参数激励耦合系统的复杂动力学行为分析

分析了耦合van der Pol振子参数共振条件下的复杂动力学行为．基于平均方程，得到了参数平面上的转迁集，这些转迁集将参数平面划分为不同的区域，在各个不同的区域对应于系统不同的解．随着参数的变化，从平衡点分岔出两类不同的周期解，根据不同的分岔特性，这两类周期解失稳后，将产生概周期解或3—D环面解，它们都会随参数的变化进一步导致混吨．发现在系统的混沌区域中，其混吨吸引子随参数的变化会突然发生变化，分解为两个对称的混吨吸引子．值得注意的是，系统首先是由于2—D环面解破裂产生混吨，该混吨吸引子破裂后演变为新的混吨吸引子，却由倒倍周期分岔走向3—D环面解，也即存在两条通向混沌的道路：倍周期分岔和环面破裂，而这两种道路产生的混吨吸引子在一定参数条件下会相互转换．

DYNAMICAL ANALYSIS OF COUPLED OSCILLATORS WITH PARAMETRICAL EXCITATION

The dynamical behavior of coupled van der Pol oscillators with 1/2 parametric resonance is studied. Based on the averaged equation, the transition boundaries are sought to divide the parameter plane into different regions, corresponding to different types of the solutions. Two types of periodic solutions may bifurcate from the trivial equilibrium. The periodic solutions may bifurcate into quasi-periodic solutions or 3-D tori according to different types of bifurcations. Both 2-D and 3-D tori may lead to chaos. It is found that, in the chaotic region, a sudden change may occur to the chaotic attractor, which leads to two symmetric chaotic attractors. It is noticed that the chaos directly results from the 2-D torus and this chaotic attractor splits into two symmetric chaotic attractors, which lead to 3-D torus via a cascading of period-doubling bifurcation. It means that there exist two roads to chaos and the chaotic attractors by different ways may change into each other with the variation of parameters.