二阶自治广义Birkhoff系统的奇点分析

SINGULAR POINTS ANALYSIS FOR SECOND ORDER AUTONOMOUS GENERALIZED BIRKHOFF SYSTEMS

建立二阶自治广义Birkhoff系统的微分方程.给出该系统的线性化方程,得到该线性方程转化为梯度系统的条件,利用梯度系统的性质对线性系统的奇点进行了分析,然后再利用Perron定理探讨了相应的非线性系统的奇点类型.结果表明,如果线性系统能成为梯度系统,那么相应的非线性系统的奇点可能是结点或者鞍点.

The differential equations of the second order autonomous generalized Birkhoff systems were firstly es- tablished. The linearized equations of this system were also given. The conditions for＂ the translation of linearized system into a gradient system were put forward. The singular points for linearized system were analyzed by the characteristic of the gradient system. Moreover, the types of singular points for the corresponding nonlinear sys- tem were studied by Perron theorem. The results show that if the linearized system can be translated into a gradi- ent system, the singular point for the corresponding nonlinear system is probably a node or a saddle point.