A New Renormalization Method for the Asymptotic Solution of Weakly Nonlinear Vector Systems

Abstract

This paper considers the asymptotic integration of a special class of initial value problems involving a nonlinear regular perturbation scaled by a small parameter $ \epsilon >0$. For $t= {\mathcal O} (1/\epsilon)$, these problems were classically solved using either the method of averaging or of multiple scales to remove secular terms that arise in the natural power series procedure. Our new ansatz is straightforward and effective. Moreover, it indicates when problems might occur in providing the asymptotic solution on very long time intervals. Other closely related problems are also attacked using renormalization.