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Numerical analysis of a degenerate generalized Hopf bifurcation

Author

Listed:
  • Miao Xue

    (Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, P. R. China)

  • Qinsheng Bi

    (Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, P. R. China)

  • Shaolong Li

    (Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, P. R. China)

  • Yibo Xia

    (Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, P. R. China)

Abstract

In this paper, we present a numeric bifurcation analysis of the normal form of degenerate Hopf bifurcation truncated up to seventh order with an equilibrium point located at the origin. By applying the genericity nondegenerate conditions and normal form theory, we study the bifurcation analysis of the codimension-3 Takens–Hopf bifurcation for the difficult case, where a rich bifurcation scenario is displayed. The third Lyapunov coefficient is used to distinguish the different cases of a codimension-3 Takens–Hopf bifurcation point, which can be efficiently computed with the aid of a software program based on the symbolic package Maple, presented in Appendix A. The normal form analysis results can be used to depict the complete bifurcation diagrams and phase portraits. In order to investigate the mechanism of the transitions between equilibrium and limit cycles, the methods of two scales in frequency domain are employed to study the evolutions.

Suggested Citation

  • Miao Xue & Qinsheng Bi & Shaolong Li & Yibo Xia, 2021. "Numerical analysis of a degenerate generalized Hopf bifurcation," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 32(08), pages 1-26, August.
    • Handle: RePEc:wsi:ijmpcx:v:32:y:2021:i:08:n:s0129183121501059
    DOI: 10.1142/S0129183121501059
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