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arXiv · 1508.02438

Global Dynamics for Steep Sigmoidal Nonlinearities in Two Dimensions

Tomas Gedeon·Shaun Harker·Hiroshi Kokubu·Konstantin Mischaikow·Hiroe Oka
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Abstract

We introduce a novel approach to obtaining mathematically rigorous results on the global dynamics of ordinary differential equations. Motivated by models of regulatory networks, we construct a state transition graph from a piecewise affine ordinary differential equation. We use efficient graph algorithms to compute an associated Morse graph that codifies the recurrent and gradient-like dynamics. We prove that for 2-dimensional systems, the Morse graph defines a Morse decomposition for the dynamics of any smooth differential equation that is sufficiently close to the original piecewise affine ordinary differential equation.

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Tomas Gedeon, Shaun Harker, Hiroshi Kokubu, Konstantin Mischaikow, Hiroe Oka. 2015-08-10. Global Dynamics for Steep Sigmoidal Nonlinearities in Two Dimensions. https://arxiv.org/abs/1508.02438

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