The averaging of systems with a hierarchy of phase rotation speeds

Abstract

An investigation is carried out of a system of ordinary equations with rotating phases whose frequencies (rotation speeds) form a hierarchy in terms of powers of a small parameter. Attention is also devoted to a more complex system, whose right-hand sides also contain terms “with zero means” (averaged over trajectories of fast motions). The scheme proposed to deal with such systems involves the successive application of a standard procedure which isolates the “fastest” variables to within a certain accuracy in terms of the small parameter. The degree of correspondence between the solutions of exact equations and those of the equations thus averaged is determined over an asymptotically large time interval, during which the “slowest” variable increases by one order of magnitude.