Variation of Parameters and the Renormalization Group Method

This paper presents a straightforward procedure for using Renormalization Group methods to solve a significant variety of perturbation problems, including some that result from applying a nonlinear version of variation of parameters. A regular perturbation procedure typically provides asymptotic solutions valid for bounded t values as a positive parameter e tends to zero. One can eliminate secular terms by introducing a slowly-varying amplitude obtained as a solution of an amplitude equation on intervals where is bounded. With sufficient stability hypotheses, the results may even hold for all . These ideas are illustrated for a number of nontrivial problems involving ordinary differential equations.