Transversal heteroclinic and homoclinic orbits in singular perturbation problems

Abstract

In this paper we give a geometric construction of heteroclinic and homoclinic orbits for singularly perturbed differential equations. By using methods from invariant manifold theory we show that transversal intersection of stable and unstable manifolds of the reduced problem implies the existence of transversal heteroclinic or homoclinic orbits of the singularly perturbed problem. We derive analytical conditions for transversality. We show how these results can be used to prove the existence of heteroclinic and homoclinic orbits in singularly perturbed problems which depend on additional parameters. We describe a configuration which implies transversal intersection of the stable and unstable manifolds of periodic orbits and the associated chaotic dynamics.