求非线性自治系统同(异)宿轨线的双曲函数摄动法  

Hyperbolic Perturbation Method for Determining Homoclinic and Heteroclinic Solutions of Certain Nonlinear Autonomous Systems

作　　者：陈树辉[1] 陈洋洋[1] 

机构地区：[1]中山大学工学院应用力学与工程系,广州510275

出　　处：《力学季刊》2009年第1期133-137,共5页Chinese Quarterly of Mechanics

基　　金：国家自然科学基金项目(10672193)

摘　　要：本文提出一个新的摄动法,称为双曲函数摄动法,它适合于求解非线性自治系统的同(异)宿轨线。具体研究具有三次非线性的自治系统+c_1x+c_3x^3=εf(μ,x,■),阐述双曲函数摄动法求解同(异)宿轨线的过程。该法在求解过程中还能同时确定存在同(异)宿轨线的参数μ。以两个广义Liénard方程为典型算例,双曲函数摄动法求得的同宿轨线与Runge-Kutta方法求得的结果非常吻合,说明了双曲函数摄动法是求非线性自治系统同(异)宿轨线的有效方法。A hyperbolic perturbation method was presented for determining the homoclinic and heteroclinic solutions of certain strongly nonlinear autonomous oscillators in the form x^-＋c1x＋c3x^3=εf（μ,x,x^·）, in which the hyperbolic functions can be employed instead of the usual periodic functions in the usual perturbation procedure. The value of μ, under which there exists a homoclinic or heteroclinic solution of oscillator, can be determined in the perturbation procedure. The generalized Lienard oscillator with f（μ, x, x^·） = （μ -μ1x^2 -μ2x^·2）x^· was studied. Comparisons with numerical simulations （Runge-Kutta method） illustrate the present method is provided with efficiency and accuracy.

关 键 词：双曲函数摄动法 非线性自治系统 同宿轨线 异宿轨线 

分 类 号：O322[理学—一般力学与力学基础]