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Heteroclinic Cycles Imply Chaos and Are Structurally Stable

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  • Xiaoying Wu
  • Abdul Qadeer Khan

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 ℠n, 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.

Suggested Citation

  • Xiaoying Wu & Abdul Qadeer Khan, 2021. "Heteroclinic Cycles Imply Chaos and Are Structurally Stable," Discrete Dynamics in Nature and Society, Hindawi, vol. 2021, pages 1-7, May.
    • Handle: RePEc:hin:jnddns:6647132
    DOI: 10.1155/2021/6647132
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