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.
Highlights
Since Li and Yorke first introduced the term “chaos” in 1975 [1], chaotic dynamics have been observed in various fields [2,3,4,5,6,7,8,9,10]
When chaotic theory was in its initial stage, Marotto generalized the results of Li and Yorke in interval mapping to multidimensional discrete systems and proved that a snapback repeller implies chaos [11]
Based on the research of Marotto, Shi and Chen raised the concept of snapback repellers in Banach spaces and complete metric spaces [14, 15]
Summary
Since Li and Yorke first introduced the term “chaos” in 1975 [1], chaotic dynamics have been observed in various fields [2,3,4,5,6,7,8,9,10]. When chaotic theory was in its initial stage, Marotto generalized the results of Li and Yorke in interval mapping to multidimensional discrete systems and proved that a snapback repeller implies chaos [11]. Some homoclinic and heteroclinic cycles imply chaos in dynamical systems [6, 16]. Lin and Chen introduced the new chaotic criteria of heteroclinic repellers in Rn [17]. In [19], Marotto showed that the systems with snapback repellers under delayed perturbations still have chaotic behaviors. Based on the concept of heteroclinic repellers in discrete dynamical systems, we introduce a new concept of heteroclinic cycles connecting expanding periodic points in Rn; by a novel method, we can construct shift invariant sets and prove that heteroclinic cycles imply chaos in the sense of Devaney.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
