周期激励下分段线性电路的动力学行为

THE DYNAMICAL BEHAVIOR OF A PIECEWISE-LINEAR ELECTRIC CIRCUIT WITH PERIODICAL EXCITATION

基于四阶自治分段线性电路的分岔特性,探讨了两种幅值周期激励下该电路系统的复杂动力学行为.给出了弱周期激励下系统共存的两种分岔模式及其产生的原因,讨论了不同分岔模式下动力学行为的演化过程及混沌吸引子相互作用机理.而随着激励幅值的增大,即强激励作用下,围绕两个原自治系统平衡点的周期轨道不再分裂,从而导致共存的分岔模式消失.指出无论在弱激励还是在强激励下,由于系统的固有频率与外激励频率存在量级上的差距,其相应的各种运动模式,诸如周期运动、概周期运动甚至混沌运动均表现出明显的快慢效应,进而从分岔的角度分析了不同快慢效应的产生机制.

Since the chaotic phenomenon in Chua＇s circuit was reported, the complicated dynamics in nonlinear circuits has been one of the key topics to attract a lot of researchers. Based on Chua＇s circuits, many modified models have been established, which exhibit rich nonlinear behaviors, such as intermittency and chaos crisis. Because of the piecewise-linear function between the current and the voltage introduced, non-smooth bifurcation may occur at the singular positions. Up to now, most of the obtained results focus on the dynamics of autonomous vector fields. However, many real electric circuits are non-autonomous, in which the time-dependent terms may come from the alternating current or the controllers. Therefore, it is very important to explore the evolution of the dynamics of such types of systems. Based on the bifurcation prosperities of a fourth-order autonomous piecewise-linear electric circuit, complicated dynamics of the oscillator with periodic excitation for two different excitation amplitudes has been investigated in details. Two coexisted bifurcation forms for weak excitation are presented. Different chaotic attractors can be observed via sequences of associated bifurcations, which may interact with each other to form an enlarged chaotic attractor. While for the relatively strong excitation, the periodic orbit circling around the original two equilibrium points does not split into two parts, resulting in the disappearance of the coexisted phenomenon. Because of the different scale between the natural frequency and the excitation frequency, fast-slow effect was obviously found on the behaviors of both the weak and strong excitation, such as periodic solutions, quasi-periodic movements, and even for chaotic oscillation. Furthermore, the mechanism of fast-slow effect has been discussed from the view point of bifurcation.