多频驱动的近哈密顿系统的Hopf分岔  

Hopf bifurcation of a near-hamiltonian system driven with multi-frequency

作　　者：符五久[1] 

机构地区：[1]东华理工大学理学院,江西抚州344000

出　　处：《振动与冲击》2012年第19期40-47,85,共9页Journal of Vibration and Shock

基　　金：江西省教育厅科技项目(GJJ09265)

摘　　要：将无扰闭轨道变量变换到作用-角变量,再将微扰变量在无扰闭轨道附近展开,获得了有微扰的作用-角变量一级近似表达式。以无扰闭轨道的周期为采样时间,用作用-角变量表达式建立了二维多频驱动的Poincar'e映射,由其中的作用变量映射定义了多频驱动的次谐Melnikov函数,并用该函数,给出了Hopf分岔条件。并应用到多频驱动的Duffing-Van der pol系统中,导出了该系统的Hopf分岔条件。按分岔条件取参数,对三频驱动的Duffing-Van der pol方程进行了数值模拟,无一例外地均出现了Hopf分岔。Variables of an unperturbed closed orbit were transformed into action-angalar variables here. Then, the first-order approximate expressions for action-angalar variables of a perturbed closed orbit were obtained by expanding its variables into Taylor series near the corresponding unperturbed closed orbit. The two-dimensional multi-frequency driven Poincare maps were established with the action-angular variable expressions using the period of the unperturbed closed orbit as the sampling time. The multi-frequency driven subharmonic Melnikov function was defined with the first-order term of action variable of Poincare maps. And Hopf bifurcation conditions were given with this sub-harmonic Melnikov function. This theory was applied into a multi-frequency driven Dulling-Van der pol system, its Hopf bifurcation conditions were derived. The parameters obtained from these conditions were used to solve a three-frequency driven Duffing-Van der pol equation, Hopf bifurcations appeared in all results.

关 键 词：多频驱动 近哈密顿系统 Melnikov函数方法 HOPF分岔 

分 类 号：O547[理学—物理]