Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 2000. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: Philip J. Morrison. Vita. Includes bibliographical references.

Orientador: Kyoko Furuya / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin / Made available in DSpace on 2018-08-12T21:50:29Z (GMT). No. of bitstreams: 1 Netto_DomeniqueVelloso_M.pdf: 11311141 bytes, checksum: 3ff0f796112fbf62feb5f0de7b56c879 (MD5) Previous issue date: 2009 / Resumo: Neste trabalho realizamos para o modelo de Jahn-Teller E Ä b, que descreve a interação de um qubit com um modo de um oscilador, um estudo do emaranhamento e o relacionamos com a bifurcação do ponto fixo do análogo clássico do modelo. Estudamos também a incidência de caos e sua variação em função dos parâmetros do sistema. Confirmamos os indícios de que no limite massivo do oscilador (m ® ¥ ) o emaranhamento do estado fundamental do sistema como função do seu parâmetro crítico apresenta uma mudançã não suave no seu comportamento quando esse parâmetro atinge o valor crítico. Confirmamos também que para esse mesmo valor crítico o análogo clássico apresenta uma bifurcação de alguns de seus pontos fixos. Sobre a possível influência do caos no emaranhamento, pudemos observar que o aparecimento de região caótica substancial no modelo clássico ocorre quando o parâmetro crítico está nas vizinhanças do valor crítico. A análise da função de Husimi no estado fundamental nos permitiu estabelecer dois aspectos: (i) uma conexão entre a localização do máximo desta função e a localização dos pontos fixos no espaço de fase, comprovando a ocorrência da bifurcaçãco no estado quãntico nesta representação; (ii) verificar que de fato há uma aproximação do limite clássico do oscilador quando aumentamos o parâmetro do modelo associado à quebra de degenerescência das energias do sistema. / Abstract: In this work we realize a study of entanglement for the Jahn-Teller E Ä b model, wich describes the interaction of a qubit with an oscillator mode, and relate it with the bifurcation of the fixed point of the classical analogue of the model. We also study qualitatively the ocurrence of chaos and its variation as a function of the parameters of the system. We have confirmed the signs that in the massive limit of the oscillator ( m ® ¥ ) the entanglement of the fundamental state of the system presents a non-smooth change (as a function of its critical parameter) in its behavior as this parameter attains the critical value. We also confirm that at this same critical value the classical analogue presents a bifurcation of some of its fixed points. Concerning the possible influence of chaos on the entanglement, we have been able to see the appearance of a substantial chaotic region in the classical phase space when the critical parameter is in the vici-nity of its critical value. The analysis of the Husimi function in the fundamental state allowed us to establish two things: (i) a connection between the localization of the maximum of this function and the localization of the fixed points in the phase space, confirming the occurrence of the bifurcation in the quantum state in this representation; (ii) verification that in fact the classical limit of the oscillator is approached when we increase the energy degeneracy breaking parameter of the system. / Mestrado / Física / Mestre em Física

In this thesis, we apply a recently developed technique to comprehensively categorize all possible families of solitary wave solutions in two models of topical interest. The models considered are: a) the Generalized Pochhammer-Chree Equations, which govern the propagation of longitudinal waves in elastic rods, and b) a generalized microstructure PDE. Limited analytic results exist for the occurrence of one family of solitary wave solutions for each of these equations. Since, as mentioned above, solitary wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions of both models here (via the normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves for each model, we find a continuum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. For the microstructure equation, the new family of solutions occur in regions of parameter space distinct from the known solitary wave solutions and are thus entirely new. Directions for future work, including the dynamics of each family of solitary waves using exponential asymptotics techniques, are also mentioned.

Rich dynamics have been demonstrated when a discrete time delay is introduced in a simple predator-prey system. For example, Hopf bifurcations and a sequence of period doubling bifurcations that appear to lead to chaotic dynamics have been observed. In this thesis we consider two different predator-prey models: the classical Gause-type predator-prey model and the chemostat predator-prey model. In both cases, we explore how different ways of modeling the time between the first contact of the predator with the prey and its eventual conversion to predator biomass affects the possible range of dynamics predicted by the models. The models we explore are systems of integro-differential equations with delay kernels from various distributions including the gamma distribution of different orders, the uniform distribution, and the Dirac delta distribution. We study the models using bifurcation theory taking the mean delay as the main bifurcation parameter. We use both an analytical approach and a computational approach using the numerical continuation software XPPAUT and DDE-BIFTOOL. First, general results common to all the models are established. Then, the differences due to the selection of particular delay kernels are considered. In particular, the differences in regions of stability of the coexistence equilibrium are investigated. Finally, the effects on the predicted range of dynamics between the classical Gause-type and the chemostat predator-prey models are compared. / Thesis / Doctor of Philosophy (PhD)

Circadian rhythms in mammals are self-sustained repeating activities driven by the circadian gene expression in cells, which is regulated at both transcriptional and posttranscriptional stages. In this work, we first used mathematical modeling to investigate the transcriptional regulation of circadian gene expression, with a focus on the mechanisms of robust genetic oscillations in the mammalian circadian core clock. Secondly, we built a coarse-grained model to study the post-transcriptional regulation of the rhythmicities of poly(A) tail length observed in hundreds of mRNAs in mouse liver. Lastly, we examined the application of Sobol indices, which is a global sensitivity analysis method, to mathematical models of biological oscillation systems, and proposed two methods tailored for the calculation of circular Sobol indices. In the first project, we modified the core negative feedback loop in a mathematical model of the mammalian genetic oscillator so that the unrealistic tight binding between the repressor PER and the activator BMAL1 is relaxed for robust oscillations. By studying the modified extended models, we found that the auxiliary positive feedback loop, rather than the auxiliary negative feedback loop, makes the oscillations more robust, yet they are similar when accounting for circadian rhythms (~24h period). In the second project, we investigated the regulation of rhythmicities in poly(A) tail length by four coupled rhythmic processes, which are transcription, deadenylation, polyadenylation, and degradation. We found that rhythmic deadenylation is the strongest contributor to the rhythmicity in poly(A) tail length and the rhythmicity in the abundance of the mRNA subpopulation with long poly(A) tails. In line with this finding, the model further showed that the experimentally observed distinct peak phases in the expression of deadenylases, regardless of other rhythmic controls, can robustly cluster the rhythmic mRNAs by their peak phases in poly(A) tail length and abundance of the long-tailed subpopulation. In the last project, we reviewed the theoretical basis of Sobol indices and identified potential problems when it is applied to mathematical models of biological oscillation systems. Based on circular statistics, we proposed two methods for the calculation of circular Sobol indices and compared their performance with the original Sobol indices in several models. We found that though the relative rankings of the contribution from parameters are the same across three methods, circular Sobol indices can better quantitatively distinguish the contribution of individual parameters. Through this work, we showed that mathematical modeling combined with sensitivity analysis can help us understand the mechanisms underlying the circadian gene expression in mammalian cells. Also, testable predictions are made for future experiments and new ideas are provided that can enable potential chronopharmacology research. / Doctor of Philosophy / Circadian rhythms are repeating biological activities with ~24h period observed in most living organisms. Disruption of circadian rhythms in humans has been found to be promote cancer, metabolic diseases, cognitive degeneration etc. In this work, we first used mathematical modeling to study the mechanisms of robust oscillations in the mammalian circadian core clock, which is a molecular regulatory network that drives circadian gene expression at transcriptional stage. Secondly, we built a coarse-grained model to investigate the post-transcriptional regulation of the rhythmicities in poly(A) tail length, which are observed in hundreds of mRNAs in mouse liver. Lastly, we examined the application of Sobol indices, which is a global sensitivity analysis method, to mathematical models of biological oscillation systems, and proposed two methods tailored for the calculation of circular Sobol indices. In the first project, we modified a previous mathematical model of the mammalian genetic oscillator so that it sustains robust oscillation with more realistic parameter values. Our analysis of the model further showed that the auxiliary positive feedback loop, rather than the auxiliary negative feedback loop, makes the oscillations more robust. In the second project, we found that rhythmic deadenylation, among the coupled transcription, polyadenylation, and degradation processes, mostly controls the rhythmicity of poly(A) tail length and mRNA subpopulation with long poly(A) tails. Lastly, we reviewed the theoretical basis of Sobol indices and found potential problems when it is applied to mathematical models of biological oscillation systems. Based on circular statistics, we proposed two circular Sobol indices, which can better distinguish the contribution of individual parameters to model outputs than the original Sobol indices. Altogether, we used mathematical modeling and sensitivity analysis to investigate the regulation of circadian gene expression in mammalian cells, providing testable predictions and new ideas for future experiments and chronopharmacology research.

The work we present in this thesis is a series of studies of how heterogeneities in coupling affect the synchronization of coupled neural oscillators. We begin by examining how heterogeneity in coupling strength affects the equilibrium phase difference of a pair of coupled, spiking neurons when compared to the case of identical coupling. This study is performed using pairs of Hodgkin-Huxley and Wang-Buzsaki neurons. We find that heterogeneity in coupling strength breaks the symmetry of the bifurcation diagrams of equilibrium phase difference versus the synaptic rate constant for weakly coupled pairs of neurons. We observe important qualitative changes such as the loss of the ubiquitous in-phase and anti-phase solutions found when the coupling is identical and regions of parameter space where no phase locked solution exists. Another type of heterogeneity can be found by having different types of coupling between oscillators. Synaptic coupling between neurons can either be exciting or inhibiting. We examine the synchronization dynamics when a pair of neurons is coupled with one excitatory and one inhibitory synapse. We also use coupled pairs of Hodgkin-Huxley neurons and Wang-Buzsaki neurons for this work. We then explore the existance of 1:n coupled states for a coupled pair of theta neurons. We do this in order to reproduce an observed effect called quantal slowing. Quantal slowing is the phenomena where jumping between different $1:n$ coupled states is observed instead of gradual changes in period as a parameter in the system is varied. All of these topics fall under the general heading of coupled, non-linear oscillators and specifically weakly coupled, neural oscillators. The audience for this thesis is most likely going to be a mixed crowd as the research reported herein is interdisciplinary. Choosing the content for the introduction proved far more challenging than expected. It might be impossible to write a maximally useful introductory portion of a thesis when it could be read by a physicist, mathematician, engineer or biologist. Undoubtedly readers will find some portion of this introduction elementary. At the risk of boring some or all of my readers we decided it was best to proceed so that enough of the mathematical (biological) background is explained in the introduction so that a biologist (mathematician) is able to appreciate the motivations for the research and the results presented. We begin with a introduction in nonlinear dynamics explaining the mathematical tools we use to characterize the excitability of individual neurons, as well as oscillations and synchrony in neural networks. The next part of the introductory material is an overview of the biology of neurons. We then describe the neuron models used in this work and finally describe the techniques we employ to study coupled neurons.

The results of an experimental study of phenomena which occur in the flow of a viscous fluid in closed domains with discrete symmetries are presented. The purpose is to investigate the role which ideas from low-dimensional dynamical systems have to play in describing qualitative changes that take place with variation of the governing parameters. Such a descriptive framework already exists for the case of the Taylor-Couette system, where the domain possesses a continuous azimuthal symmetry group. The present investigation is aimed at establishing the typicality of previously reported behaviour under progressive reductions of azimuthal symmetry. In the first investigations, the fixed outer circular cylinder of the standard system is replaced with one of square cross-section. Thus there is now discrete Ζ₄ symmetry in the azimuthal direction. Knowledge of the two-dimensional flow field is used to establish the nature of the steady three-dimensional motion equivalent to Taylor vortex flow. It is shown that similar bifurcation sequences exist in both standard and square systems for the case of very small aspect ratio where a single Taylor cell is formed. This flow develops as the result of a bifurcation which breaks the Ζ₂ symmetry that is imposed on the annulus by two solid stationary ends. The study is then extended to consider time-dependent effects in the square system. Two different oscillatory single-cell flows are identified, and it is shown that each is the result of a Hopf bifurcation. Selection of a particular dynamic mode is found to depend on the aspect ratio of the system. A low-dimensional bifurcation structure is uncovered which connects the two modes in parameter space, and involves a novel type of steady single-cell flow. Finally, observations are reported of a nontrivial type of dynamical behaviour which bears strong resemblance to motion found in a circularly symmetric Taylor-Couette system that is related to the Šilnikov mechanism for finite-dimensional chaos. A second variant on the Taylor-Couette system is considered where the outer cylinder is shaped like a stadium. The effect is to reduce further the overall symmetry of the domain to a Ζ₂ × Ζ₂ group. The two-dimensional flow field is investigated using both numerical and experimental techniques. Time-dependent phenomena are then investigated in the three-dimensional flow over a relatively wide range of aspect ratio. It is found that a sequence of a Hopf bifurcation followed by period-doubling bifurcations exists up to a certain aspect ratio, beyond which there is an apparently sudden and reversible transition between regular and irregular dynamical behaviour. Although this transition is not of a low-dimensional nature, the experimental results suggest that it exists as the result of a coalescence of the bifurcations which are found at lower values of aspect ratio.

Thesis (M.Sc. (Applied Mathematics))--Mahidol University, 1986.

Thesis (Ph. D.)--Ohio State University, 2005. / Title from first page of PDF file. Document formatted into pages; contains ix, 134 p.; also includes graphics. Includes bibliographical references (p. 130-134). Available online via OhioLINK's ETD Center