非线性动力学方程的一种级数解  

A series solution for nonlinear dynamics equations

作　　者：裘春航[1] 吕和祥[1] 

机构地区：[1]大连理工大学工程力学系,辽宁大连116023

出　　处：《计算力学学报》2001年第1期1-7,共7页Chinese Journal of Computational Mechanics

基　　金：国家自然科学基金重大项目 !(19990 510 )

摘　　要：本文针对 n维未知向量 v的一阶微分方程 v=H v +f(v,t)进行求解 ,其中 H v和 f(v,t)分别是右端项的线性齐次部分和非线性部分。首先 ,将非线性部分 f(v,t)在所论时刻 tk处展成 t- tk=τ的泰勒级数 ,并通过求原函数的方法直接给出每一项的积分 ,从而获得了待求微分方程在级数形式下的闭合解。它的具有不同精度的各次近似解可表示成τ的分段解析函数 ,便于研究非线性动力学行为与其物理参数的依赖关系。本文还用算例验证了各次近似解之间的数值关系 。In this paper we discuss a general nonlinear dynamics system governed by the equation [AKv·D5]=Hv+f(v,t),in which v is an unknown n\|dimensional vector, H is a coefficient matrix, Hv and f(v,t) are, respectively, the linear homogeneous part and nonlinear part in the right members of the equation.First the nonlinear part f(v,t) is expressed in taylor series, and then the integral for each item is evaluated directly. Thus, a series solution for the equation under consideration is obtained, and a number of approximate solutions, which can be analytically expressed in each time\|segment, are successively deduced. Several numerical examples are presented to demonstrate the performances in analyzing periodic and chaotic vibrations.

关 键 词：非线性振动 临界点 稳定性 周期吸引子 混沌吸引子 级数解 非线性动力学 

分 类 号：O322[理学—一般力学与力学基础]