Renormalization Group Method And Center Manifold Theory

Perturbed problem is divided into regular perturbation and singular perturbation. Theregular perturbation problem is simple, commonly used methods are power series method,iterative method and parameter diferential method, etc. Singular perturbation problem ismore complicated, main reason is the appearance of consistency, the asymptotic expansionis non-uniformly valid because of infinite field, diferent types of partial diferential equa-tions, small parameter multiply highrst derivative and singular point, etc. For diferenttypes of problems, Mathematicians and physicists have developed the multiple time scale,boundary layer method, WKB method, telescopic coordinates method, gradual expansionmatching method, center manifold theory and envelope theory.These methods had been widely used in physics, engineering, chemistry, astronomysince they were constructed. They are playing important roles in solving some practicalapplications. But they have their own limitations, which prevent us from getting the validglobal solution. For example, in multiple scales method, it is difcult to determine thetime scale; derivative expansion method will easily fail in the big moments of singularperturbation problems; WKB method is only applicable to linear diferential equations;boundary layer method is applicable to linear problem and the nonlinear problem, but thethickness of boundary layer and the location of the boundary layer are generally difcultto determine, and small parameter fractional power may arise. So, we cannot use thesemethods blindly, and finding a more efcient way to overcome the limitation of the originalperturbation methods is important.Based on the quantum field theory of renormalization, Chen, Goldenfeld and Oonopropose the renormalization group method. Experiments show that it can overcome thelimitation of the traditional singular perturbation method in application and is a novel method. We can naturally generated perturbation sequence in order to overcome the difculty in timescale and limitation in traditional singular perturbation method.The main process of renormalization group method is as follows, first construct theperturbation expansions of the system, then give the renormalized parametersτ, and usethe initial conditions to eliminate the direct perturbation which may appear in the long term The derivative of the renormalization parameters is zero for it has nothing to do with the problem; finally, let τ=t, and obtain the renormalization equations and the uniformly valid asymptotic expansion.In this article, we study the consistent and effective approach of the center manifold of an ordinary differential equation in fixed piont using renormalization group method where ε>0is a small parameter, B can be diagonalization and the real part of characteristic vallue of B negative, F and G are vector function, satify F(εX,εY)=O(ε2), G(εX, εY)=O(ε2).Under the scaling transformation,(1) translate into thefollowing form let We have:Theorem1Γε is the uniformly valid second order approximation of center manifold of (2) in fixed point (0,0).Finally, we consider an example we have the uniformly valid approximation of center manifold of (2) in fixed point, and it’s consistent with classical center manifold theory.