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). The package is described and showcased in [4].

Scope

This package allows modellers working with dynamical systems (autonomous ordinary differential equations) that have a polynomial RHS to perform dimensionality reduction via timescale separation semi-automatically. The crucial difference to other similar approaches is that this yields all possible reductions of a given ODE system.

Polynomial ODE systems arise frequently e.g. in mathematical chemistry or biology, where processes often evolve on different time scales, which renders this framework particularly helpful. In essence, this package may be used to derive simple conceptual models from realistic but high dimensional (i.e. mathematically complex) ones. It can also be used to study the behaviour of a larger system in certain extreme parameter regions.

Outline

Consider an ODE system of the form math \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 [5] and Fenichel [6] 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 7.

More details can be found in the documentation. A practical example can be found in the getting started section.

Requirements and installation

This package requires at least Julia 1.9 and relies on Oscar.jl, which means it must be installed in the Windows Subsystem for Linux on Windows. To install it, run ~~~ add TikhonovFenichelReductions ~~~ in Julia package Mode. More details can be found here and in the documentation for Oscar.jl.

Contribution and support

Please open an issue here, if you encounter a bug or want to share suggestions for the package. Questions or support requests can be posted in Discussions. Pull requests are also welcome, preferably after reaching out.

Unit tests can be run with ~~~ julia --project=./test ./test/runtests.jl ~~~ from the main directory. Make sure to install the dependencies first: ~~~ (TikhonovFenichelReductions/test) pkg> instantiate ~~~

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] J. Apelt and V. Liebscher, ‘Tikhonov-Fenichel Reductions and Their Application to a Novel Modelling Approach for Mutualism’, Theoretical Population Biology, pp. 16–35, Dec. 2025, doi: 10.1016/j.tpb.2025.08.004u.

[5] 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.

[6] 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

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

License

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