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 $s$. For small $\epsilon$, we prove the existence of chaotic, homoclinic and heteroclinic solutions. We use a dynamical systems approach, based on the Stretching Along Paths technique and on the Conley-Wa\.zewski's method.
Asymptotic and chaotic solutions of a singularly perturbed Nagumo-type equation
BOSCAGGIN, Alberto;DAMBROSIO, Walter;
2015-01-01
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 $s$. For small $\epsilon$, we prove the existence of chaotic, homoclinic and heteroclinic solutions. We use a dynamical systems approach, based on the Stretching Along Paths technique and on the Conley-Wa\.zewski's method.
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