translate(278,0)" style="stroke-width:3;"></path><path data-c="A0" d="" transform="translate(722,0)" style="stroke-width:3;"></path></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mtext>lc </mtext></mrow></msub></math></mjx-assistive-mml></mjx-container>, we conclude that under the new ansatz, a limit cycle solution appears as a fourfold-degenerate steady state.</p><p>The harmonic equations can now be solved using homotopy continuation to obtain all steady states. Compared to the single-harmonic ansatz however, we have significantly enlarged the polynomial system to be solved. As the number of solutions scales exponentially (<a href="https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem" target="_blank" rel="noreferrer">Bézout bound</a>), we expect vast numbers of solutions even for fairly small systems.</p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/background/limit_cycles.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/background/stability_response" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Stability and Linear Response</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/tutorials/steady_states" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Steady States</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. 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tabindex="-1" data-v-18756405><span class="visually-hidden" id="sidebar-aria-label" data-v-18756405> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0" data-v-9e426adc data-v-a4b0d9bf><!----><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/index" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Home</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/index" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Getting Started</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible has-active" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Theoretical Background</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/harmonic_balance" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Floquet expansions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/stability_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Stability and Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Tutorials</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/steady_states" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/classification" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Classifying Solutions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Transient Dynamics</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Examples</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Three Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Oscillator</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>State Dependent Perturbation</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><div 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href="/HarmonicBalance.jl/dev/introduction/citation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Bibliography</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Manual</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Entering Equations of Motion</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 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data-v-a4b0d9bf>Krylov-Bogoliubov</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Time Evolution</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Plotting</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/saving" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Saving and Loading</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-a9a9e638 data-v-91765379><div class="VPDoc has-sidebar has-aside" data-v-91765379 data-v-83890dd9><!--[--><!--]--><div class="container" data-v-83890dd9><div class="aside" data-v-83890dd9><div class="aside-curtain" data-v-83890dd9></div><div class="aside-container" data-v-83890dd9><div class="aside-content" data-v-83890dd9><div class="VPDocAside" data-v-83890dd9 data-v-6d7b3c46><!--[--><!--]--><!--[--><!--]--><nav aria-labelledby="doc-outline-aria-label" class="VPDocAsideOutline" data-v-6d7b3c46 data-v-b38bf2ff><div class="content" data-v-b38bf2ff><div class="outline-marker" data-v-b38bf2ff></div><div aria-level="2" class="outline-title" id="doc-outline-aria-label" role="heading" data-v-b38bf2ff>On this page</div><ul class="VPDocOutlineItem root" data-v-b38bf2ff data-v-3f927ebe><!--[--><!--]--></ul></div></nav><!--[--><!--]--><div class="spacer" data-v-6d7b3c46></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--]--></div></div></div></div><div class="content" data-v-83890dd9><div class="content-container" data-v-83890dd9><!--[--><!--]--><main class="main" data-v-83890dd9><div style="position:relative;" class="vp-doc _HarmonicBalance_jl_dev_background_stability_response" data-v-83890dd9><div><h1 id="linresp_background" tabindex="-1">Stability and linear response <a class="header-anchor" href="#linresp_background" aria-label="Permalink to "Stability and linear response {#linresp_background}""></a></h1><p>The core of the harmonic balance method is expressing the system's behaviour in terms of Fourier components or <em>harmonics</em>. For an <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="2.009ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 888 683" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 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data-mml-node="mo" transform="translate(20723.7,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g><g data-mml-node="mstyle" transform="translate(21112.7,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mo" transform="translate(21279.7,0)"><path data-c="2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>u</mi><mrow data-mjx-texclass="ORD"><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>v</mi><mrow data-mjx-texclass="ORD"><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>,</mo><mo>…</mo><msub><mi>u</mi><mrow data-mjx-texclass="ORD"><mi>N</mi><mo>,</mo><msub><mi>M</mi><mi>N</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>v</mi><mrow data-mjx-texclass="ORD"><mi>N</mi><mo>,</mo><msub><mi>M</mi><mi>N</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mo>,</mo></math></mjx-assistive-mml></mjx-container><p>we may obtain the <em>harmonic equations</em> (see <a href="/HarmonicBalance.jl/dev/background/harmonic_balance#Duffing_harmeq">an example of this procedure</a>)</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-1.575ex;" xmlns="http://www.w3.org/2000/svg" width="14.832ex" height="4.878ex" role="img" focusable="false" viewBox="0 -1460 6555.6 2156" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g 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45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g></g><rect width="2841" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(3358.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(4414.6,0)"><g data-mml-node="mover"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D405" d="M425 0L228 3Q63 3 51 0H39V62H147V618H39V680H644V676Q647 670 659 552T675 428V424H613Q613 433 605 477Q599 511 589 535T562 574T530 599T488 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363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(6166.6,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mrow><mi>d</mi><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mi>T</mi></mrow></mfrac><mo>=</mo><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><p>where <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.844ex" height="2.672ex" role="img" focusable="false" viewBox="0 -931 2141 1181" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D405" d="M425 0L228 3Q63 3 51 0H39V62H147V618H39V680H644V676Q647 670 659 552T675 428V424H613Q613 433 605 477Q599 511 589 535T562 574T530 599T488 612T441 617T387 618H368H304V371H333Q389 373 411 390T437 468V488H499V192H437V212Q436 244 430 263T408 292T378 305T333 309H304V62H439V0H425Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(362,241) translate(-250 0)"><path data-c="AF" d="M69 544V590H430V544H69Z" style="stroke-width:3;"></path></g></g></g><g data-mml-node="mo" transform="translate(724,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(1113,0)"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1752,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> is a nonlinear function. A steady state <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.433ex" height="1.393ex" role="img" focusable="false" viewBox="0 -450 1075.6 615.6" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub></math></mjx-assistive-mml></mjx-container> is defined by <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="9.98ex" height="2.672ex" role="img" focusable="false" viewBox="0 -931 4411.1 1181" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D405" d="M425 0L228 3Q63 3 51 0H39V62H147V618H39V680H644V676Q647 670 659 552T675 428V424H613Q613 433 605 477Q599 511 589 535T562 574T530 599T488 612T441 617T387 618H368H304V371H333Q389 373 411 390T437 468V488H499V192H437V212Q436 244 430 263T408 292T378 305T333 309H304V62H439V0H425Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(362,241) translate(-250 0)"><path data-c="AF" d="M69 544V590H430V544H69Z" style="stroke-width:3;"></path></g></g></g><g data-mml-node="mo" transform="translate(724,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(1113,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(2188.6,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2855.3,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(3911.1,0)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container>.</p><h3 id="stability" tabindex="-1">Stability <a class="header-anchor" href="#stability" aria-label="Permalink to "Stability""></a></h3><p>Let us assume that we found a steady state <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.433ex" height="1.393ex" role="img" focusable="false" viewBox="0 -450 1075.6 615.6" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub></math></mjx-assistive-mml></mjx-container>. When the system is in this state, it responds to small perturbations either by returning to <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.433ex" height="1.393ex" role="img" focusable="false" viewBox="0 -450 1075.6 615.6" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub></math></mjx-assistive-mml></mjx-container> over some characteristic timescale (<em>stable state</em>) or by evolving away from <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.433ex" height="1.393ex" role="img" focusable="false" viewBox="0 -450 1075.6 615.6" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub></math></mjx-assistive-mml></mjx-container> (<em>unstable state</em>). To analyze the stability of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.433ex" height="1.393ex" role="img" focusable="false" viewBox="0 -450 1075.6 615.6" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub></math></mjx-assistive-mml></mjx-container>, we linearize the equations of motion around <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.433ex" height="1.393ex" role="img" focusable="false" viewBox="0 -450 1075.6 615.6" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub></math></mjx-assistive-mml></mjx-container> for a small perturbation <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.375ex;" xmlns="http://www.w3.org/2000/svg" width="12.112ex" height="1.997ex" role="img" focusable="false" viewBox="0 -717 5353.6 882.6" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(444,0)"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1360.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(2416.6,0)"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(3277.8,0)"><path data-c="2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(4278,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo>−</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub></math></mjx-assistive-mml></mjx-container> to obtain</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-1.575ex;" xmlns="http://www.w3.org/2000/svg" width="26.279ex" height="4.674ex" role="img" focusable="false" viewBox="0 -1370 11615.1 2066" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mfrac"><g data-mml-node="mi" transform="translate(572,676)"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g><g data-mml-node="mrow" transform="translate(220,-686)"><g data-mml-node="mi"><path data-c="1D451" d="M366 683Q367 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621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g></g><rect width="1424" height="60" x="120" y="220"></rect></g><g data-mml-node="mrow" transform="translate(1664,0)"><g data-mml-node="mo"><path data-c="5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(278,0)"><path data-c="1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 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style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>T</mi></mrow></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mi>δ</mi><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mi>J</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub><mo stretchy="false">)</mo><mi>δ</mi><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mo>,</mo></math></mjx-assistive-mml></mjx-container><p>where <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.604ex;" xmlns="http://www.w3.org/2000/svg" width="18.18ex" height="2.711ex" role="img" focusable="false" viewBox="0 -931 8035.5 1198.1" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D43D" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(633,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(1022,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(2097.6,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2764.3,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(3820.1,0)"><g data-mml-node="mi"><path data-c="2207" d="M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(866,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(5188,0)"><g data-mml-node="mover"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D405" d="M425 0L228 3Q63 3 51 0H39V62H147V618H39V680H644V676Q647 670 659 552T675 428V424H613Q613 433 605 477Q599 511 589 535T562 574T530 599T488 612T441 617T387 618H368H304V371H333Q389 373 411 390T437 468V488H499V192H437V212Q436 244 430 263T408 292T378 305T333 309H304V62H439V0H425Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(362,241) translate(-250 0)"><path data-c="AF" d="M69 544V590H430V544H69Z" style="stroke-width:3;"></path></g></g></g><g data-mml-node="msub" transform="translate(5912,0)"><g data-mml-node="mo" transform="translate(0 -0.5)"><path data-c="7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(311,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(639,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(1417,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow></mrow></msub><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">F</mi></mrow><mo stretchy="false">¯</mo></mover></mrow><msub><mo data-mjx-texclass="ORD" stretchy="false">|</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo>=</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub></mrow></msub></math></mjx-assistive-mml></mjx-container> is the <em>Jacobian matrix</em> of the system evaluated at <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.375ex;" xmlns="http://www.w3.org/2000/svg" width="6.896ex" height="1.694ex" role="img" focusable="false" viewBox="0 -583 3048.1 748.6" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(916.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(1972.6,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo>=</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub></math></mjx-assistive-mml></mjx-container>.</p><p>The linearised system is exactly solvable for <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.803ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2565 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(444,0)"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1083,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1472,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2176,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> given an initial condition <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="6.519ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2881.6 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(444,0)"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1083,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(1472,0)"><g data-mml-node="mi"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(617,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(2492.6,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">(</mo><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>. The solution can be expanded in terms of the complex eigenvalues <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.357ex;" xmlns="http://www.w3.org/2000/svg" width="2.228ex" height="1.927ex" role="img" focusable="false" viewBox="0 -694 984.9 851.8" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><path data-c="1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mi>r</mi></msub></math></mjx-assistive-mml></mjx-container> and eigenvectors <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.357ex;" xmlns="http://www.w3.org/2000/svg" width="2.283ex" height="1.361ex" role="img" focusable="false" viewBox="0 -444 1008.9 601.8" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42F" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mi" transform="translate(640,-150) scale(0.707)"><path data-c="1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">v</mi></mrow><mi>r</mi></msub></math></mjx-assistive-mml></mjx-container> of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.626ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2486.6 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D43D" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(633,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(1022,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(2097.6,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>, namely</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-2.619ex;" xmlns="http://www.w3.org/2000/svg" width="21.681ex" height="4.769ex" role="img" focusable="false" viewBox="0 -950 9583.1 2107.8" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(444,0)"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1083,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1472,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2176,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2842.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="munder" transform="translate(3898.6,0)"><g data-mml-node="mo"><path data-c="2211" d="M60 948Q63 950 665 950H1267L1325 815Q1384 677 1388 669H1348L1341 683Q1320 724 1285 761Q1235 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316T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g></g><g data-mml-node="msub" transform="translate(5509.2,0)"><g data-mml-node="mi"><path data-c="1D450" d="M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(466,-150) scale(0.707)"><path data-c="1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mspace" transform="translate(6344.1,0)"></g><g data-mml-node="msub" transform="translate(6552.9,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42F" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mi" transform="translate(640,-150) scale(0.707)"><path data-c="1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g><g data-mml-node="msup" transform="translate(7561.9,0)"><g data-mml-node="mi"><path data-c="1D452" d="M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(499,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><path data-c="1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mi" transform="translate(984.9,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g></g></g><g data-mml-node="mo" transform="translate(9305.1,0)"><path data-c="2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>δ</mi><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></munder><msub><mi>c</mi><mi>r</mi></msub><mspace width="1mm"></mspace><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">v</mi></mrow><mi>r</mi></msub><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><msub><mi>λ</mi><mi>r</mi></msub><mi>T</mi></mrow></msup><mo>.</mo></math></mjx-assistive-mml></mjx-container><p>The dynamical behaviour near the steady states is thus governed by <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.025ex;" xmlns="http://www.w3.org/2000/svg" width="3.944ex" height="1.956ex" role="img" focusable="false" viewBox="0 -853.7 1743.2 864.7" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><path data-c="1D452" d="M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(499,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><path data-c="1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mi" transform="translate(984.9,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><msub><mi>λ</mi><mi>r</mi></msub><mi>T</mi></mrow></msup></math></mjx-assistive-mml></mjx-container>: if <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="10.806ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4776.5 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="52" d="M130 622Q123 629 119 631T103 634T60 637H27V683H202H236H300Q376 683 417 677T500 648Q595 600 609 517Q610 512 610 501Q610 468 594 439T556 392T511 361T472 343L456 338Q459 335 467 332Q497 316 516 298T545 254T559 211T568 155T578 94Q588 46 602 31T640 16H645Q660 16 674 32T692 87Q692 98 696 101T712 105T728 103T732 90Q732 59 716 27T672 -16Q656 -22 630 -22Q481 -16 458 90Q456 101 456 163T449 246Q430 304 373 320L363 322L297 323H231V192L232 61Q238 51 249 49T301 46H334V0H323Q302 3 181 3Q59 3 38 0H27V46H60Q102 47 111 49T130 61V622ZM491 499V509Q491 527 490 539T481 570T462 601T424 623T362 636Q360 636 340 636T304 637H283Q238 637 234 628Q231 624 231 492V360H289Q390 360 434 378T489 456Q491 467 491 499Z" style="stroke-width:3;"></path><path data-c="65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z" transform="translate(736,0)" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1180,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(1569,0)"><g data-mml-node="mi"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><path data-c="1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(2553.9,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(3220.7,0)"><path data-c="3C" d="M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(4276.5,0)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">Re</mi></mrow><mo stretchy="false">(</mo><msub><mi>λ</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container> for all <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.357ex;" xmlns="http://www.w3.org/2000/svg" width="2.228ex" height="1.927ex" role="img" focusable="false" viewBox="0 -694 984.9 851.8" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><path data-c="1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mi>r</mi></msub></math></mjx-assistive-mml></mjx-container>, the state <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.433ex" height="1.393ex" role="img" focusable="false" viewBox="0 -450 1075.6 615.6" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub></math></mjx-assistive-mml></mjx-container> is stable. Conversely, if <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="10.806ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4776.5 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="52" d="M130 622Q123 629 119 631T103 634T60 637H27V683H202H236H300Q376 683 417 677T500 648Q595 600 609 517Q610 512 610 501Q610 468 594 439T556 392T511 361T472 343L456 338Q459 335 467 332Q497 316 516 298T545 254T559 211T568 155T578 94Q588 46 602 31T640 16H645Q660 16 674 32T692 87Q692 98 696 101T712 105T728 103T732 90Q732 59 716 27T672 -16Q656 -22 630 -22Q481 -16 458 90Q456 101 456 163T449 246Q430 304 373 320L363 322L297 323H231V192L232 61Q238 51 249 49T301 46H334V0H323Q302 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189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(2553.9,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(3220.7,0)"><path data-c="3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(4276.5,0)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mi data-mjx-auto-op="false">Re</mi></mrow><mo stretchy="false">(</mo><msub><mi>λ</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></math></mjx-assistive-mml></mjx-container> for at least one <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.357ex;" xmlns="http://www.w3.org/2000/svg" width="2.228ex" height="1.927ex" role="img" focusable="false" viewBox="0 -694 984.9 851.8" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><path data-c="1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mi>r</mi></msub></math></mjx-assistive-mml></mjx-container>, the state is unstable - perturbations such as noise or a small applied drive will force the system away from <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.433ex" height="1.393ex" role="img" focusable="false" viewBox="0 -450 1075.6 615.6" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub></math></mjx-assistive-mml></mjx-container>.</p><h3 id="Linear-response" tabindex="-1">Linear response <a class="header-anchor" href="#Linear-response" aria-label="Permalink to "Linear response {#Linear-response}""></a></h3><p>The response of a stable steady state to an additional oscillatory force, caused by weak probes or noise, is often of interest. It can be calculated by solving for the perturbation <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.803ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2565 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(444,0)"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1083,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1472,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2176,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> in the presence of an additional drive term.</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-1.575ex;" xmlns="http://www.w3.org/2000/svg" width="34.647ex" height="4.674ex" role="img" focusable="false" 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196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(345,0)"><path data-c="3A9" d="M55 454Q55 503 75 546T127 617T197 665T272 695T337 704H352Q396 704 404 703Q527 687 596 615T666 454Q666 392 635 330T559 200T499 83V80H543Q589 81 600 83T617 93Q622 102 629 135T636 172L637 177H677V175L660 89Q645 3 644 2V0H552H488Q461 0 456 3T451 20Q451 89 499 235T548 455Q548 512 530 555T483 622T424 656T361 668Q332 668 303 658T243 626T193 560T174 456Q174 380 222 233T270 20Q270 7 263 0H77V2Q76 3 61 89L44 175V177H84L85 172Q85 171 88 155T96 119T104 93Q109 86 120 84T178 80H222V83Q206 132 162 199T87 329T55 454Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1067,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g></g></g><g data-mml-node="mstyle" transform="translate(14868.8,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mo" transform="translate(15035.8,0)"><path data-c="2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>T</mi></mrow></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mi>δ</mi><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mi>J</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub><mo stretchy="false">)</mo><mi>δ</mi><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>+</mo><mi mathvariant="bold-italic">ξ</mi><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi mathvariant="normal">Ω</mi><mi>T</mi></mrow></msup><mstyle scriptlevel="0"><mspace width="0.167em"></mspace></mstyle><mo>,</mo></math></mjx-assistive-mml></mjx-container><p>Suppose we have found an eigenvector of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.626ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2486.6 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D43D" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(633,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(1022,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(2097.6,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> such that <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="11.721ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 5180.6 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D43D" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(633,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(1022,0)"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1661,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(2050,0)"><g data-mml-node="mi"><path data-c="1D42F" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(2934.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(3990.6,0)"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(4573.6,0)"><g data-mml-node="mi"><path data-c="1D42F" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">v</mi></mrow><mo>=</mo><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">v</mi></mrow></math></mjx-assistive-mml></mjx-container>. To solve the linearised equations of motion, we insert <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="19.737ex" height="2.513ex" role="img" focusable="false" viewBox="0 -860.8 8723.8 1110.8" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(444,0)"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1083,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1472,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2176,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2842.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(3898.6,0)"><path data-c="1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(4648.6,0)"><path 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0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>A</mi><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>i</mi><mi mathvariant="normal">Ω</mi><mo>−</mo><mi>λ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mi mathvariant="bold-italic">ξ</mi><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">v</mi></mrow><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><mo stretchy="false">⟹</mo><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mi>A</mi><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi mathvariant="bold-italic">ξ</mi><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">v</mi></mrow></mrow><mrow><mo>−</mo><mtext>Re</mtext><mo stretchy="false">[</mo><mi>λ</mi><mo stretchy="false">]</mo><mo>+</mo><mi>i</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi mathvariant="normal">Ω</mi><mo>−</mo><mtext>Im</mtext><mo stretchy="false">[</mo><mi>λ</mi><mo stretchy="false">]</mo><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mfrac></math></mjx-assistive-mml></mjx-container><p>We see that each eigenvalue <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.027ex;" xmlns="http://www.w3.org/2000/svg" width="1.319ex" height="1.597ex" role="img" focusable="false" viewBox="0 -694 583 706" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></mjx-assistive-mml></mjx-container> results in a linear response that is a Lorentzian centered at <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="9.929ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4388.6 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="3A9" d="M55 454Q55 503 75 546T127 617T197 665T272 695T337 704H352Q396 704 404 703Q527 687 596 615T666 454Q666 392 635 330T559 200T499 83V80H543Q589 81 600 83T617 93Q622 102 629 135T636 172L637 177H677V175L660 89Q645 3 644 2V0H552H488Q461 0 456 3T451 20Q451 89 499 235T548 455Q548 512 530 555T483 622T424 656T361 668Q332 668 303 658T243 626T193 560T174 456Q174 380 222 233T270 20Q270 7 263 0H77V2Q76 3 61 89L44 175V177H84L85 172Q85 171 88 155T96 119T104 93Q109 86 120 84T178 80H222V83Q206 132 162 199T87 329T55 454Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(999.8,0)"><path data-c="3D" d="M56 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432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z" transform="translate(361,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(3249.6,0)"><path data-c="5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(3527.6,0)"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(4110.6,0)"><path data-c="5D" d="M22 710V750H159V-250H22V-210H119V710H22Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Ω</mi><mo>=</mo><mtext>Im</mtext><mo stretchy="false">[</mo><mi>λ</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>. Effectively, the linear response matches that of a harmonic oscillator with resonance frequency <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.278ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2333 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><path data-c="49" d="M328 0Q307 3 180 3T32 0H21V46H43Q92 46 106 49T126 60Q128 63 128 342Q128 620 126 623Q122 628 118 630T96 635T43 637H21V683H32Q53 680 180 680T328 683H339V637H317Q268 637 254 634T234 623Q232 620 232 342Q232 63 234 60Q238 55 242 53T264 48T317 46H339V0H328Z" style="stroke-width:3;"></path><path data-c="6D" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 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transform="translate(1194,0)"><path data-c="5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1472,0)"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2055,0)"><path data-c="5D" d="M22 710V750H159V-250H22V-210H119V710H22Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Im</mtext><mo stretchy="false">[</mo><mi>λ</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container> and damping <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.247ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2319 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><path data-c="52" d="M130 622Q123 629 119 631T103 634T60 637H27V683H202H236H300Q376 683 417 677T500 648Q595 600 609 517Q610 512 610 501Q610 468 594 439T556 392T511 361T472 343L456 338Q459 335 467 332Q497 316 516 298T545 254T559 211T568 155T578 94Q588 46 602 31T640 16H645Q660 16 674 32T692 87Q692 98 696 101T712 105T728 103T732 90Q732 59 716 27T672 -16Q656 -22 630 -22Q481 -16 458 90Q456 101 456 163T449 246Q430 304 373 320L363 322L297 323H231V192L232 61Q238 51 249 49T301 46H334V0H323Q302 3 181 3Q59 3 38 0H27V46H60Q102 47 111 49T130 61V622ZM491 499V509Q491 527 490 539T481 570T462 601T424 623T362 636Q360 636 340 636T304 637H283Q238 637 234 628Q231 624 231 492V360H289Q390 360 434 378T489 456Q491 467 491 499Z" style="stroke-width:3;"></path><path data-c="65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z" transform="translate(736,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1180,0)"><path data-c="5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1458,0)"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2041,0)"><path data-c="5D" d="M22 710V750H159V-250H22V-210H119V710H22Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mo stretchy="false">[</mo><mi>λ</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>.</p><p>Knowing the response of the harmonic variables <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.799ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2121 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(639,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1028,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1732,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>, what is the corresponding behaviour of the "natural" variables <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.611ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2038 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(605,-150) scale(0.707)"><path data-c="1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(899,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1288,0)"><path data-c="1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1649,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>? To find this out, we insert the perturbation back into the harmonic ansatz. Since we require real variables, let us use <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.791ex;" xmlns="http://www.w3.org/2000/svg" width="33.011ex" height="2.738ex" role="img" focusable="false" viewBox="0 -860.8 14590.9 1210.3" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(444,0)"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1083,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1472,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2176,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2842.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(3898.6,0)"><path data-c="1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(4648.6,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(5037.6,0)"><path data-c="3A9" d="M55 454Q55 503 75 546T127 617T197 665T272 695T337 704H352Q396 704 404 703Q527 687 596 615T666 454Q666 392 635 330T559 200T499 83V80H543Q589 81 600 83T617 93Q622 102 629 135T636 172L637 177H677V175L660 89Q645 3 644 2V0H552H488Q461 0 456 3T451 20Q451 89 499 235T548 455Q548 512 530 555T483 622T424 656T361 668Q332 668 303 658T243 626T193 560T174 456Q174 380 222 233T270 20Q270 7 263 0H77V2Q76 3 61 89L44 175V177H84L85 172Q85 171 88 155T96 119T104 93Q109 86 120 84T178 80H222V83Q206 132 162 199T87 329T55 454Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(5759.6,0)"><path data-c="29" d="M60 749L64 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30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(345,0)"><path data-c="3A9" d="M55 454Q55 503 75 546T127 617T197 665T272 695T337 704H352Q396 704 404 703Q527 687 596 615T666 454Q666 392 635 330T559 200T499 83V80H543Q589 81 600 83T617 93Q622 102 629 135T636 172L637 177H677V175L660 89Q645 3 644 2V0H552H488Q461 0 456 3T451 20Q451 89 499 235T548 455Q548 512 530 555T483 622T424 656T361 668Q332 668 303 658T243 626T193 560T174 456Q174 380 222 233T270 20Q270 7 263 0H77V2Q76 3 61 89L44 175V177H84L85 172Q85 171 88 155T96 119T104 93Q109 86 120 84T178 80H222V83Q206 132 162 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role="img" focusable="false" viewBox="0 -717 1083 727" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(444,0)"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow></math></mjx-assistive-mml></mjx-container> corresponding to the respective harmonics <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.666ex;" xmlns="http://www.w3.org/2000/svg" width="3.251ex" height="1.668ex" role="img" focusable="false" viewBox="0 -443 1436.9 737.2" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(345,0)"><path data-c="2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(623,0)"><path data-c="1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z" style="stroke-width:3;"></path></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container>.</p><p>We see that a motion of the harmonic variables at frequency <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="1.633ex" height="1.593ex" role="img" focusable="false" viewBox="0 -704 722 704" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="3A9" d="M55 454Q55 503 75 546T127 617T197 665T272 695T337 704H352Q396 704 404 703Q527 687 596 615T666 454Q666 392 635 330T559 200T499 83V80H543Q589 81 600 83T617 93Q622 102 629 135T636 172L637 177H677V175L660 89Q645 3 644 2V0H552H488Q461 0 456 3T451 20Q451 89 499 235T548 455Q548 512 530 555T483 622T424 656T361 668Q332 668 303 658T243 626T193 560T174 456Q174 380 222 233T270 20Q270 7 263 0H77V2Q76 3 61 89L44 175V177H84L85 172Q85 171 88 155T96 119T104 93Q109 86 120 84T178 80H222V83Q206 132 162 199T87 329T55 454Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Ω</mi></math></mjx-assistive-mml></mjx-container> appears as motion of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.615ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2482 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(444,0)"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(605,-150) scale(0.707)"><path data-c="1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1343,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1732,0)"><path data-c="1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2093,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> at frequencies <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.666ex;" xmlns="http://www.w3.org/2000/svg" width="7.65ex" height="2.258ex" role="img" focusable="false" viewBox="0 -704 3381.3 998.2" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(345,0)"><path data-c="2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(623,0)"><path data-c="1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z" style="stroke-width:3;"></path></g></g></g><g data-mml-node="mo" transform="translate(1659.1,0)"><path data-c="B1" d="M56 320T56 333T70 353H369V502Q369 651 371 655Q376 666 388 666Q402 666 405 654T409 596V500V353H707Q722 345 722 333Q722 320 707 313H409V40H707Q722 32 722 20T707 0H70Q56 7 56 20T70 40H369V313H70Q56 320 56 333Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(2659.3,0)"><path data-c="3A9" d="M55 454Q55 503 75 546T127 617T197 665T272 695T337 704H352Q396 704 404 703Q527 687 596 615T666 454Q666 392 635 330T559 200T499 83V80H543Q589 81 600 83T617 93Q622 102 629 135T636 172L637 177H677V175L660 89Q645 3 644 2V0H552H488Q461 0 456 3T451 20Q451 89 499 235T548 455Q548 512 530 555T483 622T424 656T361 668Q332 668 303 658T243 626T193 560T174 456Q174 380 222 233T270 20Q270 7 263 0H77V2Q76 3 61 89L44 175V177H84L85 172Q85 171 88 155T96 119T104 93Q109 86 120 84T178 80H222V83Q206 132 162 199T87 329T55 454Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>±</mo><mi mathvariant="normal">Ω</mi></math></mjx-assistive-mml></mjx-container>.</p><p>To make sense of this, we normalize the vector <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.023ex;" xmlns="http://www.w3.org/2000/svg" width="2.45ex" height="1.645ex" role="img" focusable="false" viewBox="0 -717 1083 727" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(444,0)"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow></math></mjx-assistive-mml></mjx-container> and use normalised components <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.666ex;" xmlns="http://www.w3.org/2000/svg" width="4.142ex" height="2.498ex" role="img" focusable="false" viewBox="0 -810 1830.9 1104.2" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(444,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 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348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z" style="stroke-width:3;"></path></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><msub><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">^</mo></mover></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container> and 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0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><msub><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">^</mo></mover></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container>. We also define the Lorentzian distribution</p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-2.62ex;" xmlns="http://www.w3.org/2000/svg" width="25.91ex" height="5.656ex" role="img" focusable="false" viewBox="0 -1342 11452 2500" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D43F" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 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0px;border:0px;display:block;overflow:hidden;width:100%;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><msub><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><msub><mi>x</mi><mn>0</mn></msub><mo>,</mo><mi>γ</mi></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>x</mi><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup></mrow></mfrac></math></mjx-assistive-mml></mjx-container><p>We see that all components of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.615ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2482 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(444,0)"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(605,-150) scale(0.707)"><path data-c="1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(1343,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1732,0)"><path data-c="1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2093,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> are proportional to <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.8ex;" xmlns="http://www.w3.org/2000/svg" width="13.009ex" height="2.497ex" role="img" focusable="false" viewBox="0 -750 5750 1103.5" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D43F" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(681,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1070,0)"><path data-c="3A9" d="M55 454Q55 503 75 546T127 617T197 665T272 695T337 704H352Q396 704 404 703Q527 687 596 615T666 454Q666 392 635 330T559 200T499 83V80H543Q589 81 600 83T617 93Q622 102 629 135T636 172L637 177H677V175L660 89Q645 3 644 2V0H552H488Q461 0 456 3T451 20Q451 89 499 235T548 455Q548 512 530 555T483 622T424 656T361 668Q332 668 303 658T243 626T193 560T174 456Q174 380 222 233T270 20Q270 7 263 0H77V2Q76 3 61 89L44 175V177H84L85 172Q85 171 88 155T96 119T104 93Q109 86 120 84T178 80H222V83Q206 132 162 199T87 329T55 454Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(1792,0)"><g data-mml-node="mo"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(422,-176.7) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mtext"><path data-c="49" d="M328 0Q307 3 180 3T32 0H21V46H43Q92 46 106 49T126 60Q128 63 128 342Q128 620 126 623Q122 628 118 630T96 635T43 637H21V683H32Q53 680 180 680T328 683H339V637H317Q268 637 254 634T234 623Q232 620 232 342Q232 63 234 60Q238 55 242 53T264 48T317 46H339V0H328Z" style="stroke-width:3;"></path><path data-c="6D" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z" transform="translate(361,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1194,0)"><path data-c="5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1472,0)"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2055,0)"><path data-c="5D" d="M22 710V750H159V-250H22V-210H119V710H22Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2333,0)"><path data-c="2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z" style="stroke-width:3;"></path></g><g data-mml-node="mtext" transform="translate(2611,0)"><path data-c="52" d="M130 622Q123 629 119 631T103 634T60 637H27V683H202H236H300Q376 683 417 677T500 648Q595 600 609 517Q610 512 610 501Q610 468 594 439T556 392T511 361T472 343L456 338Q459 335 467 332Q497 316 516 298T545 254T559 211T568 155T578 94Q588 46 602 31T640 16H645Q660 16 674 32T692 87Q692 98 696 101T712 105T728 103T732 90Q732 59 716 27T672 -16Q656 -22 630 -22Q481 -16 458 90Q456 101 456 163T449 246Q430 304 373 320L363 322L297 323H231V192L232 61Q238 51 249 49T301 46H334V0H323Q302 3 181 3Q59 3 38 0H27V46H60Q102 47 111 49T130 61V622ZM491 499V509Q491 527 490 539T481 570T462 601T424 623T362 636Q360 636 340 636T304 637H283Q238 637 234 628Q231 624 231 492V360H289Q390 360 434 378T489 456Q491 467 491 499Z" style="stroke-width:3;"></path><path data-c="65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z" transform="translate(736,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(3791,0)"><path data-c="5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(4069,0)"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(4652,0)"><path data-c="5D" d="M22 710V750H159V-250H22V-210H119V710H22Z" style="stroke-width:3;"></path></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><msub><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mtext>Im</mtext><mo stretchy="false">[</mo><mi>λ</mi><mo stretchy="false">]</mo><mo>,</mo><mtext>Re</mtext><mo stretchy="false">[</mo><mi>λ</mi><mo stretchy="false">]</mo></mrow></msub></math></mjx-assistive-mml></mjx-container>. The first and last two summands are Lorentzians centered at <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="3.394ex" height="1.593ex" role="img" focusable="false" viewBox="0 -704 1500 704" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><path data-c="B1" d="M56 320T56 333T70 353H369V502Q369 651 371 655Q376 666 388 666Q402 666 405 654T409 596V500V353H707Q722 345 722 333Q722 320 707 313H409V40H707Q722 32 722 20T707 0H70Q56 7 56 20T70 40H369V313H70Q56 320 56 333Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(778,0)"><path data-c="3A9" d="M55 454Q55 503 75 546T127 617T197 665T272 695T337 704H352Q396 704 404 703Q527 687 596 615T666 454Q666 392 635 330T559 200T499 83V80H543Q589 81 600 83T617 93Q622 102 629 135T636 172L637 177H677V175L660 89Q645 3 644 2V0H552H488Q461 0 456 3T451 20Q451 89 499 235T548 455Q548 512 530 555T483 622T424 656T361 668Q332 668 303 658T243 626T193 560T174 456Q174 380 222 233T270 20Q270 7 263 0H77V2Q76 3 61 89L44 175V177H84L85 172Q85 171 88 155T96 119T104 93Q109 86 120 84T178 80H222V83Q206 132 162 199T87 329T55 454Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo><mi mathvariant="normal">Ω</mi></math></mjx-assistive-mml></mjx-container> which oscillate at <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.666ex;" xmlns="http://www.w3.org/2000/svg" width="7.65ex" height="2.258ex" role="img" focusable="false" viewBox="0 -704 3381.3 998.2" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" 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xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>±</mo><mi mathvariant="normal">Ω</mi></math></mjx-assistive-mml></mjx-container>, respectively. From this, we can extract the linear response function in Fourier space, <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.584ex" height="2.265ex" role="img" focusable="false" viewBox="0 -751 2026 1001" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D712" d="M576 -125Q576 -147 547 -175T487 -204H476Q394 -204 363 -157Q334 -114 293 26L284 59Q283 58 248 19T170 -66T92 -151T53 -191Q49 -194 43 -194Q36 -194 31 -189T25 -177T38 -154T151 -30L272 102L265 131Q189 405 135 405Q104 405 87 358Q86 351 68 351Q48 351 48 361Q48 369 56 386T89 423T148 442Q224 442 258 400Q276 375 297 320T330 222L341 180Q344 180 455 303T573 429Q579 431 582 431Q600 431 600 414Q600 407 587 392T477 270Q356 138 353 134L362 102Q392 -10 428 -89T490 -168Q504 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1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>χ</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">~</mo></mover></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></p><mjx-container class="MathJax" jax="SVG" display="true" style="direction:ltr;display:block;text-align:center;margin:1em 0;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-6.207ex;" xmlns="http://www.w3.org/2000/svg" width="91.687ex" height="13.545ex" role="img" focusable="false" viewBox="0 -3243.4 40525.6 5986.7" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtable" 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data-mjx-texclass="ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo stretchy="false">]</mo><mtext>Re</mtext><mo stretchy="false">[</mo><mi>δ</mi><msub><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">^</mo></mover></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo stretchy="false">]</mo><mo>−</mo><mtext>Re</mtext><mo stretchy="false">[</mo><mi>δ</mi><msub><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo stretchy="false">^</mo></mover></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo stretchy="false">]</mo><mtext>Im</mtext><mo stretchy="false">[</mo><mi>δ</mi><msub><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">^</mo></mover></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo stretchy="false">]</mo><mo data-mjx-texclass="CLOSE">)</mo></mrow></math></mjx-assistive-mml></mjx-container><p>The above solution applies to every eigenvalue <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.027ex;" xmlns="http://www.w3.org/2000/svg" width="1.319ex" height="1.597ex" role="img" focusable="false" viewBox="0 -694 583 706" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 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389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mi>r</mi></msub></math></mjx-assistive-mml></mjx-container> and harmonic <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.666ex;" xmlns="http://www.w3.org/2000/svg" width="3.251ex" height="1.668ex" role="img" focusable="false" viewBox="0 -443 1436.9 737.2" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" 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stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msubsup></math></mjx-assistive-mml></mjx-container></p></li><li><p>A Lorentzian centered at <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.666ex;" xmlns="http://www.w3.org/2000/svg" width="12.204ex" height="2.363ex" role="img" focusable="false" viewBox="0 -750 5394.2 1044.2" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(655,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(345,0)"><path data-c="2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 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190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z" transform="translate(361,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(3853.3,0)"><path data-c="5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(4131.3,0)"><g data-mml-node="mi"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><path data-c="1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(5116.2,0)"><path data-c="5D" d="M22 710V750H159V-250H22V-210H119V710H22Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>+</mo><mtext>Im</mtext><mo stretchy="false">[</mo><msub><mi>λ</mi><mi>r</mi></msub><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container> with amplitude <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.991ex;" xmlns="http://www.w3.org/2000/svg" width="7.499ex" height="3.391ex" role="img" focusable="false" viewBox="0 -1060.7 3314.5 1498.8" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><path data-c="31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(722.2,0)"><path data-c="2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z" style="stroke-width:3;"></path></g><g data-mml-node="msubsup" transform="translate(1722.4,0)"><g data-mml-node="mi"><path data-c="1D6FC" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z" style="stroke-width:3;"></path></g><g data-mml-node="TeXAtom" transform="translate(673,530.4) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(389,0)"><path data-c="1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(840,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g><g data-mml-node="TeXAtom" transform="translate(673,-293.8) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(345,0)"><path data-c="2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(623,0)"><path data-c="1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z" style="stroke-width:3;"></path></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>−</mo><msubsup><mi>α</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msubsup></math></mjx-assistive-mml></mjx-container></p></li></ul><p><em>Sidenote:</em> As <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.05ex;" xmlns="http://www.w3.org/2000/svg" width="1.432ex" height="1.595ex" role="img" focusable="false" viewBox="0 -683 633 705" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D43D" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math></mjx-assistive-mml></mjx-container> a real matrix, there is an eigenvalue <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.576ex;" xmlns="http://www.w3.org/2000/svg" width="2.307ex" height="2.147ex" role="img" focusable="false" viewBox="0 -694 1019.6 948.8" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msubsup"><g data-mml-node="mi"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(616,363) scale(0.707)"><path data-c="2217" d="M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(616,-247) scale(0.707)"><path data-c="1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>λ</mi><mi>r</mi><mo>∗</mo></msubsup></math></mjx-assistive-mml></mjx-container> for each <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.357ex;" xmlns="http://www.w3.org/2000/svg" width="2.228ex" height="1.927ex" role="img" focusable="false" viewBox="0 -694 984.9 851.8" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><path data-c="1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mi>r</mi></msub></math></mjx-assistive-mml></mjx-container>. The maximum number of peaks in the linear response is thus equal to the dimensionality of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.799ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2121 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(639,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1028,0)"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1732,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>.</p><p>The linear response of the system in the state <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.433ex" height="1.393ex" role="img" focusable="false" viewBox="0 -450 1075.6 615.6" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub></math></mjx-assistive-mml></mjx-container> is thus fully specified by the complex eigenvalues and eigenvectors of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.626ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2486.6 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D43D" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(633,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z" style="stroke-width:3;"></path></g><g data-mml-node="msub" transform="translate(1022,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D42E" d="M40 442L134 446Q228 450 229 450H235V273V165Q235 90 238 74T254 52Q268 46 304 46H319Q352 46 380 67T419 121L420 123Q424 135 425 199Q425 201 425 207Q425 233 425 249V316Q425 354 423 363T410 376Q396 380 369 380H356V442L554 450V267Q554 84 556 79Q561 62 610 62H623V31Q623 0 622 0Q603 0 527 -3T432 -6Q431 -6 431 25V56L420 45Q373 6 332 -1Q313 -6 281 -6Q208 -6 165 14T109 87L107 98L106 230Q106 358 104 366Q96 380 50 380H37V442H40Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mn" transform="translate(672,-150) scale(0.707)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path></g></g><g data-mml-node="mo" transform="translate(2097.6,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">u</mi></mrow><mn>0</mn></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>. In HarmonicBalance.jl, the module <a href="/HarmonicBalance.jl/dev/manual/linear_response#linresp_man">LinearResponse</a> creates a set of plottable <a href="/HarmonicBalance.jl/dev/manual/linear_response#HarmonicBalance.LinearResponse.Lorentzian-manual-linear_response"><code>Lorentzian</code></a> objects to represent this.</p><p><a href="/HarmonicBalance.jl/dev/tutorials/linear_response#linresp_ex">Check out this example</a> of the linear response module of HarmonicBalance.jl</p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/background/stability_response.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/background/harmonic_balance" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Floquet expansions</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/background/limit_cycles" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Limit Cycles</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. 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tabindex="-1" data-v-18756405><span class="visually-hidden" id="sidebar-aria-label" data-v-18756405> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0" data-v-9e426adc data-v-a4b0d9bf><!----><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/index" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Home</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/index" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Getting Started</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Theoretical Background</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/harmonic_balance" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Floquet expansions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/stability_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Stability and Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Tutorials</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/steady_states" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/classification" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Classifying Solutions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Transient Dynamics</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Examples</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Three Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Oscillator</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>State Dependent Perturbation</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Resources</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/API" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>API</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Bibliography</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Manual</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Entering Equations of Motion</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Effective System</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/methods" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Krylov-Bogoliubov</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Time Evolution</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Plotting</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/saving" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Saving and Loading</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-a9a9e638 data-v-91765379><div class="VPDoc has-sidebar has-aside" data-v-91765379 data-v-83890dd9><!--[--><!--]--><div class="container" data-v-83890dd9><div class="aside" data-v-83890dd9><div class="aside-curtain" data-v-83890dd9></div><div class="aside-container" data-v-83890dd9><div class="aside-content" data-v-83890dd9><div class="VPDocAside" data-v-83890dd9 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data-v-68744f5e><!--[--><div class="img-box" data-v-68744f5e data-v-7654366a><a href="wave_mixing" data-v-7654366a><img src="https://nonlinearoscillations.github.io/HarmonicBalance.jl/assets/example_thumbnails/Four_wave_upconversion.jpg" height="150px" alt="" data-v-7654366a><div class="transparent-box1" data-v-7654366a><div class="caption" data-v-7654366a><h3 data-v-7654366a>Wave mixing</h3></div></div><div class="transparent-box2" data-v-7654366a><div class="subcaption" data-v-7654366a><p class="opacity-low" data-v-7654366a>Understand three and four wave mixing.</p></div></div></a></div><div class="img-box" data-v-68744f5e data-v-7654366a><a href="parametric_via_three_wave_mixing" data-v-7654366a><img src="https://nonlinearoscillations.github.io/HarmonicBalance.jl/assets/example_thumbnails/HB_thumbnail.png" height="150px" alt="" data-v-7654366a><div class="transparent-box1" data-v-7654366a><div class="caption" data-v-7654366a><h3 data-v-7654366a>Parametric three wave mixing</h3></div></div><div class="transparent-box2" data-v-7654366a><div class="subcaption" data-v-7654366a><p class="opacity-low" data-v-7654366a>Parametric excitation through three wave mixing.</p></div></div></a></div><div class="img-box" data-v-68744f5e data-v-7654366a><a href="parametron" data-v-7654366a><img src="https://nonlinearoscillations.github.io/HarmonicBalance.jl/assets/example_thumbnails/HB_thumbnail.png" height="150px" alt="" data-v-7654366a><div class="transparent-box1" data-v-7654366a><div class="caption" data-v-7654366a><h3 data-v-7654366a>Parametric oscillator</h3></div></div><div class="transparent-box2" data-v-7654366a><div class="subcaption" data-v-7654366a><p class="opacity-low" data-v-7654366a>Introduction to the parametric oscillator.</p></div></div></a></div><!--]--></div><div class="tip custom-block"><p class="custom-block-title">TIP</p><p>If you wrote an amazing tutorial showcasing <code>HarmonicBalance.jl</code> yourself, please open an issue or PR to add 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diff --git a/dev/examples/parametric_via_three_wave_mixing.html b/dev/examples/parametric_via_three_wave_mixing.html
index 9b6e3a78d..78574efcf 100644
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<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">fixed </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (α </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, β </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 2.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ω0 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, γ </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0.001</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0.005</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
<span class="line"></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">result </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(harmonic_eq2, varied, fixed)</span></span>
-<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result; y</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"v1"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><img src="/HarmonicBalance.jl/dev/assets/vcubulm.qoR36MJx.png" alt=""></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">varied </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> range</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.6</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">100</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">), F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> range</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1e-6</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.01</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">50</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span>
+<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result; y</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"v1"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><img src="/HarmonicBalance.jl/dev/assets/vcubulm.CoWwsMSb.png" alt=""></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">varied </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> range</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.6</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">100</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">), F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> range</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1e-6</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.01</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">50</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">fixed </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (α </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, β </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 2.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ω0 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, γ </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0.01</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
<span class="line"></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">method </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> TotalDegree</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">()</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">result </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(harmonic_eq2, method, varied, fixed)</span></span>
<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_phase_diagram</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result; class</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"stable"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><img src="/HarmonicBalance.jl/dev/assets/aqesnsv.B-Cc1T24.png" alt=""></p><hr><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl" target="_blank" rel="noreferrer">Literate.jl</a>.</em></p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/examples/parametric_via_three_wave_mixing.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Wave Mixing</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Parametric Oscillator</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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@@ -64,8 +64,8 @@
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">varied </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> range</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.8</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">50</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">), λ </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> range</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.001</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.6</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">50</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">result_2D </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(harmonic_eq, varied, fixed);</span></span></code></pre></div><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span></span></span>
<span class="line"><span>Solving for 2500 parameters... 80%|███████████████▉ | ETA: 0:00:00[K</span></span>
-<span class="line"><span> # parameters solved: 1988[K</span></span>
-<span class="line"><span> # paths tracked: 9940[K</span></span>
+<span class="line"><span> # parameters solved: 1989[K</span></span>
+<span class="line"><span> # paths tracked: 9945[K</span></span>
<span class="line"><span>[A</span></span>
<span class="line"><span>[A</span></span>
<span class="line"><span></span></span>
@@ -77,7 +77,7 @@
<span class="line"><span> # paths tracked: 12500[K</span></span></code></pre></div><p>Now, we count the number of solutions for each point and represent the corresponding phase diagram in parameter space. This is done using <code>plot_phase_diagram</code>. Only counting stable solutions,</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_phase_diagram</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result_2D; class</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"stable"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><img src="/HarmonicBalance.jl/dev/assets/qqpavdd.XwKutZKG.png" alt=""></p><p>In addition to phase diagrams, we can plot functions of the solution. The syntax is identical to 1D plotting. Let us overlay 2 branches into a single plot,</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># overlay branches with different colors</span></span>
<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result_2D, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u1^2 + v1^2)"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, class</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"stable"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, camera</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">60</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">40</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span>
<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result_2D, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u1^2 + v1^2)"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, class</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"stable"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, color</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">:red</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><img src="/HarmonicBalance.jl/dev/assets/ktigpzm.pYnWXlWu.png" alt=""></p><p>Note that solutions are ordered in parameter space according to their closest neighbors. Plots can again be limited to a given class (e.g stable solutions only) through the keyword argument <code>class</code>.</p><hr><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl" target="_blank" rel="noreferrer">Literate.jl</a>.</em></p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/examples/parametron.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Parametric Three Wave Mixing</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>State Dependent Perturbation</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> clim</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">crange,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> size</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">800</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">300</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">),</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><img src="/HarmonicBalance.jl/dev/assets/kkbtxcf.DyQS-0z8.png" alt=""></p><p>We see that the perturbed symmetirc mode gives the same bifurcation lines as the full system. Hence, the nonlinear normal mode coupling instantiates a new bifurcation in the system. For more detail consider reading the paper <a href="https://www.arxiv.org/abs/2501.08793" target="_blank" rel="noreferrer">Ameye et al. (arXiv:2501.08793)</a> where we explore and explain the bifurcation line of the coupled Kerr/Duffing parametric oscillators.</p><hr><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl" target="_blank" rel="noreferrer">Literate.jl</a>.</em></p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/examples/state_dependent_perturbation.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Parametric Oscillator</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/manual/API" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>API</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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@@ -49,7 +49,7 @@
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<span class="line"><span>(2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ + u1(T)*u2(T)*β - u1(T)*(ω^2) + u1(T)*(ω0^2) + v3(T)*v2(T)*β + v2(T)*v1(T)*β + v1(T)*γ*ω + u3(T)*u2(T)*β + (3//4)*(u1(T)^3)*α + (3//4)*(u1(T)^2)*u3(T)*α + (3//2)*u1(T)*(v3(T)^2)*α + (3//2)*u1(T)*v3(T)*v1(T)*α + (3//2)*u1(T)*(v2(T)^2)*α + (3//4)*u1(T)*(v1(T)^2)*α + (3//2)*u1(T)*(u3(T)^2)*α + (3//2)*u1(T)*(u2(T)^2)*α + (3//2)*v3(T)*v2(T)*u2(T)*α - (3//4)*(v2(T)^2)*u3(T)*α - (3//4)*(v1(T)^2)*u3(T)*α + (3//4)*u3(T)*(u2(T)^2)*α ~ F</span></span>
<span class="line"><span></span></span>
-<span class="line"><span>Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω + u1(T)*v2(T)*β - u1(T)*γ*ω + v3(T)*u2(T)*β - v2(T)*u3(T)*β - v1(T)*u2(T)*β - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v3(T)*α + (3//4)*(u1(T)^2)*v1(T)*α - (3//2)*u1(T)*v1(T)*u3(T)*α + (3//2)*(v3(T)^2)*v1(T)*α + (3//4)*v3(T)*(v2(T)^2)*α - (3//4)*v3(T)*(v1(T)^2)*α - (3//4)*v3(T)*(u2(T)^2)*α + (3//2)*(v2(T)^2)*v1(T)*α + (3//2)*v2(T)*u3(T)*u2(T)*α + (3//4)*(v1(T)^3)*α + (3//2)*v1(T)*(u3(T)^2)*α + (3//2)*v1(T)*(u2(T)^2)*α ~ 0</span></span>
+<span class="line"><span>Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω + u1(T)*v2(T)*β - u1(T)*γ*ω + v3(T)*u2(T)*β - v2(T)*u3(T)*β - v1(T)*u2(T)*β - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v3(T)*α + (3//4)*(u1(T)^2)*v1(T)*α - (3//2)*u1(T)*v1(T)*u3(T)*α + (3//2)*(v3(T)^2)*v1(T)*α + (3//4)*v3(T)*(v2(T)^2)*α - (3//4)*v3(T)*(v1(T)^2)*α - (3//4)*v3(T)*(u2(T)^2)*α + (3//2)*(v2(T)^2)*v1(T)*α + (3//2)*v2(T)*u3(T)*u2(T)*α + (3//4)*(v1(T)^3)*α + (3//2)*v1(T)*(u3(T)^2)*α + (3//2)*v1(T)*(u2(T)^2)*α ~ 0//1</span></span>
<span class="line"><span></span></span>
<span class="line"><span>Differential(T)(u2(T))*γ + (4//1)*Differential(T)(v2(T))*ω + (1//2)*(u1(T)^2)*β + u1(T)*u3(T)*β + v3(T)*v1(T)*β + (2//1)*v2(T)*γ*ω - (1//2)*(v1(T)^2)*β - (4//1)*u2(T)*(ω^2) + u2(T)*(ω0^2) + (3//2)*(u1(T)^2)*u2(T)*α + (3//2)*u1(T)*v3(T)*v2(T)*α + (3//2)*u1(T)*u3(T)*u2(T)*α + (3//2)*(v3(T)^2)*u2(T)*α - (3//2)*v3(T)*v1(T)*u2(T)*α + (3//4)*(v2(T)^2)*u2(T)*α + (3//2)*v2(T)*v1(T)*u3(T)*α + (3//2)*(v1(T)^2)*u2(T)*α + (3//2)*(u3(T)^2)*u2(T)*α + (3//4)*(u2(T)^3)*α ~ 0//1</span></span>
<span class="line"><span></span></span>
@@ -79,7 +79,7 @@
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">p2 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result; y</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"√(u2^2+v2^2)"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, legend</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">:best</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ylims</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">p3 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result; y</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"√(u3^2+v3^2)"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, legend</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">:best</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(p1, p2, p3; layout</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">3</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">), size</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">900</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">300</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">), margin</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">5</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">mm)</span></span></code></pre></div><p><img src="/HarmonicBalance.jl/dev/assets/sfsiicg.BWuHbhjm.png" alt=""></p><hr><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl" target="_blank" rel="noreferrer">Literate.jl</a>.</em></p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/examples/wave_mixing.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/tutorials/limit_cycles" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Limit Cycles</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Parametric Three Wave Mixing</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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diff --git a/dev/hashmap.json b/dev/hashmap.json
index 7ef32c727..7f02f3800 100644
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href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-acbfed09><!--[--><span data-v-acbfed09>Krylov-Bogoliubov</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-acbfed09><!--[--><span data-v-acbfed09>Time Evolution</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-acbfed09><!--[--><span data-v-acbfed09>Linear Response</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-acbfed09><!--[--><span data-v-acbfed09>Plotting</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/manual/saving" 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tabindex="-1" data-v-18756405><span class="visually-hidden" id="sidebar-aria-label" data-v-18756405> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0" data-v-9e426adc data-v-a4b0d9bf><!----><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/index" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Home</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/index" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Getting Started</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Theoretical Background</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/harmonic_balance" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Floquet expansions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/stability_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Stability and Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Tutorials</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/steady_states" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/classification" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Classifying Solutions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Transient Dynamics</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Examples</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Three Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Oscillator</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>State Dependent Perturbation</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible has-active" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Resources</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/API" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>API</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Bibliography</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Manual</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Entering Equations of Motion</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Effective System</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/methods" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Krylov-Bogoliubov</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Time Evolution</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Plotting</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/saving" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Saving and Loading</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-a9a9e638 data-v-91765379><div class="VPDoc has-sidebar has-aside" data-v-91765379 data-v-83890dd9><!--[--><!--]--><div class="container" data-v-83890dd9><div class="aside" data-v-83890dd9><div class="aside-curtain" data-v-83890dd9></div><div class="aside-container" data-v-83890dd9><div class="aside-content" data-v-83890dd9><div class="VPDocAside" data-v-83890dd9 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HarmonicBalance.jl in your project, we kindly ask you to cite <a href="https://doi.org/10.21468/SciPostPhysCodeb.6" target="_blank" rel="noreferrer">this paper</a>, namely:</p><blockquote><p><strong>HarmonicBalance.jl: A Julia suite for nonlinear dynamics using harmonic balance</strong>, Jan Košata, Javier del Pino, Toni L. Heugel, Oded Zilberberg, SciPost Phys. Codebases 6 (2022)</p></blockquote><p>The limit cycle finding algorithm is based on the work of <a href="https://doi.org/10.1103/PhysRevResearch.6.033180" target="_blank" rel="noreferrer">this paper</a>:</p><blockquote><p><strong>Limit cycles as stationary states of an extended harmonic balance ansatz</strong> J. del Pino, J. Košata, and O. Zilberberg, Phys. Rev. Res. 6, 033180 (2024).</p></blockquote><p>For more context and details on the type of systems studied using the package, we recommend reading <a href="https://global.oup.com/academic/product/classical-and-quantum-parametric-phenomena-9780192862709?cc=de&lang=en&" target="_blank" rel="noreferrer">this book</a>:</p><blockquote><p><strong>Classical and Quantum Parametric Phenomena</strong> A. Eichler and O. Zilberberg, Oxford University Press (2023).</p></blockquote><p>And visiting our <a href="https://www.zilberberg.uni-konstanz.de/home/index.php?id=intro" target="_blank" rel="noreferrer">group's homepage</a>.</p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/introduction/citation.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/manual/API" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>API</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Entering Equations of Motion</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. 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<span class="line"><span> of which stable: 2</span></span>
<span class="line"><span></span></span>
<span class="line"><span>Classes: stable, physical, Hopf</span></span></code></pre></div><p>The obtained steady states can be plotted as a function of the driving frequency:</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u1^2 + v1^2)"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><img src="/HarmonicBalance.jl/dev/assets/cgkxkux.B1eISI2b.png" alt=""></p><p>If you want learn more on what you can do with HarmonicBalance.jl, check out the <a href="/HarmonicBalance.jl/dev/tutorials/index#tutorials">tutorials</a>. We also have collected some <a href="/HarmonicBalance.jl/dev/examples/index#examples">examples</a> of different physical systems.</p><hr><h1 id="Setting-Up-Julia-and-Jupyter-Notebooks" tabindex="-1">Setting Up Julia and Jupyter Notebooks <a class="header-anchor" href="#Setting-Up-Julia-and-Jupyter-Notebooks" aria-label="Permalink to "Setting Up Julia and Jupyter Notebooks {#Setting-Up-Julia-and-Jupyter-Notebooks}""></a></h1><p>To ensure a smooth experience with our package, please follow these steps to set up Julia and Jupyter notebooks. Once these steps are completed, you can proceed to install and use our package seamlessly.</p><ol><li><p><strong>Download Julia</strong>: Visit the <a href="https://julialang.org" target="_blank" rel="noreferrer">Julia Downloads page</a> and download the latest stable release for your operating system.</p></li><li><p><strong>Install Julia</strong>: Follow the installation instructions for your platform:</p></li><li><p>Open Julia and enter the package manager by typing <code>]</code> in the Julia REPL.</p></li><li><p>Add the <code>IJulia</code> package, which integrates Julia with Jupyter:</p></li></ol><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">add IJulia</span></span></code></pre></div><ol start="5"><li>Once installed, <code>IJulia</code> will automatically set up Julia as a Jupyter kernel.</li></ol></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/introduction/index.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><!----></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/index" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Home</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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tabindex="-1" data-v-18756405><span class="visually-hidden" id="sidebar-aria-label" data-v-18756405> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0" data-v-9e426adc data-v-a4b0d9bf><!----><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/index" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Home</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/index" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Getting 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data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/stability_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Stability and Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Tutorials</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/steady_states" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/classification" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Classifying Solutions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div 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class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Examples</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Three Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Oscillator</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>State Dependent Perturbation</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible has-active" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Resources</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/API" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>API</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Bibliography</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Manual</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Entering Equations of Motion</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Effective System</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/methods" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Krylov-Bogoliubov</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Time Evolution</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Plotting</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/saving" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Saving and Loading</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-a9a9e638 data-v-91765379><div class="VPDoc has-sidebar has-aside" data-v-91765379 data-v-83890dd9><!--[--><!--]--><div class="container" data-v-83890dd9><div class="aside" data-v-83890dd9><div class="aside-curtain" data-v-83890dd9></div><div class="aside-container" data-v-83890dd9><div class="aside-content" data-v-83890dd9><div class="VPDocAside" data-v-83890dd9 data-v-6d7b3c46><!--[--><!--]--><!--[--><!--]--><nav aria-labelledby="doc-outline-aria-label" class="VPDocAsideOutline" data-v-6d7b3c46 data-v-b38bf2ff><div class="content" data-v-b38bf2ff><div class="outline-marker" data-v-b38bf2ff></div><div aria-level="2" class="outline-title" id="doc-outline-aria-label" role="heading" data-v-b38bf2ff>On this page</div><ul class="VPDocOutlineItem root" data-v-b38bf2ff data-v-3f927ebe><!--[--><!--]--></ul></div></nav><!--[--><!--]--><div class="spacer" data-v-6d7b3c46></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--]--></div></div></div></div><div class="content" data-v-83890dd9><div class="content-container" data-v-83890dd9><!--[--><!--]--><main class="main" data-v-83890dd9><div style="position:relative;" class="vp-doc _HarmonicBalance_jl_dev_manual_API" data-v-83890dd9><div><h1 id="doc-API" tabindex="-1">API <a class="header-anchor" href="#doc-API" aria-label="Permalink to "API {#doc-API}""></a></h1><p><strong>Table of contents</strong></p><nav class="table-of-contents"><ul><li><a href="#System-objects-and-types">System objects and types</a></li><li><a href="#Solving-and-transforming-solutions">Solving and transforming solutions</a><ul><li><a href="#methods">Methods</a></li><li><a href="#Access-solutions">Access solutions</a></li><li><a href="#classify">Classify</a></li><li><a href="#plotting">Plotting</a></li></ul></li><li><a href="#Limit-cycles">Limit cycles</a></li><li><a href="#Linear-Response">Linear Response</a></li><li><a href="#extensions">Extensions</a><ul><li><a href="#ordinarydiffeq">OrdinaryDiffEq</a></li><li><a href="#steadystatesweep">SteadyStateSweep</a></li><li><a href="#modelingtoolkit">ModelingToolkit</a></li></ul></li></ul></nav><h2 id="System-objects-and-types" tabindex="-1">System objects and types <a class="header-anchor" href="#System-objects-and-types" aria-label="Permalink to "System objects and types {#System-objects-and-types}""></a></h2><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.d" href="#HarmonicBalance.d"><span class="jlbinding">HarmonicBalance.d</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><p>The derivative of f w.r.t. x of degree deg</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicVariable.jl#L108" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.DifferentialEquation" href="#HarmonicBalance.DifferentialEquation"><span class="jlbinding">HarmonicBalance.DifferentialEquation</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> DifferentialEquation</span></span></code></pre></div><p>Holds differential equation(s) of motion and a set of harmonics to expand each variable. This is the primary input for <code>HarmonicBalance.jl</code>. After inputting the equations, the harmonics ansatz needs to be specified using <code>add_harmonic!</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>equations::OrderedCollections.OrderedDict{Num, Equation}</code>: Assigns to each variable an equation of motion.</p></li><li><p><code>harmonics::OrderedCollections.OrderedDict{Num, OrderedCollections.OrderedSet{Num}}</code>: Assigns to each variable a set of harmonics.</p></li></ul><p><strong>Example</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> t, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">y</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), ω0, ω, F, k;</span></span>
+ <div id="app"><div class="Layout" data-v-a9a9e638><!--[--><!--]--><!--[--><span tabindex="-1" data-v-492508fc></span><a href="#VPContent" class="VPSkipLink visually-hidden" data-v-492508fc>Skip to content</a><!--]--><!----><header class="VPNav" data-v-a9a9e638 data-v-f1e365da><div class="VPNavBar" data-v-f1e365da data-v-822684d1><div class="wrapper" data-v-822684d1><div class="container" data-v-822684d1><div class="title" data-v-822684d1><div class="VPNavBarTitle has-sidebar" data-v-822684d1 data-v-0f4f798b><a class="title" href="/HarmonicBalance.jl/dev/" data-v-0f4f798b><!--[--><!--]--><!--[--><img class="VPImage logo" src="/HarmonicBalance.jl/dev/logo.png" width="24" height="24" alt data-v-35a7d0b8><!--]--><span data-v-0f4f798b>HarmonicBalance.jl</span><!--[--><!--]--></a></div></div><div class="content" data-v-822684d1><div class="content-body" data-v-822684d1><!--[--><!--]--><div class="VPNavBarSearch search" data-v-822684d1><!--[--><!----><div id="local-search"><button 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class="title" data-v-48c802d0>Examples</p><!--[--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-acbfed09><!--[--><span data-v-acbfed09>Wave Mixing</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-acbfed09><!--[--><span data-v-acbfed09>Parametric Three Wave Mixing</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-acbfed09><!--[--><span data-v-acbfed09>Parametric Oscillator</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-acbfed09><!--[--><span data-v-acbfed09>State 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href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-acbfed09><!--[--><span data-v-acbfed09>Krylov-Bogoliubov</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-acbfed09><!--[--><span data-v-acbfed09>Time Evolution</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-acbfed09><!--[--><span data-v-acbfed09>Linear Response</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-acbfed09><!--[--><span data-v-acbfed09>Plotting</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/manual/saving" 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tabindex="-1" data-v-18756405><span class="visually-hidden" id="sidebar-aria-label" data-v-18756405> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0" data-v-9e426adc data-v-a4b0d9bf><!----><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/index" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Home</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/index" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Getting Started</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Theoretical Background</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/harmonic_balance" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Floquet expansions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/stability_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Stability and Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Tutorials</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/steady_states" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/classification" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Classifying Solutions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Transient Dynamics</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Examples</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Three Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Oscillator</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>State Dependent Perturbation</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible has-active" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Resources</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/API" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>API</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Bibliography</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Manual</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Entering Equations of Motion</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Effective System</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/methods" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Krylov-Bogoliubov</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Time Evolution</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Plotting</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/saving" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Saving and Loading</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-a9a9e638 data-v-91765379><div class="VPDoc has-sidebar has-aside" data-v-91765379 data-v-83890dd9><!--[--><!--]--><div class="container" data-v-83890dd9><div class="aside" data-v-83890dd9><div class="aside-curtain" data-v-83890dd9></div><div class="aside-container" data-v-83890dd9><div class="aside-content" data-v-83890dd9><div class="VPDocAside" data-v-83890dd9 data-v-6d7b3c46><!--[--><!--]--><!--[--><!--]--><nav aria-labelledby="doc-outline-aria-label" class="VPDocAsideOutline" data-v-6d7b3c46 data-v-b38bf2ff><div class="content" data-v-b38bf2ff><div class="outline-marker" data-v-b38bf2ff></div><div aria-level="2" class="outline-title" id="doc-outline-aria-label" role="heading" data-v-b38bf2ff>On this page</div><ul class="VPDocOutlineItem root" data-v-b38bf2ff data-v-3f927ebe><!--[--><!--]--></ul></div></nav><!--[--><!--]--><div class="spacer" data-v-6d7b3c46></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--]--></div></div></div></div><div class="content" data-v-83890dd9><div class="content-container" data-v-83890dd9><!--[--><!--]--><main class="main" data-v-83890dd9><div style="position:relative;" class="vp-doc _HarmonicBalance_jl_dev_manual_API" data-v-83890dd9><div><h1 id="doc-API" tabindex="-1">API <a class="header-anchor" href="#doc-API" aria-label="Permalink to "API {#doc-API}""></a></h1><p><strong>Table of contents</strong></p><nav class="table-of-contents"><ul><li><a href="#System-objects-and-types">System objects and types</a></li><li><a href="#Solving-and-transforming-solutions">Solving and transforming solutions</a><ul><li><a href="#methods">Methods</a></li><li><a href="#Access-solutions">Access solutions</a></li><li><a href="#classify">Classify</a></li><li><a href="#plotting">Plotting</a></li></ul></li><li><a href="#Limit-cycles">Limit cycles</a></li><li><a href="#Linear-Response">Linear Response</a></li><li><a href="#extensions">Extensions</a><ul><li><a href="#ordinarydiffeq">OrdinaryDiffEq</a></li><li><a href="#steadystatesweep">SteadyStateSweep</a></li><li><a href="#modelingtoolkit">ModelingToolkit</a></li></ul></li></ul></nav><h2 id="System-objects-and-types" tabindex="-1">System objects and types <a class="header-anchor" href="#System-objects-and-types" aria-label="Permalink to "System objects and types {#System-objects-and-types}""></a></h2><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.d" href="#HarmonicBalance.d"><span class="jlbinding">HarmonicBalance.d</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><p>The derivative of f w.r.t. x of degree deg</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicVariable.jl#L108" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.DifferentialEquation" href="#HarmonicBalance.DifferentialEquation"><span class="jlbinding">HarmonicBalance.DifferentialEquation</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> DifferentialEquation</span></span></code></pre></div><p>Holds differential equation(s) of motion and a set of harmonics to expand each variable. This is the primary input for <code>HarmonicBalance.jl</code>. After inputting the equations, the harmonics ansatz needs to be specified using <code>add_harmonic!</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>equations::OrderedCollections.OrderedDict{Num, Equation}</code>: Assigns to each variable an equation of motion.</p></li><li><p><code>harmonics::OrderedCollections.OrderedDict{Num, OrderedCollections.OrderedSet{Num}}</code>: Assigns to each variable a set of harmonics.</p></li></ul><p><strong>Example</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> t, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">y</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), ω0, ω, F, k;</span></span>
<span class="line"></span>
<span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># equivalent ways to enter the simple harmonic oscillator</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">d</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(x,t,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> *</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> x </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">t), x);</span></span>
@@ -34,16 +34,16 @@
<span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># two coupled oscillators, one of them driven</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">d</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(x,t,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> *</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> x </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> k</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">d</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(y,t,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> *</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> y </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> k</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">x] </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], [x,y]</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">);</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.HarmonicVariable" href="#HarmonicBalance.HarmonicVariable"><span class="jlbinding">HarmonicBalance.HarmonicVariable</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicVariable</span></span></code></pre></div><p>Holds a variable stored under <code>symbol</code> describing the harmonic <code>ω</code> of <code>natural_variable</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>symbol::Num</code>: Symbol of the variable in the HarmonicBalance namespace.</p></li><li><p><code>name::String</code>: Human-readable labels of the variable, used for plotting.</p></li><li><p><code>type::String</code>: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)</p></li><li><p><code>ω::Num</code>: The harmonic being described.</p></li><li><p><code>natural_variable::Num</code>: The natural variable whose harmonic is being described.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicVariable.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.HarmonicEquation" href="#HarmonicBalance.HarmonicEquation"><span class="jlbinding">HarmonicBalance.HarmonicEquation</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Holds a set of algebraic equations governing the harmonics of a <code>DifferentialEquation</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>equations::Vector{Equation}</code>: A set of equations governing the harmonics.</p></li><li><p><code>variables::Vector{HarmonicVariable}</code>: A set of variables describing the harmonics.</p></li><li><p><code>parameters::Vector{Num}</code>: The parameters of the equation set.</p></li><li><p><code>natural_equation::DifferentialEquation</code>: The natural equation (before the harmonic ansatz was used).</p></li><li><p><code>jacobian::Matrix{Num}</code>: The Jacobian of the natural equation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.rearrange_standard" href="#HarmonicBalance.rearrange_standard"><span class="jlbinding">HarmonicBalance.rearrange_standard</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">rearrange_standard</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">);</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.HarmonicVariable" href="#HarmonicBalance.HarmonicVariable"><span class="jlbinding">HarmonicBalance.HarmonicVariable</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicVariable</span></span></code></pre></div><p>Holds a variable stored under <code>symbol</code> describing the harmonic <code>ω</code> of <code>natural_variable</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>symbol::Num</code>: Symbol of the variable in the HarmonicBalance namespace.</p></li><li><p><code>name::String</code>: Human-readable labels of the variable, used for plotting.</p></li><li><p><code>type::String</code>: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)</p></li><li><p><code>ω::Num</code>: The harmonic being described.</p></li><li><p><code>natural_variable::Num</code>: The natural variable whose harmonic is being described.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicVariable.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.HarmonicEquation" href="#HarmonicBalance.HarmonicEquation"><span class="jlbinding">HarmonicBalance.HarmonicEquation</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Holds a set of algebraic equations governing the harmonics of a <code>DifferentialEquation</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>equations::Vector{Equation}</code>: A set of equations governing the harmonics.</p></li><li><p><code>variables::Vector{HarmonicVariable}</code>: A set of variables describing the harmonics.</p></li><li><p><code>parameters::Vector{Num}</code>: The parameters of the equation set.</p></li><li><p><code>natural_equation::DifferentialEquation</code>: The natural equation (before the harmonic ansatz was used).</p></li><li><p><code>jacobian::Matrix{Num}</code>: The Jacobian of the natural equation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.rearrange_standard" href="#HarmonicBalance.rearrange_standard"><span class="jlbinding">HarmonicBalance.rearrange_standard</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">rearrange_standard</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Rearrange <code>eom</code> to the standard form, such that the derivatives of the variables are on one side.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L194" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.rearrange_standard!" href="#HarmonicBalance.rearrange_standard!"><span class="jlbinding">HarmonicBalance.rearrange_standard!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">rearrange_standard!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
-<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">rearrange_standard!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, degree)</span></span></code></pre></div><p>Rearranges the differential equations in <code>eom</code> to standard form, where the highest derivative of each variable (specified by <code>degree</code>, default 2) appears isolated on the left-hand side. Modifies the equations in place.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L160" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">rearrange_standard!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Rearrange <code>eom</code> to the standard form, such that the derivatives of the variables are on one side.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L194" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.rearrange_standard!" href="#HarmonicBalance.rearrange_standard!"><span class="jlbinding">HarmonicBalance.rearrange_standard!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">rearrange_standard!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
+<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">rearrange_standard!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, degree)</span></span></code></pre></div><p>Rearranges the differential equations in <code>eom</code> to standard form, where the highest derivative of each variable (specified by <code>degree</code>, default 2) appears isolated on the left-hand side. Modifies the equations in place.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L160" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">rearrange_standard!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Rearrange <code>eom</code> to the standard form, such that the derivatives of the variables are on one side.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L204" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.first_order_transform!" href="#HarmonicBalance.first_order_transform!"><span class="jlbinding">HarmonicBalance.first_order_transform!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">first_order_transform!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, time)</span></span></code></pre></div><p>Transforms a higher-order differential equation system into an equivalent first-order system by introducing additional variables. Modifies the system in place. The <code>time</code> parameter specifies the independent variable used for differentiation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L202" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.is_rearranged_standard" href="#HarmonicBalance.is_rearranged_standard"><span class="jlbinding">HarmonicBalance.is_rearranged_standard</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">is_rearranged_standard</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Rearrange <code>eom</code> to the standard form, such that the derivatives of the variables are on one side.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L204" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.first_order_transform!" href="#HarmonicBalance.first_order_transform!"><span class="jlbinding">HarmonicBalance.first_order_transform!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">first_order_transform!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, time)</span></span></code></pre></div><p>Transforms a higher-order differential equation system into an equivalent first-order system by introducing additional variables. Modifies the system in place. The <code>time</code> parameter specifies the independent variable used for differentiation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L202" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.is_rearranged_standard" href="#HarmonicBalance.is_rearranged_standard"><span class="jlbinding">HarmonicBalance.is_rearranged_standard</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">is_rearranged_standard</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span>
<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">is_rearranged_standard</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> degree</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Checks if the differential equations in <code>eom</code> are arranged in standard form, where the highest derivative of each variable appears isolated on the left-hand side. The default degree is 2, corresponding to second-order differential equations.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L147" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_equations" href="#HarmonicBalance.get_equations"><span class="jlbinding">HarmonicBalance.get_equations</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_equations</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Equation}</span></span></code></pre></div><p>Return the equations of <code>eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L140" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_harmonic_equations" href="#HarmonicBalance.get_harmonic_equations"><span class="jlbinding">HarmonicBalance.get_harmonic_equations</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_harmonic_equations</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Checks if the differential equations in <code>eom</code> are arranged in standard form, where the highest derivative of each variable appears isolated on the left-hand side. The default degree is 2, corresponding to second-order differential equations.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L147" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_equations" href="#HarmonicBalance.get_equations"><span class="jlbinding">HarmonicBalance.get_equations</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_equations</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Equation}</span></span></code></pre></div><p>Return the equations of <code>eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L140" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_harmonic_equations" href="#HarmonicBalance.get_harmonic_equations"><span class="jlbinding">HarmonicBalance.get_harmonic_equations</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_harmonic_equations</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> fast_time,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> slow_time,</span></span>
@@ -71,7 +71,7 @@
<span class="line"></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F</span></span>
<span class="line"></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L295-L336" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.KrylovBogoliubov.get_krylov_equations" href="#HarmonicBalance.KrylovBogoliubov.get_krylov_equations"><span class="jlbinding">HarmonicBalance.KrylovBogoliubov.get_krylov_equations</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_krylov_equations</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L295-L336" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.KrylovBogoliubov.get_krylov_equations" href="#HarmonicBalance.KrylovBogoliubov.get_krylov_equations"><span class="jlbinding">HarmonicBalance.KrylovBogoliubov.get_krylov_equations</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_krylov_equations</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> order,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> fast_time,</span></span>
@@ -99,7 +99,7 @@
<span class="line"></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">/</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T))</span></span>
<span class="line"></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">/</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T))</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/KrylovBogoliubov.jl#L25" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.add_harmonic!" href="#HarmonicBalance.add_harmonic!"><span class="jlbinding">HarmonicBalance.add_harmonic!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">add_harmonic!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, var</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ω)</span></span></code></pre></div><p>Add the harmonic <code>ω</code> to the harmonic ansatz used to expand the variable <code>var</code> in <code>diff_eom</code>.</p><p><strong>Example</strong></p><p><strong>define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> t, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">y</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), ω0, ω, F, k;</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">/</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T))</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/KrylovBogoliubov.jl#L25" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.add_harmonic!" href="#HarmonicBalance.add_harmonic!"><span class="jlbinding">HarmonicBalance.add_harmonic!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">add_harmonic!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, var</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ω)</span></span></code></pre></div><p>Add the harmonic <code>ω</code> to the harmonic ansatz used to expand the variable <code>var</code> in <code>diff_eom</code>.</p><p><strong>Example</strong></p><p><strong>define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> t, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">y</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), ω0, ω, F, k;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> diff_eq </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">d</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(x,t,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> *</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> x </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">t), x);</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> add_harmonic!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eq, x, ��) </span><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># expand x using ω</span></span>
<span class="line"></span>
@@ -107,11 +107,11 @@
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Variables</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Harmonic ansatz</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω;</span></span>
<span class="line"></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t))) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L88" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_independent_variables" href="#HarmonicBalance.get_independent_variables"><span class="jlbinding">HarmonicBalance.get_independent_variables</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_independent_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t))) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L88" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_independent_variables" href="#HarmonicBalance.get_independent_variables"><span class="jlbinding">HarmonicBalance.get_independent_variables</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_independent_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Return the independent dependent variables of <code>diff_eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L131" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_independent_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Return the independent dependent variables of <code>diff_eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L131" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_independent_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Num}</span></span></code></pre></div><p>Return the independent variables (typically time) of <code>eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L247" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="Symbolics.get_variables" href="#Symbolics.get_variables"><span class="jlbinding">Symbolics.get_variables</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Num}</span></span></code></pre></div><p>Return the dependent variables of <code>diff_eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L113" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Num}</span></span></code></pre></div><p>Get the internal symbols of the independent variables of <code>eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L214" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Solving-and-transforming-solutions" tabindex="-1">Solving and transforming solutions <a class="header-anchor" href="#Solving-and-transforming-solutions" aria-label="Permalink to "Solving and transforming solutions {#Solving-and-transforming-solutions}""></a></h2><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_steady_states" href="#HarmonicBalance.get_steady_states"><span class="jlbinding">HarmonicBalance.get_steady_states</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(problem</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Num}</span></span></code></pre></div><p>Return the independent variables (typically time) of <code>eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L247" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="Symbolics.get_variables" href="#Symbolics.get_variables"><span class="jlbinding">Symbolics.get_variables</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Num}</span></span></code></pre></div><p>Return the dependent variables of <code>diff_eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L113" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Num}</span></span></code></pre></div><p>Get the internal symbols of the independent variables of <code>eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L214" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Solving-and-transforming-solutions" tabindex="-1">Solving and transforming solutions <a class="header-anchor" href="#Solving-and-transforming-solutions" aria-label="Permalink to "Solving and transforming solutions {#Solving-and-transforming-solutions}""></a></h2><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_steady_states" href="#HarmonicBalance.get_steady_states"><span class="jlbinding">HarmonicBalance.get_steady_states</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(problem</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> method</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalanceMethod</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> swept_parameters,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> fixed_parameters;</span></span>
@@ -140,16 +140,16 @@
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> of which real</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> of which stable</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1</span></span>
<span class="line"></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Classes</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> stable, physical, Hopf, binary_labels</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/solve_homotopy.jl#L1-L60" target="_blank" rel="noreferrer">source</a></p></details><h3 id="methods" tabindex="-1">Methods <a class="header-anchor" href="#methods" aria-label="Permalink to "Methods""></a></h3><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.WarmUp" href="#HarmonicBalance.WarmUp"><span class="jlbinding">HarmonicBalance.WarmUp</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">WarmUp</span></span></code></pre></div><p>The Warm Up method prepares a warmup system with the Total Degree method using the parameter at <code>index</code> perturbed by <code>perturbation_size</code>. The warmup system is used to perform a homotopy using all other systems in the parameter sweep. It is very efficient for systems with minimal bifurcation in the parameter sweep. The Warm Up method should in theory guarantee to find all solutions, however, if the <code>start_parameters</code> is not proper (to close to the real line) it could miss some solutions.</p><p>See<a href="https://www.juliahomotopycontinuation.org/guides/many-systems/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>warm_up_method::Union{Polyhedral{T}, TotalDegree{T}} where T</code>: Method used for the warmup system.</p></li><li><p><code>start_parameters::Vector</code>: Start parameters.</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/methods.jl#L110-L125" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.TotalDegree" href="#HarmonicBalance.TotalDegree"><span class="jlbinding">HarmonicBalance.TotalDegree</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">TotalDegree</span></span></code></pre></div><p>The Total Degree homotopy method performs a homotopy <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg 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data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">ᵢ</text></g></g></g><g data-mml-node="mo" transform="translate(5701.7,0)"><path data-c="2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(6702,0)"><path data-c="1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 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style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.452ex;" xmlns="http://www.w3.org/2000/svg" width="2.534ex" height="2.149ex" role="img" focusable="false" viewBox="0 -750 1120 950" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(520,0)"><text data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">ᵢ</text></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>ᵢ</mi></math></mjx-assistive-mml></mjx-container> determined by the <a href="https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem" target="_blank" rel="noreferrer">Bezout bound</a>. The method guarantees to find all solutions, however, it comes with a high computational cost. See <a href="https://www.juliahomotopycontinuation.org/guides/totaldegree/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>gamma::Complex</code>: Complex multiplying factor of the start system G(x) for the homotopy</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/methods.jl#L8-L20" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.Polyhedral" href="#HarmonicBalance.Polyhedral"><span class="jlbinding">HarmonicBalance.Polyhedral</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Polyhedral</span></span></code></pre></div><p>The Polyhedral homotopy method constructs a homotopy based on the polyhedral structure of the polynomial system. It is more efficient than the Total Degree method for sparse systems, meaning most of the coefficients are zero. It can be especially useful if you don't need to find the zero solutions (<code>only_non_zero = true</code>), resulting in a speed up. See <a href="https://www.juliahomotopycontinuation.org/guides/polyhedral/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>only_non_zero::Bool</code>: Boolean indicating if only non-zero solutions are considered.</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/methods.jl#L57-L69" target="_blank" rel="noreferrer">source</a></p></details><h3 id="Access-solutions" tabindex="-1">Access solutions <a class="header-anchor" href="#Access-solutions" aria-label="Permalink to "Access solutions {#Access-solutions}""></a></h3><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_single_solution" href="#HarmonicBalance.get_single_solution"><span class="jlbinding">HarmonicBalance.get_single_solution</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_single_solution</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Classes</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> stable, physical, Hopf, binary_labels</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/solve_homotopy.jl#L1-L60" target="_blank" rel="noreferrer">source</a></p></details><h3 id="methods" tabindex="-1">Methods <a class="header-anchor" href="#methods" aria-label="Permalink to "Methods""></a></h3><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.WarmUp" href="#HarmonicBalance.WarmUp"><span class="jlbinding">HarmonicBalance.WarmUp</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">WarmUp</span></span></code></pre></div><p>The Warm Up method prepares a warmup system with the Total Degree method using the parameter at <code>index</code> perturbed by <code>perturbation_size</code>. The warmup system is used to perform a homotopy using all other systems in the parameter sweep. It is very efficient for systems with minimal bifurcation in the parameter sweep. The Warm Up method should in theory guarantee to find all solutions, however, if the <code>start_parameters</code> is not proper (to close to the real line) it could miss some solutions.</p><p>See<a href="https://www.juliahomotopycontinuation.org/guides/many-systems/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>warm_up_method::Union{Polyhedral{T}, TotalDegree{T}} where T</code>: Method used for the warmup system.</p></li><li><p><code>start_parameters::Vector</code>: Start parameters.</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/methods.jl#L110-L125" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.TotalDegree" href="#HarmonicBalance.TotalDegree"><span class="jlbinding">HarmonicBalance.TotalDegree</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">TotalDegree</span></span></code></pre></div><p>The Total Degree homotopy method performs a homotopy <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg 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data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">ᵢ</text></g></g></g><g data-mml-node="mo" transform="translate(5701.7,0)"><path data-c="2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(6702,0)"><path data-c="1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(7231,0)"><text data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">ᵢ</text></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><msup><mi>ᵢ</mi><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>ᵢ</mi></mrow></msup><mo>+</mo><mi>a</mi><mi>ᵢ</mi></math></mjx-assistive-mml></mjx-container> with the maximal degree <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.452ex;" xmlns="http://www.w3.org/2000/svg" width="2.534ex" height="2.149ex" role="img" focusable="false" viewBox="0 -750 1120 950" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(520,0)"><text data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">ᵢ</text></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>ᵢ</mi></math></mjx-assistive-mml></mjx-container> determined by the <a href="https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem" target="_blank" rel="noreferrer">Bezout bound</a>. The method guarantees to find all solutions, however, it comes with a high computational cost. See <a href="https://www.juliahomotopycontinuation.org/guides/totaldegree/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>gamma::Complex</code>: Complex multiplying factor of the start system G(x) for the homotopy</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/methods.jl#L8-L20" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.Polyhedral" href="#HarmonicBalance.Polyhedral"><span class="jlbinding">HarmonicBalance.Polyhedral</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Polyhedral</span></span></code></pre></div><p>The Polyhedral homotopy method constructs a homotopy based on the polyhedral structure of the polynomial system. It is more efficient than the Total Degree method for sparse systems, meaning most of the coefficients are zero. It can be especially useful if you don't need to find the zero solutions (<code>only_non_zero = true</code>), resulting in a speed up. See <a href="https://www.juliahomotopycontinuation.org/guides/polyhedral/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>only_non_zero::Bool</code>: Boolean indicating if only non-zero solutions are considered.</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/methods.jl#L57-L69" target="_blank" rel="noreferrer">source</a></p></details><h3 id="Access-solutions" tabindex="-1">Access solutions <a class="header-anchor" href="#Access-solutions" aria-label="Permalink to "Access solutions {#Access-solutions}""></a></h3><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_single_solution" href="#HarmonicBalance.get_single_solution"><span class="jlbinding">HarmonicBalance.get_single_solution</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_single_solution</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result{S, P, D, F}</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> where</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> {D, F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">FunctionWrappers.FunctionWrapper{Array{S, 2}, Tuple{Array{S, 1}}}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">};</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> branch,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> index</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Return an ordered dictionary specifying all variables and parameters of the solution in <code>result</code> on <code>branch</code> at the position <code>index</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/transform_solutions.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.transform_solutions" href="#HarmonicBalance.transform_solutions"><span class="jlbinding">HarmonicBalance.transform_solutions</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">transform_solutions</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Return an ordered dictionary specifying all variables and parameters of the solution in <code>result</code> on <code>branch</code> at the position <code>index</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/transform_solutions.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.transform_solutions" href="#HarmonicBalance.transform_solutions"><span class="jlbinding">HarmonicBalance.transform_solutions</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">transform_solutions</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result{S, ParType, D, F}</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> where</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> {ParType</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Number</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, D, F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">FunctionWrappers.FunctionWrapper{Array{S, 2}, Tuple{Array{S, 1}}}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">},</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> func;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> branches,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> realify</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector</span></span></code></pre></div><p>Takes a <code>Result</code> object and a string <code>f</code> representing a Symbolics.jl expression. Returns an array with the values of <code>f</code> evaluated for the respective solutions. Additional substitution rules can be specified in <code>rules</code> in the format <code>("a" => val)</code> or <code>(a => val)</code></p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/transform_solutions.jl#L64" target="_blank" rel="noreferrer">source</a></p></details><h3 id="classify" tabindex="-1">Classify <a class="header-anchor" href="#classify" aria-label="Permalink to "Classify""></a></h3><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.classify_solutions!" href="#HarmonicBalance.classify_solutions!"><span class="jlbinding">HarmonicBalance.classify_solutions!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector</span></span></code></pre></div><p>Takes a <code>Result</code> object and a string <code>f</code> representing a Symbolics.jl expression. Returns an array with the values of <code>f</code> evaluated for the respective solutions. Additional substitution rules can be specified in <code>rules</code> in the format <code>("a" => val)</code> or <code>(a => val)</code></p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/transform_solutions.jl#L64" target="_blank" rel="noreferrer">source</a></p></details><h3 id="classify" tabindex="-1">Classify <a class="header-anchor" href="#classify" aria-label="Permalink to "Classify""></a></h3><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.classify_solutions!" href="#HarmonicBalance.classify_solutions!"><span class="jlbinding">HarmonicBalance.classify_solutions!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> func</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Union{Function, String}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> name</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
@@ -158,34 +158,34 @@
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">res </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(problem, swept_parameters, fixed_parameters)</span></span>
<span class="line"></span>
<span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># classify, store in result.classes["large_amplitude"]</span></span>
-<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u1^2 + v1^2) > 1.0"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> , </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"large_amplitude"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/classification.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_class" href="#HarmonicBalance.get_class"><span class="jlbinding">HarmonicBalance.get_class</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_class</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u1^2 + v1^2) > 1.0"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> , </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"large_amplitude"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/classification.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_class" href="#HarmonicBalance.get_class"><span class="jlbinding">HarmonicBalance.get_class</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_class</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int64</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> class</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Returns an array of booleans classifying <code>branch</code> in the solutions in <code>res</code> according to <code>class</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/classification.jl#L43" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_class</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Returns an array of booleans classifying <code>branch</code> in the solutions in <code>res</code> according to <code>class</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/classification.jl#L43" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_class</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> class</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector</span></span></code></pre></div><p>Returns an array of booleans classifying each branch in the solutions in <code>res</code> according to <code>class</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/classification.jl#L53" target="_blank" rel="noreferrer">source</a></p></details><h3 id="plotting" tabindex="-1">Plotting <a class="header-anchor" href="#plotting" aria-label="Permalink to "Plotting""></a></h3><details class="jldocstring custom-block" open><summary><a id="RecipesBase.plot" href="#RecipesBase.plot"><span class="jlbinding">RecipesBase.plot</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESolution</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, f</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, harm_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a function <code>f</code> of a time-dependent solution <code>soln</code> of <code>harm_eq</code>.</p><p><strong>As a function of time</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p><code>f</code> is parsed by Symbolics.jl</p><p><strong>parametric plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p>Parametric plot of f[1] against f[2]</p><p>Also callable as plot!</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/plotting.jl#L4-L22" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector</span></span></code></pre></div><p>Returns an array of booleans classifying each branch in the solutions in <code>res</code> according to <code>class</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/classification.jl#L53" target="_blank" rel="noreferrer">source</a></p></details><h3 id="plotting" tabindex="-1">Plotting <a class="header-anchor" href="#plotting" aria-label="Permalink to "Plotting""></a></h3><details class="jldocstring custom-block" open><summary><a id="RecipesBase.plot" href="#RecipesBase.plot"><span class="jlbinding">RecipesBase.plot</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESolution</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, f</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, harm_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a function <code>f</code> of a time-dependent solution <code>soln</code> of <code>harm_eq</code>.</p><p><strong>As a function of time</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p><code>f</code> is parsed by Symbolics.jl</p><p><strong>parametric plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p>Parametric plot of f[1] against f[2]</p><p>Also callable as plot!</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/plotting.jl#L4-L22" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result{S, P, D, F}</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> where</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">FunctionWrappers.FunctionWrapper{Array{S, 2}, Tuple{Array{S, 1}}}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> varargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> cut,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Plot</span></span></code></pre></div><p><strong>Plot a <code>Result</code> object.</strong></p><p>Class selection done by passing <code>String</code> or <code>Vector{String}</code> as kwarg:</p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>class : only plot solutions in this class(es) ("all" --> plot everything)</span></span>
<span class="line"><span>not_class : do not plot solutions in this class(es)</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr().</p><p>See also <code>plot!</code></p><p>The x,y,z arguments are Strings compatible with Symbolics.jl, e.g., <code>y=2*sqrt(u1^2+v1^2)</code> plots the amplitude of the first quadratures multiplied by 2.</p><p><strong>1D plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(res::Result; x::String, y::String, class="default", not_class=[], kwargs...)</span></span>
-<span class="line"><span>plot(res::Result, y::String; kwargs...) # take x automatically from Result</span></span></code></pre></div><p>Default behaviour is to plot stable solutions as full lines, unstable as dashed.</p><p>If a sweep in two parameters were done, i.e., <code>dim(res)==2</code>, a one dimensional cut can be plotted by using the keyword <code>cut</code> were it takes a <code>Pair{Num, Float}</code> type entry. For example, <code>plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2))</code> plots a cut at <code>λ = 0.2</code>.</p><hr><p><strong>2D plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)</span></span></code></pre></div><p>To make the 2d plot less chaotic it is required to specify the specific <code>branch</code> to plot, labeled by a <code>Int64</code>.</p><p>The x and y axes are taken automatically from <code>res</code></p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/plotting_Plots.jl#L11" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="RecipesBase.plot!" href="#RecipesBase.plot!"><span class="jlbinding">RecipesBase.plot!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span>plot(res::Result, y::String; kwargs...) # take x automatically from Result</span></span></code></pre></div><p>Default behaviour is to plot stable solutions as full lines, unstable as dashed.</p><p>If a sweep in two parameters were done, i.e., <code>dim(res)==2</code>, a one dimensional cut can be plotted by using the keyword <code>cut</code> were it takes a <code>Pair{Num, Float}</code> type entry. For example, <code>plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2))</code> plots a cut at <code>λ = 0.2</code>.</p><hr><p><strong>2D plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)</span></span></code></pre></div><p>To make the 2d plot less chaotic it is required to specify the specific <code>branch</code> to plot, labeled by a <code>Int64</code>.</p><p>The x and y axes are taken automatically from <code>res</code></p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/plotting_Plots.jl#L11" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="RecipesBase.plot!" href="#RecipesBase.plot!"><span class="jlbinding">RecipesBase.plot!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> varargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Plot</span></span></code></pre></div><p>Similar to <code>plot</code> but adds a plot onto an existing plot.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/plotting_Plots.jl#L62" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.plot_phase_diagram" href="#HarmonicBalance.plot_phase_diagram"><span class="jlbinding">HarmonicBalance.plot_phase_diagram</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_phase_diagram</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Plot</span></span></code></pre></div><p>Similar to <code>plot</code> but adds a plot onto an existing plot.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/plotting_Plots.jl#L62" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.plot_phase_diagram" href="#HarmonicBalance.plot_phase_diagram"><span class="jlbinding">HarmonicBalance.plot_phase_diagram</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_phase_diagram</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Plot</span></span></code></pre></div><p>Plot the number of solutions in a <code>Result</code> object as a function of the parameters. Works with 1D and 2D datasets.</p><p>Class selection done by passing <code>String</code> or <code>Vector{String}</code> as kwarg:</p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>class::String : only count solutions in this class ("all" --> plot everything)</span></span>
-<span class="line"><span>not_class::String : do not count solutions in this class</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr()</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/plotting_Plots.jl#L275" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.plot_spaghetti" href="#HarmonicBalance.plot_spaghetti"><span class="jlbinding">HarmonicBalance.plot_spaghetti</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_spaghetti</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; x, y, z, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a three dimension line plot of a <code>Result</code> object as a function of the parameters. Works with 1D and 2D datasets.</p><p>Class selection done by passing <code>String</code> or <code>Vector{String}</code> as kwarg:</p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>class::String : only count solutions in this class ("all" --> plot everything)</span></span>
-<span class="line"><span>not_class::String : do not count solutions in this class</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr()</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/plotting_Plots.jl#L342-L354" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Limit-cycles" tabindex="-1">Limit cycles <a class="header-anchor" href="#Limit-cycles" aria-label="Permalink to "Limit cycles {#Limit-cycles}""></a></h2><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LimitCycles.get_limit_cycles" href="#HarmonicBalance.LimitCycles.get_limit_cycles"><span class="jlbinding">HarmonicBalance.LimitCycles.get_limit_cycles</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_limit_cycles</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, method</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalanceMethod</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, swept, fixed, ω_lc; kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Variant of <code>get_steady_states</code> for a limit cycle problem characterised by a Hopf frequency (usually called ω_lc)</p><p>Solutions with ω_lc = 0 are labelled unphysical since this contradicts the assumption of distinct harmonic variables corresponding to distinct harmonics.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LimitCycles/gauge_fixing.jl#L106-L113" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LimitCycles.get_cycle_variables" href="#HarmonicBalance.LimitCycles.get_cycle_variables"><span class="jlbinding">HarmonicBalance.LimitCycles.get_cycle_variables</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_cycle_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span>not_class::String : do not count solutions in this class</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr()</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/plotting_Plots.jl#L275" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.plot_spaghetti" href="#HarmonicBalance.plot_spaghetti"><span class="jlbinding">HarmonicBalance.plot_spaghetti</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_spaghetti</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; x, y, z, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a three dimension line plot of a <code>Result</code> object as a function of the parameters. Works with 1D and 2D datasets.</p><p>Class selection done by passing <code>String</code> or <code>Vector{String}</code> as kwarg:</p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>class::String : only count solutions in this class ("all" --> plot everything)</span></span>
+<span class="line"><span>not_class::String : do not count solutions in this class</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr()</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/plotting_Plots.jl#L342-L354" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Limit-cycles" tabindex="-1">Limit cycles <a class="header-anchor" href="#Limit-cycles" aria-label="Permalink to "Limit cycles {#Limit-cycles}""></a></h2><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LimitCycles.get_limit_cycles" href="#HarmonicBalance.LimitCycles.get_limit_cycles"><span class="jlbinding">HarmonicBalance.LimitCycles.get_limit_cycles</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_limit_cycles</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, method</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalanceMethod</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, swept, fixed, ω_lc; kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Variant of <code>get_steady_states</code> for a limit cycle problem characterised by a Hopf frequency (usually called ω_lc)</p><p>Solutions with ω_lc = 0 are labelled unphysical since this contradicts the assumption of distinct harmonic variables corresponding to distinct harmonics.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LimitCycles/gauge_fixing.jl#L106-L113" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LimitCycles.get_cycle_variables" href="#HarmonicBalance.LimitCycles.get_cycle_variables"><span class="jlbinding">HarmonicBalance.LimitCycles.get_cycle_variables</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_cycle_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω_lc</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{HarmonicVariable}</span></span></code></pre></div><p>Return the harmonic variables which participate in the limit cycle labelled by <code>ω_lc</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LimitCycles/gauge_fixing.jl#L22" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LimitCycles.add_pairs!" href="#HarmonicBalance.LimitCycles.add_pairs!"><span class="jlbinding">HarmonicBalance.LimitCycles.add_pairs!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">add_pairs!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; ω_lc</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, n</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Add a limit cycle harmonic <code>ω_lc</code> to the system Equivalent to adding <code>n</code> pairs of harmonics ω +- ω_lc for each existing ω.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LimitCycles/gauge_fixing.jl#L9-L14" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Linear-Response" tabindex="-1">Linear Response <a class="header-anchor" href="#Linear-Response" aria-label="Permalink to "Linear Response {#Linear-Response}""></a></h2><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.plot_eigenvalues-Union{Tuple{HarmonicBalance.Result{S, P, D, F} where {D, F<:FunctionWrappers.FunctionWrapper{Matrix{S}, Tuple{Vector{S}}}}}, Tuple{P}, Tuple{S}} where {S, P}" href="#HarmonicBalance.LinearResponse.plot_eigenvalues-Union{Tuple{HarmonicBalance.Result{S, P, D, F} where {D, F<:FunctionWrappers.FunctionWrapper{Matrix{S}, Tuple{Vector{S}}}}}, Tuple{P}, Tuple{S}} where {S, P}"><span class="jlbinding">HarmonicBalance.LinearResponse.plot_eigenvalues</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_eigenvalues</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, class</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">[</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"physical"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], type</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">:imag</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, projection</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">v </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, cscheme</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">:default</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot the eigenvalues of the jacobian in the rotating frame for Result <code>res</code> on <code>branch</code>. Either the real (<code>type=:real``) or imaginary part (</code>type=:imag``) can be plotted. The<code>projection</code> function ℜᵈ → ℜ is applied to the eigenvectors and defines the color of the eigenvalues. The color scheme can be set to a custom one or to the default one.</p><p>Any kwargs are fed to Plots' gr().</p><p>Solutions not belonging to the <code>physical</code> class are ignored.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/plotting.jl#L259-L267" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.plot_linear_response-Tuple{HarmonicBalance.Result, Num}" href="#HarmonicBalance.LinearResponse.plot_linear_response-Tuple{HarmonicBalance.Result, Num}"><span class="jlbinding">HarmonicBalance.LinearResponse.plot_linear_response</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_linear_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, nat_var</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; Ω_range, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, order</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, logscale</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">false</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, show_progress</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot the linear response to white noise of the variable <code>nat_var</code> for Result <code>res</code> on <code>branch</code> for input frequencies <code>Ω_range</code>. Slow-time derivatives up to <code>order</code> are kept in the process.</p><p>Any kwargs are fed to Plots' gr().</p><p>Solutions not belonging to the <code>physical</code> class are ignored.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/plotting.jl#L120-L129" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.plot_rotframe_jacobian_response-Union{Tuple{HarmonicBalance.Result{S, P, D, F} where {D, F<:FunctionWrappers.FunctionWrapper{Matrix{S}, Tuple{Vector{S}}}}}, Tuple{P}, Tuple{S}} where {S, P}" href="#HarmonicBalance.LinearResponse.plot_rotframe_jacobian_response-Union{Tuple{HarmonicBalance.Result{S, P, D, F} where {D, F<:FunctionWrappers.FunctionWrapper{Matrix{S}, Tuple{Vector{S}}}}}, Tuple{P}, Tuple{S}} where {S, P}"><span class="jlbinding">HarmonicBalance.LinearResponse.plot_rotframe_jacobian_response</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_rotframe_jacobian_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; Ω_range, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, logscale</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, damping_mod </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, show_progress</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot the linear response to white noise in the rotating frame for Result <code>res</code> on <code>branch</code> for input frequencies <code>Ω_range</code>. 'damping_mod' gets multiplied by the real part of the eigenvalues of the Jacobian in order to be able to make peaks with similar frequency separately identifiable.</p><p>Any kwargs are fed to Plots' gr().</p><p>Solutions not belonging to the <code>physical</code> class are ignored.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/plotting.jl#L214-L222" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_Jacobian" href="#HarmonicBalance.get_Jacobian"><span class="jlbinding">HarmonicBalance.get_Jacobian</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_Jacobian</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom)</span></span></code></pre></div><p>Obtain the symbolic Jacobian matrix of <code>eom</code>. This is the linearised left-hand side of F(u) = du/dT.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/Jacobian.jl#L59" target="_blank" rel="noreferrer">source</a></p><p>Obtain a Jacobian from a <code>DifferentialEquation</code> by first converting it into a <code>HarmonicEquation</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/Jacobian.jl#L77" target="_blank" rel="noreferrer">source</a></p><p>Get the Jacobian of a set of equations <code>eqs</code> with respect to the variables <code>vars</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/Jacobian.jl#L86" target="_blank" rel="noreferrer">source</a></p></details><h2 id="extensions" tabindex="-1">Extensions <a class="header-anchor" href="#extensions" aria-label="Permalink to "Extensions""></a></h2><h3 id="ordinarydiffeq" tabindex="-1">OrdinaryDiffEq <a class="header-anchor" href="#ordinarydiffeq" aria-label="Permalink to "OrdinaryDiffEq""></a></h3><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.AdiabaticSweep" href="#HarmonicBalance.AdiabaticSweep"><span class="jlbinding">HarmonicBalance.AdiabaticSweep</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><p>Represents a sweep of one or more parameters of a <code>HarmonicEquation</code>. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.</p><p>Sweeps of different variables can be combined using <code>+</code>.</p><p><strong>Fields</strong></p><ul><li><code>functions::Dict{Num, Function}</code>: Maps each swept parameter to a function.</li></ul><p><strong>Examples</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># create a sweep of parameter a from 0 to 1 over time 0 -> 100</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{HarmonicVariable}</span></span></code></pre></div><p>Return the harmonic variables which participate in the limit cycle labelled by <code>ω_lc</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LimitCycles/gauge_fixing.jl#L22" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LimitCycles.add_pairs!" href="#HarmonicBalance.LimitCycles.add_pairs!"><span class="jlbinding">HarmonicBalance.LimitCycles.add_pairs!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">add_pairs!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; ω_lc</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, n</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Add a limit cycle harmonic <code>ω_lc</code> to the system Equivalent to adding <code>n</code> pairs of harmonics ω +- ω_lc for each existing ω.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LimitCycles/gauge_fixing.jl#L9-L14" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Linear-Response" tabindex="-1">Linear Response <a class="header-anchor" href="#Linear-Response" aria-label="Permalink to "Linear Response {#Linear-Response}""></a></h2><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.plot_eigenvalues-Union{Tuple{HarmonicBalance.Result{S, P, D, F} where {D, F<:FunctionWrappers.FunctionWrapper{Matrix{S}, Tuple{Vector{S}}}}}, Tuple{P}, Tuple{S}} where {S, P}" href="#HarmonicBalance.LinearResponse.plot_eigenvalues-Union{Tuple{HarmonicBalance.Result{S, P, D, F} where {D, F<:FunctionWrappers.FunctionWrapper{Matrix{S}, Tuple{Vector{S}}}}}, Tuple{P}, Tuple{S}} where {S, P}"><span class="jlbinding">HarmonicBalance.LinearResponse.plot_eigenvalues</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_eigenvalues</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, class</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">[</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"physical"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], type</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">:imag</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, projection</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">v </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, cscheme</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">:default</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot the eigenvalues of the jacobian in the rotating frame for Result <code>res</code> on <code>branch</code>. Either the real (<code>type=:real``) or imaginary part (</code>type=:imag``) can be plotted. The<code>projection</code> function ℜᵈ → ℜ is applied to the eigenvectors and defines the color of the eigenvalues. The color scheme can be set to a custom one or to the default one.</p><p>Any kwargs are fed to Plots' gr().</p><p>Solutions not belonging to the <code>physical</code> class are ignored.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/plotting.jl#L259-L267" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.plot_linear_response-Tuple{HarmonicBalance.Result, Num}" href="#HarmonicBalance.LinearResponse.plot_linear_response-Tuple{HarmonicBalance.Result, Num}"><span class="jlbinding">HarmonicBalance.LinearResponse.plot_linear_response</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_linear_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, nat_var</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; Ω_range, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, order</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, logscale</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">false</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, show_progress</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot the linear response to white noise of the variable <code>nat_var</code> for Result <code>res</code> on <code>branch</code> for input frequencies <code>Ω_range</code>. Slow-time derivatives up to <code>order</code> are kept in the process.</p><p>Any kwargs are fed to Plots' gr().</p><p>Solutions not belonging to the <code>physical</code> class are ignored.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/plotting.jl#L120-L129" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.plot_rotframe_jacobian_response-Union{Tuple{HarmonicBalance.Result{S, P, D, F} where {D, F<:FunctionWrappers.FunctionWrapper{Matrix{S}, Tuple{Vector{S}}}}}, Tuple{P}, Tuple{S}} where {S, P}" href="#HarmonicBalance.LinearResponse.plot_rotframe_jacobian_response-Union{Tuple{HarmonicBalance.Result{S, P, D, F} where {D, F<:FunctionWrappers.FunctionWrapper{Matrix{S}, Tuple{Vector{S}}}}}, Tuple{P}, Tuple{S}} where {S, P}"><span class="jlbinding">HarmonicBalance.LinearResponse.plot_rotframe_jacobian_response</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_rotframe_jacobian_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; Ω_range, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, logscale</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, damping_mod </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, show_progress</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot the linear response to white noise in the rotating frame for Result <code>res</code> on <code>branch</code> for input frequencies <code>Ω_range</code>. 'damping_mod' gets multiplied by the real part of the eigenvalues of the Jacobian in order to be able to make peaks with similar frequency separately identifiable.</p><p>Any kwargs are fed to Plots' gr().</p><p>Solutions not belonging to the <code>physical</code> class are ignored.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/plotting.jl#L214-L222" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_Jacobian" href="#HarmonicBalance.get_Jacobian"><span class="jlbinding">HarmonicBalance.get_Jacobian</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_Jacobian</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom)</span></span></code></pre></div><p>Obtain the symbolic Jacobian matrix of <code>eom</code>. This is the linearised left-hand side of F(u) = du/dT.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/Jacobian.jl#L59" target="_blank" rel="noreferrer">source</a></p><p>Obtain a Jacobian from a <code>DifferentialEquation</code> by first converting it into a <code>HarmonicEquation</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/Jacobian.jl#L77" target="_blank" rel="noreferrer">source</a></p><p>Get the Jacobian of a set of equations <code>eqs</code> with respect to the variables <code>vars</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/Jacobian.jl#L86" target="_blank" rel="noreferrer">source</a></p></details><h2 id="extensions" tabindex="-1">Extensions <a class="header-anchor" href="#extensions" aria-label="Permalink to "Extensions""></a></h2><h3 id="ordinarydiffeq" tabindex="-1">OrdinaryDiffEq <a class="header-anchor" href="#ordinarydiffeq" aria-label="Permalink to "OrdinaryDiffEq""></a></h3><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.AdiabaticSweep" href="#HarmonicBalance.AdiabaticSweep"><span class="jlbinding">HarmonicBalance.AdiabaticSweep</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><p>Represents a sweep of one or more parameters of a <code>HarmonicEquation</code>. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.</p><p>Sweeps of different variables can be combined using <code>+</code>.</p><p><strong>Fields</strong></p><ul><li><code>functions::Dict{Num, Function}</code>: Maps each swept parameter to a function.</li></ul><p><strong>Examples</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># create a sweep of parameter a from 0 to 1 over time 0 -> 100</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> a,b;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(a </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">100</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">));</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep[a](</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">50</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
@@ -197,22 +197,22 @@
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">([a </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], b </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">]], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">100</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span></code></pre></div><p>Successive sweeps can be combined,</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep1 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.95</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep2 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(λ </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.05</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.01</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">4e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep1 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep2</span></span></code></pre></div><p>multiple parameters can be swept simultaneously,</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">([ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.95</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], λ </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">5e-2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1e-2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">]], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span></code></pre></div><p>and custom sweep functions may be used.</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#6F42C1;--shiki-dark:#B392F0;">ωfunc</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ωfunc)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/types.jl#L9-L48" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.follow_branch" href="#HarmonicBalance.follow_branch"><span class="jlbinding">HarmonicBalance.follow_branch</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">follow_branch</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ωfunc)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/types.jl#L9-L48" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.follow_branch" href="#HarmonicBalance.follow_branch"><span class="jlbinding">HarmonicBalance.follow_branch</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">follow_branch</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> starting_branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int64</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> y,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> tf,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ϵ</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Tuple{Any, Any}</span></span></code></pre></div><p>Return the indexes and values following stable branches along a 1D sweep. When a no stable solutions are found (e.g. in a bifurcation), the next stable solution is calculated by time evolving the previous solution (quench).</p><p><strong>Keyword arguments</strong></p><ul><li><p><code>y</code>: Dependent variable expression (parsed into Symbolics.jl) to evaluate the followed solution branches on .</p></li><li><p><code>sweep</code>: Direction for the sweeping of solutions. A <code>right</code> (<code>left</code>) sweep proceeds from the first (last) solution, ordered as the sweeping parameter.</p></li><li><p><code>tf</code>: time to reach steady</p></li><li><p><code>ϵ</code>: small random perturbation applied to quenched solution, in a bifurcation in order to favour convergence in cases where multiple solutions are identically accessible (e.g. symmetry breaking into two equal amplitude states)</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/hysteresis_sweep.jl#L15" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.plot_1D_solutions_branch" href="#HarmonicBalance.plot_1D_solutions_branch"><span class="jlbinding">HarmonicBalance.plot_1D_solutions_branch</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><p>1D plot with the followed branch highlighted</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/plotting.jl#L62-L64" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.follow_branch-Tuple{Int64, HarmonicBalance.Result}-manual-API" href="#HarmonicBalance.follow_branch-Tuple{Int64, HarmonicBalance.Result}-manual-API"><span class="jlbinding">HarmonicBalance.follow_branch</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">follow_branch</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Tuple{Any, Any}</span></span></code></pre></div><p>Return the indexes and values following stable branches along a 1D sweep. When a no stable solutions are found (e.g. in a bifurcation), the next stable solution is calculated by time evolving the previous solution (quench).</p><p><strong>Keyword arguments</strong></p><ul><li><p><code>y</code>: Dependent variable expression (parsed into Symbolics.jl) to evaluate the followed solution branches on .</p></li><li><p><code>sweep</code>: Direction for the sweeping of solutions. A <code>right</code> (<code>left</code>) sweep proceeds from the first (last) solution, ordered as the sweeping parameter.</p></li><li><p><code>tf</code>: time to reach steady</p></li><li><p><code>ϵ</code>: small random perturbation applied to quenched solution, in a bifurcation in order to favour convergence in cases where multiple solutions are identically accessible (e.g. symmetry breaking into two equal amplitude states)</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/hysteresis_sweep.jl#L15" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.plot_1D_solutions_branch" href="#HarmonicBalance.plot_1D_solutions_branch"><span class="jlbinding">HarmonicBalance.plot_1D_solutions_branch</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><p>1D plot with the followed branch highlighted</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/plotting.jl#L62-L64" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.follow_branch-Tuple{Int64, HarmonicBalance.Result}-manual-API" href="#HarmonicBalance.follow_branch-Tuple{Int64, HarmonicBalance.Result}-manual-API"><span class="jlbinding">HarmonicBalance.follow_branch</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">follow_branch</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> starting_branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int64</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> y,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> tf,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ϵ</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Tuple{Any, Any}</span></span></code></pre></div><p>Return the indexes and values following stable branches along a 1D sweep. When a no stable solutions are found (e.g. in a bifurcation), the next stable solution is calculated by time evolving the previous solution (quench).</p><p><strong>Keyword arguments</strong></p><ul><li><p><code>y</code>: Dependent variable expression (parsed into Symbolics.jl) to evaluate the followed solution branches on .</p></li><li><p><code>sweep</code>: Direction for the sweeping of solutions. A <code>right</code> (<code>left</code>) sweep proceeds from the first (last) solution, ordered as the sweeping parameter.</p></li><li><p><code>tf</code>: time to reach steady</p></li><li><p><code>ϵ</code>: small random perturbation applied to quenched solution, in a bifurcation in order to favour convergence in cases where multiple solutions are identically accessible (e.g. symmetry breaking into two equal amplitude states)</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/hysteresis_sweep.jl#L15" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-API" href="#RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-API"><span class="jlbinding">RecipesBase.plot</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESolution</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, f</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, harm_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a function <code>f</code> of a time-dependent solution <code>soln</code> of <code>harm_eq</code>.</p><p><strong>As a function of time</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p><code>f</code> is parsed by Symbolics.jl</p><p><strong>parametric plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p>Parametric plot of f[1] against f[2]</p><p>Also callable as plot!</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/plotting.jl#L4-L22" target="_blank" rel="noreferrer">source</a></p></details><h3 id="steadystatesweep" tabindex="-1">SteadyStateSweep <a class="header-anchor" href="#steadystatesweep" aria-label="Permalink to "SteadyStateSweep""></a></h3><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.steady_state_sweep" href="#HarmonicBalance.steady_state_sweep"><span class="jlbinding">HarmonicBalance.steady_state_sweep</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">steady_state_sweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(prob</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">SteadyStateProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, alg</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DynamicSS</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; varied</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Pair</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Sweeps through a range of parameter values using a dynamic steady state solver <code>DynamicSS</code> of the <code>SteadyStateDiffEq.jl</code> package. Given a steady state problem and a parameter to vary, computes the steady state solution for each value in the sweep range. The solutions are returned as a vector where each element corresponds to the steady state found at that parameter value.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/SteadyStateDiffEqExt.jl#L10-L18" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">steady_state_sweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(prob_np</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">NonlinearProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, prob_ss</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">SteadyStateProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> alg_np, alg_ss</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DynamicSS</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; varied</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Pair</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Performs a parameter sweep by combining nonlinear root <code>alg_np</code> and steady state solvers <code>alg_ss</code>. For each parameter value, it first attempts a direct nonlinear root solver and checks its stability. If the solution is unstable or not found, it switches to a dynamic steady state solver. This hybrid approach is much faster then only using a steady state solver.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/SteadyStateDiffEqExt.jl#L37-L45" target="_blank" rel="noreferrer">source</a></p></details><h3 id="modelingtoolkit" tabindex="-1">ModelingToolkit <a class="header-anchor" href="#modelingtoolkit" aria-label="Permalink to "ModelingToolkit""></a></h3><details class="jldocstring custom-block" open><summary><a id="SciMLBase.ODEProblem" href="#SciMLBase.ODEProblem"><span class="jlbinding">SciMLBase.ODEProblem</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODEProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Tuple{Any, Any}</span></span></code></pre></div><p>Return the indexes and values following stable branches along a 1D sweep. When a no stable solutions are found (e.g. in a bifurcation), the next stable solution is calculated by time evolving the previous solution (quench).</p><p><strong>Keyword arguments</strong></p><ul><li><p><code>y</code>: Dependent variable expression (parsed into Symbolics.jl) to evaluate the followed solution branches on .</p></li><li><p><code>sweep</code>: Direction for the sweeping of solutions. A <code>right</code> (<code>left</code>) sweep proceeds from the first (last) solution, ordered as the sweeping parameter.</p></li><li><p><code>tf</code>: time to reach steady</p></li><li><p><code>ϵ</code>: small random perturbation applied to quenched solution, in a bifurcation in order to favour convergence in cases where multiple solutions are identically accessible (e.g. symmetry breaking into two equal amplitude states)</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/hysteresis_sweep.jl#L15" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-API" href="#RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-API"><span class="jlbinding">RecipesBase.plot</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESolution</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, f</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, harm_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a function <code>f</code> of a time-dependent solution <code>soln</code> of <code>harm_eq</code>.</p><p><strong>As a function of time</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p><code>f</code> is parsed by Symbolics.jl</p><p><strong>parametric plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p>Parametric plot of f[1] against f[2]</p><p>Also callable as plot!</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/plotting.jl#L4-L22" target="_blank" rel="noreferrer">source</a></p></details><h3 id="steadystatesweep" tabindex="-1">SteadyStateSweep <a class="header-anchor" href="#steadystatesweep" aria-label="Permalink to "SteadyStateSweep""></a></h3><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.steady_state_sweep" href="#HarmonicBalance.steady_state_sweep"><span class="jlbinding">HarmonicBalance.steady_state_sweep</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">steady_state_sweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(prob</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">SteadyStateProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, alg</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DynamicSS</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; varied</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Pair</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Sweeps through a range of parameter values using a dynamic steady state solver <code>DynamicSS</code> of the <code>SteadyStateDiffEq.jl</code> package. Given a steady state problem and a parameter to vary, computes the steady state solution for each value in the sweep range. The solutions are returned as a vector where each element corresponds to the steady state found at that parameter value.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/SteadyStateDiffEqExt.jl#L10-L18" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">steady_state_sweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(prob_np</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">NonlinearProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, prob_ss</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">SteadyStateProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> alg_np, alg_ss</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DynamicSS</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; varied</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Pair</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Performs a parameter sweep by combining nonlinear root <code>alg_np</code> and steady state solvers <code>alg_ss</code>. For each parameter value, it first attempts a direct nonlinear root solver and checks its stability. If the solution is unstable or not found, it switches to a dynamic steady state solver. This hybrid approach is much faster then only using a steady state solver.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/SteadyStateDiffEqExt.jl#L37-L45" target="_blank" rel="noreferrer">source</a></p></details><h3 id="modelingtoolkit" tabindex="-1">ModelingToolkit <a class="header-anchor" href="#modelingtoolkit" aria-label="Permalink to "ModelingToolkit""></a></h3><details class="jldocstring custom-block" open><summary><a id="SciMLBase.ODEProblem" href="#SciMLBase.ODEProblem"><span class="jlbinding">SciMLBase.ODEProblem</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODEProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> fixed_parameters;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep,</span></span>
@@ -220,27 +220,27 @@
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> timespan,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> perturb_initial,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Creates an ODEProblem object used by OrdinaryDiffEqTsit5.jl from the equations in <code>eom</code> to simulate time-evolution within <code>timespan</code>. <code>fixed_parameters</code> must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If <code>u0</code> is specified, it is used as an initial condition; otherwise the values from <code>fixed_parameters</code> are used.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/ODEProblem.jl#L3-L9" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODEProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Creates an ODEProblem object used by OrdinaryDiffEqTsit5.jl from the equations in <code>eom</code> to simulate time-evolution within <code>timespan</code>. <code>fixed_parameters</code> must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If <code>u0</code> is specified, it is used as an initial condition; otherwise the values from <code>fixed_parameters</code> are used.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/ODEProblem.jl#L3-L9" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODEProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Union{DifferentialEquation, HarmonicEquation}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> u0,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> tspan</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Tuple</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> p</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">AbstractDict</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> in_place,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.ODEProblem from a DifferentialEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/ModelingToolkitExt.jl#L100" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="ModelingToolkit.ODESystem" href="#ModelingToolkit.ODESystem"><span class="jlbinding">ModelingToolkit.ODESystem</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESystem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.ODESystem from a HarmonicEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/ModelingToolkitExt.jl#L39" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESystem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.ODESystem from a DifferentialEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/ModelingToolkitExt.jl#L69" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="SciMLBase.SteadyStateProblem" href="#SciMLBase.SteadyStateProblem"><span class="jlbinding">SciMLBase.SteadyStateProblem</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">SteadyStateProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.ODEProblem from a DifferentialEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/ModelingToolkitExt.jl#L100" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="ModelingToolkit.ODESystem" href="#ModelingToolkit.ODESystem"><span class="jlbinding">ModelingToolkit.ODESystem</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESystem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.ODESystem from a HarmonicEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/ModelingToolkitExt.jl#L39" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESystem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.ODESystem from a DifferentialEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/ModelingToolkitExt.jl#L69" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="SciMLBase.SteadyStateProblem" href="#SciMLBase.SteadyStateProblem"><span class="jlbinding">SciMLBase.SteadyStateProblem</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">SteadyStateProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> u0,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> p</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">AbstractDict</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> in_place,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.SteadyStateProblem from a DifferentialEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/ModelingToolkitExt.jl#L135" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="SciMLBase.NonlinearProblem" href="#SciMLBase.NonlinearProblem"><span class="jlbinding">SciMLBase.NonlinearProblem</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">NonlinearProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.SteadyStateProblem from a DifferentialEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/ModelingToolkitExt.jl#L135" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="SciMLBase.NonlinearProblem" href="#SciMLBase.NonlinearProblem"><span class="jlbinding">SciMLBase.NonlinearProblem</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">NonlinearProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> u0,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> p</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">AbstractDict</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> in_place,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.NonlinearProblem from a DifferentialEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/ModelingToolkitExt.jl#L123" target="_blank" rel="noreferrer">source</a></p></details></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/API.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>State Dependent Perturbation</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Bibliography</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Creates and ModelingToolkit.NonlinearProblem from a DifferentialEquation.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/ModelingToolkitExt.jl#L123" target="_blank" rel="noreferrer">source</a></p></details></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/API.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>State Dependent Perturbation</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Bibliography</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">/</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T))</span></span>
<span class="line"></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">/</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T))</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/KrylovBogoliubov.jl#L25" target="_blank" rel="noreferrer">source</a></p></details><p>For further information and a detailed understanding of this method, refer to <a href="https://en.wikipedia.org/wiki/Krylov%E2%80%93Bogoliubov_averaging_method" target="_blank" rel="noreferrer">Krylov-Bogoliubov averaging method on Wikipedia</a>.</p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/Krylov-Bogoliubov_method.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav 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+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">/</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T))</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/KrylovBogoliubov.jl#L25" target="_blank" rel="noreferrer">source</a></p></details><p>For further information and a detailed understanding of this method, refer to <a href="https://en.wikipedia.org/wiki/Krylov%E2%80%93Bogoliubov_averaging_method" target="_blank" rel="noreferrer">Krylov-Bogoliubov averaging method on Wikipedia</a>.</p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/Krylov-Bogoliubov_method.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav 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Started</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Theoretical Background</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/harmonic_balance" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Floquet expansions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" 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class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/steady_states" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/classification" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Classifying Solutions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Transient Dynamics</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Examples</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Three Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Oscillator</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>State Dependent Perturbation</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><div 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link link" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Bibliography</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible has-active" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Manual</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Entering Equations of Motion</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Effective System</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/methods" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Krylov-Bogoliubov</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Time Evolution</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Plotting</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/saving" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Saving and Loading</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-a9a9e638 data-v-91765379><div class="VPDoc has-sidebar has-aside" data-v-91765379 data-v-83890dd9><!--[--><!--]--><div class="container" data-v-83890dd9><div class="aside" data-v-83890dd9><div class="aside-curtain" data-v-83890dd9></div><div class="aside-container" data-v-83890dd9><div class="aside-content" data-v-83890dd9><div class="VPDocAside" data-v-83890dd9 data-v-6d7b3c46><!--[--><!--]--><!--[--><!--]--><nav aria-labelledby="doc-outline-aria-label" class="VPDocAsideOutline" data-v-6d7b3c46 data-v-b38bf2ff><div class="content" data-v-b38bf2ff><div class="outline-marker" data-v-b38bf2ff></div><div aria-level="2" class="outline-title" id="doc-outline-aria-label" role="heading" data-v-b38bf2ff>On this page</div><ul class="VPDocOutlineItem root" data-v-b38bf2ff data-v-3f927ebe><!--[--><!--]--></ul></div></nav><!--[--><!--]--><div class="spacer" data-v-6d7b3c46></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--]--></div></div></div></div><div class="content" data-v-83890dd9><div class="content-container" data-v-83890dd9><!--[--><!--]--><main class="main" data-v-83890dd9><div style="position:relative;" class="vp-doc _HarmonicBalance_jl_dev_manual_entering_eom" data-v-83890dd9><div><h1 id="Entering-equations-of-motion" tabindex="-1">Entering equations of motion <a class="header-anchor" href="#Entering-equations-of-motion" aria-label="Permalink to "Entering equations of motion {#Entering-equations-of-motion}""></a></h1><p>The struct <code>DifferentialEquation</code> is the primary input method; it holds an ODE or a coupled system of ODEs composed of terms with harmonic time-dependence The dependent variables are specified during input, any other symbols are identified as parameters. Information on which variable is to be expanded in which harmonic is specified using <code>add_harmonic!</code>.</p><p><code>DifferentialEquation.equations</code> stores a dictionary assigning variables to equations. This information is necessary because the harmonics belonging to a variable are later used to Fourier-transform its corresponding ODE.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.d-manual-entering_eom" href="#HarmonicBalance.d-manual-entering_eom"><span class="jlbinding">HarmonicBalance.d</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><p>The derivative of f w.r.t. x of degree deg</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicVariable.jl#L108" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.DifferentialEquation-manual-entering_eom" href="#HarmonicBalance.DifferentialEquation-manual-entering_eom"><span class="jlbinding">HarmonicBalance.DifferentialEquation</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> DifferentialEquation</span></span></code></pre></div><p>Holds differential equation(s) of motion and a set of harmonics to expand each variable. This is the primary input for <code>HarmonicBalance.jl</code>. After inputting the equations, the harmonics ansatz needs to be specified using <code>add_harmonic!</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>equations::OrderedCollections.OrderedDict{Num, Equation}</code>: Assigns to each variable an equation of motion.</p></li><li><p><code>harmonics::OrderedCollections.OrderedDict{Num, OrderedCollections.OrderedSet{Num}}</code>: Assigns to each variable a set of harmonics.</p></li></ul><p><strong>Example</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> t, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">y</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), ω0, ω, F, k;</span></span>
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class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/steady_states" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/classification" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Classifying Solutions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Transient Dynamics</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Examples</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Three Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Oscillator</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>State Dependent Perturbation</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible has-active" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Resources</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/API" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>API</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Bibliography</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible has-active" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Manual</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Entering Equations of Motion</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Effective System</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/methods" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Krylov-Bogoliubov</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Time Evolution</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Plotting</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/saving" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Saving and Loading</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-a9a9e638 data-v-91765379><div class="VPDoc has-sidebar has-aside" data-v-91765379 data-v-83890dd9><!--[--><!--]--><div class="container" data-v-83890dd9><div class="aside" data-v-83890dd9><div class="aside-curtain" data-v-83890dd9></div><div class="aside-container" data-v-83890dd9><div class="aside-content" data-v-83890dd9><div class="VPDocAside" data-v-83890dd9 data-v-6d7b3c46><!--[--><!--]--><!--[--><!--]--><nav aria-labelledby="doc-outline-aria-label" class="VPDocAsideOutline" data-v-6d7b3c46 data-v-b38bf2ff><div class="content" data-v-b38bf2ff><div class="outline-marker" data-v-b38bf2ff></div><div aria-level="2" class="outline-title" id="doc-outline-aria-label" role="heading" data-v-b38bf2ff>On this page</div><ul class="VPDocOutlineItem root" data-v-b38bf2ff data-v-3f927ebe><!--[--><!--]--></ul></div></nav><!--[--><!--]--><div class="spacer" data-v-6d7b3c46></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--]--></div></div></div></div><div class="content" data-v-83890dd9><div class="content-container" data-v-83890dd9><!--[--><!--]--><main class="main" data-v-83890dd9><div style="position:relative;" class="vp-doc _HarmonicBalance_jl_dev_manual_entering_eom" data-v-83890dd9><div><h1 id="Entering-equations-of-motion" tabindex="-1">Entering equations of motion <a class="header-anchor" href="#Entering-equations-of-motion" aria-label="Permalink to "Entering equations of motion {#Entering-equations-of-motion}""></a></h1><p>The struct <code>DifferentialEquation</code> is the primary input method; it holds an ODE or a coupled system of ODEs composed of terms with harmonic time-dependence The dependent variables are specified during input, any other symbols are identified as parameters. Information on which variable is to be expanded in which harmonic is specified using <code>add_harmonic!</code>.</p><p><code>DifferentialEquation.equations</code> stores a dictionary assigning variables to equations. This information is necessary because the harmonics belonging to a variable are later used to Fourier-transform its corresponding ODE.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.d-manual-entering_eom" href="#HarmonicBalance.d-manual-entering_eom"><span class="jlbinding">HarmonicBalance.d</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><p>The derivative of f w.r.t. x of degree deg</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicVariable.jl#L108" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.DifferentialEquation-manual-entering_eom" href="#HarmonicBalance.DifferentialEquation-manual-entering_eom"><span class="jlbinding">HarmonicBalance.DifferentialEquation</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> DifferentialEquation</span></span></code></pre></div><p>Holds differential equation(s) of motion and a set of harmonics to expand each variable. This is the primary input for <code>HarmonicBalance.jl</code>. After inputting the equations, the harmonics ansatz needs to be specified using <code>add_harmonic!</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>equations::OrderedCollections.OrderedDict{Num, Equation}</code>: Assigns to each variable an equation of motion.</p></li><li><p><code>harmonics::OrderedCollections.OrderedDict{Num, OrderedCollections.OrderedSet{Num}}</code>: Assigns to each variable a set of harmonics.</p></li></ul><p><strong>Example</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> t, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">y</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), ω0, ω, F, k;</span></span>
<span class="line"></span>
<span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># equivalent ways to enter the simple harmonic oscillator</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">d</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(x,t,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> *</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> x </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">t), x);</span></span>
@@ -34,7 +34,7 @@
<span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># two coupled oscillators, one of them driven</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">d</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(x,t,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> *</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> x </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> k</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">d</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(y,t,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> *</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> y </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> k</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">x] </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], [x,y]</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">);</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.add_harmonic!-manual-entering_eom" href="#HarmonicBalance.add_harmonic!-manual-entering_eom"><span class="jlbinding">HarmonicBalance.add_harmonic!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">add_harmonic!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, var</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ω)</span></span></code></pre></div><p>Add the harmonic <code>ω</code> to the harmonic ansatz used to expand the variable <code>var</code> in <code>diff_eom</code>.</p><p><strong>Example</strong></p><p><strong>define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> t, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">y</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), ω0, ω, F, k;</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">);</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.add_harmonic!-manual-entering_eom" href="#HarmonicBalance.add_harmonic!-manual-entering_eom"><span class="jlbinding">HarmonicBalance.add_harmonic!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">add_harmonic!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, var</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ω)</span></span></code></pre></div><p>Add the harmonic <code>ω</code> to the harmonic ansatz used to expand the variable <code>var</code> in <code>diff_eom</code>.</p><p><strong>Example</strong></p><p><strong>define the simple harmonic oscillator and specify that x(t) oscillates with frequency ω</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> t, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">y</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t), ω0, ω, F, k;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> diff_eq </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">d</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(x,t,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> *</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> x </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">t), x);</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> add_harmonic!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eq, x, ��) </span><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># expand x using ω</span></span>
<span class="line"></span>
@@ -42,10 +42,10 @@
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Variables</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Harmonic ansatz</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω;</span></span>
<span class="line"></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t))) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L88" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="Symbolics.get_variables-Tuple{DifferentialEquation}-manual-entering_eom" href="#Symbolics.get_variables-Tuple{DifferentialEquation}-manual-entering_eom"><span class="jlbinding">Symbolics.get_variables</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Num}</span></span></code></pre></div><p>Return the dependent variables of <code>diff_eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L113" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}-manual-entering_eom" href="#HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}-manual-entering_eom"><span class="jlbinding">HarmonicBalance.get_independent_variables</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_independent_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">x</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t))) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L88" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="Symbolics.get_variables-Tuple{DifferentialEquation}-manual-entering_eom" href="#Symbolics.get_variables-Tuple{DifferentialEquation}-manual-entering_eom"><span class="jlbinding">Symbolics.get_variables</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector{Num}</span></span></code></pre></div><p>Return the dependent variables of <code>diff_eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L113" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}-manual-entering_eom" href="#HarmonicBalance.get_independent_variables-Tuple{DifferentialEquation}-manual-entering_eom"><span class="jlbinding">HarmonicBalance.get_independent_variables</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_independent_variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> diff_eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Return the independent dependent variables of <code>diff_eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/DifferentialEquation.jl#L131" target="_blank" rel="noreferrer">source</a></p></details></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/entering_eom.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Bibliography</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Computing Effective System</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Return the independent dependent variables of <code>diff_eom</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/DifferentialEquation.jl#L131" target="_blank" rel="noreferrer">source</a></p></details></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/entering_eom.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Bibliography</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Computing Effective System</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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@@ -53,17 +53,17 @@
<span class="line"></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F</span></span>
<span class="line"></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L295-L336" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.harmonic_ansatz-manual-extracting_harmonics" href="#HarmonicBalance.harmonic_ansatz-manual-extracting_harmonics"><span class="jlbinding">HarmonicBalance.harmonic_ansatz</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">harmonic_ansatz</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, time</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; coordinates</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"Cartesian"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Expand each variable of <code>diff_eom</code> using the harmonics assigned to it with <code>time</code> as the time variable. For each harmonic of each variable, instance(s) of <code>HarmonicVariable</code> are automatically created and named.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L72-L78" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.slow_flow-manual-extracting_harmonics" href="#HarmonicBalance.slow_flow-manual-extracting_harmonics"><span class="jlbinding">HarmonicBalance.slow_flow</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">slow_flow</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; fast_time</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, slow_time</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, degree</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Removes all derivatives w.r.t <code>fast_time</code> (and their products) in <code>eom</code> of power <code>degree</code>. In the remaining derivatives, <code>fast_time</code> is replaced by <code>slow_time</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L139-L144" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.fourier_transform-manual-extracting_harmonics" href="#HarmonicBalance.fourier_transform-manual-extracting_harmonics"><span class="jlbinding">HarmonicBalance.fourier_transform</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">fourier_transform</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">v1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">//</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ω</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Differential</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">u1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(T)) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">~</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L295-L336" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.harmonic_ansatz-manual-extracting_harmonics" href="#HarmonicBalance.harmonic_ansatz-manual-extracting_harmonics"><span class="jlbinding">HarmonicBalance.harmonic_ansatz</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">harmonic_ansatz</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, time</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; coordinates</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"Cartesian"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Expand each variable of <code>diff_eom</code> using the harmonics assigned to it with <code>time</code> as the time variable. For each harmonic of each variable, instance(s) of <code>HarmonicVariable</code> are automatically created and named.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L72-L78" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.slow_flow-manual-extracting_harmonics" href="#HarmonicBalance.slow_flow-manual-extracting_harmonics"><span class="jlbinding">HarmonicBalance.slow_flow</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">slow_flow</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; fast_time</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, slow_time</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, degree</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Removes all derivatives w.r.t <code>fast_time</code> (and their products) in <code>eom</code> of power <code>degree</code>. In the remaining derivatives, <code>fast_time</code> is replaced by <code>slow_time</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L139-L144" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.fourier_transform-manual-extracting_harmonics" href="#HarmonicBalance.fourier_transform-manual-extracting_harmonics"><span class="jlbinding">HarmonicBalance.fourier_transform</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">fourier_transform</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> time</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Extract the Fourier components of <code>eom</code> corresponding to the harmonics specified in <code>eom.variables</code>. For each non-zero harmonic of each variable, 2 equations are generated (cos and sin Fourier coefficients). For each zero (constant) harmonic, 1 equation is generated <code>time</code> does not appear in the resulting equations anymore.</p><p>Underlying assumption: all time-dependences are harmonic.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L256" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.ExprUtils.drop_powers-manual-extracting_harmonics" href="#HarmonicBalance.ExprUtils.drop_powers-manual-extracting_harmonics"><span class="jlbinding">HarmonicBalance.ExprUtils.drop_powers</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">drop_powers</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(expr, vars, deg)</span></span></code></pre></div><p>Remove parts of <code>expr</code> where the combined power of <code>vars</code> is => <code>deg</code>.</p><p><strong>Example</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> x,y;</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Extract the Fourier components of <code>eom</code> corresponding to the harmonics specified in <code>eom.variables</code>. For each non-zero harmonic of each variable, 2 equations are generated (cos and sin Fourier coefficients). For each zero (constant) harmonic, 1 equation is generated <code>time</code> does not appear in the resulting equations anymore.</p><p>Underlying assumption: all time-dependences are harmonic.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L256" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.ExprUtils.drop_powers-manual-extracting_harmonics" href="#HarmonicBalance.ExprUtils.drop_powers-manual-extracting_harmonics"><span class="jlbinding">HarmonicBalance.ExprUtils.drop_powers</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">drop_powers</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(expr, vars, deg)</span></span></code></pre></div><p>Remove parts of <code>expr</code> where the combined power of <code>vars</code> is => <code>deg</code>.</p><p><strong>Example</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> x,y;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">drop_powers</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, x, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> +</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">drop_powers</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, [x,y], </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">drop_powers</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">((x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> +</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y)</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">3</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, [x,y], </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">3</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> +</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> y</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> +</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/ExprUtils/drop_powers.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><h2 id="HarmonicVariable-and-HarmonicEquation-types" tabindex="-1">HarmonicVariable and HarmonicEquation types <a class="header-anchor" href="#HarmonicVariable-and-HarmonicEquation-types" aria-label="Permalink to "HarmonicVariable and HarmonicEquation types {#HarmonicVariable-and-HarmonicEquation-types}""></a></h2><p>The equations governing the harmonics are stored using the two following structs. 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xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>, which is needed to later reconstruct <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.611ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2038 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 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id="HarmonicBalance.HarmonicVariable-manual-extracting_harmonics" href="#HarmonicBalance.HarmonicVariable-manual-extracting_harmonics"><span class="jlbinding">HarmonicBalance.HarmonicVariable</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicVariable</span></span></code></pre></div><p>Holds a variable stored under <code>symbol</code> describing the harmonic <code>ω</code> of <code>natural_variable</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>symbol::Num</code>: Symbol of the variable in the HarmonicBalance namespace.</p></li><li><p><code>name::String</code>: Human-readable labels of the variable, used for plotting.</p></li><li><p><code>type::String</code>: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)</p></li><li><p><code>ω::Num</code>: The harmonic being described.</p></li><li><p><code>natural_variable::Num</code>: The natural variable whose harmonic is being described.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicVariable.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><p>When the full set of equations of motion is expanded using the harmonic ansatz, the result is stored as a <code>HarmonicEquation</code>. For an initial equation of motion consisting of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="2.378ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 1051 683" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D440" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></mjx-assistive-mml></mjx-container> variables, each expanded in <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="2.009ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 888 683" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></mjx-assistive-mml></mjx-container> harmonics, the resulting <code>HarmonicEquation</code> holds <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="5.518ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 2439 683" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(500,0)"><path data-c="1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1388,0)"><path data-c="1D440" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>N</mi><mi>M</mi></math></mjx-assistive-mml></mjx-container> equations of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="5.518ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 2439 683" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><path data-c="32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(500,0)"><path data-c="1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1388,0)"><path data-c="1D440" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>N</mi><mi>M</mi></math></mjx-assistive-mml></mjx-container> variables. Each symbol not corresponding to a variable is identified as a parameter.</p><p>A <code>HarmonicEquation</code> can be either parsed into a steady-state <code>Problem</code> or solved using a dynamical ODE solver.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.HarmonicEquation-manual-extracting_harmonics" href="#HarmonicBalance.HarmonicEquation-manual-extracting_harmonics"><span class="jlbinding">HarmonicBalance.HarmonicEquation</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Holds a set of algebraic equations governing the harmonics of a <code>DifferentialEquation</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>equations::Vector{Equation}</code>: A set of equations governing the harmonics.</p></li><li><p><code>variables::Vector{HarmonicVariable}</code>: A set of variables describing the harmonics.</p></li><li><p><code>parameters::Vector{Num}</code>: The parameters of the equation set.</p></li><li><p><code>natural_equation::DifferentialEquation</code>: The natural equation (before the harmonic ansatz was used).</p></li><li><p><code>jacobian::Matrix{Num}</code>: The Jacobian of the natural equation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/HarmonicEquation.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><p>MarkdownAST.LineBreak()</p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/extracting_harmonics.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Entering Equations of Motion</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/manual/methods" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Computing Steady States</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. 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+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> +</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> y</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">^</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> +</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 2</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">x</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">*</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">y</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/ExprUtils/drop_powers.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><h2 id="HarmonicVariable-and-HarmonicEquation-types" tabindex="-1">HarmonicVariable and HarmonicEquation types <a class="header-anchor" href="#HarmonicVariable-and-HarmonicEquation-types" aria-label="Permalink to "HarmonicVariable and HarmonicEquation types {#HarmonicVariable-and-HarmonicEquation-types}""></a></h2><p>The equations governing the harmonics are stored using the two following structs. 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id="HarmonicBalance.HarmonicVariable-manual-extracting_harmonics" href="#HarmonicBalance.HarmonicVariable-manual-extracting_harmonics"><span class="jlbinding">HarmonicBalance.HarmonicVariable</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicVariable</span></span></code></pre></div><p>Holds a variable stored under <code>symbol</code> describing the harmonic <code>ω</code> of <code>natural_variable</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>symbol::Num</code>: Symbol of the variable in the HarmonicBalance namespace.</p></li><li><p><code>name::String</code>: Human-readable labels of the variable, used for plotting.</p></li><li><p><code>type::String</code>: Type of the variable (u or v for quadratures, a for a constant, Hopf for Hopf etc.)</p></li><li><p><code>ω::Num</code>: The harmonic being described.</p></li><li><p><code>natural_variable::Num</code>: The natural variable whose harmonic is being described.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicVariable.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><p>When the full set of equations of motion is expanded using the harmonic ansatz, the result is stored as a <code>HarmonicEquation</code>. For an initial equation of motion consisting of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="2.378ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 1051 683" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D440" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></mjx-assistive-mml></mjx-container> variables, each expanded in <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="2.009ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 888 683" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></mjx-assistive-mml></mjx-container> harmonics, the resulting <code>HarmonicEquation</code> holds <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="5.518ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 2439 683" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g 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0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>N</mi><mi>M</mi></math></mjx-assistive-mml></mjx-container> variables. Each symbol not corresponding to a variable is identified as a parameter.</p><p>A <code>HarmonicEquation</code> can be either parsed into a steady-state <code>Problem</code> or solved using a dynamical ODE solver.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.HarmonicEquation-manual-extracting_harmonics" href="#HarmonicBalance.HarmonicEquation-manual-extracting_harmonics"><span class="jlbinding">HarmonicBalance.HarmonicEquation</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> HarmonicEquation</span></span></code></pre></div><p>Holds a set of algebraic equations governing the harmonics of a <code>DifferentialEquation</code>.</p><p><strong>Fields</strong></p><ul><li><p><code>equations::Vector{Equation}</code>: A set of equations governing the harmonics.</p></li><li><p><code>variables::Vector{HarmonicVariable}</code>: A set of variables describing the harmonics.</p></li><li><p><code>parameters::Vector{Num}</code>: The parameters of the equation set.</p></li><li><p><code>natural_equation::DifferentialEquation</code>: The natural equation (before the harmonic ansatz was used).</p></li><li><p><code>jacobian::Matrix{Num}</code>: The Jacobian of the natural equation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/HarmonicEquation.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><p>MarkdownAST.LineBreak()</p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/extracting_harmonics.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Entering Equations of Motion</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/manual/methods" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Computing Steady States</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. 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href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-acbfed09><!--[--><span data-v-acbfed09>Krylov-Bogoliubov</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-acbfed09><!--[--><span data-v-acbfed09>Time Evolution</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link active" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-acbfed09><!--[--><span data-v-acbfed09>Linear Response</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-acbfed09><!--[--><span data-v-acbfed09>Plotting</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/manual/saving" 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tabindex="-1" data-v-18756405><span class="visually-hidden" id="sidebar-aria-label" data-v-18756405> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0" data-v-9e426adc data-v-a4b0d9bf><!----><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/index" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Home</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/index" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Getting Started</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Theoretical Background</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/harmonic_balance" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Floquet expansions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/stability_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Stability and Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Tutorials</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/steady_states" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/classification" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Classifying Solutions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Transient Dynamics</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Examples</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Three Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Oscillator</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>State Dependent Perturbation</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible has-active" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Resources</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/API" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>API</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Bibliography</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible has-active" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Manual</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Entering Equations of Motion</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Effective System</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/methods" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Krylov-Bogoliubov</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Time Evolution</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Plotting</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/saving" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Saving and Loading</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-a9a9e638 data-v-91765379><div class="VPDoc has-sidebar has-aside" data-v-91765379 data-v-83890dd9><!--[--><!--]--><div class="container" data-v-83890dd9><div class="aside" data-v-83890dd9><div class="aside-curtain" data-v-83890dd9></div><div class="aside-container" data-v-83890dd9><div class="aside-content" data-v-83890dd9><div class="VPDocAside" data-v-83890dd9 data-v-6d7b3c46><!--[--><!--]--><!--[--><!--]--><nav aria-labelledby="doc-outline-aria-label" class="VPDocAsideOutline" data-v-6d7b3c46 data-v-b38bf2ff><div class="content" data-v-b38bf2ff><div class="outline-marker" data-v-b38bf2ff></div><div aria-level="2" class="outline-title" id="doc-outline-aria-label" role="heading" data-v-b38bf2ff>On this page</div><ul class="VPDocOutlineItem root" data-v-b38bf2ff data-v-3f927ebe><!--[--><!--]--></ul></div></nav><!--[--><!--]--><div class="spacer" data-v-6d7b3c46></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--]--></div></div></div></div><div class="content" data-v-83890dd9><div class="content-container" data-v-83890dd9><!--[--><!--]--><main class="main" data-v-83890dd9><div style="position:relative;" class="vp-doc _HarmonicBalance_jl_dev_manual_linear_response" data-v-83890dd9><div><h1 id="linresp_man" tabindex="-1">Linear response (WIP) <a class="header-anchor" href="#linresp_man" aria-label="Permalink to "Linear response (WIP) {#linresp_man}""></a></h1><p>This module currently has two goals. One is calculating the first-order Jacobian, used to obtain stability and approximate (but inexpensive) the linear response of steady states. The other is calculating the full response matrix as a function of frequency; this is more accurate but more expensive.</p><p>The methodology used is explained in <a href="https://www.doi.org/10.3929/ethz-b-000589190" target="_blank" rel="noreferrer">Jan Kosata phd thesis</a>.</p><h2 id="stability" tabindex="-1">Stability <a class="header-anchor" href="#stability" aria-label="Permalink to "Stability""></a></h2><p>The Jacobian is used to evaluate stability of the solutions. It can be shown explicitly,</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_Jacobian-manual-linear_response" href="#HarmonicBalance.get_Jacobian-manual-linear_response"><span class="jlbinding">HarmonicBalance.get_Jacobian</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_Jacobian</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom)</span></span></code></pre></div><p>Obtain the symbolic Jacobian matrix of <code>eom</code>. This is the linearised left-hand side of F(u) = du/dT.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/Jacobian.jl#L59" target="_blank" rel="noreferrer">source</a></p><p>Obtain a Jacobian from a <code>DifferentialEquation</code> by first converting it into a <code>HarmonicEquation</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/Jacobian.jl#L77" target="_blank" rel="noreferrer">source</a></p><p>Get the Jacobian of a set of equations <code>eqs</code> with respect to the variables <code>vars</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/Jacobian.jl#L86" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Linear-response" tabindex="-1">Linear response <a class="header-anchor" href="#Linear-response" aria-label="Permalink to "Linear response {#Linear-response}""></a></h2><p>The response to white noise can be shown with <code>plot_linear_response</code>. Depending on the <code>order</code> argument, different methods are used.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.plot_linear_response-manual-linear_response" href="#HarmonicBalance.LinearResponse.plot_linear_response-manual-linear_response"><span class="jlbinding">HarmonicBalance.LinearResponse.plot_linear_response</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_linear_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, nat_var</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; Ω_range, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, order</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, logscale</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">false</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, show_progress</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot the linear response to white noise of the variable <code>nat_var</code> for Result <code>res</code> on <code>branch</code> for input frequencies <code>Ω_range</code>. Slow-time derivatives up to <code>order</code> are kept in the process.</p><p>Any kwargs are fed to Plots' gr().</p><p>Solutions not belonging to the <code>physical</code> class are ignored.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/plotting.jl#L120-L129" target="_blank" rel="noreferrer">source</a></p></details><h3 id="First-order" tabindex="-1">First order <a class="header-anchor" href="#First-order" aria-label="Permalink to "First order {#First-order}""></a></h3><p>The simplest way to extract the linear response of a steady state is to evaluate the Jacobian of the harmonic equations. Each of its eigenvalues <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.027ex;" xmlns="http://www.w3.org/2000/svg" width="1.319ex" height="1.597ex" role="img" focusable="false" viewBox="0 -694 583 706" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></mjx-assistive-mml></mjx-container> describes a Lorentzian peak in the response; <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.247ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2319 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><path data-c="52" d="M130 622Q123 629 119 631T103 634T60 637H27V683H202H236H300Q376 683 417 677T500 648Q595 600 609 517Q610 512 610 501Q610 468 594 439T556 392T511 361T472 343L456 338Q459 335 467 332Q497 316 516 298T545 254T559 211T568 155T578 94Q588 46 602 31T640 16H645Q660 16 674 32T692 87Q692 98 696 101T712 105T728 103T732 90Q732 59 716 27T672 -16Q656 -22 630 -22Q481 -16 458 90Q456 101 456 163T449 246Q430 304 373 320L363 322L297 323H231V192L232 61Q238 51 249 49T301 46H334V0H323Q302 3 181 3Q59 3 38 0H27V46H60Q102 47 111 49T130 61V622ZM491 499V509Q491 527 490 539T481 570T462 601T424 623T362 636Q360 636 340 636T304 637H283Q238 637 234 628Q231 624 231 492V360H289Q390 360 434 378T489 456Q491 467 491 499Z" style="stroke-width:3;"></path><path data-c="65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z" transform="translate(736,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1180,0)"><path data-c="5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1458,0)"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2041,0)"><path data-c="5D" d="M22 710V750H159V-250H22V-210H119V710H22Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mo stretchy="false">[</mo><mi>λ</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container> gives its center and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.278ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2333 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><path data-c="49" d="M328 0Q307 3 180 3T32 0H21V46H43Q92 46 106 49T126 60Q128 63 128 342Q128 620 126 623Q122 628 118 630T96 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46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z" transform="translate(361,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1194,0)"><path data-c="5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1472,0)"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2055,0)"><path data-c="5D" d="M22 710V750H159V-250H22V-210H119V710H22Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Im</mtext><mo stretchy="false">[</mo><mi>λ</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container> its width. Transforming the harmonic variables into the non-rotating frame (that is, inverting the harmonic ansatz) then gives the response as it would be observed in an experiment.</p><p>The advantage of this method is that for a given parameter set, only one matrix diagonalization is needed to fully describe the response spectrum. However, the method is inaccurate for response frequencies far from the frequencies used in the harmonic ansatz (it relies on the response oscillating slowly in the rotating frame).</p><p>Behind the scenes, the spectra are stored using the dedicated structs <code>Lorentzian</code> and <code>JacobianSpectrum</code>.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.JacobianSpectrum-manual-linear_response" href="#HarmonicBalance.LinearResponse.JacobianSpectrum-manual-linear_response"><span class="jlbinding">HarmonicBalance.LinearResponse.JacobianSpectrum</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> JacobianSpectrum{T</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Real</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}</span></span></code></pre></div><p>Holds a set of <code>Lorentzian</code> objects belonging to a variable.</p><p><strong>Fields</strong></p><ul><li><code>peaks::Array{HarmonicBalance.LinearResponse.Lorentzian{T}, 1} where T<:Real</code></li></ul><p><strong>Constructor</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">JacobianSpectrum</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; index</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/types.jl#L21" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.Lorentzian-manual-linear_response" href="#HarmonicBalance.LinearResponse.Lorentzian-manual-linear_response"><span class="jlbinding">HarmonicBalance.LinearResponse.Lorentzian</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Lorentzian{T</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Real</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}</span></span></code></pre></div><p>Holds the three parameters of a Lorentzian peak, defined as A / sqrt((ω-ω0)² + Γ²).</p><p><strong>Fields</strong></p><ul><li><p><code>ω0::Real</code></p></li><li><p><code>Γ::Real</code></p></li><li><p><code>A::Real</code></p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/types.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><h3 id="Higher-orders" tabindex="-1">Higher orders <a class="header-anchor" href="#Higher-orders" aria-label="Permalink to "Higher orders {#Higher-orders}""></a></h3><p>Setting <code>order > 1</code> increases the accuracy of the response spectra. However, unlike for the Jacobian, here we must perform a matrix inversion for each response frequency.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.ResponseMatrix-manual-linear_response" href="#HarmonicBalance.LinearResponse.ResponseMatrix-manual-linear_response"><span class="jlbinding">HarmonicBalance.LinearResponse.ResponseMatrix</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ResponseMatrix</span></span></code></pre></div><p>Holds the compiled response matrix of a system.</p><p><strong>Fields</strong></p><ul><li><p><code>matrix::Matrix{Function}</code>: The response matrix (compiled).</p></li><li><p><code>symbols::Vector{Num}</code>: Any symbolic variables in <code>matrix</code> to be substituted at evaluation.</p></li><li><p><code>variables::Vector{HarmonicVariable}</code>: The frequencies of the harmonic variables underlying <code>matrix</code>. These are needed to transform the harmonic variables to the non-rotating frame.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/types.jl#L40" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.get_response-manual-linear_response" href="#HarmonicBalance.LinearResponse.get_response-manual-linear_response"><span class="jlbinding">HarmonicBalance.LinearResponse.get_response</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
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Started</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Theoretical Background</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/harmonic_balance" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Floquet expansions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" 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class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/steady_states" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/classification" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Classifying Solutions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Transient Dynamics</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><section 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href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Three Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Oscillator</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>State Dependent Perturbation</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><div 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link link" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Bibliography</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible has-active" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Manual</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Entering Equations of Motion</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Effective System</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/methods" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Krylov-Bogoliubov</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Time Evolution</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Plotting</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/saving" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Saving and Loading</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-a9a9e638 data-v-91765379><div class="VPDoc has-sidebar has-aside" data-v-91765379 data-v-83890dd9><!--[--><!--]--><div class="container" data-v-83890dd9><div class="aside" data-v-83890dd9><div class="aside-curtain" data-v-83890dd9></div><div class="aside-container" data-v-83890dd9><div class="aside-content" data-v-83890dd9><div class="VPDocAside" data-v-83890dd9 data-v-6d7b3c46><!--[--><!--]--><!--[--><!--]--><nav aria-labelledby="doc-outline-aria-label" class="VPDocAsideOutline" data-v-6d7b3c46 data-v-b38bf2ff><div class="content" data-v-b38bf2ff><div class="outline-marker" data-v-b38bf2ff></div><div aria-level="2" class="outline-title" id="doc-outline-aria-label" role="heading" data-v-b38bf2ff>On this page</div><ul class="VPDocOutlineItem root" data-v-b38bf2ff data-v-3f927ebe><!--[--><!--]--></ul></div></nav><!--[--><!--]--><div class="spacer" data-v-6d7b3c46></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--]--></div></div></div></div><div class="content" data-v-83890dd9><div class="content-container" data-v-83890dd9><!--[--><!--]--><main class="main" data-v-83890dd9><div style="position:relative;" class="vp-doc _HarmonicBalance_jl_dev_manual_linear_response" data-v-83890dd9><div><h1 id="linresp_man" tabindex="-1">Linear response (WIP) <a class="header-anchor" href="#linresp_man" aria-label="Permalink to "Linear response (WIP) {#linresp_man}""></a></h1><p>This module currently has two goals. One is calculating the first-order Jacobian, used to obtain stability and approximate (but inexpensive) the linear response of steady states. The other is calculating the full response matrix as a function of frequency; this is more accurate but more expensive.</p><p>The methodology used is explained in <a href="https://www.doi.org/10.3929/ethz-b-000589190" target="_blank" rel="noreferrer">Jan Kosata phd thesis</a>.</p><h2 id="stability" tabindex="-1">Stability <a class="header-anchor" href="#stability" aria-label="Permalink to "Stability""></a></h2><p>The Jacobian is used to evaluate stability of the solutions. It can be shown explicitly,</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_Jacobian-manual-linear_response" href="#HarmonicBalance.get_Jacobian-manual-linear_response"><span class="jlbinding">HarmonicBalance.get_Jacobian</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_Jacobian</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(eom)</span></span></code></pre></div><p>Obtain the symbolic Jacobian matrix of <code>eom</code>. This is the linearised left-hand side of F(u) = du/dT.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/Jacobian.jl#L59" target="_blank" rel="noreferrer">source</a></p><p>Obtain a Jacobian from a <code>DifferentialEquation</code> by first converting it into a <code>HarmonicEquation</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/Jacobian.jl#L77" target="_blank" rel="noreferrer">source</a></p><p>Get the Jacobian of a set of equations <code>eqs</code> with respect to the variables <code>vars</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/Jacobian.jl#L86" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Linear-response" tabindex="-1">Linear response <a class="header-anchor" href="#Linear-response" aria-label="Permalink to "Linear response {#Linear-response}""></a></h2><p>The response to white noise can be shown with <code>plot_linear_response</code>. Depending on the <code>order</code> argument, different methods are used.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.plot_linear_response-manual-linear_response" href="#HarmonicBalance.LinearResponse.plot_linear_response-manual-linear_response"><span class="jlbinding">HarmonicBalance.LinearResponse.plot_linear_response</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_linear_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, nat_var</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; Ω_range, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, order</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, logscale</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">false</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, show_progress</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot the linear response to white noise of the variable <code>nat_var</code> for Result <code>res</code> on <code>branch</code> for input frequencies <code>Ω_range</code>. Slow-time derivatives up to <code>order</code> are kept in the process.</p><p>Any kwargs are fed to Plots' gr().</p><p>Solutions not belonging to the <code>physical</code> class are ignored.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/plotting.jl#L120-L129" target="_blank" rel="noreferrer">source</a></p></details><h3 id="First-order" tabindex="-1">First order <a class="header-anchor" href="#First-order" aria-label="Permalink to "First order {#First-order}""></a></h3><p>The simplest way to extract the linear response of a steady state is to evaluate the Jacobian of the harmonic equations. Each of its eigenvalues <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.027ex;" xmlns="http://www.w3.org/2000/svg" width="1.319ex" height="1.597ex" role="img" focusable="false" viewBox="0 -694 583 706" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></mjx-assistive-mml></mjx-container> describes a Lorentzian peak in the response; <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.247ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2319 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><path data-c="52" d="M130 622Q123 629 119 631T103 634T60 637H27V683H202H236H300Q376 683 417 677T500 648Q595 600 609 517Q610 512 610 501Q610 468 594 439T556 392T511 361T472 343L456 338Q459 335 467 332Q497 316 516 298T545 254T559 211T568 155T578 94Q588 46 602 31T640 16H645Q660 16 674 32T692 87Q692 98 696 101T712 105T728 103T732 90Q732 59 716 27T672 -16Q656 -22 630 -22Q481 -16 458 90Q456 101 456 163T449 246Q430 304 373 320L363 322L297 323H231V192L232 61Q238 51 249 49T301 46H334V0H323Q302 3 181 3Q59 3 38 0H27V46H60Q102 47 111 49T130 61V622ZM491 499V509Q491 527 490 539T481 570T462 601T424 623T362 636Q360 636 340 636T304 637H283Q238 637 234 628Q231 624 231 492V360H289Q390 360 434 378T489 456Q491 467 491 499Z" style="stroke-width:3;"></path><path data-c="65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z" transform="translate(736,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1180,0)"><path data-c="5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1458,0)"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2041,0)"><path data-c="5D" d="M22 710V750H159V-250H22V-210H119V710H22Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mo stretchy="false">[</mo><mi>λ</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container> gives its center and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.278ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2333 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><path data-c="49" d="M328 0Q307 3 180 3T32 0H21V46H43Q92 46 106 49T126 60Q128 63 128 342Q128 620 126 623Q122 628 118 630T96 635T43 637H21V683H32Q53 680 180 680T328 683H339V637H317Q268 637 254 634T234 623Q232 620 232 342Q232 63 234 60Q238 55 242 53T264 48T317 46H339V0H328Z" style="stroke-width:3;"></path><path data-c="6D" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z" transform="translate(361,0)" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(1194,0)"><path data-c="5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(1472,0)"><path data-c="1D706" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2055,0)"><path data-c="5D" d="M22 710V750H159V-250H22V-210H119V710H22Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Im</mtext><mo stretchy="false">[</mo><mi>λ</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container> its width. Transforming the harmonic variables into the non-rotating frame (that is, inverting the harmonic ansatz) then gives the response as it would be observed in an experiment.</p><p>The advantage of this method is that for a given parameter set, only one matrix diagonalization is needed to fully describe the response spectrum. However, the method is inaccurate for response frequencies far from the frequencies used in the harmonic ansatz (it relies on the response oscillating slowly in the rotating frame).</p><p>Behind the scenes, the spectra are stored using the dedicated structs <code>Lorentzian</code> and <code>JacobianSpectrum</code>.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.JacobianSpectrum-manual-linear_response" href="#HarmonicBalance.LinearResponse.JacobianSpectrum-manual-linear_response"><span class="jlbinding">HarmonicBalance.LinearResponse.JacobianSpectrum</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">mutable struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> JacobianSpectrum{T</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Real</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}</span></span></code></pre></div><p>Holds a set of <code>Lorentzian</code> objects belonging to a variable.</p><p><strong>Fields</strong></p><ul><li><code>peaks::Array{HarmonicBalance.LinearResponse.Lorentzian{T}, 1} where T<:Real</code></li></ul><p><strong>Constructor</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">JacobianSpectrum</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; index</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Int</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/types.jl#L21" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.Lorentzian-manual-linear_response" href="#HarmonicBalance.LinearResponse.Lorentzian-manual-linear_response"><span class="jlbinding">HarmonicBalance.LinearResponse.Lorentzian</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Lorentzian{T</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Real</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}</span></span></code></pre></div><p>Holds the three parameters of a Lorentzian peak, defined as A / sqrt((ω-ω0)² + Γ²).</p><p><strong>Fields</strong></p><ul><li><p><code>ω0::Real</code></p></li><li><p><code>Γ::Real</code></p></li><li><p><code>A::Real</code></p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/types.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><h3 id="Higher-orders" tabindex="-1">Higher orders <a class="header-anchor" href="#Higher-orders" aria-label="Permalink to "Higher orders {#Higher-orders}""></a></h3><p>Setting <code>order > 1</code> increases the accuracy of the response spectra. However, unlike for the Jacobian, here we must perform a matrix inversion for each response frequency.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.ResponseMatrix-manual-linear_response" href="#HarmonicBalance.LinearResponse.ResponseMatrix-manual-linear_response"><span class="jlbinding">HarmonicBalance.LinearResponse.ResponseMatrix</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ResponseMatrix</span></span></code></pre></div><p>Holds the compiled response matrix of a system.</p><p><strong>Fields</strong></p><ul><li><p><code>matrix::Matrix{Function}</code>: The response matrix (compiled).</p></li><li><p><code>symbols::Vector{Num}</code>: Any symbolic variables in <code>matrix</code> to be substituted at evaluation.</p></li><li><p><code>variables::Vector{HarmonicVariable}</code>: The frequencies of the harmonic variables underlying <code>matrix</code>. These are needed to transform the harmonic variables to the non-rotating frame.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/types.jl#L40" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.get_response-manual-linear_response" href="#HarmonicBalance.LinearResponse.get_response-manual-linear_response"><span class="jlbinding">HarmonicBalance.LinearResponse.get_response</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> rmat</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.LinearResponse.ResponseMatrix</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> s</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">OrderedCollections.OrderedDict</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Ω</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>For <code>rmat</code> and a solution dictionary <code>s</code>, calculate the total response to a perturbative force at frequency <code>Ω</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/response.jl#L63" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.get_response_matrix-manual-linear_response" href="#HarmonicBalance.LinearResponse.get_response_matrix-manual-linear_response"><span class="jlbinding">HarmonicBalance.LinearResponse.get_response_matrix</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_response_matrix</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, freq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; order</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Obtain the symbolic linear response matrix of a <code>diff_eq</code> corresponding to a perturbation frequency <code>freq</code>. This routine cannot accept a <code>HarmonicEquation</code> since there, some time-derivatives are already dropped. <code>order</code> denotes the highest differential order to be considered.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/modules/LinearResponse/response.jl#L1-L8" target="_blank" rel="noreferrer">source</a></p></details></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/linear_response.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Time Evolution</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Plotting</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>For <code>rmat</code> and a solution dictionary <code>s</code>, calculate the total response to a perturbative force at frequency <code>Ω</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/response.jl#L63" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.LinearResponse.get_response_matrix-manual-linear_response" href="#HarmonicBalance.LinearResponse.get_response_matrix-manual-linear_response"><span class="jlbinding">HarmonicBalance.LinearResponse.get_response_matrix</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_response_matrix</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(diff_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">DifferentialEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, freq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Num</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; order</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Obtain the symbolic linear response matrix of a <code>diff_eq</code> corresponding to a perturbation frequency <code>freq</code>. This routine cannot accept a <code>HarmonicEquation</code> since there, some time-derivatives are already dropped. <code>order</code> denotes the highest differential order to be considered.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/modules/LinearResponse/response.jl#L1-L8" target="_blank" rel="noreferrer">source</a></p></details></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/linear_response.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Time Evolution</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Plotting</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/steady_states" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/classification" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Classifying Solutions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Transient Dynamics</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Examples</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Three Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Oscillator</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>State Dependent Perturbation</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible has-active" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Resources</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/API" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>API</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Bibliography</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible has-active" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Manual</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Entering Equations of Motion</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Effective System</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/methods" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Krylov-Bogoliubov</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Time Evolution</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Plotting</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/saving" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Saving and Loading</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-a9a9e638 data-v-91765379><div class="VPDoc has-sidebar has-aside" data-v-91765379 data-v-83890dd9><!--[--><!--]--><div class="container" data-v-83890dd9><div class="aside" data-v-83890dd9><div class="aside-curtain" data-v-83890dd9></div><div class="aside-container" data-v-83890dd9><div class="aside-content" data-v-83890dd9><div class="VPDocAside" data-v-83890dd9 data-v-6d7b3c46><!--[--><!--]--><!--[--><!--]--><nav aria-labelledby="doc-outline-aria-label" class="VPDocAsideOutline" data-v-6d7b3c46 data-v-b38bf2ff><div class="content" data-v-b38bf2ff><div class="outline-marker" data-v-b38bf2ff></div><div aria-level="2" class="outline-title" id="doc-outline-aria-label" role="heading" data-v-b38bf2ff>On this page</div><ul class="VPDocOutlineItem root" data-v-b38bf2ff data-v-3f927ebe><!--[--><!--]--></ul></div></nav><!--[--><!--]--><div class="spacer" data-v-6d7b3c46></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--]--></div></div></div></div><div class="content" data-v-83890dd9><div class="content-container" data-v-83890dd9><!--[--><!--]--><main class="main" data-v-83890dd9><div style="position:relative;" class="vp-doc _HarmonicBalance_jl_dev_manual_methods" data-v-83890dd9><div><h1 id="methods" tabindex="-1">Methods <a class="header-anchor" href="#methods" aria-label="Permalink to "Methods""></a></h1><p>We offer several methods for solving the nonlinear algebraic equations that arise from the harmonic balance procedure. Each method has different tradeoffs between speed, robustness, and completeness.</p><h2 id="Total-Degree-Method" tabindex="-1">Total Degree Method <a class="header-anchor" href="#Total-Degree-Method" aria-label="Permalink to "Total Degree Method {#Total-Degree-Method}""></a></h2><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.TotalDegree-manual-methods" href="#HarmonicBalance.TotalDegree-manual-methods"><span class="jlbinding">HarmonicBalance.TotalDegree</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">TotalDegree</span></span></code></pre></div><p>The Total Degree homotopy method performs a homotopy <mjx-container 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transform="translate(4004.6,0)"><g data-mml-node="mi"><text data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">ᵢ</text></g><g data-mml-node="TeXAtom" transform="translate(633,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g><g 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jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.452ex;" xmlns="http://www.w3.org/2000/svg" width="2.534ex" height="2.149ex" role="img" focusable="false" viewBox="0 -750 1120 950" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(520,0)"><text data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">ᵢ</text></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>ᵢ</mi></math></mjx-assistive-mml></mjx-container> determined by the <a href="https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem" target="_blank" rel="noreferrer">Bezout bound</a>. The method guarantees to find all solutions, however, it comes with a high computational cost. See <a href="https://www.juliahomotopycontinuation.org/guides/totaldegree/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>gamma::Complex</code>: Complex multiplying factor of the start system G(x) for the homotopy</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/methods.jl#L8-L20" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Polyhedral-Method" tabindex="-1">Polyhedral Method <a class="header-anchor" href="#Polyhedral-Method" aria-label="Permalink to "Polyhedral Method {#Polyhedral-Method}""></a></h2><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.Polyhedral-manual-methods" href="#HarmonicBalance.Polyhedral-manual-methods"><span class="jlbinding">HarmonicBalance.Polyhedral</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Polyhedral</span></span></code></pre></div><p>The Polyhedral homotopy method constructs a homotopy based on the polyhedral structure of the polynomial system. It is more efficient than the Total Degree method for sparse systems, meaning most of the coefficients are zero. It can be especially useful if you don't need to find the zero solutions (<code>only_non_zero = true</code>), resulting in a speed up. See <a href="https://www.juliahomotopycontinuation.org/guides/polyhedral/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>only_non_zero::Bool</code>: Boolean indicating if only non-zero solutions are considered.</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/methods.jl#L57-L69" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Warm-Up-Method" tabindex="-1">Warm Up Method <a class="header-anchor" href="#Warm-Up-Method" aria-label="Permalink to "Warm Up Method {#Warm-Up-Method}""></a></h2><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.WarmUp-manual-methods" href="#HarmonicBalance.WarmUp-manual-methods"><span class="jlbinding">HarmonicBalance.WarmUp</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">WarmUp</span></span></code></pre></div><p>The Warm Up method prepares a warmup system with the Total Degree method using the parameter at <code>index</code> perturbed by <code>perturbation_size</code>. The warmup system is used to perform a homotopy using all other systems in the parameter sweep. It is very efficient for systems with minimal bifurcation in the parameter sweep. The Warm Up method should in theory guarantee to find all solutions, however, if the <code>start_parameters</code> is not proper (to close to the real line) it could miss some solutions.</p><p>See<a href="https://www.juliahomotopycontinuation.org/guides/many-systems/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>warm_up_method::Union{Polyhedral{T}, TotalDegree{T}} where T</code>: Method used for the warmup system.</p></li><li><p><code>start_parameters::Vector</code>: Start parameters.</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/methods.jl#L110-L125" target="_blank" rel="noreferrer">source</a></p></details></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/methods.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Computing Effective System</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Krylov-Bogoliubov</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. 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Started</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Theoretical Background</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/harmonic_balance" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Floquet expansions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/stability_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Stability and Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Tutorials</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/steady_states" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/classification" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Classifying Solutions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Transient Dynamics</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Examples</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Three Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Oscillator</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>State Dependent Perturbation</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible has-active" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Resources</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/API" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>API</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Bibliography</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible has-active" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Manual</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Entering Equations of Motion</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Effective System</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/methods" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Krylov-Bogoliubov</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Time Evolution</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Plotting</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/saving" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Saving and Loading</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-a9a9e638 data-v-91765379><div class="VPDoc has-sidebar has-aside" data-v-91765379 data-v-83890dd9><!--[--><!--]--><div class="container" data-v-83890dd9><div class="aside" data-v-83890dd9><div class="aside-curtain" data-v-83890dd9></div><div class="aside-container" data-v-83890dd9><div class="aside-content" data-v-83890dd9><div class="VPDocAside" data-v-83890dd9 data-v-6d7b3c46><!--[--><!--]--><!--[--><!--]--><nav aria-labelledby="doc-outline-aria-label" class="VPDocAsideOutline" data-v-6d7b3c46 data-v-b38bf2ff><div class="content" data-v-b38bf2ff><div class="outline-marker" data-v-b38bf2ff></div><div aria-level="2" class="outline-title" id="doc-outline-aria-label" role="heading" data-v-b38bf2ff>On this page</div><ul class="VPDocOutlineItem root" data-v-b38bf2ff data-v-3f927ebe><!--[--><!--]--></ul></div></nav><!--[--><!--]--><div class="spacer" data-v-6d7b3c46></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--]--></div></div></div></div><div class="content" data-v-83890dd9><div class="content-container" data-v-83890dd9><!--[--><!--]--><main class="main" data-v-83890dd9><div style="position:relative;" class="vp-doc _HarmonicBalance_jl_dev_manual_methods" data-v-83890dd9><div><h1 id="methods" tabindex="-1">Methods <a class="header-anchor" href="#methods" aria-label="Permalink to "Methods""></a></h1><p>We offer several methods for solving the nonlinear algebraic equations that arise from the harmonic balance procedure. Each method has different tradeoffs between speed, robustness, and completeness.</p><h2 id="Total-Degree-Method" tabindex="-1">Total Degree Method <a class="header-anchor" href="#Total-Degree-Method" aria-label="Permalink to "Total Degree Method {#Total-Degree-Method}""></a></h2><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.TotalDegree-manual-methods" href="#HarmonicBalance.TotalDegree-manual-methods"><span class="jlbinding">HarmonicBalance.TotalDegree</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">TotalDegree</span></span></code></pre></div><p>The Total Degree homotopy method performs a homotopy <mjx-container 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jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.452ex;" xmlns="http://www.w3.org/2000/svg" width="2.534ex" height="2.149ex" role="img" focusable="false" viewBox="0 -750 1120 950" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(520,0)"><text data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">ᵢ</text></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>ᵢ</mi></math></mjx-assistive-mml></mjx-container> determined by the <a href="https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem" target="_blank" rel="noreferrer">Bezout bound</a>. The method guarantees to find all solutions, however, it comes with a high computational cost. See <a href="https://www.juliahomotopycontinuation.org/guides/totaldegree/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>gamma::Complex</code>: Complex multiplying factor of the start system G(x) for the homotopy</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/methods.jl#L8-L20" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Polyhedral-Method" tabindex="-1">Polyhedral Method <a class="header-anchor" href="#Polyhedral-Method" aria-label="Permalink to "Polyhedral Method {#Polyhedral-Method}""></a></h2><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.Polyhedral-manual-methods" href="#HarmonicBalance.Polyhedral-manual-methods"><span class="jlbinding">HarmonicBalance.Polyhedral</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Polyhedral</span></span></code></pre></div><p>The Polyhedral homotopy method constructs a homotopy based on the polyhedral structure of the polynomial system. It is more efficient than the Total Degree method for sparse systems, meaning most of the coefficients are zero. It can be especially useful if you don't need to find the zero solutions (<code>only_non_zero = true</code>), resulting in a speed up. See <a href="https://www.juliahomotopycontinuation.org/guides/polyhedral/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>only_non_zero::Bool</code>: Boolean indicating if only non-zero solutions are considered.</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/methods.jl#L57-L69" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Warm-Up-Method" tabindex="-1">Warm Up Method <a class="header-anchor" href="#Warm-Up-Method" aria-label="Permalink to "Warm Up Method {#Warm-Up-Method}""></a></h2><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.WarmUp-manual-methods" href="#HarmonicBalance.WarmUp-manual-methods"><span class="jlbinding">HarmonicBalance.WarmUp</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">WarmUp</span></span></code></pre></div><p>The Warm Up method prepares a warmup system with the Total Degree method using the parameter at <code>index</code> perturbed by <code>perturbation_size</code>. The warmup system is used to perform a homotopy using all other systems in the parameter sweep. It is very efficient for systems with minimal bifurcation in the parameter sweep. The Warm Up method should in theory guarantee to find all solutions, however, if the <code>start_parameters</code> is not proper (to close to the real line) it could miss some solutions.</p><p>See<a href="https://www.juliahomotopycontinuation.org/guides/many-systems/" target="_blank" rel="noreferrer">HomotopyContinuation.jl</a> for more information.</p><p><strong>Fields</strong></p><ul><li><p><code>warm_up_method::Union{Polyhedral{T}, TotalDegree{T}} where T</code>: Method used for the warmup system.</p></li><li><p><code>start_parameters::Vector</code>: Start parameters.</p></li><li><p><code>thread::Bool</code>: Boolean indicating if threading is enabled.</p></li><li><p><code>tracker_options::HomotopyContinuation.TrackerOptions</code>: Options for the tracker.</p></li><li><p><code>endgame_options::HomotopyContinuation.EndgameOptions</code>: Options for the endgame.</p></li><li><p><code>compile::Union{Bool, Symbol}</code>: Compilation options.</p></li><li><p><code>seed::UInt32</code>: Seed for random number generation.</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/methods.jl#L110-L125" target="_blank" rel="noreferrer">source</a></p></details></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/methods.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Computing Effective System</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Krylov-Bogoliubov</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. 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@@ -30,24 +30,24 @@
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> func;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> branches,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> realify</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector</span></span></code></pre></div><p>Takes a <code>Result</code> object and a string <code>f</code> representing a Symbolics.jl expression. Returns an array with the values of <code>f</code> evaluated for the respective solutions. Additional substitution rules can be specified in <code>rules</code> in the format <code>("a" => val)</code> or <code>(a => val)</code></p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/transform_solutions.jl#L64" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Plotting-solutions" tabindex="-1">Plotting solutions <a class="header-anchor" href="#Plotting-solutions" aria-label="Permalink to "Plotting solutions {#Plotting-solutions}""></a></h2><p>The function <code>plot</code> is multiple-dispatched to plot 1D and 2D datasets. In 1D, the solutions are colour-coded according to the branches obtained by <code>sort_solutions</code>.</p><details class="jldocstring custom-block" open><summary><a id="RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}-manual-plotting" href="#RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}-manual-plotting"><span class="jlbinding">RecipesBase.plot</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Vector</span></span></code></pre></div><p>Takes a <code>Result</code> object and a string <code>f</code> representing a Symbolics.jl expression. Returns an array with the values of <code>f</code> evaluated for the respective solutions. Additional substitution rules can be specified in <code>rules</code> in the format <code>("a" => val)</code> or <code>(a => val)</code></p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/transform_solutions.jl#L64" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Plotting-solutions" tabindex="-1">Plotting solutions <a class="header-anchor" href="#Plotting-solutions" aria-label="Permalink to "Plotting solutions {#Plotting-solutions}""></a></h2><p>The function <code>plot</code> is multiple-dispatched to plot 1D and 2D datasets. In 1D, the solutions are colour-coded according to the branches obtained by <code>sort_solutions</code>.</p><details class="jldocstring custom-block" open><summary><a id="RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}-manual-plotting" href="#RecipesBase.plot-Tuple{HarmonicBalance.Result, Vararg{Any}}-manual-plotting"><span class="jlbinding">RecipesBase.plot</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result{S, P, D, F}</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"> where</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">FunctionWrappers.FunctionWrapper{Array{S, 2}, Tuple{Array{S, 1}}}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> varargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> cut,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Plot</span></span></code></pre></div><p><strong>Plot a <code>Result</code> object.</strong></p><p>Class selection done by passing <code>String</code> or <code>Vector{String}</code> as kwarg:</p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>class : only plot solutions in this class(es) ("all" --> plot everything)</span></span>
<span class="line"><span>not_class : do not plot solutions in this class(es)</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr().</p><p>See also <code>plot!</code></p><p>The x,y,z arguments are Strings compatible with Symbolics.jl, e.g., <code>y=2*sqrt(u1^2+v1^2)</code> plots the amplitude of the first quadratures multiplied by 2.</p><p><strong>1D plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(res::Result; x::String, y::String, class="default", not_class=[], kwargs...)</span></span>
-<span class="line"><span>plot(res::Result, y::String; kwargs...) # take x automatically from Result</span></span></code></pre></div><p>Default behaviour is to plot stable solutions as full lines, unstable as dashed.</p><p>If a sweep in two parameters were done, i.e., <code>dim(res)==2</code>, a one dimensional cut can be plotted by using the keyword <code>cut</code> were it takes a <code>Pair{Num, Float}</code> type entry. For example, <code>plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2))</code> plots a cut at <code>λ = 0.2</code>.</p><hr><p><strong>2D plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)</span></span></code></pre></div><p>To make the 2d plot less chaotic it is required to specify the specific <code>branch</code> to plot, labeled by a <code>Int64</code>.</p><p>The x and y axes are taken automatically from <code>res</code></p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/plotting_Plots.jl#L11" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="RecipesBase.plot!-manual-plotting" href="#RecipesBase.plot!-manual-plotting"><span class="jlbinding">RecipesBase.plot!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span>plot(res::Result, y::String; kwargs...) # take x automatically from Result</span></span></code></pre></div><p>Default behaviour is to plot stable solutions as full lines, unstable as dashed.</p><p>If a sweep in two parameters were done, i.e., <code>dim(res)==2</code>, a one dimensional cut can be plotted by using the keyword <code>cut</code> were it takes a <code>Pair{Num, Float}</code> type entry. For example, <code>plot(res, y="sqrt(u1^2+v1^2), cut=(λ => 0.2))</code> plots a cut at <code>λ = 0.2</code>.</p><hr><p><strong>2D plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(res::Result; z::String, branch::Int64, class="physical", not_class=[], kwargs...)</span></span></code></pre></div><p>To make the 2d plot less chaotic it is required to specify the specific <code>branch</code> to plot, labeled by a <code>Int64</code>.</p><p>The x and y axes are taken automatically from <code>res</code></p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/plotting_Plots.jl#L11" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="RecipesBase.plot!-manual-plotting" href="#RecipesBase.plot!-manual-plotting"><span class="jlbinding">RecipesBase.plot!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> varargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Plot</span></span></code></pre></div><p>Similar to <code>plot</code> but adds a plot onto an existing plot.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/plotting_Plots.jl#L62" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Plotting-phase-diagrams" tabindex="-1">Plotting phase diagrams <a class="header-anchor" href="#Plotting-phase-diagrams" aria-label="Permalink to "Plotting phase diagrams {#Plotting-phase-diagrams}""></a></h2><p>In many problems, rather than in any property of the solutions themselves, we are interested in the phase diagrams, encoding the number of (stable) solutions in different regions of the parameter space. <code>plot_phase_diagram</code> handles this for 1D and 2D datasets.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.plot_phase_diagram-manual-plotting" href="#HarmonicBalance.plot_phase_diagram-manual-plotting"><span class="jlbinding">HarmonicBalance.plot_phase_diagram</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_phase_diagram</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Plot</span></span></code></pre></div><p>Similar to <code>plot</code> but adds a plot onto an existing plot.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/plotting_Plots.jl#L62" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Plotting-phase-diagrams" tabindex="-1">Plotting phase diagrams <a class="header-anchor" href="#Plotting-phase-diagrams" aria-label="Permalink to "Plotting phase diagrams {#Plotting-phase-diagrams}""></a></h2><p>In many problems, rather than in any property of the solutions themselves, we are interested in the phase diagrams, encoding the number of (stable) solutions in different regions of the parameter space. <code>plot_phase_diagram</code> handles this for 1D and 2D datasets.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.plot_phase_diagram-manual-plotting" href="#HarmonicBalance.plot_phase_diagram-manual-plotting"><span class="jlbinding">HarmonicBalance.plot_phase_diagram</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_phase_diagram</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Plots</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">Plot</span></span></code></pre></div><p>Plot the number of solutions in a <code>Result</code> object as a function of the parameters. Works with 1D and 2D datasets.</p><p>Class selection done by passing <code>String</code> or <code>Vector{String}</code> as kwarg:</p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>class::String : only count solutions in this class ("all" --> plot everything)</span></span>
-<span class="line"><span>not_class::String : do not count solutions in this class</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr()</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/plotting_Plots.jl#L275" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Plot-spaghetti-plot" tabindex="-1">Plot spaghetti plot <a class="header-anchor" href="#Plot-spaghetti-plot" aria-label="Permalink to "Plot spaghetti plot {#Plot-spaghetti-plot}""></a></h2><p>Sometimes, it is useful to plot the quadratures of the steady states (u, v) in function of a swept parameter. This is done with <code>plot_spaghetti</code>.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.plot_spaghetti-manual-plotting" href="#HarmonicBalance.plot_spaghetti-manual-plotting"><span class="jlbinding">HarmonicBalance.plot_spaghetti</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_spaghetti</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; x, y, z, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a three dimension line plot of a <code>Result</code> object as a function of the parameters. Works with 1D and 2D datasets.</p><p>Class selection done by passing <code>String</code> or <code>Vector{String}</code> as kwarg:</p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>class::String : only count solutions in this class ("all" --> plot everything)</span></span>
-<span class="line"><span>not_class::String : do not count solutions in this class</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr()</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/plotting_Plots.jl#L342-L354" target="_blank" rel="noreferrer">source</a></p></details></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/plotting.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Linear Response</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/manual/saving" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Saving and Loading</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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+<span class="line"><span>not_class::String : do not count solutions in this class</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr()</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/plotting_Plots.jl#L275" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Plot-spaghetti-plot" tabindex="-1">Plot spaghetti plot <a class="header-anchor" href="#Plot-spaghetti-plot" aria-label="Permalink to "Plot spaghetti plot {#Plot-spaghetti-plot}""></a></h2><p>Sometimes, it is useful to plot the quadratures of the steady states (u, v) in function of a swept parameter. This is done with <code>plot_spaghetti</code>.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.plot_spaghetti-manual-plotting" href="#HarmonicBalance.plot_spaghetti-manual-plotting"><span class="jlbinding">HarmonicBalance.plot_spaghetti</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_spaghetti</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; x, y, z, kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a three dimension line plot of a <code>Result</code> object as a function of the parameters. Works with 1D and 2D datasets.</p><p>Class selection done by passing <code>String</code> or <code>Vector{String}</code> as kwarg:</p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>class::String : only count solutions in this class ("all" --> plot everything)</span></span>
+<span class="line"><span>not_class::String : do not count solutions in this class</span></span></code></pre></div><p>Other kwargs are passed onto Plots.gr()</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/plotting_Plots.jl#L342-L354" target="_blank" rel="noreferrer">source</a></p></details></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/plotting.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Linear Response</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/manual/saving" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Saving and Loading</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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tabindex="-1" data-v-18756405><span class="visually-hidden" id="sidebar-aria-label" data-v-18756405> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0" data-v-9e426adc data-v-a4b0d9bf><!----><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/index" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Home</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/index" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Getting 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data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/stability_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Stability and Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Tutorials</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/steady_states" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/classification" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Classifying Solutions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Transient Dynamics</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Examples</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Three Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Oscillator</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>State Dependent Perturbation</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible has-active" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Resources</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/API" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>API</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Bibliography</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible has-active" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Manual</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Entering Equations of Motion</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Effective System</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/methods" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Krylov-Bogoliubov</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Time Evolution</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Plotting</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/saving" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Saving and Loading</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-a9a9e638 data-v-91765379><div class="VPDoc has-sidebar has-aside" data-v-91765379 data-v-83890dd9><!--[--><!--]--><div class="container" data-v-83890dd9><div class="aside" data-v-83890dd9><div class="aside-curtain" data-v-83890dd9></div><div class="aside-container" data-v-83890dd9><div class="aside-content" data-v-83890dd9><div class="VPDocAside" data-v-83890dd9 data-v-6d7b3c46><!--[--><!--]--><!--[--><!--]--><nav aria-labelledby="doc-outline-aria-label" class="VPDocAsideOutline" data-v-6d7b3c46 data-v-b38bf2ff><div class="content" data-v-b38bf2ff><div class="outline-marker" data-v-b38bf2ff></div><div aria-level="2" class="outline-title" id="doc-outline-aria-label" role="heading" data-v-b38bf2ff>On this page</div><ul class="VPDocOutlineItem root" data-v-b38bf2ff data-v-3f927ebe><!--[--><!--]--></ul></div></nav><!--[--><!--]--><div class="spacer" data-v-6d7b3c46></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--]--></div></div></div></div><div class="content" data-v-83890dd9><div class="content-container" data-v-83890dd9><!--[--><!--]--><main class="main" data-v-83890dd9><div style="position:relative;" class="vp-doc _HarmonicBalance_jl_dev_manual_saving" data-v-83890dd9><div><h1 id="Saving-and-loading" tabindex="-1">Saving and loading <a class="header-anchor" href="#Saving-and-loading" aria-label="Permalink to "Saving and loading {#Saving-and-loading}""></a></h1><p>All of the types native to <code>HarmonicBalance.jl</code> can be saved into a <code>.jld2</code> file using <code>save</code> and loaded using <code>load</code>. Most of the saving/loading is performed using the package <code>JLD2.jl</code>, with the addition of reinstating the symbolic variables in the <code>HarmonicBalance</code> namespace (needed to parse expressions used in the plotting functions) and recompiling stored functions (needed to evaluate Jacobians). As a consequence, composite objects such as <code>Result</code> can be saved and loaded with no loss of information.</p><p>The function <code>export_csv</code> saves a .csv file which can be plot elsewhere.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.save-manual-saving" href="#HarmonicBalance.save-manual-saving"><span class="jlbinding">HarmonicBalance.save</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">save</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(filename, object)</span></span></code></pre></div><p>Saves <code>object</code> into <code>.jld2</code> file <code>filename</code> (the suffix is added automatically if not entered). The resulting file contains a dictionary with a single entry.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/saving.jl#L1-L7" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.load-manual-saving" href="#HarmonicBalance.load-manual-saving"><span class="jlbinding">HarmonicBalance.load</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">load</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(filename)</span></span></code></pre></div><p>Loads an object from <code>filename</code>. For objects containing symbolic expressions such as <code>HarmonicEquation</code>, the symbolic variables are reinstated in the <code>HarmonicBalance</code> namespace.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/saving.jl#L22-L28" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.export_csv-manual-saving" href="#HarmonicBalance.export_csv-manual-saving"><span class="jlbinding">HarmonicBalance.export_csv</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">export_csv</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(filename, res, branch)</span></span></code></pre></div><p>Saves into <code>filename</code> a specified solution <code>branch</code> of the Result <code>res</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/saving.jl#L77-L81" target="_blank" rel="noreferrer">source</a></p></details><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span></span></span></code></pre></div></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/saving.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Plotting</span><!--]--></a></div><div class="pager" data-v-4f9813fa><!----></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. 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class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Transient Dynamics</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Examples</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Three Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Oscillator</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>State Dependent Perturbation</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible has-active" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Resources</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/API" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>API</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Bibliography</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible has-active" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Manual</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Entering Equations of Motion</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Effective System</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/methods" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Krylov-Bogoliubov</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Time Evolution</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Plotting</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/saving" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Saving and Loading</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-a9a9e638 data-v-91765379><div class="VPDoc has-sidebar has-aside" data-v-91765379 data-v-83890dd9><!--[--><!--]--><div class="container" data-v-83890dd9><div class="aside" data-v-83890dd9><div class="aside-curtain" data-v-83890dd9></div><div class="aside-container" data-v-83890dd9><div class="aside-content" data-v-83890dd9><div class="VPDocAside" data-v-83890dd9 data-v-6d7b3c46><!--[--><!--]--><!--[--><!--]--><nav aria-labelledby="doc-outline-aria-label" class="VPDocAsideOutline" data-v-6d7b3c46 data-v-b38bf2ff><div class="content" data-v-b38bf2ff><div class="outline-marker" data-v-b38bf2ff></div><div aria-level="2" class="outline-title" id="doc-outline-aria-label" role="heading" data-v-b38bf2ff>On this page</div><ul class="VPDocOutlineItem root" data-v-b38bf2ff data-v-3f927ebe><!--[--><!--]--></ul></div></nav><!--[--><!--]--><div class="spacer" data-v-6d7b3c46></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--]--></div></div></div></div><div class="content" data-v-83890dd9><div class="content-container" data-v-83890dd9><!--[--><!--]--><main class="main" data-v-83890dd9><div style="position:relative;" class="vp-doc _HarmonicBalance_jl_dev_manual_saving" data-v-83890dd9><div><h1 id="Saving-and-loading" tabindex="-1">Saving and loading <a class="header-anchor" href="#Saving-and-loading" aria-label="Permalink to "Saving and loading {#Saving-and-loading}""></a></h1><p>All of the types native to <code>HarmonicBalance.jl</code> can be saved into a <code>.jld2</code> file using <code>save</code> and loaded using <code>load</code>. Most of the saving/loading is performed using the package <code>JLD2.jl</code>, with the addition of reinstating the symbolic variables in the <code>HarmonicBalance</code> namespace (needed to parse expressions used in the plotting functions) and recompiling stored functions (needed to evaluate Jacobians). As a consequence, composite objects such as <code>Result</code> can be saved and loaded with no loss of information.</p><p>The function <code>export_csv</code> saves a .csv file which can be plot elsewhere.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.save-manual-saving" href="#HarmonicBalance.save-manual-saving"><span class="jlbinding">HarmonicBalance.save</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">save</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(filename, object)</span></span></code></pre></div><p>Saves <code>object</code> into <code>.jld2</code> file <code>filename</code> (the suffix is added automatically if not entered). The resulting file contains a dictionary with a single entry.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/saving.jl#L1-L7" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.load-manual-saving" href="#HarmonicBalance.load-manual-saving"><span class="jlbinding">HarmonicBalance.load</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">load</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(filename)</span></span></code></pre></div><p>Loads an object from <code>filename</code>. For objects containing symbolic expressions such as <code>HarmonicEquation</code>, the symbolic variables are reinstated in the <code>HarmonicBalance</code> namespace.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/saving.jl#L22-L28" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.export_csv-manual-saving" href="#HarmonicBalance.export_csv-manual-saving"><span class="jlbinding">HarmonicBalance.export_csv</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">export_csv</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(filename, res, branch)</span></span></code></pre></div><p>Saves into <code>filename</code> a specified solution <code>branch</code> of the Result <code>res</code>.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/saving.jl#L77-L81" target="_blank" rel="noreferrer">source</a></p></details><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span></span></span></code></pre></div></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/saving.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Plotting</span><!--]--></a></div><div class="pager" data-v-4f9813fa><!----></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. 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@@ -30,7 +30,7 @@
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> swept</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">AbstractDict</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> fixed</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">AbstractDict</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> compute_Jacobian</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Bool</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/Problem.jl#L3" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_steady_states-manual-solving_harmonics" href="#HarmonicBalance.get_steady_states-manual-solving_harmonics"><span class="jlbinding">HarmonicBalance.get_steady_states</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(problem</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/Problem.jl#L3" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.get_steady_states-manual-solving_harmonics" href="#HarmonicBalance.get_steady_states-manual-solving_harmonics"><span class="jlbinding">HarmonicBalance.get_steady_states</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(problem</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> method</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalanceMethod</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> swept_parameters,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> fixed_parameters;</span></span>
@@ -59,7 +59,7 @@
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> of which real</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> of which stable</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1</span></span>
<span class="line"></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Classes</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> stable, physical, Hopf, binary_labels</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/solve_homotopy.jl#L1-L60" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.Result-manual-solving_harmonics" href="#HarmonicBalance.Result-manual-solving_harmonics"><span class="jlbinding">HarmonicBalance.Result</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Result{SolType</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Number</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ParType</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Number</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, D, F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">FunctionWrappers.FunctionWrapper{Array{SolType<:Number, 2}, Tuple{Array{SolType<:Number, 1}}}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}</span></span></code></pre></div><p>Stores the steady states of a HarmonicEquation.</p><p><strong>Fields</strong></p><ul><li><p><code>solutions::Array{Array{Vector{SolType}, 1}} where SolType<:Number</code>: The variable values of steady-state solutions.</p></li><li><p><code>swept_parameters::OrderedCollections.OrderedDict{Num, Vector{ParType}} where ParType<:Number</code>: Values of all parameters for all solutions.</p></li><li><p><code>fixed_parameters::OrderedCollections.OrderedDict{Num, ParType} where ParType<:Number</code>: The parameters fixed throughout the solutions.</p></li><li><p><code>problem::HarmonicBalance.Problem{ParType, F} where {SolType<:Number, ParType<:Number, F<:FunctionWrappers.FunctionWrapper{Matrix{SolType}, Tuple{Vector{SolType}}}}</code>: The <code>Problem</code> used to generate this.</p></li><li><p><code>classes::Dict{String, Array{BitVector, D}} where D</code>: Maps strings such as "stable", "physical" etc to arrays of values, classifying the solutions (see method <code>classify_solutions!</code>).</p></li><li><p><code>binary_labels::Array{Int64}</code>: Create binary classification of the solutions, such that each solution point receives an identifier based on its permutation of stable branches (allows to distinguish between different phases, which may have the same number of stable solutions). It works by converting each bitstring <code>[is_stable(solution_1), is_stable(solution_2), ...,]</code> into unique labels.</p></li><li><p><code>jacobian::FunctionWrappers.FunctionWrapper{Matrix{SolType}, Tuple{Vector{SolType}}} where SolType<:Number</code>: The Jacobian function with <code>fixed_parameters</code> already substituted. Accepts a vector specifying the solution. If problem.jacobian is a symbolic matrix, this holds a compiled function.</p></li><li><p><code>seed::UInt32</code>: Seed used for the solver</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/Result.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Classifying-solutions" tabindex="-1">Classifying solutions <a class="header-anchor" href="#Classifying-solutions" aria-label="Permalink to "Classifying solutions {#Classifying-solutions}""></a></h2><p>The solutions in <code>Result</code> are accompanied by similarly-sized boolean arrays stored in the dictionary <code>Result.classes</code>. The classes can be used by the plotting functions to show/hide/label certain solutions.</p><p>By default, classes "physical", "stable" and "binary_labels" are created. User-defined classification is possible with <code>classify_solutions!</code>.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.classify_solutions!-manual-solving_harmonics" href="#HarmonicBalance.classify_solutions!-manual-solving_harmonics"><span class="jlbinding">HarmonicBalance.classify_solutions!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Classes</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">:</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> stable, physical, Hopf, binary_labels</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/solve_homotopy.jl#L1-L60" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.Result-manual-solving_harmonics" href="#HarmonicBalance.Result-manual-solving_harmonics"><span class="jlbinding">HarmonicBalance.Result</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">struct</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Result{SolType</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Number</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ParType</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Number</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, D, F</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;"><:</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">FunctionWrappers.FunctionWrapper{Array{SolType<:Number, 2}, Tuple{Array{SolType<:Number, 1}}}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}</span></span></code></pre></div><p>Stores the steady states of a HarmonicEquation.</p><p><strong>Fields</strong></p><ul><li><p><code>solutions::Array{Array{Vector{SolType}, 1}} where SolType<:Number</code>: The variable values of steady-state solutions.</p></li><li><p><code>swept_parameters::OrderedCollections.OrderedDict{Num, Vector{ParType}} where ParType<:Number</code>: Values of all parameters for all solutions.</p></li><li><p><code>fixed_parameters::OrderedCollections.OrderedDict{Num, ParType} where ParType<:Number</code>: The parameters fixed throughout the solutions.</p></li><li><p><code>problem::HarmonicBalance.Problem{ParType, F} where {SolType<:Number, ParType<:Number, F<:FunctionWrappers.FunctionWrapper{Matrix{SolType}, Tuple{Vector{SolType}}}}</code>: The <code>Problem</code> used to generate this.</p></li><li><p><code>classes::Dict{String, Array{BitVector, D}} where D</code>: Maps strings such as "stable", "physical" etc to arrays of values, classifying the solutions (see method <code>classify_solutions!</code>).</p></li><li><p><code>binary_labels::Array{Int64}</code>: Create binary classification of the solutions, such that each solution point receives an identifier based on its permutation of stable branches (allows to distinguish between different phases, which may have the same number of stable solutions). It works by converting each bitstring <code>[is_stable(solution_1), is_stable(solution_2), ...,]</code> into unique labels.</p></li><li><p><code>jacobian::FunctionWrappers.FunctionWrapper{Matrix{SolType}, Tuple{Vector{SolType}}} where SolType<:Number</code>: The Jacobian function with <code>fixed_parameters</code> already substituted. Accepts a vector specifying the solution. If problem.jacobian is a symbolic matrix, this holds a compiled function.</p></li><li><p><code>seed::UInt32</code>: Seed used for the solver</p></li></ul><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/Result.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Classifying-solutions" tabindex="-1">Classifying solutions <a class="header-anchor" href="#Classifying-solutions" aria-label="Permalink to "Classifying solutions {#Classifying-solutions}""></a></h2><p>The solutions in <code>Result</code> are accompanied by similarly-sized boolean arrays stored in the dictionary <code>Result.classes</code>. The classes can be used by the plotting functions to show/hide/label certain solutions.</p><p>By default, classes "physical", "stable" and "binary_labels" are created. User-defined classification is possible with <code>classify_solutions!</code>.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.classify_solutions!-manual-solving_harmonics" href="#HarmonicBalance.classify_solutions!-manual-solving_harmonics"><span class="jlbinding">HarmonicBalance.classify_solutions!</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> func</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Union{Function, String}</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> name</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
@@ -68,12 +68,12 @@
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">res </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(problem, swept_parameters, fixed_parameters)</span></span>
<span class="line"></span>
<span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># classify, store in result.classes["large_amplitude"]</span></span>
-<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u1^2 + v1^2) > 1.0"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> , </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"large_amplitude"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/classification.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Sorting-solutions" tabindex="-1">Sorting solutions <a class="header-anchor" href="#Sorting-solutions" aria-label="Permalink to "Sorting solutions {#Sorting-solutions}""></a></h2><p>Solving a steady-state problem over a range of parameters returns a solution set for each parameter. For a continuous change of parameters, each solution in a set usually also changes continuously; it is said to form a ''solution branch''. For an example, see the three colour-coded branches for the Duffing oscillator in Example 1.</p><p>For stable states, the branches describe a system's behaviour under adiabatic parameter changes.</p><p>Therefore, after solving for a parameter range, we want to order each solution set such that the solutions' order reflects the branches.</p><p>The function <code>sort_solutions</code> goes over the the raw output of <code>get_steady_states</code> and sorts each entry such that neighboring solution sets minimize Euclidean distance.</p><p>Currently, <code>sort_solutions</code> is compatible with 1D and 2D arrays of solution sets.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.sort_solutions-manual-solving_harmonics" href="#HarmonicBalance.sort_solutions-manual-solving_harmonics"><span class="jlbinding">HarmonicBalance.sort_solutions</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">sort_solutions</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(res, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u1^2 + v1^2) > 1.0"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> , </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"large_amplitude"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/classification.jl#L1" target="_blank" rel="noreferrer">source</a></p></details><h2 id="Sorting-solutions" tabindex="-1">Sorting solutions <a class="header-anchor" href="#Sorting-solutions" aria-label="Permalink to "Sorting solutions {#Sorting-solutions}""></a></h2><p>Solving a steady-state problem over a range of parameters returns a solution set for each parameter. For a continuous change of parameters, each solution in a set usually also changes continuously; it is said to form a ''solution branch''. For an example, see the three colour-coded branches for the Duffing oscillator in Example 1.</p><p>For stable states, the branches describe a system's behaviour under adiabatic parameter changes.</p><p>Therefore, after solving for a parameter range, we want to order each solution set such that the solutions' order reflects the branches.</p><p>The function <code>sort_solutions</code> goes over the the raw output of <code>get_steady_states</code> and sorts each entry such that neighboring solution sets minimize Euclidean distance.</p><p>Currently, <code>sort_solutions</code> is compatible with 1D and 2D arrays of solution sets.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.sort_solutions-manual-solving_harmonics" href="#HarmonicBalance.sort_solutions-manual-solving_harmonics"><span class="jlbinding">HarmonicBalance.sort_solutions</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">sort_solutions</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> solutions</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Union</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">{Array{Array{Array{T, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}, Array{Array{Array{T, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">}};</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sorting,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> show_progress</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Sorts <code>solutions</code> into branches according to the specified <code>sorting</code> method.</p><p><code>solutions</code> is an n-dimensional array of <code>Vector{Vector}</code>. Each element describes a set of solutions for a given parameter set. The output is a similar array, with each solution set rearranged such that neighboring solution sets have the smallest Euclidean distance.</p><p>The <code>sorting</code> keyword argument specifies the method used to get continuous solution branches. Options are <code>"hilbert"</code> (1D sorting along a Hilbert curve), <code>"nearest"</code> (nearest-neighbor sorting), and <code>"none"</code>. The <code>show_progress</code> keyword argument indicates whether a progress bar should be displayed.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/sorting.jl#L1-L13" target="_blank" rel="noreferrer">source</a></p></details></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/solving_harmonics.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><!----></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/index" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Home</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Sorts <code>solutions</code> into branches according to the specified <code>sorting</code> method.</p><p><code>solutions</code> is an n-dimensional array of <code>Vector{Vector}</code>. Each element describes a set of solutions for a given parameter set. The output is a similar array, with each solution set rearranged such that neighboring solution sets have the smallest Euclidean distance.</p><p>The <code>sorting</code> keyword argument specifies the method used to get continuous solution branches. Options are <code>"hilbert"</code> (1D sorting along a Hilbert curve), <code>"nearest"</code> (nearest-neighbor sorting), and <code>"none"</code>. The <code>show_progress</code> keyword argument indicates whether a progress bar should be displayed.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/sorting.jl#L1-L13" target="_blank" rel="noreferrer">source</a></p></details></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/solving_harmonics.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><!----></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/index" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Home</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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System\",\"link\":\"/manual/extracting_harmonics\"},{\"text\":\"Computing Steady States\",\"link\":\"/manual/methods\"},{\"text\":\"Krylov-Bogoliubov\",\"link\":\"/manual/Krylov-Bogoliubov_method\"},{\"text\":\"Time Evolution\",\"link\":\"/manual/time_dependent\"},{\"text\":\"Linear Response\",\"link\":\"/manual/linear_response\"},{\"text\":\"Plotting\",\"link\":\"/manual/plotting\"},{\"text\":\"Saving and Loading\",\"link\":\"/manual/saving\"}]}]}],\"editLink\":{\"pattern\":\"https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/:path\"},\"socialLinks\":[{\"icon\":\"github\",\"link\":\"https://github.com/NonlinearOscillations/HarmonicBalance.jl\"},{\"icon\":\"twitter\",\"link\":\"https://x.com/Zilberberg_Phys\"}],\"footer\":{\"message\":\"Made with <a href=\\\"https://documenter.juliadocs.org/\\\" target=\\\"_blank\\\"><strong>Documenter.jl</strong></a>, <a href=\\\"https://vitepress.dev\\\" target=\\\"_blank\\\"><strong>VitePress</strong></a> and <a href=\\\"https://luxdl.github.io/DocumenterVitepress.jl/\\\" target=\\\"_blank\\\"><strong>DocumenterVitepress.jl</strong>\",\"copyright\":\"© Copyright 2025. 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@@ -33,7 +33,7 @@
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> timespan,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> perturb_initial,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Creates an ODEProblem object used by OrdinaryDiffEqTsit5.jl from the equations in <code>eom</code> to simulate time-evolution within <code>timespan</code>. <code>fixed_parameters</code> must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If <code>u0</code> is specified, it is used as an initial condition; otherwise the values from <code>fixed_parameters</code> are used.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/ODEProblem.jl#L3-L9" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.AdiabaticSweep-manual-time_dependent" href="#HarmonicBalance.AdiabaticSweep-manual-time_dependent"><span class="jlbinding">HarmonicBalance.AdiabaticSweep</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><p>Represents a sweep of one or more parameters of a <code>HarmonicEquation</code>. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.</p><p>Sweeps of different variables can be combined using <code>+</code>.</p><p><strong>Fields</strong></p><ul><li><code>functions::Dict{Num, Function}</code>: Maps each swept parameter to a function.</li></ul><p><strong>Examples</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># create a sweep of parameter a from 0 to 1 over time 0 -> 100</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Creates an ODEProblem object used by OrdinaryDiffEqTsit5.jl from the equations in <code>eom</code> to simulate time-evolution within <code>timespan</code>. <code>fixed_parameters</code> must be a dictionary mapping parameters+variables to numbers (possible to use a solution index, e.g. solutions[x][y] for branch y of solution x). If <code>u0</code> is specified, it is used as an initial condition; otherwise the values from <code>fixed_parameters</code> are used.</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/ODEProblem.jl#L3-L9" target="_blank" rel="noreferrer">source</a></p></details><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.AdiabaticSweep-manual-time_dependent" href="#HarmonicBalance.AdiabaticSweep-manual-time_dependent"><span class="jlbinding">HarmonicBalance.AdiabaticSweep</span></a> <span class="VPBadge info jlObjectType jlType"><!--[-->Type<!--]--></span></summary><p>Represents a sweep of one or more parameters of a <code>HarmonicEquation</code>. During a sweep, the selected parameters vary linearly over some timespan and are constant elsewhere.</p><p>Sweeps of different variables can be combined using <code>+</code>.</p><p><strong>Fields</strong></p><ul><li><code>functions::Dict{Num, Function}</code>: Maps each swept parameter to a function.</li></ul><p><strong>Examples</strong></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># create a sweep of parameter a from 0 to 1 over time 0 -> 100</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> @variables</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> a,b;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(a </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">100</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">));</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep[a](</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">50</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
@@ -45,18 +45,18 @@
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">julia</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">([a </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], b </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">]], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">100</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span></code></pre></div><p>Successive sweeps can be combined,</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep1 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.95</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep2 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(λ </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.05</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.01</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">4e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep1 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">+</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> sweep2</span></span></code></pre></div><p>multiple parameters can be swept simultaneously,</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">([ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.95</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], λ </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> [</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">5e-2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1e-2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">]], (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span></code></pre></div><p>and custom sweep functions may be used.</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#6F42C1;--shiki-dark:#B392F0;">ωfunc</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> cos</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(t)</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ωfunc)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/types.jl#L9-L48" target="_blank" rel="noreferrer">source</a></p></details><h2 id="plotting" tabindex="-1">Plotting <a class="header-anchor" href="#plotting" aria-label="Permalink to "Plotting""></a></h2><details class="jldocstring custom-block" open><summary><a id="RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-time_dependent" href="#RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-time_dependent"><span class="jlbinding">RecipesBase.plot</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESolution</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, f</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, harm_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a function <code>f</code> of a time-dependent solution <code>soln</code> of <code>harm_eq</code>.</p><p><strong>As a function of time</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p><code>f</code> is parsed by Symbolics.jl</p><p><strong>parametric plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p>Parametric plot of f[1] against f[2]</p><p>Also callable as plot!</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/plotting.jl#L4-L22" target="_blank" rel="noreferrer">source</a></p></details><h2 id="miscellaneous" tabindex="-1">Miscellaneous <a class="header-anchor" href="#miscellaneous" aria-label="Permalink to "Miscellaneous""></a></h2><p>Using a time-dependent simulation can verify solution stability in cases where the Jacobian is too expensive to compute.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.is_stable-manual-time_dependent" href="#HarmonicBalance.is_stable-manual-time_dependent"><span class="jlbinding">HarmonicBalance.is_stable</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">is_stable</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ωfunc)</span></span></code></pre></div><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/types.jl#L9-L48" target="_blank" rel="noreferrer">source</a></p></details><h2 id="plotting" tabindex="-1">Plotting <a class="header-anchor" href="#plotting" aria-label="Permalink to "Plotting""></a></h2><details class="jldocstring custom-block" open><summary><a id="RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-time_dependent" href="#RecipesBase.plot-Tuple{ODESolution, Any, HarmonicEquation}-manual-time_dependent"><span class="jlbinding">RecipesBase.plot</span></a> <span class="VPBadge info jlObjectType jlMethod"><!--[-->Method<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">ODESolution</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, f</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">String</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, harm_eq</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">; kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Plot a function <code>f</code> of a time-dependent solution <code>soln</code> of <code>harm_eq</code>.</p><p><strong>As a function of time</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::String, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p><code>f</code> is parsed by Symbolics.jl</p><p><strong>parametric plots</strong></p><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>plot(soln::ODESolution, f::Vector{String}, harm_eq::HarmonicEquation; kwargs...)</span></span></code></pre></div><p>Parametric plot of f[1] against f[2]</p><p>Also callable as plot!</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/plotting.jl#L4-L22" target="_blank" rel="noreferrer">source</a></p></details><h2 id="miscellaneous" tabindex="-1">Miscellaneous <a class="header-anchor" href="#miscellaneous" aria-label="Permalink to "Miscellaneous""></a></h2><p>Using a time-dependent simulation can verify solution stability in cases where the Jacobian is too expensive to compute.</p><details class="jldocstring custom-block" open><summary><a id="HarmonicBalance.is_stable-manual-time_dependent" href="#HarmonicBalance.is_stable-manual-time_dependent"><span class="jlbinding">HarmonicBalance.is_stable</span></a> <span class="VPBadge info jlObjectType jlFunction"><!--[-->Function<!--]--></span></summary><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">is_stable</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">OrderedCollections.OrderedDict</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> eom</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicEquation</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> timespan,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> tol,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> perturb_initial</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Numerically investigate the stability of a solution <code>soln</code> of <code>eom</code> within <code>timespan</code>. The initial condition is displaced by <code>perturb_initial</code>.</p><p>Return <code>true</code> the solution evolves within <code>tol</code> of the initial value (interpreted as stable).</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/ext/TimeEvolution/ODEProblem.jl#L61" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">is_stable</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p>Numerically investigate the stability of a solution <code>soln</code> of <code>eom</code> within <code>timespan</code>. The initial condition is displaced by <code>perturb_initial</code>.</p><p>Return <code>true</code> the solution evolves within <code>tol</code> of the initial value (interpreted as stable).</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/ext/TimeEvolution/ODEProblem.jl#L61" target="_blank" rel="noreferrer">source</a></p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">is_stable</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> soln</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">OrderedCollections.OrderedDict</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> res</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">::</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">HarmonicBalance.Result</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> kwargs</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">...</span></span>
-<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Returns true if the solution <code>soln</code> of the Result <code>res</code> is stable. Stable solutions are real and have all Jacobian eigenvalues Re(λ) <= 0. <code>im_tol</code> : an absolute threshold to distinguish real/complex numbers. <code>rel_tol</code>: Re(λ) considered <=0 if real.(λ) < rel_tol*abs(λmax)</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/f8ef5541f30bbaa3aea6f43c0ae257b23a4fcb4f/src/classification.jl#L77" target="_blank" rel="noreferrer">source</a></p></details></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/time_dependent.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Krylov-Bogoliubov</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Linear Response</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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+<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">) </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">-></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> Any</span></span></code></pre></div><p>Returns true if the solution <code>soln</code> of the Result <code>res</code> is stable. Stable solutions are real and have all Jacobian eigenvalues Re(λ) <= 0. <code>im_tol</code> : an absolute threshold to distinguish real/complex numbers. <code>rel_tol</code>: Re(λ) considered <=0 if real.(λ) < rel_tol*abs(λmax)</p><p><a href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/blob/3908971c8fcdcd6a613eb08682e7be59a24a438f/src/classification.jl#L77" target="_blank" rel="noreferrer">source</a></p></details></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/manual/time_dependent.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Krylov-Bogoliubov</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Linear Response</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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<span class="line"><span>Classes: zero, stable, physical, Hopf</span></span></code></pre></div><p>We can visualize the zero amplitude solution:</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_phase_diagram</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result_2D, class</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">[</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"zero"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"stable"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">])</span></span></code></pre></div><p><img src="/HarmonicBalance.jl/dev/assets/opefkjc.BuyRM-Sv.png" alt=""></p><p>This shows that inside the Mathieu lobe the zero amplitude solution becomes unstable due to the parametric drive being resonant with the oscillator.</p><p>We can also visualize the equi-amplitude curves of the solutions:</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">classify_solutions!</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result_2D, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u1^2 + v1^2) > 0.12"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"large amplitude"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot_phase_diagram</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result_2D, class</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">[</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"large amplitude"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"stable"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">])</span></span></code></pre></div><p><img src="/HarmonicBalance.jl/dev/assets/itgikab.CHo32oEM.png" alt=""></p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/tutorials/classification.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/tutorials/steady_states" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Steady States</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/tutorials/linear_response" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Linear Response</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-acbfed09><!--[--><span data-v-acbfed09>Krylov-Bogoliubov</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-acbfed09><!--[--><span data-v-acbfed09>Time Evolution</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-acbfed09><!--[--><span data-v-acbfed09>Linear Response</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-acbfed09><!--[--><span data-v-acbfed09>Plotting</span><!--]--></a></div><!--]--><!--[--><div class="VPMenuLink" data-v-48c802d0 data-v-acbfed09><a class="VPLink link" href="/HarmonicBalance.jl/dev/manual/saving" 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tabindex="-1" data-v-18756405><span class="visually-hidden" id="sidebar-aria-label" data-v-18756405> Sidebar Navigation </span><!--[--><!--]--><!--[--><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0" data-v-9e426adc data-v-a4b0d9bf><!----><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/index" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Home</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/index" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Getting Started</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Theoretical Background</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/harmonic_balance" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Floquet expansions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/stability_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Stability and Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/background/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Tutorials</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/steady_states" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/classification" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Classifying Solutions</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Transient Dynamics</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/tutorials/limit_cycles" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Limit Cycles</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Examples</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametric_via_three_wave_mixing" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Three Wave Mixing</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/parametron" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Parametric Oscillator</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/examples/state_dependent_perturbation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>State Dependent Perturbation</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><div class="no-transition group" data-v-9e426adc><section class="VPSidebarItem level-0 collapsible" data-v-9e426adc data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h2 class="text" data-v-a4b0d9bf>Resources</h2><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/API" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>API</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-1 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/introduction/citation" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Bibliography</p><!--]--></a><!----></div><!----></div><section class="VPSidebarItem level-1 collapsible" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" role="button" tabindex="0" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><h3 class="text" data-v-a4b0d9bf>Manual</h3><div class="caret" role="button" aria-label="toggle section" tabindex="0" data-v-a4b0d9bf><span class="vpi-chevron-right caret-icon" data-v-a4b0d9bf></span></div></div><div class="items" data-v-a4b0d9bf><!--[--><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/entering_eom" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Entering Equations of Motion</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/extracting_harmonics" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Effective System</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/methods" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Computing Steady States</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/Krylov-Bogoliubov_method" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Krylov-Bogoliubov</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/time_dependent" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Time Evolution</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/linear_response" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Linear Response</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/plotting" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Plotting</p><!--]--></a><!----></div><!----></div><div class="VPSidebarItem level-2 is-link" data-v-a4b0d9bf data-v-a4b0d9bf><div class="item" data-v-a4b0d9bf><div class="indicator" data-v-a4b0d9bf></div><a class="VPLink link link" href="/HarmonicBalance.jl/dev/manual/saving" data-v-a4b0d9bf><!--[--><p class="text" data-v-a4b0d9bf>Saving and Loading</p><!--]--></a><!----></div><!----></div><!--]--></div></section><!--]--></div></section></div><!--]--><!--[--><!--]--></nav></aside><div class="VPContent has-sidebar" id="VPContent" data-v-a9a9e638 data-v-91765379><div class="VPDoc has-sidebar has-aside" data-v-91765379 data-v-83890dd9><!--[--><!--]--><div class="container" data-v-83890dd9><div class="aside" data-v-83890dd9><div class="aside-curtain" data-v-83890dd9></div><div class="aside-container" data-v-83890dd9><div class="aside-content" data-v-83890dd9><div class="VPDocAside" data-v-83890dd9 data-v-6d7b3c46><!--[--><!--]--><!--[--><!--]--><nav aria-labelledby="doc-outline-aria-label" class="VPDocAsideOutline" data-v-6d7b3c46 data-v-b38bf2ff><div class="content" data-v-b38bf2ff><div class="outline-marker" data-v-b38bf2ff></div><div aria-level="2" class="outline-title" id="doc-outline-aria-label" role="heading" data-v-b38bf2ff>On this page</div><ul class="VPDocOutlineItem root" data-v-b38bf2ff data-v-3f927ebe><!--[--><!--]--></ul></div></nav><!--[--><!--]--><div class="spacer" data-v-6d7b3c46></div><!--[--><!--]--><!----><!--[--><!--]--><!--[--><!--]--></div></div></div></div><div class="content" data-v-83890dd9><div class="content-container" data-v-83890dd9><!--[--><!--]--><main class="main" data-v-83890dd9><div style="position:relative;" class="vp-doc _HarmonicBalance_jl_dev_tutorials_" data-v-83890dd9><div><h1 id="tutorials" tabindex="-1">Tutorials <a class="header-anchor" href="#tutorials" aria-label="Permalink to "Tutorials {#tutorials}""></a></h1><p>We show the capabilities of the package by providing a series of tutorials. Examples of other systems can be found in the <a href="/HarmonicBalance.jl/dev/examples/index#examples">examples</a> tab.</p><div class="gallery-image" data-v-68744f5e><!--[--><div class="img-box" data-v-68744f5e data-v-7654366a><a href="steady_states" data-v-7654366a><img src="https://raw.githubusercontent.com/NonlinearOscillations/HarmonicBalance.jl/gh-pages/v0.10.2/assets/simple_Duffing/response_single.png" height="150px" alt="" data-v-7654366a><div class="transparent-box1" data-v-7654366a><div class="caption" data-v-7654366a><h3 data-v-7654366a>Steady states</h3></div></div><div class="transparent-box2" data-v-7654366a><div class="subcaption" data-v-7654366a><p class="opacity-low" data-v-7654366a>How to get the steady states of the harmonic equations.</p></div></div></a></div><div class="img-box" data-v-68744f5e data-v-7654366a><a href="classification" data-v-7654366a><img src="https://raw.githubusercontent.com/NonlinearOscillations/HarmonicBalance.jl/gh-pages/v0.10.2/assets/parametron/2d_phase_diagram.png" height="150px" alt="" data-v-7654366a><div class="transparent-box1" data-v-7654366a><div class="caption" data-v-7654366a><h3 data-v-7654366a>Classifying solutions</h3></div></div><div class="transparent-box2" data-v-7654366a><div class="subcaption" data-v-7654366a><p class="opacity-low" data-v-7654366a>Learn how to add different types of drives.</p></div></div></a></div><div class="img-box" data-v-68744f5e data-v-7654366a><a href="linear_response" data-v-7654366a><img src="https://raw.githubusercontent.com/NonlinearOscillations/HarmonicBalance.jl/gh-pages/v0.10.2/assets/linear_response/nonlin_F_noise.png" height="150px" alt="" data-v-7654366a><div class="transparent-box1" data-v-7654366a><div class="caption" data-v-7654366a><h3 data-v-7654366a>Linear response</h3></div></div><div class="transparent-box2" data-v-7654366a><div class="subcaption" data-v-7654366a><p class="opacity-low" data-v-7654366a>Learn how to compute the linear response of a steady state.</p></div></div></a></div><div class="img-box" data-v-68744f5e data-v-7654366a><a href="time_dependent" data-v-7654366a><img src="https://raw.githubusercontent.com/NonlinearOscillations/HarmonicBalance.jl/gh-pages/v0.10.2/assets/time_dependent/evo_to_steady.png" height="150px" alt="" data-v-7654366a><div class="transparent-box1" data-v-7654366a><div class="caption" data-v-7654366a><h3 data-v-7654366a>Stroboscopic evolution</h3></div></div><div class="transparent-box2" data-v-7654366a><div class="subcaption" data-v-7654366a><p class="opacity-low" data-v-7654366a>Learn how to investigate stroboscopic time evolution.</p></div></div></a></div><div class="img-box" data-v-68744f5e data-v-7654366a><a href="limit_cycles" data-v-7654366a><img src="https://raw.githubusercontent.com/NonlinearOscillations/HarmonicBalance.jl/gh-pages/v0.10.2/assets/limit_cycles/vdp_degenerate.png" height="150px" alt="" data-v-7654366a><div class="transparent-box1" data-v-7654366a><div class="caption" data-v-7654366a><h3 data-v-7654366a>Limit cycles</h3></div></div><div class="transparent-box2" data-v-7654366a><div class="subcaption" data-v-7654366a><p class="opacity-low" data-v-7654366a>Learn how to find the limit cycles of your system.</p></div></div></a></div><!--]--></div></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/tutorials/index.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><!----></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/index" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Home</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. 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@@ -115,7 +115,7 @@
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">time_evo </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> solve</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(time_problem, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Tsit5</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(), saveat</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">100</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">);</span></span></code></pre></div><p>Inspecting the amplitude as a function of time,</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(time_evo, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u1^2 + v1^2)"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, harmonic_eq)</span></span></code></pre></div><p><img src="/HarmonicBalance.jl/dev/assets/iujazrq.BkrvM7FY.png" alt=""></p><p>we see that initially the sweep is adiabatic as it proceeds along the steady-state branch 1. At around <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.927ex" height="1.717ex" role="img" focusable="false" viewBox="0 -677 3503.6 759" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 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21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>2</mn><mi>e</mi><mn>6</mn></math></mjx-assistive-mml></mjx-container>, an instability occurs and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.635ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2490.6 1000" aria-hidden="true"><g stroke="currentColor" fill="currentColor" 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55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(2101.6,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> starts to rapidly oscillate. At that point, the sweep is stopped. Under free time evolution, the system then settles into a limit-cycle solution where the coordinates move along closed trajectories.</p><p>By plotting the <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math></mjx-assistive-mml></mjx-container> and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.097ex" height="1.027ex" role="img" focusable="false" viewBox="0 -443 485 454" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math></mjx-assistive-mml></mjx-container> variables against each other, we observe the limit cycle shapes in phase space,</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">p1 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(time_evo, [</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"u1"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"v1"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], harmonic_eq)</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">p2 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(time_evo, [</span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"u2"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"v2"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], harmonic_eq)</span></span>
<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(p1, p2)</span></span></code></pre></div><p><img src="/HarmonicBalance.jl/dev/assets/glgfsny.BJogq1ob.png" alt=""></p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/tutorials/limit_cycles.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Transient Dynamics</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/examples/wave_mixing" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Wave Mixing</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. Released under the MIT License.</p></div></footer><!--[--><!--]--></div></div>
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<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> plot_linear_response</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result, x, branch</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, Ω_range</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">range</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.9</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">300</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">), logscale</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">true</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, xscale</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">:log</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">),</span></span>
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<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><img src="/HarmonicBalance.jl/dev/assets/tevosus.DaP9_FvO.png" alt=""></p><p>We see that for low <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="1.695ex" height="1.538ex" role="img" focusable="false" viewBox="0 -680 749 680" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D439" d="M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math></mjx-assistive-mml></mjx-container>, quasi-linear behaviour with a single Lorentzian response occurs, while for larger <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:0;" xmlns="http://www.w3.org/2000/svg" width="1.695ex" height="1.538ex" role="img" focusable="false" viewBox="0 -680 749 680" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D439" d="M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math></mjx-assistive-mml></mjx-container>, two peaks form in the noise response. The two peaks are strongly unequal in magnitude, which is an example of internal squeezing (See supplemental material of <a href="https://doi.org/10.1103/PhysRevX.10.021066" target="_blank" rel="noreferrer">Huber et al.</a>).</p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/tutorials/linear_response.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/tutorials/classification" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Classifying Solutions</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/tutorials/time_dependent" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Transient Dynamics</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. 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@@ -41,7 +41,7 @@
<span class="line"><span></span></span>
<span class="line"><span>(2//1)*Differential(T)(v1(T))*ω + Differential(T)(u1(T))*γ - u1(T)*(ω^2) + u1(T)*(ω0^2) + v1(T)*γ*ω + (3//4)*(u1(T)^3)*α + (3//4)*u1(T)*(v1(T)^2)*α ~ F</span></span>
<span class="line"><span></span></span>
-<span class="line"><span>Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v1(T)^3)*α ~ 0</span></span></code></pre></div><p>The variables <code>u1</code> and <code>v1</code> were declared automatically to construct the harmonic ansatz. The <em>slow time</em> variable <code>T</code> describes variation of the quadratures on timescales much slower than <code>ω</code>. For a steady state, all derivatives w.r.t <code>T</code> vanish, leaving only algebraic equations to be solved.</p><p>We are ready to start plugging in numbers! Let us find steady states by solving <code>harmonic_eq</code> for numerical parameters. Homotopy continuation is especially suited to <a href="https://www.juliahomotopycontinuation.org/guides/parameter-homotopies/" target="_blank" rel="noreferrer">solving over a range of parameter values</a>. Here we will solve over a range of driving frequencies <code>ω</code> – these are stored as <code>Pairs{Sym, Vector{Float}}</code>:</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">varied </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> range</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.9</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">100</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">); </span><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># range of parameter values</span></span></code></pre></div><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>ω => 0.9:0.0030303030303030303:1.2</span></span></code></pre></div><p>The other parameters we be fixed – these are declared as <code>Pairs{Sym, Float}</code> pairs:</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">fixed </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (α </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ω0 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0.01</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, γ </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0.01</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">); </span><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># fixed parameters</span></span></code></pre></div><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>(α => 1.0, ω0 => 1.0, F => 0.01, γ => 0.01)</span></span></code></pre></div><p>Now everything is ready to crank the handle. <code>get_steady_states</code> solves our <code>harmonic_eq</code> using the varied and fixed parameters:</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">result </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(harmonic_eq, varied, fixed)</span></span></code></pre></div><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>A steady state result for 100 parameter points</span></span>
+<span class="line"><span>Differential(T)(v1(T))*γ - (2//1)*Differential(T)(u1(T))*ω - u1(T)*γ*ω - v1(T)*(ω^2) + v1(T)*(ω0^2) + (3//4)*(u1(T)^2)*v1(T)*α + (3//4)*(v1(T)^3)*α ~ 0//1</span></span></code></pre></div><p>The variables <code>u1</code> and <code>v1</code> were declared automatically to construct the harmonic ansatz. The <em>slow time</em> variable <code>T</code> describes variation of the quadratures on timescales much slower than <code>ω</code>. For a steady state, all derivatives w.r.t <code>T</code> vanish, leaving only algebraic equations to be solved.</p><p>We are ready to start plugging in numbers! Let us find steady states by solving <code>harmonic_eq</code> for numerical parameters. Homotopy continuation is especially suited to <a href="https://www.juliahomotopycontinuation.org/guides/parameter-homotopies/" target="_blank" rel="noreferrer">solving over a range of parameter values</a>. Here we will solve over a range of driving frequencies <code>ω</code> – these are stored as <code>Pairs{Sym, Vector{Float}}</code>:</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">varied </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> range</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.9</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.2</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">100</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">); </span><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># range of parameter values</span></span></code></pre></div><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>ω => 0.9:0.0030303030303030303:1.2</span></span></code></pre></div><p>The other parameters we be fixed – these are declared as <code>Pairs{Sym, Float}</code> pairs:</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">fixed </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (α </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1.</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, ω0 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 1.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, F </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0.01</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, γ </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> 0.01</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">); </span><span style="--shiki-light:#6A737D;--shiki-dark:#6A737D;"># fixed parameters</span></span></code></pre></div><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>(α => 1.0, ω0 => 1.0, F => 0.01, γ => 0.01)</span></span></code></pre></div><p>Now everything is ready to crank the handle. <code>get_steady_states</code> solves our <code>harmonic_eq</code> using the varied and fixed parameters:</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">result </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> get_steady_states</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(harmonic_eq, varied, fixed)</span></span></code></pre></div><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>A steady state result for 100 parameter points</span></span>
<span class="line"><span></span></span>
<span class="line"><span>Solution branches: 3</span></span>
<span class="line"><span> of which real: 3</span></span>
@@ -77,7 +77,7 @@
<span class="line"><span>Classes: stable, physical, Hopf</span></span></code></pre></div><p>Although 9 branches were found in total, only 3 remain physical (real-valued). Let us visualise the amplitudes corresponding to the two harmonics, <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-1.287ex;" xmlns="http://www.w3.org/2000/svg" width="10.781ex" height="4.208ex" role="img" focusable="false" viewBox="0 -1291.1 4765.1 1860" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msqrt"><g transform="translate(1020,0)"><g data-mml-node="msubsup"><g data-mml-node="mi"><path data-c="1D448" d="M107 637Q73 637 71 641Q70 643 70 649Q70 673 81 682Q83 683 98 683Q139 681 234 681Q268 681 297 681T342 682T362 682Q378 682 378 672Q378 670 376 658Q371 641 366 638H364Q362 638 359 638T352 638T343 637T334 637Q295 636 284 634T266 623Q265 621 238 518T184 302T154 169Q152 155 152 140Q152 86 183 55T269 24Q336 24 403 69T501 205L552 406Q599 598 599 606Q599 633 535 637Q511 637 511 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class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">p1 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u1^2 + v1^2)"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, legend</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">false</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">p2 </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u2^2 + v2^2)"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(p1, p2)</span></span></code></pre></div><p><img src="/HarmonicBalance.jl/dev/assets/vccluua.BgmpR74-.png" alt=""></p><p>The contributions of <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.407ex" height="1.027ex" role="img" focusable="false" viewBox="0 -443 622 454" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math></mjx-assistive-mml></mjx-container> and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.05ex;" xmlns="http://www.w3.org/2000/svg" width="2.538ex" height="1.554ex" role="img" focusable="false" viewBox="0 -665 1122 687" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><path data-c="33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z" style="stroke-width:3;"></path></g><g data-mml-node="mi" transform="translate(500,0)"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>ω</mi></math></mjx-assistive-mml></mjx-container> are now comparable and the system shows some fairly complex behaviour! This demonstrates how an exact solution within an extended Fourier subspace goes beyond a perturbative treatment.</p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/tutorials/steady_states.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" data-v-4f9813fa></span> Edit this page<!--]--></a></div><!----></div><nav class="prev-next" aria-labelledby="doc-footer-aria-label" data-v-4f9813fa><span class="visually-hidden" id="doc-footer-aria-label" data-v-4f9813fa>Pager</span><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link prev" href="/HarmonicBalance.jl/dev/background/limit_cycles" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Previous page</span><span class="title" data-v-4f9813fa>Limit Cycles</span><!--]--></a></div><div class="pager" data-v-4f9813fa><a class="VPLink link pager-link next" href="/HarmonicBalance.jl/dev/tutorials/classification" data-v-4f9813fa><!--[--><span class="desc" data-v-4f9813fa>Next page</span><span class="title" data-v-4f9813fa>Classifying Solutions</span><!--]--></a></div></nav></footer><!--[--><!--]--></div></div></div><!--[--><!--]--></div></div><footer class="VPFooter has-sidebar" data-v-a9a9e638 data-v-c970a860><div class="container" data-v-c970a860><p class="message" data-v-c970a860>Made with <a href="https://documenter.juliadocs.org/" target="_blank"><strong>Documenter.jl</strong></a>, <a href="https://vitepress.dev" target="_blank"><strong>VitePress</strong></a> and <a href="https://luxdl.github.io/DocumenterVitepress.jl/" target="_blank"><strong>DocumenterVitepress.jl</strong></p><p class="copyright" data-v-c970a860>© Copyright 2025. 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@@ -62,7 +62,7 @@
<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(result, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u1^2 + v1^2)"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span></code></pre></div><p><img src="/HarmonicBalance.jl/dev/assets/savwtiv.dPeTlm0F.png" alt=""></p><p>Clearly when evolving from <code>u0 = [0., 0.]</code>, the system ends up in the low-amplitude branch 2. With <code>u0 = [0.2, 0.2]</code>, the system ends up in branch 3.</p><h2 id="Adiabatic-parameter-sweeps" tabindex="-1">Adiabatic parameter sweeps <a class="header-anchor" href="#Adiabatic-parameter-sweeps" aria-label="Permalink to "Adiabatic parameter sweeps {#Adiabatic-parameter-sweeps}""></a></h2><p>Experimentally, the primary means of exploring the steady state landscape is an adiabatic sweep one or more of the system parameters. This takes the system along a solution branch. If this branch disappears or becomes unstable, a jump occurs.</p><p>The object <a href="/HarmonicBalance.jl/dev/manual/API#HarmonicBalance.AdiabaticSweep"><code>AdiabaticSweep</code></a> specifies a sweep, which is then used as an optional <code>sweep</code> keyword in the <code>ODEProblem</code> constructor.</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> AdiabaticSweep</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ω </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=></span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;"> (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.9</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">,</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">1.1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">), (</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span></code></pre></div><div class="language- vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang"></span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span>AdiabaticSweep(Dict{Num, Function}(ω => TimeEvolution.var"#f#1"{Tuple{Float64, Float64}, Float64, Int64}((0.9, 1.1), 20000.0, 0)))</span></span></code></pre></div><p>The sweep linearly interpolates between <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.316ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 3233.6 748" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(899.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(1955.6,0)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" style="stroke-width:3;"></path><path data-c="2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z" transform="translate(500,0)" style="stroke-width:3;"></path><path data-c="39" d="M352 287Q304 211 232 211Q154 211 104 270T44 396Q42 412 42 436V444Q42 537 111 606Q171 666 243 666Q245 666 249 666T257 665H261Q273 665 286 663T323 651T370 619T413 560Q456 472 456 334Q456 194 396 97Q361 41 312 10T208 -22Q147 -22 108 7T68 93T121 149Q143 149 158 135T173 96Q173 78 164 65T148 49T135 44L131 43Q131 41 138 37T164 27T206 22H212Q272 22 313 86Q352 142 352 280V287ZM244 248Q292 248 321 297T351 430Q351 508 343 542Q341 552 337 562T323 588T293 615T246 625Q208 625 181 598Q160 576 154 546T147 441Q147 358 152 329T172 282Q197 248 244 248Z" transform="translate(778,0)" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>0.9</mn></math></mjx-assistive-mml></mjx-container> at time 0 and <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.316ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 3233.6 748" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(899.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(1955.6,0)"><path data-c="31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z" style="stroke-width:3;"></path><path data-c="2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z" transform="translate(500,0)" style="stroke-width:3;"></path><path data-c="31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z" transform="translate(778,0)" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>1.1</mn></math></mjx-assistive-mml></mjx-container> at time 2e4. For earlier/later times, <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.407ex" height="1.027ex" role="img" focusable="false" viewBox="0 -443 622 454" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math></mjx-assistive-mml></mjx-container> is constant.</p><p>Let us now define a new <code>ODEProblem</code> which incorporates <code>sweep</code> and again use <code>solve</code>:</p><div class="language-julia vp-adaptive-theme"><button title="Copy Code" class="copy"></button><span class="lang">julia</span><pre class="shiki shiki-themes github-light github-dark vp-code" tabindex="0"><code><span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">ode_problem </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> ODEProblem</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(harmonic_eq, fixed, sweep</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">sweep, u0</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">[</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.1</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">;</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0.0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">], timespan</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">0</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">2e4</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">))</span></span>
<span class="line"><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">time_evo </span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;"> solve</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(ode_problem, </span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">Tsit5</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(), saveat</span><span style="--shiki-light:#D73A49;--shiki-dark:#F97583;">=</span><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">100</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">)</span></span>
<span class="line"><span style="--shiki-light:#005CC5;--shiki-dark:#79B8FF;">plot</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">(time_evo, </span><span style="--shiki-light:#032F62;--shiki-dark:#9ECBFF;">"sqrt(u1^2 + v1^2)"</span><span style="--shiki-light:#24292E;--shiki-dark:#E1E4E8;">, harmonic_eq)</span></span></code></pre></div><p><img src="/HarmonicBalance.jl/dev/assets/ufurxsz.Bl1qALVt.png" alt=""></p><p>We see the system first evolves from the initial condition towards the low-amplitude steady state. The amplitude increases as the sweep proceeds, with a jump occurring around <mjx-container class="MathJax" jax="SVG" style="direction:ltr;position:relative;"><svg style="overflow:visible;min-height:1px;min-width:1px;vertical-align:-0.186ex;" xmlns="http://www.w3.org/2000/svg" width="8.447ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 3733.6 748" aria-hidden="true"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z" style="stroke-width:3;"></path></g><g data-mml-node="mo" transform="translate(899.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z" style="stroke-width:3;"></path></g><g data-mml-node="mn" transform="translate(1955.6,0)"><path data-c="31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z" style="stroke-width:3;"></path><path data-c="2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z" transform="translate(500,0)" style="stroke-width:3;"></path><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z" transform="translate(778,0)" style="stroke-width:3;"></path><path data-c="38" d="M70 417T70 494T124 618T248 666Q319 666 374 624T429 515Q429 485 418 459T392 417T361 389T335 371T324 363L338 354Q352 344 366 334T382 323Q457 264 457 174Q457 95 399 37T249 -22Q159 -22 101 29T43 155Q43 263 172 335L154 348Q133 361 127 368Q70 417 70 494ZM286 386L292 390Q298 394 301 396T311 403T323 413T334 425T345 438T355 454T364 471T369 491T371 513Q371 556 342 586T275 624Q268 625 242 625Q201 625 165 599T128 534Q128 511 141 492T167 463T217 431Q224 426 228 424L286 386ZM250 21Q308 21 350 55T392 137Q392 154 387 169T375 194T353 216T330 234T301 253T274 270Q260 279 244 289T218 306L210 311Q204 311 181 294T133 239T107 157Q107 98 150 60T250 21Z" transform="translate(1278,0)" style="stroke-width:3;"></path></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline" style="top:0px;left:0px;clip:rect(1px, 1px, 1px, 1px);-webkit-touch-callout:none;-webkit-user-select:none;-khtml-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;position:absolute;padding:1px 0px 0px 0px;border:0px;display:block;width:auto;overflow:hidden;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>1.08</mn></math></mjx-assistive-mml></mjx-container> (i.e., time 18000).</p></div></div></main><footer class="VPDocFooter" data-v-83890dd9 data-v-4f9813fa><!--[--><!--]--><div class="edit-info" data-v-4f9813fa><div class="edit-link" data-v-4f9813fa><a class="VPLink link vp-external-link-icon no-icon edit-link-button" href="https://github.com/NonlinearOscillations/HarmonicBalance.jl/edit/master/docs/src/tutorials/time_dependent.md" target="_blank" rel="noreferrer" data-v-4f9813fa><!--[--><span class="vpi-square-pen edit-link-icon" 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