Analysis of a Renormalization Group Method for Solving Perturbed Ordinary Differential Equations

For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. E. 49:4502-4511,1994] has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system. It has been applied to a variety of problems traditionally analyzed using disparate methods, including the method of multiple scales, boundary layer theory, the WKBJ method, the Poincaré-Lindstedt method, the method of averaging, and others. In this work, we examine the mathematical basis of this RG method. We analyze a simplified algorithm for the method and show that its crucial step is a near-identity change of coordinates equivalent to that of normal form theory. This is done in the context of two classes of singularly perturbed differential equations which depend on a small parameter . For systems with autonomous perturbations, we extend the RG method up to second order and show it is equivalent to the classical Poincaré-Birkhoff normal form up to and including terms of O( 2). This analysis may be generalized to higher order. For systems with nonautonomous perturbations, the RG method is equivalent to a time-asymptotic normal form theory which we also present here. Finally, we establish how well the solution to the RG equations approximate the solution of the original equations on timescales of O(1/ ). PACS numbers: 05.10.Cc, 05.45.-a, 02.30.Hq