五次非线性自激系统的同宿解及其分岔

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振动与冲击 ›› 2013, Vol. 32 ›› Issue (11) : 117-125.

论文

五次非线性自激系统的同宿解及其分岔

陈洋洋 1，燕乐纬 2，陈树辉 3

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Homoclinic Solution and Bifurcation of Self-Excited System with Quintic Strong Nonlinearity

Chen Yangyang 1 Yan Lewei 2 Chen Shuhui 3

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摘要

推广了双曲函数Lindstedt-Poincaré (L-P)法摄动步骤，定量求解一类派生系统含五次强非线性项的自激振子的同宿解及其分岔值。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显式摄动解，推导出任意阶解和同宿分岔点判别的一般表达式。应用该法具体分析了一类推广的Liénard振子的同宿解和同宿分岔问题，指出方法的优点和存在的问题。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较吻合，相应的同宿分岔点判定值也具备较好的精度。该方法可以进一步研究推广应用于分析其它形式更一般的系统的同(异)宿解和同(异)宿分岔问题。

Abstract

The hyperbolic Lindstedt-Poincaré (L-P) perturbation procedure is extended for homoclinic solution and homoclinic bifurcation analysis of self-excited system, in which the generating system is with quintic strong nonlinearity. In the procedure, the homoclinic bifurcation value for limit cycle is expanded in power of perturbation parameter, the definition of secular terms for homoclinic perturbation solutions is given. And then the homoclinic bifurcation values can be determined by eliminating secular terms. The explicit homoclinic solutions by which the homoclinic conditions can strictly satisfy are obtained. The general solution formula up to arbitrary perturbation order can also be derived. By the present method, the homoclinic bifurcation of a general Liénard oscillator is studied in detail, in which the advantage and problems to be solved are discussed. Phase portraits and bifurcation values of typical examples are presented. 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. Base on the procedure and idea, the present method can be extended to deal with homoclinic (heteroclinic) solution and homoclinic (heteroclinic) bifurcation problems for more general systems.

导出引用

陈洋洋;燕乐纬;陈树辉 . 五次非线性自激系统的同宿解及其分岔[J]. 振动与冲击, 2013, 32(11): 117-125

Chen Yangyang Yan Lewei Chen Shuhui . Homoclinic Solution and Bifurcation of Self-Excited System with Quintic Strong Nonlinearity[J]. Journal of Vibration and Shock, 2013, 32(11): 117-125