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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.

Thesis (Ph.D.)--Aberdeen University, 2008. / Title from web page (viewed on July 1, 2009). Includes bibliographical references.

The nonlinear dynamics of PWM DC-DC switching regulators operating in the continuous conduction mode are investigated. A quick review of the existing analysis techniques and their limitations is first presented. A discrete nonlinear time-domain model is derived for open-loop DC-DC converters. This model is then extended for closed-loop regulator systems implementing any type of compensation scheme. The equilibrium solutions of the closed-loop system are calculated and their stability is determined. The methods developed are used to study the dynamic behavior of a DC-DC buck regulator implementing different types of compensation design: proportional, integral, proportional-integral, and proportional-integral-derivative feedback control. A detailed bifurcation analysis of the dynamic solutions as a design or a control parameter is changed is presented. A period-doubling route to chaos is shown to exist in voltage-mode regulators, depending on the values of the parameters of the compensator and the input voltage. An investigation of the behavior of the converter in the instability regions has been carried out to shed light on its bifurcations. The interactions of input filters with DC-DC switching-mode regulators are investigated as well. It is shown that the small-signal averaged model widely used in the design of DC-DC regulators does not provide a complete understanding of the stability of the filter-regulator system. It can only provide the local borders of small-signal stable operation. The large-signal time-domain nonlinear averaged model is used to further understand the interaction on the slow scale using nonlinear analysis techniques. No fast scale interactions, however, can be predicted using this model. A complete nonlinear switching model is thus used to investigate the interaction of the filter and the regulator on all scales: fast and slow. / Master of Science

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This thesis is concerned with long-range sound propagation in deep water.  The main area of interest is the stability of acoustic ray paths in wave guides in which there is a transition from single to double duct sound speed profiles, or vice-versa.  Sound propagation is modelled within a ray theoretical framework, which facilitates a dynamical systems approach of understanding long-range propagation phenomena, and the use of its tools of analysis. Alternative reduction techniques to the Poincaré sections are presented, by which the stability of acoustic rays can be graphically determined.  Beyond periodic driving, these techniques prove to be useful in case of the simplest quasiperiodic driving of the ray equations.  One of the techniques facilitates a special representation of ray trajectories for periodic driving. Namely, the space of sectioned trajectories is partitioned into nonintersecting regular and chaotic regions as with the Poincaré sections, when quasiperiodic and chaotic trajectories are represented by curve segments and area filling points, respectively.  In case of the simplest quasiperiodic driving – speaking about the same technique – regular trajectories are represented by curves similar to Lissajous curves, which are opened or closed depending on whether the two driving frequencies involved make relative primes or not. It is confirmed for a perturbed canonical profile that the background sound speed structure controls ray stability. It is also demonstrated for a particular double duct profile, when the singularity of the nonlinearity parameter for the homoclinic trajectory associated with this profile refers to the strong instability of corresponding perturbed trajectories.

The nonlinear dynamics of three physical systems has been investigated. Diode resonator systems are experimentally shown to display a period doubling route to chaos, quasiperiodic states, periodic locking states, and Hopf bifurcation to chaos. Particularly, the transition from quasiperiodic states to chaos in line-coupled systems agrees well with the Curry-Yorke model. The SPICE program has been modified to give realistic models for the diode resonator systems.

Control algorithms which exploit the unique properties of chaos can vastly improve the design and performance of many practical and useful systems. The program Perfect Moment is built around such an algorithm. Given two points in the system's state space, it autonomously maps the space, chooses a set of trajectory segments from the maps, uses them to construct a composite path between the points, then causes the system to follow that path. This program is illustrated with two practical examples: the driven single pendulum and its electronic analog, the phase-locked loop. Strange attractor bridges, which alter the reachability of different state space points, can be used to increase the capture range of the circuit.