具有一对纯虚及单零特征根系统的最简规范形

Simplest Normal Form for the Singularity of a Pair of Pure Imaginary and a Zero Eigenvalue System

最简规范形在分析高余维非线性系统分岔及稳定性等动力学特性方面具有重要的研究价值.为简化系统最简规范形的求解过程,采用复规范形理论,以复数运算替代原有的实数形式矩阵分析过程,获得了具有一对纯虚和单零特征根(Hopf-zero)分岔系统的最简规范形,归纳出了该类系统高阶关键方程的一般形式,并且重新定义了新的非线性变换表达式.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 simplest normal form （SNF） plays an important role in researching into the dynamical characteristic of high co-dimensional system, such as the bifurcation and stability. The SNF of the bifurcation system for the singularity of a pair of pure imaginary and a zero eigenvalue （Hopf-zero） was obtained by using the complex normal form method, in which, all matrix deduction processes were substituted by complex operation to simplify the process of obtaining the SNF. The general expressions of the high order key equations and the newly defined corresponding nonlinear transformation （NF） were obtained. An actual example indicated the validity of the SNF theory in the reduction of the conventional normal form results.