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Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol Oscillators

Author

Listed:
  • Chun Zhang

    (School of Mathematical Science, Huaiyin Normal University, Huai’an 223300, China)

  • Qiaoxia Tang

    (School of Mathematical Science, Huaiyin Normal University, Huai’an 223300, China)

  • Zhixiang Wang

    (School of Mathematical Science, Huaiyin Normal University, Huai’an 223300, China)

Abstract

By introducing a switching scheme related to the state and time, a typical switched model alternating between a Duffing oscillator and van der Pol oscillator is established to explore the typical dynamical behaviors as well as the mechanism of the switched system. Shooting methods to locate the limit cycle and specify bifurcation sets are described by defining an appropriate Poincaré map. Different types of multiple-Focus/Cycle and single-Focus/Cycle period oscillations in the system can be observed. Symmetry-breaking, period-doubling, and grazing bifurcation curves are obtained in the plane of bifurcation parameters, dividing the parameters plane into several regions corresponding to different kinds of oscillations. Meanwhile, based on the numerical simulation and bifurcation analysis, the mechanisms of several typical dynamical behaviors observed in different regions are presented.

Suggested Citation

  • Chun Zhang & Qiaoxia Tang & Zhixiang Wang, 2022. "Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol Oscillators," Mathematics, MDPI, vol. 10(5), pages 1-10, February.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:5:p:772-:d:760702
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    References listed on IDEAS

    as
    1. Jamiree Harrison & Enoch Yeung, 2021. "Stability Analysis of Parameter Varying Genetic Toggle Switches Using Koopman Operators," Mathematics, MDPI, vol. 9(23), pages 1-25, December.
    2. Fucheng Liao & Lijie Cui & Yanrong Lu & Jiamei Deng, 2020. "Preview Tracking Control of Linear Periodic Switched Systems with Dwell Time," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-9, October.
    3. Vincent Wagner & Nicole Erika Radde, 2021. "SiCaSMA: An Alternative Stochastic Description via Concatenation of Markov Processes for a Class of Catalytic Systems," Mathematics, MDPI, vol. 9(10), pages 1-13, May.
    4. Li, Feng & Liu, Yuanyuan, 2016. "Limit cycles in a class of switching system with a degenerate singular point," Chaos, Solitons & Fractals, Elsevier, vol. 92(C), pages 86-90.
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