一类Mathieu方程的混沌分析及控制

Analysis and Control of Chaos for a Sort of Mathieu Equation

利用数值仿真的方法,对一类Mathieu方程的混沌运动及其控制进行了研究.利用分岔图、Lyapunov指数谱和相图等揭示了该系统经由倍周期分岔通向混沌的路径.采用二次分段函数作为非线性反馈控制器,通过控制后方程的分岔图选择适当的控制参数,对这一类Mathieu方程中的混沌行为进行有效的控制.

By means of numerical simulations, the bifurcation and the chaotic behaviors of a nonlinear Mathieu function are studied. The route to chaos is revealed by global bifurcation diagram. Phase portraits, bifurcation diagram and Lyapunov - exponent spectrum are also presented to analyze nonlinear dynamics of the system. The method,which is a subsection square function, is adopted as the nonlinear feedback controller to control chaos in this paper. By selecting proper controlling parameters according to bifurcation diagrams, the chaotic motions of the system can be successfully converted to different stable periodic orbits. The results are demonstrated and illustrated by reverse periodic - doubling bifurcation diagrams and phase portraits.