In this thesis we investigate cosmological models more general than the isotropic and homogeneous Friedmann-Lemaître models. We focus on cosmologies with one spatial degree of freedom, whose matter content consists of a perfect fluid and the cosmological constant. We formulate the Einstein field equations as a system of quasilinear first order partial differential equations, using scale-invariant variables. The primary goal is to study the dynamics in the two asymptotic regimes, i. e. near the initial singularity and at late times. We highlight the role of spatially homogeneous dynamics as the background dynamics, and analyze the inhomogeneous aspect of the dynamics. We perform a variety of numerical simulations to support our analysis and to explore new phenomena.

In this thesis we investigate cosmological models more general than the isotropic and homogeneous Friedmann-Lemaître models. We focus on cosmologies with one spatial degree of freedom, whose matter content consists of a perfect fluid and the cosmological constant. We formulate the Einstein field equations as a system of quasilinear first order partial differential equations, using scale-invariant variables. The primary goal is to study the dynamics in the two asymptotic regimes, i. e. near the initial singularity and at late times. We highlight the role of spatially homogeneous dynamics as the background dynamics, and analyze the inhomogeneous aspect of the dynamics. We perform a variety of numerical simulations to support our analysis and to explore new phenomena.

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

Thesis (Ph.D.)--University of Central Florida, 2009. / Adviser: Zhihua Qu. Includes bibliographical references (p. 138-143).

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This thesis consists of two parts. In the first part, which is expository, abstract theory of one-parameter operator is studied semi-groups. We develop in detail the necessary Banach space and Banach algebra theories of integration, differentiation, and series, and then give a careful rigorous proof of the exponential function characterization of continuous one-parameter operator semigroups. In the second part, which is applied and has new result, we discuss some related topics in dynamical systems. In general the linearizations give a reliable description of the non-linear orbits near the equilibrium points (the Hartman-Grobman theorem), thus illustrating the importance of linear semigroups. The aim of qualitative analysis of differential equations (DE) is to understand the qualitative behaviour (such as, for example, the long-term behaviour as $t\rightarrow \infty$) of typical solutions of the DE. The flow in the direction of increasing time defines a semigroup. As an application we study Einstein-Aether Cosmological models using dynamical systems theory.

The field of mathematical neuroscience is concerned with the modeling and interpretation of neuronal dynamics and associated phenomena. Neurons can be modeled individually, in small groups, or collectively as a large network. Mathematical models of single neurons typically involve either differential equations, discrete maps, or some combination of both. A number of two-dimensional spiking neuron models that combine continuous dynamics with an instantaneous reset have been introduced in the literature. The models are capable of reproducing a variety of experimentally observed spiking patterns, and also have the advantage of being mathematically tractable. Here an analysis of the transverse stability of orbits in the phase plane leads to sufficient conditions on the model parameters for regular spiking to occur. The application of this method is illustrated by three examples, taken from existing models in the neuroscience literature. In the first two examples the model has no equilibrium states, and regular spiking follows directly. In the third example there are equilibrium points, and some additional quantitative arguments are given to prove that regular spiking occurs. / Graduate

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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.

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