Heteroclinic Cycles Imply Chaos and Are Structurally Stable - Xiaoying Wu (Discrete Dynamics in Nature and Society)
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 ℝn, then g has heteroclinic cycles with h−gC1 being sufficiently small. The results demonstrate C1 structural stability of heteroclinic cycles. In the end, two examples are given to illustrate our theoretical results and applications.