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

Abstract

We deal with the singularly perturbed Nagumo-type equation $${\epsilon }^2{u}^{\prime \prime }+u(1-u)(u-a(s))=0,$$ 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.

Cited by in corpus (1)

• Multiple positive solutions to elliptic boundary blow-up problems