A Geometric Approach to Systems with Multiple Time Scales

Physical systems often involve several processes that are evolving on di erent time scales. The resulting equations have a speci c structure which can be exploited to great e ect and the kind of results that can be obtained, at least with a certain goal in mind, are the subject of this paper. On account of the physical motivation for the di erent scales, there are many areas in which applications can be found. Apart from their physical relevance, there is another important reason for developing a theory tailored to multiple time scale systems. Systems in low dimensions, here one should think of dimensions 1 and 2, can often be analyzed by well-established techniques. Indeed, the asymptotic behavior is determined by either periodic orbits and/or critical points, as follows from the celebrated Poincar e-Bendixson Theorem in 2 dimensions and purely elementary considerations in 1 dimension. If the underlying dimension of some system under consideration is higher then we are faced with two di culties: rst, the techniques are largely unavailable for performing the analysis and, secondly, the phenomena occurring can be considerably more complicated, involving, for instance, chaotic behavior. To resolve this apparent intractability of higher-dimensional problems, techniques involving reductions to lower dimensional phase spaces are very attractive. Such techniques are often found through the presence of conserved quantities or symmetries. While these techniques do make certain higher dimensional problems susceptible to analysis, the resulting behavior must necessarily be characteristic of the lower dimensional system and thus the richness of motion available in higher dimensional systems is not re ected in systems for which this approach works. Systems with multiple time scales, however, o er a method of reduction which preserves the higher dimensional nature of the behavior. This idea seems counter-intuitive but is not unreasonable when one considers the fact that at least two separate reductions are being used and, although each leads to a lower dimensional system, the combined e ect is to allow behavior characteristic of the full dimensional space. It is a consequence of the Recti cation Theorem, see Arnold [1], which states that a ow is locally trivial except near a critical point, that interesting behavior in a dynamical system will either occur near a critical point or be a result of recurrent motion. It is then not surprising