强非线性自激振子同宿轨道的摄动分析方法

在双曲函数摄动法的基础上，推广双曲函数Lindstedt-Poincaré (L-P)法的适用范围，使之适用于定量分析一类含五次强非线性项的自激振子的同宿分岔和同宿解问题。以双曲函数系为基础推导出适用于高次非线性系统的摄动步骤，对极限环的同宿分岔参数进行摄动展开，给出同宿摄动解奇异项的定义，以消除同宿摄动解奇异项作为确定极限环同宿分岔点的条件，给出能够严格满足同宿条件的同宿轨道摄动解。算例表明，在相平面内该方法的结果与Runge-Kutta法数值周期轨道的逼近结果比较吻合。

Perturbation Method for Analyzing the Homoclinic Orbit ofa High-power Strongly Non-linear Self-excited Oscillator

Base on the previous studies on hyperbolic perturbation methods, the hyperbolic Lindstedt-Poincaré (L-P) method is extended for homoclinic solution and bifurcation analysis of quintic strongly nonlinear self-excited oscillator. By adopting the hyperbolic functions instead of traditional periodic functions in the L-P method, the perturbation procedure for high power strongly nonlinear system can be derived. In the present method, the homoclinic bifurcation value for limit cycle is expanded in power of perturbation parameter, the secular terms of perturbation homoclinic solutions are defined. The homoclinic bifurcation values can be determined by eliminating the secular terms. The homoclinic solutions which satisfy the homoclinic conditions is given. The general solution formula up to arbitrary perturbation order is also derived. The homoclinic bifurcation of a general Liénard oscillator is studied in detail. Phase portraits and bifurcation values of typical examples are obtained. Comparisons of results between the present method and the Runge-Kutta numerical method are made to illustrate the accuracy and efficiency of the present method.