两自由度碰撞振动系统的Lyapunov指数谱分析

Spectral analysis of Lyapunov exponents for a two-degree-of-freedom vibro-impact system

针对一类两自由度碰撞振动系统,取碰撞前瞬时的定相位面为Poincar啨截面,引入局部映射,构造Poincar啨映射并给出其Jacobi矩阵。利用Gram-Schmidt正交化、范数归一化和迭代的方法,得出两自由度碰振系统Lya-punov指数谱的计算方法。利用数值模拟,讨论系统周期吸引子和混沌吸引子的Lyapunov指数谱的收敛序列,序列的收敛性很好.为了验证该计算方法的正确性和有效性,分析当系统参数在大范围内变化时,相应的最大Lyapunov指数的变化规律。

A two-degree-of-freedom vibro-impact system was investigated.By using a constant phase surface during the pre-impact instant as Poincaré section and introducing a local map,Poincaré map was constructed and the corresponding Jacobi matrix was obtained.Using Gram-Schmidt ortho-normalization and the iterative method,the method for calculating the spectra of Lyapunov exponents of the vibro-impact system was obtained.The numerical simulation showed that the sequences of the spectra of Lyapunov exponents of the periodic attractor and the chaotic attractor of the system can converge well.The numerical simulation also showed that the largest Lyapunov exponent agrees with the attractor behavior observed in the corresponding global bifurcation diagram.