What kinds of motion can occur in classical mechanics? We address this questionby looking at the structures traced out by trajectories in phase space; the most orderly,completely integrable systems are characterized by phase trajectories confined to low-dimensional,invariant tori. The KAM theory examines what happens to the tori whenan integrable system is subjected to a small perturbation and finds that, for smallenough perturbations, most of them survive.
The KAM theory is mute about the disrupted tori, but, for two-dimensional systems,Aubry and Mather discovered an astonishing picture: the broken tori are replacedby "cantori," tattered, Cantor-set remnants of the original invariant curves.We seek to extend Aubry and Mather's picture to higher dimensional systems andreport two kinds of studies; both concern perturbations of a completely integrable,four-dimensional symplectic map. In the first study we compute some numerical approximationsto Birkhoff periodic orbits; sequences of such orbits should approximateany higher dimensional analogs of the cantori. In the second study we prove converseKAM theorems; that is, we use a combination of analytic arguments and rigorous,machine-assisted computations to find perturbations so large that no KAM tori survive.We are able to show that the last few of our Birkhoff orbits exist in a regimewhere there are no tori.
