时变动力系统的高阶乘法摄动方法

针对时变线性动力系统,提出了一种高阶乘法摄动方法.首先用不大的步长将时间域离散,在每个时间段上将动力系统的系数矩阵分解为一个大量和一个小量之和,后者为该段上相对时间坐标的一阶小量;然后利用变量变换,将原系统转换为一阶摄动系统.对于一阶摄动系统,仍然将系数矩阵分解为大量与高一阶小量之和,再利用变量变换将其化为更高阶的摄动系统.最后的高阶摄动系统在舍弃系数矩阵的高阶小量后可解析求解,然后由一系列反变换,便可确定原问题的解答.由于本方法确定的传递矩阵为一系列指数矩阵之积,可利用精细积分法精确计算,故本方法具有极高的精度和效率,以及良好的稳定性.对于哈密顿系统,该方法实际为一种高阶保辛摄动方法.算例结果表明,即使选取较大的时间步长,本方法也能给出较好的精度,并且随着摄动次数的增加,摄动解答能迅速趋向于精确解.

The high order multiplication perturbation method for time-varying dynamic system

A high order multiplication perturbation method for linear time-varying dynamic system is presented.Firstly the time domain is dispersed with small time interval and the coefficient matrix of the dynamic system is decomposed into a ＂large amount＂ and a ＂small amount＂ in each time interval.One order perturbation system is then presented from the original dynamic system by a variable transformation which is also a linear time-varying dynamic system.The coefficient matrix of the new system is also decomposed into a ＂large amount＂ and a ＂small amount＂ and then higher order perturbation system can be obtained by another variable transformation.The final perturbation system can be solved exactly after abandoning the ＂small amount＂ of the coefficient matrix in the final system,and then the answer to the original problem can be determined through a series of inverse transform.Since the transfer matrix is the product of a series of exponential matrix which can be calculated accurately by the precise time integration method,so the proposed method has fine accuracy,efficiency and stability.The proposed method actually is a high order symplectic conservative perturbation method for the Hamiltonian system.The examples show that the proposed method can also give good results even though a large time step selected,and with the increase of the perturbation number,the perturbation solutions can tend to exact solutions rapidly.