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What kinds of motion can occur in classical mechanics? We address thisquestion by looking at the structures traced out by trajectories in phasespace; the most orderely, completely integrable systems, are charactrized byphase trajectories confined to low-dimensional, invariant tori. The KAM theoryexamines what happens to the tori when an integrable system is subjected to asmall perturbation and finds that, for small enough perturbations, most of themsurvive.  The KAM theory is mute on the suject of the disrupted tori, but, for twodimensional systems, Aubry and Mather discovered an astonishing picture: thebroken tori are replaced by "cantori", tattered, Cantor-set remnants of theoriginal invariant curves. We seek to extend Aubry and Mather's picture tohigher dimensional systems and report two kinds of studies; both concernperturbations of a completely integrable, four-dimensional symplectic map. Inthe first study we compute some numerical approximations to Birkhoff periodicorbits; sequences of such orbits should approximate any higher dimensionalanalogs of the cantori. In the second study we prove converse KAM theorems;that is, we use a combination of analytic arguments and rigorous,machine-assisted computations to find perturbations so large that no KAM torisurvive. We are able to show that the last few of our Birkhoff orbits exist ina regime where there are no tori.

Ghosts of Order on the Frontier of Chaos