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The Melnikov method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a switching manifold

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A Correction to this article was published on 04 May 2022

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Abstract

In this paper, we extend the classical Melnikov method for smooth systems to a class of planar hybrid piecewise-smooth system subjected to a time-periodic perturbation. In this class, we suppose there exists a switching manifold with a more general form such that the plane is divided into two zones, and the dynamics in each zone is governed by a smooth system. Furthermore, we assume that the unperturbed system is a general planar piecewise-smooth system with non-zero trace and possesses a piecewise-smooth homoclinic orbit transversally crossing the switching manifold. We also define a reset map to describe the instantaneous impact rule on the switching manifold when a trajectory arrives at the switching manifold. Through a series of geometrical analysis and perturbation techniques, we obtain a Melnikov-type function to measure the separation of the unstable manifold and stable manifold under the effect of the time-periodic perturbations and the reset map. Finally, we use the presented Melnikov function to study global bifurcations and chaotic dynamics for a concrete planar piecewise-linear oscillator.

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Acknowledgements

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grant Nos. 11672326, 11472298, 11290152, 11472056, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB), the Fundamental Research Funds for the Central Universities through Grant No. ZXH2012K004.

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Authors and Affiliations

  1. College of Science, Civil Aviation University of China, Tianjin, 300300, China

    Shuangbao Li & Xiaojun Gong

  2. Beijing Key Laboratory on Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing, 100022, China

    Wei Zhang

  3. College of Mechanical Engineering, Beijing Information Science and Technology University, Beijing, 100192, China

    Yuxin Hao

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  1. Shuangbao Li
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  2. Xiaojun Gong
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  3. Wei Zhang
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  4. Yuxin Hao
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Correspondence to Shuangbao Li.

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Li, S., Gong, X., Zhang, W. et al. The Melnikov method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a switching manifold. Nonlinear Dyn 89, 939–953 (2017). https://doi.org/10.1007/s11071-017-3493-2

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  • DOI: https://doi.org/10.1007/s11071-017-3493-2

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