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

[pt] Este trabalho visa estudar a teoria das bifurcações associada a sistemas de equações diferenciais não lineares com aplicação aos estudos de estabilidade em sistemas elétricos de potência. Como aplicação desta teoria analisamos o problema de operação ótima em sistemas de potência. Novas restrições são derivadas desta teoria que, incluídas às restrições tradicionais de um fluxo de potência ótimo, garantem a estabilidade do ponto de operação obtido. / [en] This work aims at studying bufurcation theory associated to non-linear differential equations systems with application on stability of electrical power systems. As an application of this theory, the problem of optimal operation in power systems is analyzed. New restriction to optimum power flow are found and together with traditional ones, assure the stability of the operation point.

Made available in DSpace on 2014-06-11T19:22:32Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-08-28Bitstream added on 2014-06-13T19:28:04Z : No. of bitstreams: 1 batista_mf_me_ilha.pdf: 3972180 bytes, checksum: 619117071d2971bec37b4fbde27e7368 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Este trabalho apresenta um estudo sobre a relação entre os principais tipos de bifurcações que ocorrem em sistemas elétricos de potência e em quais ocasiões elas podem ocorrer em máquinas síncronas com ou sem RAT (Regulador Automático de Tensão). Para explorar tais fenômenos, primeiramente o sistema é modelado, sendo utilizado para o caso MBI (Máquina - Barramento In nito) o modelo um eixo e, então, a matriz de estado é calculada para a análise dos autovalores. Para os sistemas multimáquinas estudados, são incluídos dois enrolamentos amortecedores nos eixos d ¡ q. São então apresentados os métodos de análise de estabilidade transitória convencionais, amplamente utilizados, conhecidos como método Tradicional e Método Direto. As condições para a ocorrência de bifurcações são analisadas utilizando os coe - cientes linearizados do modelo de He ron-Phillips para o caso MBI, onde é mostrado que se espera perder a estabilidade para o sistema com regulador automático de tensão através de uma bifurcação de Hopf e para o caso sem RAT através de uma bifurcação Sela-Nó. Por m, é analisado o ciclo-limite para o caso de uma máquina - barramento in nito e para sistemas multimáquinas através do modelo não-linear. A região de estabilidade é analisada no plano de fase, sendo mostrada a necessidade de incluir a variação de uxo no enrolamento de campo para uma análise correta da estabilidade. É também mostrado que este ciclo-limite pode reduzir a fronteira de estabilidade calculada pelo método convencional. / The aim of this study is the relation among main types of bifurcations that occur in electrical power systems and the circumstances they can happen with the synchronous machines considered with or without AVR (Automatic Voltage Regulator). To explore such phenomena, the system is rst modeled with the synchronous machines described by the one axis model for the MIB (Machine - In nite Bus) case , and so the state matrix is computed for the analysis of its eigenvalues. For multimachine systems case two windings dampers are included in d-q axes. The conditions for the occurrence of bifurcations are analyzed using the coe cients of the He ron-Phillips model for MIB case, where it is shown that one expects the system with automatic voltage regulator lose synchronism through a Hopf bifurcation and for the case without RAT through a Saddle- Node Bifurcation. Finally, the nonlinear model is accounted for in order to consider the limit-cycle for the case of one machine - in nite bus case as well as for multimachine system. Since internal voltage a ects the boundary of the stability region it must be considered. Then the phase portrait does not su ce and the trajectories must to be observed in a sub space de ned with the internal voltage. It is also shown that this limit-cycle can reduce the boundary of stability calculated by means of the direct method.

Orientador: Marco Antonio Teixeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T09:35:17Z (GMT). No. of bitstreams: 1 AchireQuispe_JesusEnrique_D.pdf: 1199686 bytes, checksum: 2c263eb351ad3dfa30b13e0fecc5282b (MD5) Previous issue date: 2014 / Resumo: Recentemente, a Teoria de campos descontínuos (Non-Smooth Dynamic Systems) tem-se desenvolvido rapidamente, motivado principalmente pelas aplicações na física e nas engenharias, e também pela atraente beleza matemática. Neste trabalho, consideraremos campos de vetores suaves por partes, denominados campos de Filippov, e usamos o método convexo de Filippov para definir órbita solução deste tipo de campo. Assim, órbitas soluções passando por um ponto qualquer sempre existem. Há duas principais diferenças com o clássico caso diferenciável: a primeira é que as órbitas neste caso são curvas suaves por partes, enquanto que no caso diferenciável são curvas suaves. A segunda é que as órbitas soluções não tem a propriedade da unicidade, ou seja, podem existir duas ou mais órbitas passando pelo mesmo ponto. São esses fatos que fazem essa teoria um pouco diferente da teoria clássica de campos diferenciáveis. Estamos interessados em estudar qualitativamente os campos de Filippov, especialmente os que são genéricos e estruturalmente estáveis. Assim, nesta tese descrevemos propriedades genéricas necessárias para um campo de Filippov ser estruturalmente estável. Particularmente analisamos estabilidade estrutural local de singularidades tangenciais tais como o rabo de andorinha, a dobradobra,e dobra-cúspide, e adicionalmente pseudoequilíbrios e órbitas fechadas / Abstract: Recently, the Theory of Non-smooth Dynamic Systems has been developed, motivated mostly by their applications in physics and engineering, and also by its attractive mathematical beauty. In this work, we consider piecewise-smooth vector fields, called Filippov's vector fields, and we use the Filippov's convex method to define orbits solutions of this type of vector fields. Thus, orbit solution through any point always exists. But, there are two main differences with the classic differentiable case: the first is that orbits in this case are piecewise smooth curves while that in the differentiable case they are smooth curves. The second is that there is not uniqueness of solutions, this is, it may exist two or more than two orbits passing through a point. We are interested in to study qualitatively the Filippov's vector fields, especially those thatare generic and structurally stable. Thus, in this text we describe generic properties necessaryfor a vector field to be structurally stable. In particular, we analyze local structural stability attangential singularities, such as swallowtail-regular, fold-fold, fold-cusp, and additionally pseudoequilibriumsand closed orbits / Doutorado / Matematica / Doutor em Matemática