一种基于摄动理论的不连续系统Lyapunov指数算法

Lyapunov exponent algorithm based on perturbation theory for discontinuous systems

Lyapunov指数是识别系统非线性动力学特征的重要标志,但是目前的算法通用性不足且计算流程复杂.本文在经典的Lyapunov指数算法的基础上,基于摄动理论提出了一种适用于不连续系统的Lyapunov指数计算方法.首先,以系统状态参数初始值和沿相空间每个基本矢量的扰动量为初始条件,确定相轨迹.其次,采取差商近似导数法,获得Jacobi矩阵的近似矩阵.然后,对Jacobi矩阵进行特征值提取,得到系统的Lyapunov指数谱.最后,将新算法应用到二自由度干摩擦冲击振荡器系统实例中,并将计算结果与同步方法的计算结果进行对比,对新算法的有效性进行验证.该算法不仅适用于离散系统和连续时间系统,而且能够快速计算复杂不连续系统的Lyapunov指数,为确定复杂不连续系统的动力学行为提供了新思路.

Lyapunov exponent is a significant symbol to identify the nonlinear dynamic characteristics of the system.However, most of algorithms are not universal enough and complex. According to the classic Lyapunov exponent algorithm and perturbation theory, in this paper we propose a new algorithm which can be used to compute Lyapunov exponents for discontinuous systems. Firstly, the initial value of the system state parameter and the disturbance of each basic vector along the phase space are taken as initial conditions to determine the phase trajectory. Secondly, the method of difference quotient approximate derivative is adopted to obtain the Jacobi matrix. Thirdly, the eigenvalues of the Jacobi matrix are calculated to obtain the Lyapunov exponent spectrum of the system. Finally, the algorithm in a two-degree-of-freedom system with impacts and friction is used, showing its effectiveness and correctness by comparing its results with the counterparts from the synchronization method. The algorithm can not only be used for discrete systems and continuous-time dynamic systems, but also quickly calculate the Lyapunov exponent of complex discontinuous systems, which provides a new idea for determining the dynamic behavior of complex discontinuous systems.