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Complex mixed-mode vibration types triggered by the pitchfork bifurcation delay in a driven van der Pol-Duffing oscillator

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  • Ma, Xindong
  • Yu, Yue
  • Wang, Lifeng

Abstract

The intent of this work is to explore the generation mechanism of four mixed-mode vibration types triggered by the pitchfork bifurcation delay phenomenon in a driven van der Pol-Duffing oscillator. The four mixed-mode vibration types can be named “delayed sup-pitchfork/sup-Hopf” type for Homoclinic connection, symmetric “delayed sup-pitchfork/sup-Hopf” type, “point/point” type for delayed sup-pitchfork and “delayed sup-pitchfork/delayed Homoclinic” type, respectively. Our study proves that the pitchfork bifurcation delay phenomenon deserves a prominent role in the occurrence of the four mixed-mode vibration types, which enriches the dynamical behaviors of the bifurcation delay phenomenon as well as the possible routes to mixed-mode vibrations.

Suggested Citation

  • Ma, Xindong & Yu, Yue & Wang, Lifeng, 2021. "Complex mixed-mode vibration types triggered by the pitchfork bifurcation delay in a driven van der Pol-Duffing oscillator," Applied Mathematics and Computation, Elsevier, vol. 411(C).
    • Handle: RePEc:eee:apmaco:v:411:y:2021:i:c:s0096300321006111
    DOI: 10.1016/j.amc.2021.126522
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    References listed on IDEAS

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    1. Xu, Changjin & Liao, Maoxin & Li, Peiluan & Guo, Ying & Xiao, Qimei & Yuan, Shuai, 2019. "Influence of multiple time delays on bifurcation of fractional-order neural networks," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 565-582.
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    3. Xu, Changjin & Liu, Zixin & Liao, Maoxin & Li, Peiluan & Xiao, Qimei & Yuan, Shuai, 2021. "Fractional-order bidirectional associate memory (BAM) neural networks with multiple delays: The case of Hopf bifurcation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 471-494.
    4. Bilazeroğlu, Ş. & Merdan, H., 2021. "Hopf bifurcations in a class of reaction-diffusion equations including two discrete time delays: An algorithm for determining Hopf bifurcation, and its applications," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
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    Cited by:

    1. Song, Jian & Liu, Shenquan & Wen, Qixiang, 2022. "Geometric analysis of the spontaneous electrical activity in anterior pituitary corticotrophs," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    2. Chun Zhang & Qiaoxia Tang, 2022. "Complex Periodic Mixed-Mode Oscillation Patterns in a Filippov System," Mathematics, MDPI, vol. 10(5), pages 1-11, February.
    3. Danjin Zhang & Youhua Qian, 2023. "Bursting Oscillations in General Coupled Systems: A Review," Mathematics, MDPI, vol. 11(7), pages 1-16, April.

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