Date of Award

Spring 1-1-2013

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

First Advisor

James D. Meiss

Second Advisor

Juan Restrepo

Third Advisor

Keith Julien

Fourth Advisor

Elizabeth Bradley

Fifth Advisor

James Curry

Abstract

Invariant rotational tori play an important role in the dynamics of volume-preserving maps. When integrable, all orbits lie on these tori and KAM theory guarantees the persistence of some tori upon perturbation. When these tori have codimension-one they act as boundaries to transport, and therefore play a prominent role in the global stability of the system. For the area-preserving case, Greene's residue criterion is often used to predict the destruction of tori from the properties of nearby periodic orbits. Even though KAM theory applies to the three-dimensional case, the robustness of tori in such systems is still poorly understood. This dissertation begins by extending Greene's residue criterion to three-dimensional, reversible, volume-preserving maps.

The application of Greene's residue criterion requires the repeated computation of periodic orbits, which is costly if the system is nonreversible. We describe a quasi-Newton, Fourier-based scheme to numerically compute the conjugacy of a torus and demonstrate how the growth of the Sobolev norm or singular values of this conjugacy can be used to predict criticality. We will then use this method to study both reversible and nonreversible volume-preserving maps in two and three dimensions. The near-critical conjugacies, and the gaps that form within them, will be explored in the context of Aubry-Mather and Anti-Integrability theory, when applicable. This dissertation will conclude by exploring the locally and globally most robust tori in area-preserving maps.

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