Asymptotic and chaotic solutions of a singularly perturbed Nagumo-type equation

TL;DR

This paper investigates the existence of chaotic, homoclinic, and heteroclinic solutions in a singularly perturbed Nagumo-type equation using dynamical systems techniques, particularly for small perturbation parameters.

Contribution

It introduces a novel application of the Stretching Along Paths and Conley-Wazewski's methods to establish complex solution structures in a singularly perturbed nonlinear equation.

Findings

Existence of chaotic solutions for small epsilon

Presence of homoclinic and heteroclinic orbits

Application of dynamical systems methods to singular perturbations

Abstract

We deal with the singularly perturbed Nagumo-type equation $$ \epsilon^2 u'' + u(1-u)(u-a(s)) = 0, $$ where $\epsilon > 0$ is a real parameter and $a: \mathbb{R} \to \mathbb{R}$ is a piecewise constant function satisfying $0 < a(s) < 1$ for all $s$. We prove the existence of chaotic, homoclinic and heteroclinic solutions, when $\epsilon$ is small enough. We use a dynamical systems approach, based on the Stretching Along Paths method and on the Conley-Wazewski's method.