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Thesis (Ph. D.)--Oregon State University, 2009. / Printout. Includes bibliographical references (leaves 63-64). Also available on the World Wide Web.

In this thesis we consider three-dimensional dynamical systems in the neighbourhood of a singular point with rank-one and rank-two resonant eigenvalues. We first introduce and generalize here a new technique extending previous work which was described by Aziz an Christopher (2012), where a second first integral of a 3D system can be found if the system has a Darboux-analytic first integral and an inverse Jacobi multiplier. We use this new technique to find two independent first integrals one of which contains logarithmic terms, allowing for non-zero resonant terms in the formal normal form of vector field. We also consider sufficient conditions for the existence of one analytic first integral for three dimensional vector fields around a singularity. Starting from the generalized Lotka-Volterra system with rank-one resonant eigenvalues, using the normal form method, we find an inverse Jacobi multiplier of the system under suitable conditions. Moreover, these conditions are sufficient conditions for the existence of one analytic first integral of the system. We apply this to demonstrate the sufficiency of the conditions in Aziz and Christopher (2014). In the case of two-dimensional systems, Christopher et al (2003) addressed the question of orbital normalizability, integrability, normalizability and linearizability of a complex differential system in the neighbourhood at a critical point. We here address the question of normalizability, orbital normalizability, and integrability of three-dimensional systems in the neighbourhood at the origin for rank-one resonance system. We consider the case when the eigenvalues of three-dimensional systems have rank-one resonance satisfying the condition the sum of eigenvalues is equal to zero a typical example, and we use a further change of coordinates to bring the formal normal form for three-dimensional systems into a reduced normal form which contains a finite number of resonant monomials. By using this technique, we can find two independent first integrals formally. The first one of these first integrals is of Darboux-analytic type, and other first integral contains logarithmic terms corresponding to non-zero resonant monomials of the original system. We introduce the monodromy map in three-dimensional vector fields by using these two independent first integrals to study a relationship between normalizability and integrability of systems. In the case of rank-one resonant eigenvalues, we get a monodromy map which is in normal form, and then in the same way as the case of vector fields, we use a further change of coordinates to reduce this map into a reduced map which contains only a finite number of resonant monomials. This thesis also examines briefly the case of rank-two resonant eigenvalues of three-dimensional systems. The normal form in this case contains an infinite number of resonant monomials, we were not able to find a reduced normal form with a finite number of resonant monomials. This situation is therefore much more complex than the rank-one case. Thus, we simplify the investigation by truncating the 3D system to a 3D homogeneous cubic system as a first step to understanding the general case. Even though we can find two independent first integrals, the second one involves the hypergeometric function, leading to some interesting topics for further investigation.

No abstract available. Copyright 2008, University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. Please cite as follows: Beyers, FJC 2008, The Szemerédi property in noncommutative dynamical systems, PhD thesis, University of Pretoria, Pretoria, viewed yymmdd < http://upetd.up.ac.za/thesis/available/etd-05242009-145506/ > D620/ag / Thesis (PhD)--University of Pretoria, 2009. / Mathematics and Applied Mathematics / unrestricted

We introduced a method presented in Information Field Theory (IFT) [Abramovich et al., 2007] to improve interpolation in Smoothed Particle Hydrodynamics (SPH) in cases of less smooth flow. The method makes use of wavelet theory combined with B-splines for interpolation. The idea is to identify any jumps a function may have and then reconstruct the smoother segments between the jumps. The results of our work demonstrated superior capability when compared to a particular challenging SPH application, to better conserve jumps and more accurately interpolate the smoother segments of the function. The results of our work also demonstrated increased computational efficiency with limited loss in accuracy as number of multiplications and execution time were reduced. Similar benefits were observed for functions with spikes analyzed by the same method. Lesser, but similar effects were also demonstrated for real life data sets of less smooth nature. SPH is widely used in modeling and simulation of flow of matters. SPH presents advantages compared to grid based methods both in terms of computational efficiency and accuracy, in particular when dealing with less smooth flow. The results we achieved through our research is an improvement to the model in cases of less smooth flow, in particular flow with jumps and spikes. Up until now such improvements have been sought through modifications to the models' physical equations and/or kernel functions and have only partially been able to address the issue. This research, as it introduced wavelet theory and IFT to a field of science that, to our knowledge, not currently are utilizing these methods, did lay the groundwork for future research ideas to benefit SPH. Among those ideas are further development of criteria for wavelet selection, use of smoothing splines for SPH interpolation and incorporation of Bayesian field theory. Improving the method's accuracy, stability and efficiency under more challenging conditions such as flow with jumps and spikes, will benefit applications in a wide area of science. Just in medicine alone, such improvements will further increase real time diagnostics, treatments and training opportunities because jumps and spikes are often the characteristics of significant physiological and anatomic conditions such as pulsatile blood flow, peristaltic intestine contractions and organs' edges appearance in imaging.

A comparison of viscous hydrodynamic force models is presented, with application on an autonomous underwater vehicle (AUV). The models considered here are \emph{quasi-steady}, meaning that force is expressed as a function of instantaneous vehicle state. This is in contrast to physical reality, where the force applied to a rigid body moving through a viscous fluid is history-dependent. As a result, the comparison of models is restricted to how well they are able to recreate a force history, rather than how closely they represent the underlying physics. Of the models under consideration, no single model performs significantly better than the others, but several perform worse. Each viscous hydrodynamic force model presented here is expressed as a linear combination of basis functions, which are nonlinear functions of body-relative velocity. The greater dynamical model is presented in a rigid-body framework with six degrees of freedom, with terms which account for inviscid fluid flow, restoring forces due to gravity, and control forces due to actuator motion. The models are selected from several that have been proposed in the literature, which include empirically-derived and physics-based models. Some models assume that the relationship between force and velocity is fundamentally linear or quadratic in nature, or make assumptions about coupled motion. The models are compared by their relative complexities, and also by their ability to reproduce data sets generated from field experiments. The complete dynamical equations are presented for each model, including coefficients suitable for use with the Virginia Tech 690 AUV. / Master of Science

A common problem to all applications of linear finite dynamical systems is analyzing the dynamics without enumerating every possible state transition. Of particular interest is the long term dynamical behaviour, and if every element eventually converges on fixed points. In this paper, we study the number of iterations needed for a system to settle on a fixed set of elements. As our main result, we present two upper bounds on iterations needed, and each one may be readily applied to a fixed point system test. The bounds are based on submodule properties of iterated images and reduced systems modulo a prime.

The development and use of cocycles for analysis of non-autonomous behaviour is a technique that has been known for several years. Initially developed as an extension to semi-group theory for studying rion-autonornous behaviour, it was extensively used in analysing random dynamical systems [2, 9, 10, 12]. Many of the results regarding asymptotic behaviour developed for random dynamical systems, including the concept of cocycle attractors were successfully transferred and reinterpreted for deterministic non-autonomous systems primarily by P. Kloeden and B. Schmalfuss [20, 21, 28, 29]. The theory concerning cocycle attractors was later developed in various contexts specific to particular classes of dynamical systems [6, 7, 13], although a comprehensive understanding of cocycle attractors (redefined as pullback attractors within this thesis) and their role in the stability of non-autonomous dynamical systems was still at this stage incomplete. It was this purpose that motivated Chapters 1-3 to define and formalise the concept of stability within non-autonomous dynamical systems. The approach taken incorporates the elements of classical asymptotic theory, and refines the notion of pullback attraction with further development towards a study of pull-back stability arid pullback asymptotic stability. In a comprehensive manner, it clearly establishes both pullback and forward (classical) stability theory as fundamentally unique and essential components of non-autonomous stability. Many of the introductory theorems and examples highlight the key properties arid differences between pullback and forward stability. The theory also cohesively retains all the properties of classical asymptotic stability theory in an autonomous environment. These chapters are intended as a fundamental framework from which further research in the various fields of non-autonomous dynamical systems may be extended. A preliminary version of a Lyapunov-like theory that characterises pullback attraction is created as a tool for examining non-autonomous behaviour in Chapter 5. The nature of its usefulness however is at this stage restricted to the converse theorem of asymptotic stability. Chapter 7 introduces the theory of Loci Dynamics. A transformation is made to an alternative dynamical system where forward asymptotic (classical asymptotic) behaviour characterises pullback attraction to a particular point in the original dynamical system. This has the advantage in that certain conventional techniques for a forward analysis may be applied. The remainder of the thesis, Chapters 4, 6 and Section 7.3, investigates the effects of perturbations and discretisations on non-autonomous dynamical systems known to possess structures that exhibit some form of stability or attraction. Chapter 4 investigates autonomous systems with semi-group attractors, that have been non-autonomously perturbed, whilst Chapter 6 observes the effects of discretisation on non-autonomous dynamical systems that exhibit properties of forward asymptotic stability. Chapter 7 explores the same problem of discretisation, but for pullback asymptotically stable systems. The theory of Loci Dynamics is used to analyse the nature of the discretisation, but establishment of results directly analogous to those discovered in Chapter 6 is shown to be unachievable. Instead a case by case analysis is provided for specific classes of dynamical systems, for which the results generate a numerical approximation of the pullback attraction in the original continuous dynamical system. The nature of the results regarding discretisation provide a non-autonomous extension to the work initiated by A. Stuart and J. Humphries [34, 35] for the numerical approximation of semi-group attractors within autonomous systems. . Of particular importance is the effect on the system's asymptotic behaviour over non-finite intervals of discretisation.

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<p>Recent experiment in our laboratory demonstrates that an increase in slag foamingwith carbon injection rate is limited by slag volume. The current work has identified arelationship between foam height, carbon injection rate and slag volumes, whichpredicts the critical injection rate above which foaming become inefficient. Theprediction of critical injection rate employs an extension of understanding mechanismof bubble movement in the foam by estimating average/steady-state bubble size andwall thickness. The carbon gasification model developed in our laboratory by King etal., which has been extended to include greater consideration of gas bubble burstingwhen to predict bubble size, and further improvement for calculating how fast bubblecan burst instantaneously in carbon-gas-slag halo system, has found that has importantinfluence on the predicting foaming parameters in King’s model, which is crucial taskfor continuous development in future.</p> / Master of Applied Science (MASc)