切换系统Lyapunov指数的算法及应用

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.

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

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.
Keywords：switching systemsLyapunov exponentsJacobi matrixswitching manifold