On the multiple scales perturbation method for difference equations
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Abstract
In the classical multiple scales perturbation method for ordinary difference equations (O Delta Es) as developed in 1977 by Hoppensteadt and Miranker, difference equations (describing the slow dynamics of the problem) are replaced at a certain moment in the perturbation procedure by ordinary differential equations (ODEs). Taking into account the possibly different behavior of the solutions of an O Delta E and of the solutions of a nearby ODE, one cannot always be sure that the constructed approximations by the Hoppensteadt-Miranker method indeed reflect the behavior of the exact solutions of the O Delta Es. For that reason, a version of the multiple scales perturbation method for O Delta Es will be presented and formulated in this paper completely in terms of difference equations. The goal of this paper is not only to present this method, but also to show how this method can be applied to regularly perturbed O Delta Es and to singularly perturbed, linear O Delta Es.