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Singular cycles connecting saddle periodic orbit and saddle equilibrium in piecewise smooth systems

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Abstract

For flows, the singular cycles connecting saddle periodic orbit and saddle equilibrium can potentially result in the so-called singular horseshoe, which means the existence of a non-uniformly hyperbolic chaotic invariant set. However, it is very hard to find a specific dynamical system that exhibits such singular cycles in general. In this paper, the existence of the singular cycles involving saddle periodic orbits is studied by two types of piecewise smooth systems: One is the piecewise smooth systems having an admissible saddle point with only real eigenvalues and an admissible saddle periodic orbit, and the other is the piecewise smooth systems having an admissible saddle-focus and an admissible saddle periodic orbit. Several kinds of sufficient conditions are obtained for the existence of only one heteroclinic cycle or only two heteroclinic cycles in the two types of piecewise smooth systems, respectively. In addition, some examples are presented to illustrate the results.

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Acknowledgements

The first author takes this opportunity to thank Prof. Sebastian Walther (Department of Mathematics, RWTH Aachen University, Germany) for providing a good research environment and office space during the visit in the RWTH Aachen University from July 2018 to July 2019. This study was supported by the National Natural Science Foundation of China (Grant Numbers 11702077 and 11472111), the Natural Science Foundation of Anhui Province (Grant Number 1708085QA12), the Overseas Visiting and Training Foundation of Outstanding Young Talents in the Universities of Anhui Province (Grant Number gxgwfx2018070) and the Talent Fund of Hefei University (Grant Numbers 18-19RC58 and 2018xs03).

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  1. Department of Mathematics and Physics, Hefei University, Hefei, 230601, China

    Lei Wang

  2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Xiao-Song Yang

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  1. Lei Wang
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Correspondence to Xiao-Song Yang.

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Wang, L., Yang, XS. Singular cycles connecting saddle periodic orbit and saddle equilibrium in piecewise smooth systems. Nonlinear Dyn 97, 2469–2481 (2019). https://doi.org/10.1007/s11071-019-05142-y

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  • DOI: https://doi.org/10.1007/s11071-019-05142-y

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