切换电路系统的振荡行为及其非光滑分岔机理

探讨了周期时间开关及控制阈值下在两个Rayleigh型子系统之间切换的电路系统随参数变化的复杂动力学演化过程, 通过对子系统平衡点的分析, 给出了参数空间中Fold分岔和Hopf分岔的条件, 考察了切换面处广义Jacobian矩阵特征值随辅助参数变化的分布情况, 得到了切换面处系统可能存在的各种分岔行为, 进而讨论了系统不同行为的产生机理, 指出系统的相轨迹存在分别由周期开关和控制阈值决定的两类不同的分界点, 而系统轨迹与非光滑分界面的多次碰撞将导致系统由周期倍化分岔导致混沌振荡.

The oscillations of a switching electrical circuit and the mechanism of non-smooth bifurcations

The complicated dynamical evolution of a circuit system composed of two Rayleigh-types subsystems, which are switched by a periodic switch and a threshold controller, is investigated. Through the analysis of the subsystem equilibrium points, the conditions for Fold bifurcation and Hopf bifurcation in the parameter space are given respectively. The distribution of the generalized Jacobian eigenvalues varying with auxiliary parameter at the switching boundary is presented. Then the possible bifurcation behaviors of the system at the switching boundary are obtained. The mechanisms of the different behaviors of the system are discussed. It is pointed that the trajectories of the system have two kinds of turning points, which are determined by the periodic switch and the threshold controller respectively. Meanwhile, the multiple collisions between the trajectories and the non-smooth boundary may lead the system to change from chaos to period-adding bifurcation.