Method of multiple scales with three time scales

Dept. Mathematics, University of Illinois at Urbana-Champaign, M/C 382, Urbana, IL 61801Some confusion appears in the literature regarding the extension of the method of multiple scales to three or more timescales. While the work of Murdock and Wang [1] correctly indicates some obstructions to such an extension in some typesof problems, other work suggests that the extension should almost never be possible [2]. These pessimistic results generallyfollow from the imposition of additional restrictions in the calculation, which really are not necessary [3]. We will showon some simple ODE models how a systematic implementation of the method of multiple scales can succeed in correctlycapturing three active time scales, though more calculation is required than one might expect from naive considerations. Onthe other hand, an extension of the method of averaging to three time scales produces an incorrect approximation in theseproblems [4].