Asymptotic normal form theory for nonautonomous equations and its connections with a renormalization group method

For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono has been shown to be an effective general approach. 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 PoincareLindstedt method, the method of averaging, and others. In this work, we examine the mathematical basis of this RG method. In particular, we develop asymptotic normal form theory for a large class of nonlinear, nonautonomous ordinary differential equations, and we show that this normal form theory is equivalent to the RG method. Both approaches are carried out explicitly up to and including second order, and it is shown how the higher order theory may be carried out. A number of perturbation problems with secularities and with slowly-varying terms are used as illustrative examples of both methods.