切换系统Lyapunov指数的算法及应用

Algorithm for calculating the Lyapunov exponents of switching system and its application

Lyapunov指数是判定系统非线性行为的重要工具,然而目前的大多算法并不适用于切换系统.在传统Jacobi法的基础上,提出了一种新算法,可以直接计算得到n维切换系统的n个Lyapunov指数.首先,根据切换面处相邻轨线的动态变化规律,从相空间几何推导出切换面处轨线变化的Jacobi矩阵;然后,对该矩阵进行QR分解,从而利用R的对角线元素实现Lyapunov指数的切换补偿;最后,将新算法应用到平面双螺旋混沌系统、Glass网络和航天器供电系统三个实例中,并将计算结果与Poincaré映射方法的计算结果进行比较,对新算法的有效性进行验证.

Lyapunov characteristic exponent is significant for analyzing nonlinear dynamics. However, most algorithms are not applicable for the switching system. According to the traditional Jacobi method, in this paper we propose a new algorithm which can be used to compute n Lyapunov exponents for an n-dimensional switching system. We first study the geometric dynamics of two adjacent trajectories near the switching manifold, and obtain a compensation Jacobi matrix caused by switching. Then with QR-decomposition of this matrix, we compensate for the diagonal vector of R to realize the Lyapunov exponent expansion. Finally, we use the algorithm in a two-dimensional double-scrolls system, the Glass network and a spacecraft power system, and show its correctness and effectiveness by comparing the results with the Poincaré-map method.