Introduction and Notation

The typical averaging problem may be defined as follows: One considers an unperturbed problem in which the slow variables remain fixed. Upon perturbation, a slow drift appears in these variables which one would like to approximate independently of the fast variables. This situation may be described with the aid of a fiber bundle in which the base represents the slow variables and the fibers represent the fast variables. This may seem unnecessarily pedantic, but in fact it provides a convenient and precise language which supports the intuitive image of the “mixing” that occurs among the fast variables, as opposed to the drift of the slow variables on the base. For these reasons, we shall sometimes employ this geometrical language when appropriate but we will not make use of any nontrivial (i.e. global) property of fiber bundles in this work.