Ghosts of Order on the Frontier of Chaos

Muldoon, Mark (1989) Ghosts of Order on the Frontier of Chaos. Doctoral thesis, California Institute of Technology.

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Abstract

What kinds of motion can occur in classical mechanics? We address this question by looking at the structures traced out by trajectories in phase space; the most orderely, completely integrable systems are characterized by phase trajectories confined to low-dimensional, invariant tori. The KAM theory examines what happens to the tori when an integrable system is subjected to a small perturbation and finds that, for small enough 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 replaced by �cantori,� tattered, Cantor-set remnants of the original invariant curves. We seek to extend Aubry and Mather�s picture to higher dimensional systems and report two kinds of studies; both concern perturbations of a completely integrable, four-dimensional symplectic map. In the first study we compute some numerical approximations to Birkhoff periodic orbits; sequences of such orbits should approximate any higher dimensional analogs 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 tori survive. We are able to show that the last few of our Birkhoff orbits exist in a regime where there are no tori.

Item Type: Thesis (Doctoral)
Additional Information: This version of my thesis differs from the one I submitted in 1989 in that I have typeset it afresh, using scanned versions of the original figures. I also updated the code for the machine-assisted, converse KAM proofs and have included sources for Lloyd Zusman's arbitrary precision arithmetic library (APM), which was the foundation of my interval-arithmetic calculations.
Uncontrolled Keywords: Converse KAM theorems, machine-assisted theorem proving, Froschlé map, cantori, dynamical systems
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 37 Dynamical systems and ergodic theory
MSC 2010, the AMS's Mathematics Subject Classification > 70 Mechanics of particles and systems
Depositing User: Dr Mark Muldoon
Date Deposited: 09 Sep 2016
Last Modified: 20 Oct 2017 14:13
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2500

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