复规范形法求解非共振双Hopf分岔系统的最简规范形

Simplest normal form of non-resonance double Hopf bifurcation system with the complex normal form method

提出了一种应用复规范形理论获取非共振双Hopf分岔系统最简规范形的有效方法,以简化最简规范形的求解过程.建立了复坐标下非共振双Hopf分岔系统的规范形及非线性变换,采用复数运算替代原有实数形式矩阵表示法的矩阵推导过程,获得了系统高阶关键方程的一般形式,简化了非线性变换的表达式,并且由此推导出了此类系统的最简规范形表达式.所附算例验证了最简规范形理论对于简化传统规范形结果的有效性.

The complex normal form method is brought forward to simplify the process of finding the simplest normal form （SNF） of the non-resonance double Hopf bifurcation system. The normal form of the non-resonance system and the associated nonlinear transformation are constructed in a complex ordinate form. During the course of computing the key equations, all matrix deduction process of the common matrix representation method are substituted by the complex operation. Thus the iterative formulas of these equations are found and nonlinear transformation up to the nth-order can be rewritten more compactly. With the help of that the SNF of the non-resonance double Hopf bifurcation system is obtained. An actual example attached indicates the validity of the SNF theory in the reduction of the normal form results.