The Structural Stability of Maps with Heteroclinic Repellers

Abstract

This note is concerned with the effect of small [Formula: see text] perturbations on a discrete dynamical system [Formula: see text], which has heteroclinic repellers. The question to be addressed is whether such perturbed system [Formula: see text] has heteroclinic repellers. It will be shown that if [Formula: see text] is small enough, [Formula: see text] has heteroclinic repellers, which implies that it is chaotic in the sense of Devaney. In addition, if [Formula: see text] and [Formula: see text] has regular nondegenerate heteroclinic repellers, then [Formula: see text] has regular nondegenerate heteroclinic repellers, where [Formula: see text] is a small Lipschitz perturbation of [Formula: see text]. Three examples are presented to validate the theoretical conclusions.