Heteroclinic cycles imply chaos and are structurally stable (Q2039203)

Summary: 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 \(\mathbb{R}^n\), then \(g\) has heteroclinic cycles with \(\left\| h - g\right\|_{C^1}\) 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.