In this thesis I make extensive use of the Lyapunov analysis formalism to unravel fundamental mechanisms of instability in two different systems : the Kuramoto model of globally coupled phase-oscillators and the Lorenz 96 (L96) atmospheric "toy" model, portraying the evolution of a physical quantity along a latitude circle. I start by introducing the relevant theoretical background, with special attention on the main tools I have been using throughout this work : Lyapunov Exponents (LEs), which quantify the asymptotic growth rates of infinitesimal perturbations in a system, and by extension, its degree of chaoticity, and Covariant Lyapunov Vectors (CLVs), which indicate the phase space direction (or the geometry) associated with these growth rates. The Kuramoto model is central in the study of synchronization among oscillatory units characterized by their various natural frequencies, but little is known on its chaotic dynamics in the unsynchronized state. I thus investigate the scaling behavior of the first LE, upon different assumptions on the natural frequencies, and make use of educated structural simplifications to analyze the origin of chaos in the finite size model. On the other hand, the L96 model has been devised to gather the main dynamical ingredients of atmospheric dynamics, namely advection, damping, external (solar) forcing and transfers across different scales of motion, in a minimalist and functional way. It features two coupled dynamical layers : the large scale variables, representing synoptic scale atmospheric dynamics, and the small scale variables, faster and more numerous, associated with convective scale dynamics. The core of the study revolves around geometrical properties of CLVs, in the aim of understanding the processes underlying the observed multiscale chaoticity, and an exhaustive study of a non-trivial ensemble of CLVs featuring relevant projection on the slow subspace.

Graduation date: 2003

The Melnikov method is applied to a model of parametrically generated cross-waves in a long rectangular channel in order to determine if these cross-waves are chaotic. A great deal of preparation is involved in order to obtain a suitable form for the application of the Melnikov method. The Lagrangian for water waves, which consists of the volume integrals of the kinetic energy density, potential energy density, and a dynamic pressure component, is transformed to surface integrals in order to avoid constant conjugate momenta. The Lagrangian is simplified by subtracting the zero variation integrals and written in terms of generalized coordinates, the time dependent components of the crosswave and progressive wave velocity potentials. The conjugate momenta are calculated after expanding the Lagrangian in a Taylor series. The Hamiltonian is then determined by a Legendre transformation of the Lagrangian. Ordinarily, the first order evolution equations obtained from derivatives of the Hamiltonian are suitable for applications of the Melnikov method. However, the crosswave model results in extremely complicated evolution equations which must be simplified before a Melnikov analysis is possible. A sequence of seven canonical transformations are applied and yield a final set of evolution equations in fairly simple form. The unperturbed system is analyzed to determine hyperbolic fixed points and the equations describing the heteroclinic orbits for near resonance cases. The Melnikov function is calculated for the perturbed system which must also satisfy KAM conditions. The Melnikov results indicate the system is chaotic near resonance. Furthermore, the heteroclinic orbits, about which chaotic motions occur, are transformed back to the original set of variables and found to be extremely complicated; this orbit would be impossible to determine analytically without the canonical transformations. The theoretical results were verified by experiments. Poincare maps obtained from measurements of the free surface displacement indicate both quasi-periodic and chaotic motions of the water surface. Power spectra and time series of the water surface displacement are also analyzed for chaotic behavior, with less conclusive results. Stability diagrams of cross-wave generation confirm behavior consistent with parametric excitation. / Graduation date: 1994

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 280-297). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (M.S.)--Naval Postgraduate School, 2002. / Thesis advisor(s): John Arquilla, George Lober. Includes bibliographical references (p. 99-102). Also available online.

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

Thesis (M.S.)--Naval Postgraduate School, 2002. / Title from title screen (viewed July 18, 2003). "June 2002." Includes bibliographical references (p. 99-102). Also issued in paper format.

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