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On the periodic orbit bifurcating from one single non-hyperbolic equilibrium in a chaotic jerk system

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Abstract

This paper proposes a chaotic jerk system coexisting with only one non-hyperbolic equilibrium with one zero eigenvalue and a pair of complex conjugate eigenvalues. The system has no classical Hopf bifurcations and belongs to a newly category of chaotic systems. Based on the averaging theory, an analytic proof of the existence of zero-Hopf bifurcation is exhibited. Moreover, unstable periodic orbits from the zero-Hopf bifurcation are obtained. This approach may be useful to clarify chaotic attractors with non-hyperbolic equilibrium hidden behind complicated phenomena.

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Acknowledgments

The authors acknowledge the referees and the editor for carefully reading this paper and suggesting many helpful comments. This work was supported by the Natural Science Foundation of China (11401543, 11290152, 11427802, 41230637), the Natural Science Foundation of Hubei Province (No. 2014CFB897), Beijing Postdoctoral Research Foundation (2015ZZ17), the China Postdoctoral Science Foundation (No. 2014M560028, 2015T80029), the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (No. CUGL150419), and the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB).

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Authors and Affiliations

  1. School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, People’s Republic of China

    Zhouchao Wei

  2. Mathematical Institute, University of Oxford, Oxford, OX2 6GG, England

    Zhouchao Wei

  3. College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Wei Zhang & Minghui Yao

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  1. Zhouchao Wei
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  2. Wei Zhang
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  3. Minghui Yao
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Correspondence to Wei Zhang.

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Wei, Z., Zhang, W. & Yao, M. On the periodic orbit bifurcating from one single non-hyperbolic equilibrium in a chaotic jerk system. Nonlinear Dyn 82, 1251–1258 (2015). https://doi.org/10.1007/s11071-015-2230-y

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  • DOI: https://doi.org/10.1007/s11071-015-2230-y

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