| In this paper, there axe three parts, the outline of the paper is as follows. Chapter 1 is a introduction. First, we introduce the historical background of renormalization group approach which is mainly used to research the problem that we interest in here.In 1971, the American scientist K. G. Wilson applied the renormalization group approach in the quantum field theory to the study of the critical phenomena,which was a major breakthrough. Since then renormalization group aroused more people's interest. Later, Chen, Goldenfeld and Oono introduced the perturbationrenormalization group approach as a uniformly tool for a global and asymptotic analysis of ordinary and partial differential equations. There are a lot of research work on the renormalization group and the applications of the renormalization group approach are also very broad and successful. In the first chapter, we show a outline of the investigation results of a few mathematicians, such as L. Y. Chen, Goldenfeld, M. Ziane, B. Mudavanhu and R. E. O'Malley, K. Nozaki and Y. Oono, S. Ei, K. Fujii, T. Kuniniro, and others in the past. Their accomplishments in the singular perturbation and reductive perturbation have the representative significance.In the second chapter, we present the process solving the Mathieu equation transition curves by the classic method of multiple scales. In the singular pertur- bation, there are many other methods, such as deformation parameters, Whittaker,the averaging method, can be applied to this issue. In the use of multi-scale method for solving Mathieu equation transition curves, the main feature is the introduction of all necessary time scales. But when we use the renormalization group method to solve the problem, all the necessary time scales appear naturally in the RG equations. This is a major distinguishing feature about renormalization group approach, so it is much more direct and effective than multi-scale approach in determining the slow time scales.Chapter 3 is the main part of this paper. First of all, we describe the renormalization procedure in detail below. The three steps are:1. We write a naive perturbation expansion (perturbation sequence without a priori knowledge), which contains (in general) secular terms;2. We introduce a free parameterμand use a near identity transformation in order to remove the possible secular terms;3. The RG equations are derived using the fact that the approximate solutionof the perturbed problem should not depend on the free parameterμ. At last, the renormalization amplitude and phase equations should be solved asymptotically.Next, we consider the following second-order linear homogeneous ordinary differential equation with almost constant coefficientsu" + [λ+εf(x)]u = 0, x∈R, (1)whereλ,εare positive parameters,ε<<1, both u and f are scalar functions with the independent variable x. when taking f(x) = 2 cosx, the equationu" + [λ+ 2εcosx]u = 0is the famous Mathieu equation, which describes the system with single degree of freedom and the incentive parameters. In the last chapter, it has been carefully studied.In a number of assumptions, on the equation (1), we have the following conclusions:Lemma 1 When the parameterλ> 0,λ≠n~2/4,n = 1,2,3,…, for sufficientlysmall e, the solutions of equation (1) are stable.Theorem 1 For the system(1), nearλ= n~2/4, there exists many transition curves that separate the stability regions and instability regions, and nearε= 0, to order e, the transition curves are given byλ=n~2/4±ε/2(?)+ O(ε~2),ε→0,where n is a arbitrary positive integer, constants a_n, b_n are the n- th coefficients in f(x) 's Fourier series.The lemma 1 is necessary, and we present a brief proof. In chapter 3, we give a detailed proof of the theorem 1. At the same time, we have demonstrated the application procedure of the RG techniques. The results we obtained here are compared and analyzed. |
PDF Full Text Request