非线性切换系统的振荡行为及其Lyapunov指数计算

Oscillated behavior and Lyapunov exponent calculation of nonlinear switching system

讨论了2组不同系数下的Chen系统经过周期切换生成的一类三维非线性切换系统的动力学行为及其演化过程.由平衡点的局部分岔行为分析,得到子系统不同分岔,如Fold分岔、Hopf分岔的临界条件和相关稳态解.两子系统的不同稳态解之间,如焦点与焦点、焦点与极限环之间,通过周期切换,呈现出丰富的振荡行为.随系统参数变化,切换系统会出现非光滑分岔,导致诸如混沌等复杂的非线性现象.利用Poincaré映射分析方法,计算了周期切换系统的Lyapunov指数.通过与相应的分岔图比对,验证了算法的有效性.以Lyapunov指数为判据,可以有效揭示此类混杂系统由倍周期分岔通向混沌的道路.

The complicated behaviors of the model in 3D nonlinear switching system were investigated in details.Based on the local analysis,the critical conditions of Fold bifurcation and Hopf bifurcation were derived to explore the bifurcations of compound systems with different stable solutions of focus or stable cycle in the two subsystems.With the change of parameters,different types of non-smooth bifurcations occurred in the switching system to result in chaotic oscillations.By Poincaré mapping,the Lyapunov exponent of the switching system was calculated.Compared with bifurcation diagram,the effectiveness of the algorithm was verified.The results show that with Lyapunov index as criterion,the route to chaos via period-doubling bifurcations in such compound system is revealed explicitly.