The Renormalization Group Method And Generalized Lienard Equation

This article mainly studies the following generalized Lienard equation with single pa-rameter x + ?f(x)x + x = 0,and generalized Lienard equation with two small parameters x + ?f(x)x + x + ?g(x)= 0,Where ?,? arc positive small parameters,f(?)1+ ? k=1 Nakx2k,g(x)=? n=1 N bnx2n+1.We first use the renormalization group method established by Chen,Goldenfeld and Oono to calculate the uniformly valid asymptotic solution of the corresponding problems.Then,the dynamic behavior of the renormalized group equation is used to analyze the dy-namic behavior of the original equation and the existence and non existence conditions of the limit cycle of the original equation are obtained.This article is divided into three parts.The first chapter is the introduction,briefly introducing the background of the singular perturbation method,the renormalization group method and the Lienard equation.The second chapter mainly introduces how to use the renormalization group equation method to obtain the uniformly valid asymptotic solution of the initial value problem of differential equations,The relationship between the dynamical behavior of the renormalization group equation and the dynamical behavior of the original equation is also discussed.The third chapter is the main result of this paper.First,we use the renormalization group method to obtain the uniformly valid asymptotic solution of the two kind of generalized Lienard equation,and then analyze the dynamic behavior of the original equation by the renormalization equation.