摘要将无扰闭轨道变量变换到作用-角变量，再将微扰闭轨道变量在无扰闭轨道附近展开，获得了微扰闭轨道作用-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，并用该次谐Melnikov函数,给出了Hopf分岔条件。将这些理论应用到多频驱动的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，均出现了Hopf分岔。
Abstract： A non-perturbed closed orbit variables were transformed into the action-angle variables in this paper. Then got the first-order approximate expressions for the perturbed action-angle variables by the variables were expanded into Taylor series at the non-perturbed variables near. The two-dimensional multi-frequency driven Poincare maps were established from the action-angle variable expressions by using the period of non-perturbed closed orbit qua the sampling time.The multi-frequency driven subharmonic Melnikov function was defined by the first-order term of action variable in this Poincare maps, And the Hopf bifurcation conditions were given by this harmonic Melnikov function.Apply our theory to multi-frequency driven Duffing-Van der pol system, which Hopf bifurcation conditions were derived. Apply the parameters that which were obtained From these conditions to solved three-frequency driven Duffing-Van der pol equation, result, without exception, have there is Hopf bifurcation.
2012, Vol. 31 
