Averaging, passage through resonances, and capture into resonance in two-frequency systems

Applying small perturbations to an integrable system leads to its slow evolution. For an approximate description of this evolution the classical averaging method prescribes averaging the rate of evolution over all the phases of the unperturbed motion. This simple recipe does not always produce correct results, because of resonances arising in the process of evolution. The phenomenon of capture into resonance consists in the system starting to evolve in such a way as to preserve the resonance property once it has arisen. This paper is concerned with application of the averaging method to a description of evolution in two-frequency systems. It is assumed that the trajectories of the averaged system intersect transversally the level surfaces of the frequency ratio and that certain other conditions of general position are satisfied. The rate of evolution is characterized by a small parameter ε. The main content of the paper is a proof of the following result: outside a set of initial data with measure of order √ε the averaging method describes the evolution to within O(√ε |ln ε|) for periods of time of order 1/ε. This estimate is sharp. The exceptional set of measure √ε contains the initial data for phase points captured into resonance. A description of the motion of such phase points is given, along with a survey of related results on averaging. Examples of capture into resonance are presented for some problems in the dynamics of charged particles. Several open problems are stated. Bibliography: 65 titles