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Bursting behaviors induced by the bifurcation delay in a generalized parametrically forced van der Pol-Duffing system

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Abstract

Different bursting patterns and the generation principles are investigated in a generalized parametrically forced van der Pol-Duffing system. Six bursting patterns induced by the bifurcation delay, namely bursting of “delayed pitchfork/pitchfork” form, bursting of symmetric “delayed pitchfork/pitchfork” form, bursting of “delayed pitchfork/delayed sup-Hopf” form via “delayed pitchfork/pitchfork” hysteresis loop, bursting of symmetric “delayed pitchfork/delayed sup-Hopf” form via “delayed pitchfork/pitchfork” hysteresis loop, bursting of “delayed pitchfork/delayed sup-Hopf/Homoclinic” form via “delayed pitchfork/pitchfork” hysteresis loop and bursting of symmetric “delayed pitchfork/delayed sup-Hopf/Homoclinic” form via “delayed pitchfork/pitchfork” hysteresis loop, are analyzed. First, considering the parametrically forced term as a slow-changing state variable, a time invariant continuous smooth system is exhibited. Then, with the help of the calculation of the characteristic equation and bifurcation map, the critical conditions of pitchfork bifurcation, Hopf bifurcation and Homoclinic bifurcation are presented. In addition, two bifurcation delay behaviors named supercritical Hopf bifurcation delay and pitchfork bifurcation delay are proposed. Based on that, the generation mechanisms of the bursting dynamics triggered by the bifurcation delay phenomenon are revealed. The present study shows that the delayed dynamical behaviors act a crucial part in the generation of different bursting oscillations, since the delay dynamics occurs in distinct parameter intervals, which results in distinct repetitive excited state forms. Finally, the numerical simulations are provided to support the correctness of the results proposed in the paper.

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Acknowledgements

This paper is supported by the National Natural Science Foundation of China (Grant Nos. 12002134 and 11972173).

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Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, 200016, China

    X. D. Ma & L. F. Wang

  2. Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang, 212013, China

    X. D. Ma & Q. S. Bi

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  1. X. D. Ma
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  2. L. F. Wang
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Ma, X.D., Wang, L.F. & Bi, Q.S. Bursting behaviors induced by the bifurcation delay in a generalized parametrically forced van der Pol-Duffing system. Indian J Phys 96, 4269–4282 (2022). https://doi.org/10.1007/s12648-022-02367-3

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