Method of multiple scales with three time scales

Abstract

Some confusion appears in the literature regarding the extension of the method of multiple scales to three or more time scales. While the work of Murdock and Wang [1] correctly indicates some obstructions to such an extension in some types of problems, other work suggests that the extension should almost never be possible [2]. These pessimistic results generally follow from the imposition of additional restrictions in the calculation, which really are not necessary [3]. We will show on some simple ODE models how a systematic implementation of the method of multiple scales can succeed in correctly capturing three active time scales, though more calculation is required than one might expect from naive considerations. On the other hand, an extension of the method of averaging to three time scales produces an incorrect approximation in these problems [4]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)