Mathematics Department, Imperial College, London SW7 2BZ, United Kingdom

摘要

This paper describes the representation and breakdown of the invariant Kol'mogorov-Arnol'd-Moser (KAM) tori of the driven particle in an infinite square well in terms of the periodic trajectories of the system. The periodic cycles are characterized analytically and numerically and their stability as the amplitude of the driving field increases is determined numerically. A representation of the zoning number, analogous to the winding number of the standard map, is developed for the system. It is shown that a KAM surface can be approximated by high-order periodic cycles with winding numbers corresponding to continued fraction approximates of the KAM surface's irrational zoning number. The zoning numbers of the most robust KAM tori between primary resonances are related to the golden mean and approximated accordingly by periodic cycles. The critical fields at which the invariant tori break down and the accompanying transition from local to global stochasticity occurs is estimated from the breakdown fields of the cyclic approximates.