The Conley index for fast-slow systems. I: One-dimensional slow variable
The family of differential equations \[ \dot x=f(x,\lambda),\quad \dot \lambda=\varepsilon g(x,\lambda), \] with \(f:\mathbb{R}^n\times \mathbb{R}\to \mathbb{R}^n\) and \(g:\mathbb{R}^n\times \mathbb{R}\to \mathbb{R}\), is considered. The main theorems of the paper are based on the Conley index theory and assert the existence of heteroclinic connections and periodic solutions for the system at \(\varepsilon >0\) small enough provided there exist isolating neighborhoods of some special forms for the system at \(\varepsilon=0\) and restricted values of \(\lambda\). The concepts introduced and the obtained results are explained using the Nagumo equation.
scientific article; zbMATH DE number 1334590