A new method for evaluating the Lyapunov exponent for a Hamiltonian system involves a spatial evaluation, rather than a numerical time integration. The introduction of a novel vector field to the phase space allows the Lyapunov exponent to be expressed in a form that does not involve time. The Lyapunov exponent is seen to be a property of the geometry and topology of ergodic regions of phase space. As a result it has a more regular behavior than previously thought. The Lyapunov exponent is found to be a differentiable function of the perturbation coupling in regions where it was previously thought to be discontinuous. Properties of the Lyapunov function once taken for granted are shown to be artifacts of the traditional computation methods. The technique is discussed with examples from a system of coupled quartic oscillators. / Graduation date: 1996

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--University of Oregon, 2003. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 184-187). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2009. / Committee Chair: Gangbo, Wilfrid; Committee Member: Loss, Michael; Committee Member: Pan, Ronghua; Committee Member: Swiech, Andrzej; Committee Member: Tannenbaum, Allen. Part of the SMARTech Electronic Thesis and Dissertation Collection.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references (leaves 115-121). Available also in a digital version from Dissertation Abstracts.

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The Hamiltonian cycle problem (HCP) can be succinctly stated as: "Given a graph, find a cycle that passes through every single vertex exactly once, or determine that this cannot be achieved". Such a cycle is called a Hamiltonian cycle. The HCP is a special case of the better known Travelling salesman problem, both of which are computationally difficult to solve. An efficient solution to the HCP would help solving the TSP effectively, and therefore would have a great impact in various fields such as computer science and operations research. In this thesis, we obtain novel theoretical results that approach this discrete, deterministic, problem using tools from stochastic processes, matrix analysis, and graph theory. / PhD Doctorate