diff --git a/Manifest.toml b/Manifest.toml
index b20aadd8..a87f5a04 100644
--- a/Manifest.toml
+++ b/Manifest.toml
@@ -46,9 +46,9 @@ version = "0.1.38"
 
 [[deps.Adapt]]
 deps = ["LinearAlgebra", "Requires"]
-git-tree-sha1 = "6a55b747d1812e699320963ffde36f1ebdda4099"
+git-tree-sha1 = "d80af0733c99ea80575f612813fa6aa71022d33a"
 uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
-version = "4.0.4"
+version = "4.1.0"
 weakdeps = ["StaticArrays"]
 
     [deps.Adapt.extensions]
@@ -226,9 +226,9 @@ version = "0.7.6"
 
 [[deps.ColorSchemes]]
 deps = ["ColorTypes", "ColorVectorSpace", "Colors", "FixedPointNumbers", "PrecompileTools", "Random"]
-git-tree-sha1 = "b5278586822443594ff615963b0c09755771b3e0"
+git-tree-sha1 = "13951eb68769ad1cd460cdb2e64e5e95f1bf123d"
 uuid = "35d6a980-a343-548e-a6ea-1d62b119f2f4"
-version = "3.26.0"
+version = "3.27.0"
 
 [[deps.ColorTypes]]
 deps = ["FixedPointNumbers", "Random"]
@@ -804,6 +804,12 @@ git-tree-sha1 = "f31929b9e67066bee48eec8b03c0df47d31a74b3"
 uuid = "d2c73de3-f751-5644-a686-071e5b155ba9"
 version = "0.73.8+0"
 
+[[deps.GenericLinearAlgebra]]
+deps = ["LinearAlgebra", "Printf", "Random", "libblastrampoline_jll"]
+git-tree-sha1 = "f47136cac29a9b7a8a88dbce1195394978091edb"
+uuid = "14197337-ba66-59df-a3e3-ca00e7dcff7a"
+version = "0.3.13"
+
 [[deps.GenericSchur]]
 deps = ["LinearAlgebra", "Printf"]
 git-tree-sha1 = "af49a0851f8113fcfae2ef5027c6d49d0acec39b"
@@ -1232,9 +1238,9 @@ version = "1.11.0"
 
 [[deps.LoggingExtras]]
 deps = ["Dates", "Logging"]
-git-tree-sha1 = "c1dd6d7978c12545b4179fb6153b9250c96b0075"
+git-tree-sha1 = "f02b56007b064fbfddb4c9cd60161b6dd0f40df3"
 uuid = "e6f89c97-d47a-5376-807f-9c37f3926c36"
-version = "1.0.3"
+version = "1.1.0"
 
 [[deps.LoopVectorization]]
 deps = ["ArrayInterface", "CPUSummary", "CloseOpenIntervals", "DocStringExtensions", "HostCPUFeatures", "IfElse", "LayoutPointers", "LinearAlgebra", "OffsetArrays", "PolyesterWeave", "PrecompileTools", "SIMDTypes", "SLEEFPirates", "Static", "StaticArrayInterface", "ThreadingUtilities", "UnPack", "VectorizationBase"]
@@ -1311,21 +1317,23 @@ uuid = "a63ad114-7e13-5084-954f-fe012c677804"
 version = "1.11.0"
 
 [[deps.ModelingToolkit]]
-deps = ["AbstractTrees", "ArrayInterface", "BlockArrays", "Combinatorics", "Compat", "ConstructionBase", "DataStructures", "DiffEqBase", "DiffEqCallbacks", "DiffEqNoiseProcess", "DiffRules", "Distributed", "Distributions", "DocStringExtensions", "DomainSets", "DynamicQuantities", "ExprTools", "Expronicon", "FindFirstFunctions", "ForwardDiff", "FunctionWrappers", "FunctionWrappersWrappers", "Graphs", "InteractiveUtils", "JuliaFormatter", "JumpProcesses", "Latexify", "Libdl", "LinearAlgebra", "MLStyle", "NaNMath", "NonlinearSolve", "OffsetArrays", "OrderedCollections", "PrecompileTools", "RecursiveArrayTools", "Reexport", "RuntimeGeneratedFunctions", "SciMLBase", "SciMLStructures", "Serialization", "Setfield", "SimpleNonlinearSolve", "SparseArrays", "SpecialFunctions", "StaticArrays", "SymbolicIndexingInterface", "SymbolicUtils", "Symbolics", "URIs", "UnPack", "Unitful"]
-git-tree-sha1 = "0d145af02e12be68c04412f31abfdadc69a2e173"
+deps = ["AbstractTrees", "ArrayInterface", "BlockArrays", "Combinatorics", "CommonSolve", "Compat", "ConstructionBase", "DataStructures", "DiffEqBase", "DiffEqCallbacks", "DiffEqNoiseProcess", "DiffRules", "Distributed", "Distributions", "DocStringExtensions", "DomainSets", "DynamicQuantities", "ExprTools", "Expronicon", "FindFirstFunctions", "ForwardDiff", "FunctionWrappers", "FunctionWrappersWrappers", "Graphs", "InteractiveUtils", "JuliaFormatter", "JumpProcesses", "Latexify", "Libdl", "LinearAlgebra", "MLStyle", "NaNMath", "NonlinearSolve", "OffsetArrays", "OrderedCollections", "PrecompileTools", "RecursiveArrayTools", "Reexport", "RuntimeGeneratedFunctions", "SciMLBase", "SciMLStructures", "Serialization", "Setfield", "SimpleNonlinearSolve", "SparseArrays", "SpecialFunctions", "StaticArrays", "SymbolicIndexingInterface", "SymbolicUtils", "Symbolics", "URIs", "UnPack", "Unitful"]
+git-tree-sha1 = "154c6d719b221a4f9f95eaec2720f42678970847"
 uuid = "961ee093-0014-501f-94e3-6117800e7a78"
-version = "9.46.1"
+version = "9.47.0"
 
     [deps.ModelingToolkit.extensions]
     MTKBifurcationKitExt = "BifurcationKit"
     MTKChainRulesCoreExt = "ChainRulesCore"
     MTKDeepDiffsExt = "DeepDiffs"
+    MTKHomotopyContinuationExt = "HomotopyContinuation"
     MTKLabelledArraysExt = "LabelledArrays"
 
     [deps.ModelingToolkit.weakdeps]
     BifurcationKit = "0f109fa4-8a5d-4b75-95aa-f515264e7665"
     ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
     DeepDiffs = "ab62b9b5-e342-54a8-a765-a90f495de1a6"
+    HomotopyContinuation = "f213a82b-91d6-5c5d-acf7-10f1c761b327"
     LabelledArrays = "2ee39098-c373-598a-b85f-a56591580800"
 
 [[deps.MozillaCACerts_jll]]
@@ -1518,10 +1526,10 @@ uuid = "becaefa8-8ca2-5cf9-886d-c06f3d2bd2c4"
 version = "1.2.0"
 
 [[deps.OrdinaryDiffEqFIRK]]
-deps = ["DiffEqBase", "FastBroadcast", "LinearAlgebra", "LinearSolve", "MuladdMacro", "OrdinaryDiffEqCore", "OrdinaryDiffEqDifferentiation", "OrdinaryDiffEqNonlinearSolve", "RecursiveArrayTools", "Reexport", "SciMLOperators"]
-git-tree-sha1 = "795221c662698851328cb7787965ab4a180d9468"
+deps = ["DiffEqBase", "FastBroadcast", "FastPower", "GenericLinearAlgebra", "GenericSchur", "LinearAlgebra", "LinearSolve", "MuladdMacro", "OrdinaryDiffEqCore", "OrdinaryDiffEqDifferentiation", "OrdinaryDiffEqNonlinearSolve", "Polynomials", "RecursiveArrayTools", "Reexport", "RootedTrees", "SciMLOperators", "Symbolics"]
+git-tree-sha1 = "5735f4c094dff311f5064d1a351da9669e4647e3"
 uuid = "5960d6e9-dd7a-4743-88e7-cf307b64f125"
-version = "1.1.1"
+version = "1.2.0"
 
 [[deps.OrdinaryDiffEqFeagin]]
 deps = ["DiffEqBase", "FastBroadcast", "MuladdMacro", "OrdinaryDiffEqCore", "Polyester", "RecursiveArrayTools", "Reexport", "Static"]
@@ -1754,6 +1762,24 @@ git-tree-sha1 = "645bed98cd47f72f67316fd42fc47dee771aefcd"
 uuid = "1d0040c9-8b98-4ee7-8388-3f51789ca0ad"
 version = "0.2.2"
 
+[[deps.Polynomials]]
+deps = ["LinearAlgebra", "RecipesBase", "Requires", "Setfield", "SparseArrays"]
+git-tree-sha1 = "1a9cfb2dc2c2f1bd63f1906d72af39a79b49b736"
+uuid = "f27b6e38-b328-58d1-80ce-0feddd5e7a45"
+version = "4.0.11"
+
+    [deps.Polynomials.extensions]
+    PolynomialsChainRulesCoreExt = "ChainRulesCore"
+    PolynomialsFFTWExt = "FFTW"
+    PolynomialsMakieCoreExt = "MakieCore"
+    PolynomialsMutableArithmeticsExt = "MutableArithmetics"
+
+    [deps.Polynomials.weakdeps]
+    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
+    FFTW = "7a1cc6ca-52ef-59f5-83cd-3a7055c09341"
+    MakieCore = "20f20a25-4f0e-4fdf-b5d1-57303727442b"
+    MutableArithmetics = "d8a4904e-b15c-11e9-3269-09a3773c0cb0"
+
 [[deps.PooledArrays]]
 deps = ["DataAPI", "Future"]
 git-tree-sha1 = "36d8b4b899628fb92c2749eb488d884a926614d3"
@@ -1959,6 +1985,16 @@ git-tree-sha1 = "58cdd8fb2201a6267e1db87ff148dd6c1dbd8ad8"
 uuid = "f50d1b31-88e8-58de-be2c-1cc44531875f"
 version = "0.5.1+0"
 
+[[deps.RootedTrees]]
+deps = ["LaTeXStrings", "Latexify", "LinearAlgebra", "Preferences", "RecipesBase", "Requires"]
+git-tree-sha1 = "c0c464d3063e46e4128d21fd677ca575ace44fdc"
+uuid = "47965b36-3f3e-11e9-0dcf-4570dfd42a8c"
+version = "2.23.1"
+weakdeps = ["Plots"]
+
+    [deps.RootedTrees.extensions]
+    PlotsExt = "Plots"
+
 [[deps.RuntimeGeneratedFunctions]]
 deps = ["ExprTools", "SHA", "Serialization"]
 git-tree-sha1 = "04c968137612c4a5629fa531334bb81ad5680f00"
@@ -2152,9 +2188,9 @@ version = "2.23.0"
 
 [[deps.SparseMatrixColorings]]
 deps = ["ADTypes", "DataStructures", "DocStringExtensions", "LinearAlgebra", "Random", "SparseArrays"]
-git-tree-sha1 = "ccc32032d8f6790ad198c99fb8ef20d8100a0de4"
+git-tree-sha1 = "f37f046636f8dc353a39279abfefe296db212171"
 uuid = "0a514795-09f3-496d-8182-132a7b665d35"
-version = "0.4.7"
+version = "0.4.8"
 weakdeps = ["Colors"]
 
     [deps.SparseMatrixColorings.extensions]
@@ -2201,9 +2237,9 @@ weakdeps = ["OffsetArrays", "StaticArrays"]
 
 [[deps.StaticArrays]]
 deps = ["LinearAlgebra", "PrecompileTools", "Random", "StaticArraysCore"]
-git-tree-sha1 = "eeafab08ae20c62c44c8399ccb9354a04b80db50"
+git-tree-sha1 = "777657803913ffc7e8cc20f0fd04b634f871af8f"
 uuid = "90137ffa-7385-5640-81b9-e52037218182"
-version = "1.9.7"
+version = "1.9.8"
 weakdeps = ["ChainRulesCore", "Statistics"]
 
     [deps.StaticArrays.extensions]