﻿ 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

Vol.03 No.02(2014), Article ID:13471,7 pages
10.12677/AAM.2014.32009

Explicit Computations of Hopf and Bautin Bifurcations in -Symmetric Systems

Guojun Peng, Xianfa Fu

School of Computer Science, Guangdong Polytechnic Normal University, Guangzhou

Email: pgjatsin@sina.com

Received: Mar. 20th, 2014; revised: Apr. 21st, 2014; accepted: Apr. 29th, 2014

ABSTRACT

By using a homogical method, we drive out computational formulae for normal forms of the Hopf and Bautin bifurcations in -symmetric systems. For practical bifurcation analysis of Hopf and Bautin in a -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 -symmetric systems.

Keywords:Hopf Bifurcation, Bautin Bifurcation, -Symmetric, Normal Form, Homogical Method

Email: pgjatsin@sina.com

1. 引言

(1.1)

(1.2)

(1.3)

(1.4)

，则Hopf分岔退化为余维以上的情形，对应余维情形的Bautin分岔的规范型为 [1] [2] ：

(1.5)

Figure 1. Hopf bifurcation diagram of (1.4) with

(1.6)

2. 临界规范型系数的计算公式

(2.1)

(2.2)

(2.3)

Figure 2. Bautin bifurcation diagram of (1.6) with, , ,

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

(2.10)

(2.11)

,则需要计算第二Lyapunov系统来判断分岔余维是否为。此时，由引理2.1得到，，将其和(2.11)一起代入(2.10)后，通过比较两端项的系数得奇线性方程：

3. 广义开折参数的计算公式

(3.1)

(3.2)

(3.3)

Hopf分岔的余维是，其规范型 (1.3)中的开折参数

(3.4)

(3.5)

4. 应用实例

(4.1)

。再利

Explicit Computations of Hopf and Bautin Bifurcations in Z2-Symmetric Systems. 应用数学进展,02,54-61. doi: 10.12677/AAM.2014.32009

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