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.

Heteroclinic Cycles Imply Chaos and Are Structurally Stable

“…Since Li and Yorke first introduced the term "chaos" in 1975 [1], chaotic dynamics have been observed in various fields [2][3][4][5][6][7][8][9][10]. When chaotic theory was in its initial stage, Marotto generalized the results of Li and Yorke in interval mapping to multidimensional discrete systems and proved that a snapback repeller implies chaos [11].…”

“…Chen and Li provided a sufficient condition of a high-dimensional difference equation having symbolic embedding for enough small C 1 perturbations [22]. In 2020, Chen and Wu et al showed that the system with heteroclinic repellers has the structural stability in [4,16].…”