  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.
