一类非线性振荡电路中的Lyapunov指数分析

Analysis of Lyapunov exponents in class of nonlinear electrical oscillator

通过Duffing方程研究了一类非线性振荡电路中的复杂动力学行为,分析了带有激振力的Duffing方程在参数改变时对系统动力学行为的影响。当系统的分岔参数有微小的改变时,系统呈现出非常丰富多样的动力学行为。分岔图显示有周期泡现象产生。利用Poincar啨映射图分析了系统混沌吸引子的特性,通过仿真系统的分岔图准确的刻画出系统的周期运动和混沌运动,通过计算Duffing方程时间序列的Lyapunov指数谱和维数谱分析了系统混沌特性,揭示了此类系统通向混沌的过程与系统的动力学行为的复杂性,验证了该系统的分岔图与Lyapunov指数谱图和维数谱图的一致性。此项研究得到了一些具有理论和工程价值的结论,为其他系统的研究提供了可靠的理论依据和有效的数值方法。

This paper is an effort to probe into the complex dynamics behaviors of a class of nonlinear electrical oscillator described by Dulling equation. The influence on the global dynamics behaviors by the change of the parameters of Duffing equation with force excitation was investigated. The very rich and multiplex nonlinear dynamics of the Dulling equation was investigated by theoretical and numerical simulation with the tiny change of the system parameters, the periodic bubble in the bifurcation diagram, which follows the sequence of P-1→P-2→P-1. could be observed. The characteristics of chaos attractors of the systems were analyzed by the Poincare map. periodic and chaos motions under the presented parameters were demonstrated exactly by simulating the bifurcation diagrams. The chaos characteristics of the systems were analyzed by computing time series＇ Lyapunov exponents and Lyapunov dimensions of Dulling equation. Routes from doubling-periodic bifurcation of periodic motion to chaos and the complex dynamics behaviors of the Systems were discussed. In addition, the paper justifies the conformity of the Lyapunov exponents and Lyapunov dimensions and bifurcation diagrams of the systems. By studying the theoretical and numerical simulation, it is possible to provide reliable theory and effective numerical method for other systems. In addition, the methods and conclusions would be useful for electrical designers.