Heteroclinic Cycles Imply Chaos and Are Structurally Stable

Abstract

This paper is concerned with the chaos of discrete dynamical systems. A new concept of heteroclinic cycles connecting expanding periodic points is raised, and by a novel method, we prove an invariant subsystem is topologically conjugate to the one-side symbolic system. Thus, heteroclinic cycles imply chaos in the sense of Devaney. In addition, if a continuous differential map $h$ has heteroclinic cycles in ${ℝ}^n$ , then $g$ has heteroclinic cycles with ${‖h-g‖}_{C^1}$ being sufficiently small. The results demonstrate $C^1$ structural stability of heteroclinic cycles. In the end, two examples are given to illustrate our theoretical results and applications.