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

Abstract

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.