显式计算Z₂对称系统的Hopf和Bautin分岔

Explicit Computations of Hopf and Bautin Bifurcations in Z₂-Symmetric Systems

本文用同调方法导出任意维Z2对称系统的Hopf和Bautin分岔规范型的计算公式。在对实际Z2对称系统进行Hopf和Bautin分岔分析时,可以直接利用这些公式来计算第一和第二Lyapunov系数,从而判断对应分岔是否退化。进一步地,还可以利用开折参数的计算公式来确定系统参数在临界值附近扰动时,对应的拓扑结构。因此,实际建立了任意维Z2对称系统Hopf和Bautin分岔的参数和其结构之间的对应关系。

By using a homogical method, we drive out computational formulae for normal forms of the Hopf and Bautin bifurcations in Z2-symmetric systems. For practical bifurcation analysis of Hopf and Bautin in a Z2-symmetric system, we can use these formulae to compute the first and the second Lyapunov coefficients, and check whether the bifurcation is degenerate. Furthermore, we can use the formulae of unfolding parameters to decide the topological structures when parameters perturb in a neighborhood of the critical values. So, we construct the relation between the parameters and the structures for Hopf and Bautin bifurcations in any Z2-symmetric systems.