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 ε representing the ratio of time scales. It is well known that for all sufficiently small ε>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 ε ∈ (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 ε ∈ [1.5 × 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 ε=0.0015 the classical interval Newton-Moore method applied to a sequence of Poincar\'e maps already succeeds. Techniques described in this paper can be adapted to other fast-slow systems of similar structure.