The dynamical behavior of two coupled parametrically excited van der pol oscillators is investigated in this paper. Based on the averaged equations, the transition boundaries are sought to divide the parameter space into a set of regions, which correspond to dilerent types of solutions. Two types of periodic solutions may bifurcate from the initial equilibrium. The periodic solutions may lose their stabilities via a generalized static bifurcation, which leads to stable quasi-periodic solutions, or via a generalized Hopf bifurcation, which leads to stable 3D tori. The instabilities of both the quasi-periodic solutions and the 3D tori may directly lead to chaos with the variation of the parameters. Two symmetric chaotic attractors are observed and for certain values of the parameters, the two attractors may interact with each other to form another enlarged chaotic attractor.

通过对非线性Duffing方程解的稳定性进行研究，得到了其周期一解失稳的转迁集的解析表达式，同时应用广义牛顿法，得到了Duffing方程对称破缺分岔转迁集的解析表达式，与Ueda用模拟计算机的方法和A. Y. T. Leung用增量谐波平衡数值方法的结果吻合良好，克服了用模拟计算机或数字计算机确定物理参数平面上的转迁集计算工作量十分大的困难。

In this paper, the dynamic behaviour of a double pendulum system in the vicinity of several compound critical points is explored through both analytical and numerical approaches. Four types of critical points are considered, which are characterized by a double zero eigenvalue, a simple zero and a pair of pure imaginary eigenvalues, and two pairs of pure imaginary eigenvalues including resonant and non!resonant cases. With the aid of normal form theory, the explicit expressions for the critical bifurcation lines leading to incipient and secondary bifurcations are obtained. Possible bifurcations leading to 2-D and 3-D tori are also investigated. Closed form stability conditions of the bifurcation solutions are presented. A time integration scheme is used to and the numerical solutions for these bifurcation cases, which agree with the analytic results. Finally, numerical simulation is also applied to obtain double-period cascading bifurcations leading to chaos.

引入不可分偶数维不变流形的概念来定义非线性模态，在此基础上，揭示出了一种新的模态——耦合非线性模态，并对实际系统中各种可能的模态进行了分类。这种分类可能是新的构筑非线性模态理论的框架。用此方法构造非线性模态，得到的模态振子具有范式的形式，形式最简、却能反映原系统在平衡点附近的主要动力学行为，且易于得到非线性频率及非线性稳定性等方面的信息。不仅适用于分析一般的多自由度系统，还可用于分析奇数维系统；不仅可构造内共振系统的非耦合模态，还可用于构造内共振耦合模态. 从掌握的资料看，以前的方法还不能解决上述所有问题。

本文通过引入非线性频率，利用Floquet理论及解通过转迁集时的特性，研究了不可通约两周期激励作用下的Duffing方程在一次近似下的各种分岔模式及其转迁集，并指出其通向混沌可能的途径。