五次非线性自激系统同宿解及分岔

Homoclinic solution and bifurcation of a self-excited system with quintic strong nonlinearity

推广双曲函数Lindstedt-Poincaré(L-P)法摄动步骤,定量求解派生系统含五次强非线性项自激振子的同宿解及分岔值。对极限环同宿分岔参数进行摄动展开,给出同宿摄动解奇异项定义,消除同宿摄动解奇异项作为确定极限环同宿分岔点条件,给出能严格满足同宿条件的同宿轨道显式摄动解,推导出任意阶解及同宿分岔点判别的一般表达式。应用该法具体分析推广的Liénard振子同宿解及同宿分岔问题,并指出方法的优点与存在问题。算例表明,在相平面内该方法结果与Runge-Kutta法数值周期轨道逼近结果较吻合,同宿分岔点判定值亦具较好精度。该方法可研究推广应用于分析其它形式系统的同(异)宿解及同(异)宿分岔问题。

The hyperbolic Lindstedt-Poincare´ (L-P) perturbation procedure was extended for homoclinic solution and homoclinic bifurcation analysis of a self-excited system with quintic strong nonlinearity. In the procedure, the homoclinic bifurcation value for limit cycle was expanded in a power form of perturbation parameter, the definition of secular terms for homoclinic perturbation solutions was given. And then, the homoclinic bifurcation values could be determined by eliminating secular terms. The explicit homoclinic solutions that could strictly satisfy homoclinic conditions were obtained. The general solution formula to arbitrary perturbation order could also be derived. With the proposed method, the homoclinic bifurcation of a general Lie´nard oscillator was studied in detail, the advantage and the existing problems of the method were discussed. Phase portraits and bifurcation values of typical examples were presented. Comparisons between the results with the presented method and the Runge-Kutta numerical method were made to illustrate the accuracy and efficiency of the presented method. Finally, the proposed method could be extended to deal with homoclinic (heteroclinic) solution and homoclinic (heteroclinic) bifurcation problems of more general systems.