控制欧拉动屈曲问题混沌的两种方法

Two Methods to Control Chaos of Euler's Dynamically Buckling Problems

对一种特殊的Mathieu方程———欧拉动屈曲问题,通过数值仿真的方法,得到其全局分岔图,以此来揭示系统由周期通向混沌的道路.另用时间响应图、相图和庞加莱截面图来表明系统的非线性状态.在此基础上,当系统处于混沌状态时,通过分岔图来选择适当的控制参数,利用耦合控制法和周期激振力法分别对欧拉动屈曲问题中的混沌行为进行了有效的控制.通过控制后的全局分岔图来判断控制后的效果.用控制后的相图和时间响应图来与分析和研究控制后系统的非线性状态.结果表明,通过这两种方法,可以控制系统的混沌运动而得到稳定的周期振动结果.

In this paper, a kind of special Mathieu function - Euler＇ s dynamically buckling problems was studied. The global bifurcation graph of the Euler＇ s bucking problems was got by means of computer simulation. The route to chaos was revealed by global bifurcation graph. Phase diagrams, time responses and Poincar6 section were presented to analysis dynamic behavior of the system. Controlling parameters according to the bifurcation graph were selected when the system was chaos. The chaotic behaviors in Euler＇ s dynamically buckling system were controlled by means of the coupled feedback controlling and periodic force excitation. The behaviors of the system were judged by the global bifurcation graph after controlling. The dynamic behaviors of the system were analyzed by the phase diagrams and the time responses. The chaotic motions of the system can be successfully converted to the stable periodic orbits after the two methods were used to control chaos.