Thesis (M.S.)--State University of New York at Binghamton, Physics Department, 2007. / Includes bibliographical references.

Thesis (Ph. D.)--West Virginia University, 2004. / Title from document title page. Document formatted into pages; contains vi, 166 p. : ill. Includes abstract. Includes bibliographical references (p. 142-160).

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 95-98). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 98-102). Available also in a digital version from Dissertation Abstracts.

Synchronization of chaos in coupled systems of ordinary differential equations is an area of mathematics which has attracted much attention in recent years, in particular for the potential technological applications such systems have in engineering and industry. The motivation for this research was to understand mathematically, synchronization observed in systems of two and three solid state lasers studied by collaborators at the Georgia Institute of Technology. The main objectives of this thesis are to understand more clearly some of the dynamical phenomena associated with the synchronization of chaos, and to develop new techniques for the analysis of dynamical systems with symmetry; with a view to applying these techniques to models of solid state laser systems and other applications. First we introduce the main ideas of chaotic synchronization and some useful tools for the analysis of dynamical systems with symmetry. We then introduce a model for a solid state Nd:YAG laser and examine the types of dynamics which may be exhibited. Subsequently we look at systems of two and three coupled solid state lasers and examine the onset of synchronization in such systems, both in a fully symmetric system and in the case of two coupled lasers, the case of broken symmetry. We then contrast these results with those of a modified Rossler system and observe similar results in both cases. We examine how chaotic systems may be used for communication purposes and develop a new scheme for the communication of a signal using the synchronization of chaos. Finally we introduce a new definition of attractor and using topological and measure theoretic properties of sets, we reexamine the concepts of basin riddling and are able in certain situations to determine the presence or otherwise of riddling.

Self-organising processes occurring in isotropic turbulence and homogeneous magnetohydrodynamic (MHD) turbulence are investigated in relation to the stability of helical flow structures. A stability analysis of helical triad interactions shows that compared to hydrodynamics, equilibria of the triadic evolution equations have more instabilities with respect to perturbations on scales larger than the characteristic scale of the system. Some of these instabilities can be mapped to Stretch-Twist-Fold dynamo action and others to the inverse cascade of magnetic helicity. High levels of cross-helicity are found to constrain small-scale instabilities more than large scale instabilities and are thus expected to have an asymmetric damping effect on forward and inverse energy transfer. Results from a numerical investigation into the influence of helicity on energy transfer and dissipation are consistent with this observation. The numerical work also confirms the predictions of an approximate method describing the Reynolds number dependence of the dimensionless dissipation coefficient for MHD turbulence. These predictions are complemented by the derivation of mathematically rigorous upper bounds on the dissipation rates of total energy and cross-helicity in terms of applied external forces. Large-scale helical flows are also found to emerge in relaminarisation events in direct numerical simulations of isotropic hydrodynamic turbulence at low Reynolds number, where the turbulent fluctuations suddenly collapse in favour of a large-scale helical flow, which was identified as a phase-shifted ABC-flow. A statistical investigation shows similarities to relaminarisation of localised turbulence in wall-bounded parallel shear flows. The turbulent states have an exponential survival probability indicating a memoryless process with a characteristic lifetime, which is found to depend super-exponentially on Reynolds number akin to well-established results for pipe and plane Couette flow. These and further similarites suggest that the phase space dynamics of isotropic turbulence and wall-bounded shear flows are qualitatively similar and that the relaminarisation of isotropic turbulence can also be explained by the escape from a chaotic saddle.

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M.Sc. / Please refer to full text to view abstract

M.Sc. / This thesis is a meeting of two disciplines: Chaos theory and psychopathology. Chaos theory developed out of mathematics, it aims to explain what is called a 'Chaotic system'. This is a system in which small changes lead to large effects: it is unstable, complex, and in continuous interaction with elements both within and outside of itself. According to this definition, human beings are inherently Chaotic systems. Psychopathology is the study of psychological disorder of human beings, including descriptions, etiologies and treatments. In the past, psychopathology was viewed as a 'modernist' science, seeking one-to-one relationships between cause, effect, symptom and treatment. With the rise of postmodernism, many theorists have criticised this view and sought out a more integrative, context-driven approach to understanding disorder. Although in its infancy, one of these approaches is the application of Chaos theory. In this thesis, the two disciplines meet around a theoretical analysis, and an exploration of a case study of Susan -a 'patient' diagnosed as having Bipolar Mood Disorder. Through exploring the life-story of Susan within the context of Chaos theory, we discover a new, integrative way of looking at 'disorder'1 its manifestations and our reactions to it. This thesis does not aim to give a definitive perspedive of Susan's life-story, or even of the two disciplines. Rather it aims to provide an academic framework for an application of Chaos theory to psychopathology. The thesis concludes that Chaos theory is a useful analogy in constructing a meaning and interpretation of psychopathology.

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