TikhonovFenichelReductions.jl

A Julia package for computing Tikhonov-Fenichel Parameter Values (TFPVs) for polynomial ODE systems and their corresponding reductions (see [1-3] for details).

Outline

Consider an ODE system of the form

$$\dot{x} = f(x,\pi, \varepsilon), \quad x(0)=x_0, x \in U\subseteq\mathbb{R}^n, \pi \in \Pi \subseteq \mathbb{R}^m,$$

where $f \in \mathbb{R}[x,\pi]$ is polynomial and $\varepsilon \geq 0$ is a small parameter. The results from [1-3] allow us to compute a reduced system for $\varepsilon \to 0$ in the sense of Tikhonov [4] and Fenichel [5] using methods from commutative algebra and algebraic geometry.

TikhonovFenichelReductions.jl implements methods for finding all possible TFPVs. It also includes functions to simplify the computation of the corresponding reduced systems. Note that this approach yields all possible timescale separations of rates instead of components as in Tikhonov's theorem [6].

Installation

Run

add https://github.com/jo-ap/TikhonovFenichelReductions.jl

in Julia package Mode.

Example

Have a look at the file example.jl, which uses this package to compute all TFPVs as discussed in [7] and yields the Rosenzweig-MacArthur model as a reduction from a three-dimensional system.

Dependencies

This packages builds up on Oscar.jl.

References

[1] A. Goeke and S. Walcher, ‘Quasi-Steady State: Searching for and Utilizing Small Parameters’, in Recent Trends in Dynamical Systems, A. Johann, H.-P. Kruse, F. Rupp, and S. Schmitz, Eds., in Springer Proceedings in Mathematics & Statistics, vol. 35. Basel: Springer Basel, 2013, pp. 153–178. doi: 10.1007/978-3-0348-0451-6_8.

[2] A. Goeke and S. Walcher, ‘A constructive approach to quasi-steady state reductions’, J Math Chem, vol. 52, no. 10, pp. 2596–2626, Nov. 2014, doi: 10.1007/s10910-014-0402-5.

[3] A. Goeke, S. Walcher, and E. Zerz, ‘Determining “small parameters” for quasi-steady state’, Journal of Differential Equations, vol. 259, no. 3, pp. 1149–1180, Aug. 2015, doi: 10.1016/j.jde.2015.02.038.

[4] A. N. Tikhonov, ‘Systems of differential equations containing small parameters in the derivatives’, Mat. Sb. (N.S.), vol. 73, no. 3, pp. 575--586, 1952, https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=5548&option_lang=eng.

[5] N. Fenichel, ‘Geometric singular perturbation theory for ordinary differential equations’, Journal of Differential Equations, vol. 31, no. 1, pp. 53–98, Jan. 1979, doi: 10.1016/0022-0396(79)90152-9

[6] F. Verhulst, ‘Singular perturbation methods for slow–fast dynamics’, Nonlinear Dynamics, vol. 50, pp. 747–753, 2007, doi: 10.1007/s11071-007-9236-z

[7] N. Kruff, C. Lax, V. Liebscher, and S. Walcher, ‘The Rosenzweig–MacArthur system via reduction of an individual based model’, J. Math. Biol., vol. 78, no. 1–2, pp. 413–439, Jan. 2019, doi: 10.1007/s00285-018-1278-y.

License

GNU GENERAL PUBLIC LICENSE, Version 3 or later (see LICENSE)