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We present a new approach for finding the asymptotic solution of certain weakly nonlinear oscillator equations. In particular, we develop a simplified version of the renormalization group method of Chen, et al. to obtain higherorder approximations on longer time intervals than are typically provided by averaging and two-timing. The technique, which has much greater potential, is illustrated on three challenging examples from the literature for which better than usual asymptotic solutions are obtained.

This paper surveys, compares and updates techniques to obtain the asymptotic solution of the weakly nonlinear oscillator equationÿ + y + f (y,ẏ) = 0 as → 0 and for corresponding first-order vector systems. The solutions found by the regular perturbation method generally feature resonance and so break down as t → ∞. The classical methods of averaging and multiple scales eliminate such secular behavior and provide asymptotic solutions valid for time intervals of length t = O( −1 ). The renormalization group method proposed by Chen et al. [Phys. Rev. E 54 (1996) 376-394] gives equivalent results. Several well-known examples are solved with these methods to demonstrate the respective techniques and the equivalency of the approximations produced. Finally, an amplitude-equation method is derived that incorporates the best features of all these techniques. This method is both straightforward to automate with a computer-algebra system and flexible enough to allow the forcing f to depend on the small parameter.This paper is dedicated to Jerry Kevorkian with sincere appreciation for his long career of dedicated teaching at the University of Washington and for his substantial contributions to multiscale asymptotics.

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