Existence of periodic solutions of the FitzHugh-Nagumo equations for an explicit range of the small parameter

Abstract

The FitzHugh-Nagumo model describing propagation of nerve impulses in axon is given by fast-slow reaction-diffusion equations, with dependence on a parameter $\epsilon$ representing the ratio of time scales. It is well known that for all sufficiently small $\epsilon >0$ the system possesses a periodic traveling wave. With aid of computer-assisted rigorous computations, we prove the existence of this periodic orbit in the traveling wave equation for an explicit range $\epsilon \in (0,0.0015]$. Our approach is based on a novel method of combination of topological techniques of covering relations and isolating segments, for which we provide a self-contained theory. We show that the range of existence is wide enough, so the upper bound can be reached by standard validated continuation procedures. In particular, for the range $\epsilon \in [1.5\times 10^{-4},0.0015]$ we perform a rigorous continuation based on covering relations and not specifically tailored to the fast-slow setting. Moreover, we confirm that for $\epsilon =0.0015$ the classical interval Newton-Moore method applied to a sequence of Poincaré maps already succeeds. Techniques described in this paper can be adapted to other fast-slow systems of similar structure.

References in corpus (1)

• Homoclinic Orbits of the FitzHugh-Nagumo Equation: The Singular-Limit

Cited by in corpus (2)

• Traveling wave solutions for the FPU chain: a constructive approach
• Instability of small-amplitude periodic waves from fold-Hopf bifurcation