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

We deal with the singularly perturbed Nagumo-type equation $$ \epsilon^2 u’’

• u(1-u)(u-a(s)) = 0, $$where$\epsilon > 0$isarealparameterand$a: $\mathbb{R}$ \to $\mathbb{R}$$isapiecewiseconstantfunctionsatisfying$0 < a(s) < 1$forall$s$. We prove the existence of chaotic, homoclinic and heteroclinic solutions, when $ϵ$ is small enough. We use a dynamical systems approach, based on the Stretching Along Paths method and on the Conley-Wazewski’s method.