Convergence, nonconvergence and adiabatic transitions in fully coupled averaging

Systems which combine fast and slow motions can only be described by complicated two scale equations and in order to simplify the study one may rely on the averaging principle which suggests approximation of the slow motion by averaging in fast variables. On the time scale 1/ε this prescription usually works for all or almost all initial conditions when the fast motion does not depend on the slow one. On the other hand, when the slow and fast motions depend on each other (fully coupled), as is usually the case, the averaging approximation does not always work and when it is valid then only in the weaker sense of convergence in measure (or in average) with respect to initial conditions. We will discuss the corresponding convergence results and nonconvergence examples and formulate problems connected with the latter. For chaotic fast motions such as axiom A (hyperbolic) flows and diffeomorphisms as well as expanding transformations it is possible sometimes to describe the very long time (of order e^c/ε, c > 0) behaviour of the slow motion (which is natural to call adiabatic) but there is no complete understanding in this direction either.