Numerical analysis of a degenerate generalized Hopf bifurcation

Miao Xue, Qinsheng Bi, Shaolong Li and Yibo Xia
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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

International Journal of Modern Physics C (IJMPC), 2021, vol. 32, issue 08, 1-26

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.

Keywords: Normal form; bifurcation scenario; multiple scale method; Maple (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0129183121501059

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