Working with multiscale asymptotics

Abstract

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.

Variation of Parameters and the Renormalization Group Method

“…as is immediately found by directly seeking a solution as the product x = e Ft A. WhenȦ is 2π -periodic in t, a renormalization group solution was given in Mudavanhu et al [18]. The solution could also be obtained by averaging (cf.…”

Examples Illustrating the Use of Renormalization Techniques for Singularly Perturbed Differential Equations

“…Chen, Goldenfeld and Oono consider the equation 3 y + (1 + + 2 )y + (1 + + 2 )y + y = 0 (33) on t ≥ 0 with appropriate initial conditions so that the exact solution…”

“…Special attention should, in particular, be given to the recent paper of Veysey and Goldenfeld [2]. Applied mathematicians have likewise related the method to averaging, normal forms, and Lie group techniques though often restricting the class of problems under consideration [3][4][5]. A simplification of the ideas involved in the renormalization procedure is gradually taking place.…”