Application Of Renormalization Group Method To Coupled Mathieu Equations

As an important branch of differential equation research,singular perturbation theory has been widely and effectively applied in the study of nonlinear problems in the fields of astromechanics,fluid mechanics and cybernetics,and has been highly concerned by mathematicians and physicists.So far,many singular perturbation methods have been developed,such as matched asymptotic expansions,WKB,averaging method,and renormalization group method.In the early 1990 s,Chen,Goldenfeld and Oono applied the renormalization group idea in quantum mechanics to the problem of solving approximate solutions to singularly perturbed differential equations.They established a singularly perturbed renormalization group method,and applied it to Rayleigh equation,Mathieu equation,boundary layer,center manifolds and other problems,and obtained uniformly effective results.In 1999,Ziane et al.gave a renormalization scheme for a class of semilinear singularly perturbed problems,obtained uniformly valid approximate solutions for the problem strictly.In this paper,we study the approximate solution and transition curve problem of a class of coupled Mathieu equations.First,we consider a class of general linear perturbation problems of second order and obtain uniformly valid approximate solutions satisfying initial value conditions by means of renormalization group method.On the basis,the uniform effective approximate solution of a class of coupled Mathieu equations is further studied,and the transition curve of coupled Mathieu equations is analyzed by using the dynamic properties of renormalization group equations.This paper is divided into four chapters.The first chapter is the introduction,which briefly introduces the research background of singular perturbation method,renormalization group method and Mathieu equation.In the second chapter,the renormalization schemes of renormalization group methods under two kinds of equations and the corresponding dynamic properties are briefly introduced.In chapter 3,we study the system of linear perturbation with second order,and obtain the uniformly valid approximate solution of the initial value problem by renormalization group method,and give strict error estimate.The fourth chapter is the main content.Using the results of the third chapter,the approximate solution of a class of coupled Mathieu equations is obtained and the transition curve is further analyzed.