ln this paper, a double pendulum system is studied for analyzing the dynamic behaviour near a critical point characterized b9 non-semisimple 1:1 resonance. Based on normal from theory, it is shown that two phase-locked periodic solutions may bofircate from an initial equilibrium, one of them is unstable and the other may be stable for certain values of parameters. A secondary bifurcation from the stable periodic solution yields a family of quasi-periodic solutions lying on a two dimensional torus, further cascading bifurcnations from the quasi-periodic motions lead to two chaos via period-doublin9 route. It is shown that all the solutions and chaotic motions are obtained under positive dampin9.

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.

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。

本文研究了已具有静变形的受周期激励作用下浅拱在1:2内共振条件下的分岔特性，进而按系统的运动形式将整个参数平面分成不同的区域，得到了物理参数平面上浅拱的定常运动分布情况，结合数值分析方法详细分析了系统在各个区域内特别是Hopf分岔区域内系统的动力学特性，指出系统模态相互作用的规律及其通向混沌的过程。

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。