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Qualitative analysis in a delayed Van der Pol oscillator

Author

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  • Peng, Miao
  • Zhang, Zhengdi
  • Qu, Zifang
  • Bi, Qinsheng

Abstract

The main purpose of this manuscript is to introduce and compare two methods for discussing the nature of Hopf bifurcation, namely, a research method combining the normal form theory and the center manifold theorem and the other method of multiple scales. Taking a delayed Van der Pol oscillator as an example, the local stability as well as the occurrence of Hopf bifurcation are explored via choosing time delay as the bifurcation parameter. On the basis of two different methods respectively, the characters of Hopf bifurcation which includes the direction of Hopf bifurcation as well as the stability of bifurcating periodic solutions are analyzed. Finally, numerical simulations supporting the theoretical findings are given, and it is observed that the results of two methods are consistent.

Suggested Citation

  • Peng, Miao & Zhang, Zhengdi & Qu, Zifang & Bi, Qinsheng, 2020. "Qualitative analysis in a delayed Van der Pol oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 544(C).
    • Handle: RePEc:eee:phsmap:v:544:y:2020:i:c:s0378437119319442
    DOI: 10.1016/j.physa.2019.123482
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    References listed on IDEAS

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    1. Yu, Jinchen & Peng, Mingshu, 2016. "Stability and bifurcation analysis for the Kaldor–Kalecki model with a discrete delay and a distributed delay," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 460(C), pages 66-75.
    2. Hao, Pengmiao & Wang, Xuechen & Wei, Junjie, 2018. "Hopf bifurcation analysis of a diffusive single species model with stage structure and strong Allee effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 153(C), pages 1-14.
    3. Wang, Xuedi & Peng, Miao & Liu, Xiuyu, 2015. "Stability and Hopf bifurcation analysis of a ratio-dependent predator–prey model with two time delays and Holling type III functional response," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 496-508.
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    Citations

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    Cited by:

    1. Cai, Chengcai & Shen, Yongjun & Wen, Shaofang, 2023. "Simultaneously primary and super-harmonic resonance of a van der Pol oscillator with fractional-order derivative," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    2. Yani Chen & Youhua Qian, 2021. "Stability Switches and Double Hopf Bifurcation Analysis on Two-Degree-of-Freedom Coupled Delay van der Pol Oscillator," Mathematics, MDPI, vol. 9(19), pages 1-17, October.

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