The Structural Stability of Maps with Heteroclinic Repellers

This note is concerned with the effect of small C-1 perturbations on a discrete dynamical system (X,f), which has heteroclinic repellers. The question to be addressed is whether such perturbed system (X,g) has heteroclinic repellers. It will be shown that if parallel to f - g parallel to(C1) is small enough, (X,g) has heteroclinic repellers, which implies that it is chaotic in the sense of Devaney. In addition, if X = R-n and (X,f) has regular nondegenerate heteroclinic repellers, then (X,g) has regular nondegenerate heteroclinic repellers, where g is a small Lipschitz perturbation of f. Three examples are presented to validate the theoretical conclusions.