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Ghosts of order on the frontier of chaos


Muldoon, Mark (1989) Ghosts of order on the frontier of chaos. Dissertation (Ph.D.), California Institute of Technology.


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 orderly, 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 (Dissertation (Ph.D.))
Subject Keywords:Physics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Physics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Katok, Anatole
Thesis Committee:
  • Unknown, Unknown
Defense Date:30 May 1989
Record Number:CaltechTHESIS:06052013-085416894
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7839
Deposited By: Benjamin Perez
Deposited On:05 Jun 2013 16:10
Last Modified:02 Dec 2020 01:54

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