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.

中文翻译：

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简并广义Hopf分岔的数值分析

在本文中，我们对简并 Hopf 分岔的正常形式进行了数值分岔分析，该分岔被截断到七阶，平衡点位于原点。通过应用泛性非退化条件和范式理论，我们研究了codimension-3 Takens-Hopf分岔在困难情况下的分岔分析，其中显示了丰富的分岔场景。第三个 Lyapunov 系数用于区分 codimension-3 Takens-Hopf 分岔点的不同情况，可以借助基于符号包 Maple 的软件程序有效计算，如附录 A 所示。范式分析结果可用于描绘完整的分岔图和相图。