φ^6-DVP振子在三势阱参数下的分岔分析

Bifurcation analysis of φ^6-DVP oscillator with triple-well parameters

利用Melnikov方法分析了含有5次方项的Φ6-Duffing-Van der Pol(Φ6-DVP)系统在三势阱参数下发生混沌的必要条件。通过Poincaré截面图、分岔图、时间序列中的Lyapunov指数谱和Lyapunov维数等,较直观地反映振动系统随周期激励信号强弱变化的动态特性,阐明了系统运动随周期激励信号强弱变化的动态特性、复杂性和系统的非线性特征,揭示了Φ6-DVP振子方程的分岔形式以及通向混沌运动的道路。结果表明:由于系统的混沌特性以及本身对称性,导致系统在通向混沌的道路上和较窄的混沌带中,对称地出现了多种类型的分岔形式。

A set of parameters, with which φ^6- DVP oscillator can conduce to chaos, is selected by the Melnikov method. Some basic properties, routes to chaos, periodic windows and periodic bubbles are studied numerically. Nonlinear dynamics of the φ^6-DVP oscillator with triple-well parameters is investigated by theoretical and numerical simulation with the tiny change of the external forced excitation. The characteristics of the system are demonstrated by Poincare maps, bifurcation diagram, Lyapunov exponents and so on. The results indicate that there are many kinds of bifurcation types to emerge symmetrically, both on the route to chaos and in the very narrow chaotic bands due to the chaotic characteristics and the self-symmetry of the φ^6-DVP system.