Heteroclinic Cycles Imply Chaos and Are Structurally Stable

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.