非对称动力系统混沌分析的Melnikov方法

Melnikov Method in Chaos Analysis of Asymmetric Dynamic System

以Helmholtz-Duffing双势阱非对称动力系统为研究对象,首先分析了不同非对称参数σ下退化Hamilton系统的势函数与相平面,然后对于受谐和激励的非自治系统根据Melnikov理论推导了系统左右同宿轨道出现混沌运动的阈值条件,最后用数值模拟得到系统的安全池并观察了安全池随激励幅值变化的侵蚀现象,从而验证了解析结果的正确性.理论分析和数值计算表明:0<σ<1时,系统左半部分受主要影响;σ>1时,右半部分受主要影响;Melnikov方法可有效预估非对称系统混沌阈值,并且对于同一非对称参数σ,存在一个临界频率使得系统左右两部分的混沌阈值在这一频率点相等.

Based on Helmholtz-Duffing oscillator which is an asymmetric double-potential-well dynamical system, the potential function and phase plane of degenerated Hamilton system at different asymmetric parameters of a were analyzed. Then according to the Melnikov theory, the threshold values of the left and right homoclinic orbits of the nonautonomous system excited by the harmonic force were deduced. By vir- tue of numerical simulation, the safe basin was obtained and the erosion phenomenon of safe basin along with the variation of excitation amplitude was observed, which verified the correctness of the analytical results. The research shows that when a is between 0 and 1, the left half of the system is mainly affected, and the condition is on the contrary when a is larger than 1. For the same asymmetric parameter σ, there exists a critical frequency at which the threshold value of both left and right half of system are equal.