Existence of infinitely many connecting orbits in a singularly perturbed ordinary differential equation

Abstract

We consider a one-parameter family of two-dimensional ordinary differential equations with a slow parameter drift. Our equation assumes that when there is no parameter drift, there are two invariant curves consisting of equilibria, one of which is normally hyperbolic and whose stable and unstable manifolds intersect transversely. The slow parameter drift is introduced in a way that it creates two hyperbolic equilibria in the invariant normally hyperbolic curve that is persistent under perturbation. In this situation, we prove that the number of distinct orbits which connects these two equilibria changes from finite to infinite depending on the direction of the slow parameter drift. The proof uses the Conley index theory. The relation to a singular boundary value problem studied by W Kath is also discussed.