基于增量谐波平衡法的Mathieu-Duffing振子分岔及通往混沌道路分析

Bifurcations and Analyses of Route to Chaos of Mathieu-Duffing Oscillator by the Incremental Harmonic Balance Method

提出了一种分析非线性系统分岔及通往混沌道路的新方法,以增量谐波平衡法为基础,求得特定参数状态下的周期解;根据Floquet理论,判定周期解的稳定性,分析周期解的分岔类型及参数的分岔值。求得分岔值后,根据周期解的分岔类型,构造下一级分岔周期解的谐波函数,计算下一级的分岔点。重复上述过程,可获得周期解分岔的一系列临界值及混沌产生的近似阈值。通过该方法,可以了解动力系统混沌产生的分岔过程。应用该法分析了Mathieu-Duffing振子的倍周期分岔,得到其周期倍化的系列分岔点及混沌产生的近似阈值,所得结果与数值模拟基本一致。

A systematic procedure is presented for analyzing the bifurcations and the route to chaos of nonlinear systems. In this procedure, the periodic solutions are obtained by the Incremental Harmonic Balance （IHB） method. The stability of the periodic solutions and bifurcations are examined according to Floquet theory. After the bifurcation value is determined, the new basic harmonic functions are constructed according to the types of bifurcations, then the IHB method and Floquet theory are employed to calculate and determine the periodic solutions and its stability until the next bifurcation point is detected. By repeating the procedure, a series bifurcation points in the transition to chaos can be determined one by one and therefore the route to chaos can be demonstrated. The bifurcation analysis of Mathieu-Duffing oscillator is taken as an example to show the efficiency and accuracy. A series period-doubling points and the threshold value at the onset of chaos are obtained. They are in good agreement with those obtained by Mathematica simulation.