Complex dynamics in a nerve fiber model with periodic coefficients

TL;DR

This paper investigates a nonlinear second-order ODE modeling nerve fibers with periodic coefficients, demonstrating the existence of infinitely many solutions and chaotic dynamics through topological methods, especially for step-like coefficient functions.

Contribution

It shows that for step or near-step periodic coefficients, the nerve fiber model admits infinitely many solutions and chaotic behavior, extending previous results on solution existence.

Findings

Existence of infinitely many periodic solutions.

Presence of chaotic dynamics due to topological horseshoe.

Applicability to step functions and small perturbations.

Abstract

We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo type equation $v_{xx} - g v + n(x) F(v) = 0,$ previously considered by Grindrod and Sleeman and by Chen and Bell in the study of nerve fiber models. In some recent works we discussed the case of nonexistence of nontrivial solutions as well as the case in which many positive periodic solutions may arise, the different situations depending by threshold parameters related to the weight function $n(x).$ Here we show that for a step function $n(x)$ (or for small perturbations of it) it is possible to obtain infinitely many periodic solutions and chaotic dynamics, due to the presence of a topological horseshoe (according to Kennedy and Yorke).