A stochastic-delay differential equation (SDDE) model of a small neural network with recurrent inhibition is presented and analyzed. The model exhibits unexpected transient behavior: oscillations that occur at the boundary of the basins of attraction when the system is bistable. These are known as delay-induced transitory oscillations (DITOs). This behavior is analyzed in the context of stochastic resonance, an unintuitive, though widely researched phenomenon in physical bistable systems where noise can play in constructive role in strengthening an input signal. A method for modeling the dynamics using a probabilistic three-state model is proposed, and supported with numerical evidence. The potential implications of this dynamical phenomenon to nocturnal frontal lobe epilepsy (NFLE) are also discussed.

The advances in computational and experimental modeling in the area of structural mechanics have stimulated research in a class of hybrid problems that require both of these modeling capabilities to be combined to achieve certain objectives. Real-time substructure (RTS) testing, structural system identification (SSI) and active control techniques fall in the category of hybrid problems that need efficient tools in both computational and experimental phases for their successful implementation. RTS is a hybrid testing method, which aims to overcome the scaling problems associated with the conventional dynamic testing methods (such as shake table test, effective force test and pseudo dynamic test) by testing the critical part of the structure experimentally with minimum compromise on spatio-temporal scaling, while modeling the remaining part numerically. The problem of SSI constitutes an important component within the broader framework of problems of structural health monitoring where, based on the in-situ measurements on the loading and a subset of critical responses of the structure, the system parameters are estimated with a view to detecting damage/degradation. Active control techniques are employed to maintain the functionality of important structures, especially under extreme dynamic loading. The work reported in the present thesis contributes to the areas of RTS, SSI and active control of nonlinear systems, the main focus being the computational aspects, i.e., in developing numerical strategies to address some of the unsolved issues, although limited efforts have also been made to undertake laboratory experimental investigations in the area of nonlinear SSI. The thesis is organized into seven chapters and five appendices. The first chapter contains an overview of the state of the art techniques in dynamic testing, SSI and structural control. The topics covered include effective force test, pseudo dynamic test, RTS test, time and frequency domain methods of nonlinear system identification, dynamic state estimation techniques with emphasis on particle filters, Rao-Blackwellization, structural control methods, control algorithms and active control of nonlinear systems. The review identifies a set of open problems that are subsequently addressed, to an extent, in the thesis. Chapter 2 focuses on the development of a time domain coupling technique, involving combined computational and experimental modeling, for vibration analysis of structures built-up of linear/nonlinear substructures. The numerical and experimental substructures are allowed to interact in real-time. The equation of motion of the numerical substructure is integrated using a step-by-step procedure that is formulated in the state space. For systems with nonlinear substructures, a multi-step transversal linearization method is used to integrate the equations of motion; and, a multi-step extrapolation scheme, based on the reproducing kernel particle method, is employed to handle the time delays that arise while accounting for the interaction between the substructures. Numerical illustrations on a few low dimensional vibrating structures are presented and these examples are fashioned after problems of seismic qualification testing of engineering structures using RTS testing techniques. The concept of substructuring is extended in Chapter 3 for implementing Rao-Blackwellization, a technique of combining particle filters with analytical computation through Kalman filters, for state and parameter estimations of a class of nonlinear dynamical systems with additive Gaussian process/observation noises. The strategy is based on decomposing the system to be estimated into mutually coupled linear and nonlinear substructures and then putting in place a rational framework to account for coupling between the substructures. While particle filters are applied to the nonlinear substructures, estimation of linear substructures proceeds using a bank of Kalman filters. Numerical illustrations for state/parameter estimations of a few linear and nonlinear oscillators with noise in both the process and measurements are provided to demonstrate the potential of the Rao-Blackwellized particle filter (RBPF) with substructuring. In Chapter 4, the concept of Rao-Blackwellization is extended to handle more general nonlinear systems, using two different schemes of linearization. A semi-analytical filter and a conditionally linearized filter, within the framework of Monte Carlo simulations, are proposed for state and parameter estimations of nonlinear dynamical systems with additively Gaussian process/observation noises. The first filter uses a local linearization of the nonlinear drift fields in the process/observation equations based on explicit Ito-Taylor expansions to transform the given nonlinear system into a family of locally linearized systems. Using the most recent observation, conditionally Gaussian posterior density functions of the linearized systems are analytically obtained through the Kalman filter. In the second filter, the marginalized posterior distribution of an appropriately chosen subset of the state vector is obtained using a particle filter. Samples of these marginalized states are then used to construct a family of conditionally linearized system of equations to obtain the posterior distribution of the states using a bank of Kalman filters. The potential of the proposed filters in state/parameter estimations is demonstrated through numerical illustrations on a few nonlinear oscillators. The problem of active control of nonlinear structural dynamical systems, in the presence of both process and measurement noises, is considered in Chapter 5. The focus of the study is on the exploitability of particle filters for state estimation in feedback control algorithms for nonlinear structures, when a limited number of noisy output measurements are available. The control design is done using the state dependent Riccati equation (SDRE) method. The Bayesian bootstrap filter and another based on sequential importance sampling are employed for state estimation. Numerical illustrations are provided for a few typically nonlinear oscillators of interest in structural engineering. The experimental validation of the RBPF using substructuring (developed in Chapter 3) and the conditionally linearized Monte Carlo filter (developed in Chapter 4), for parameter estimation, is reported in Chapter 6. Measured data available through laboratory experiments on simple building frame models subjected to harmonic base motions is processed using the proposed algorithms to identify the unknown parameters of the model. A brief summary of the contributions made in this thesis, together with a few suggestions for future research, are presented in Chapter 7. Appendix A provides an account of the multi-step transversal linearization method. The derivation of the reproducing kernel shape functions are presented in Appendix B. Appendix C provides the details of the stochastic Taylor expansion and derivation of the covariance structure of Gaussian MSI-s. The performance of a particle filtering algorithm (bootstrap filter) and Kalman filter in the state estimation of a linear system is provided in Appendix D and Appendix E contains the theoretical details of the Rao-Blackwellized particle filter.

The mathematical analysis of dynamical systems covers a wide range of challenging problems related to the time evolution, transient and asymptotic behavior, or regulation and control of physical systems. A large part of my work has been motivated by new mathematical questions arising from biological systems, especially signaling and genetic regulatory networks, where the classical methods usually don't directly apply. Problems include parameter estimation, robustness of the system, model reduction, or model assembly from smaller modules, or control of a system towards a desired state. Although many different formalisms and methodologies can be used to study these problems, in the past decade my work has focused on discrete and hybrid modeling frameworks with the goal of developing intuitive, computationally amenable, and mathematically rigorous, methods of analysis. Discrete (and, in particular, Boolean) models involve a high degree of abstraction and provide a qualitative description of the systems' dynamics. Such models are often suitable to represent the known interactions in gene regulatory networks and their advantage is that a large range of theoretical analysis tools are available using, for instance, graph theoretical concepts. Hybrid (piecewise affine) models have discontinuous vector fields but provide a continuous and more quantitative description of the dynamics. These systems can be analytically studied in each region of an appropriate partition of the state space, and the full solution given as a concatenation of the solutions in each region. Here, I will introduce the two formalisms and then, using several examples, illustrate how a combination of different formalisms permits comparison of results, as well as gaining quantitative knowledge and predictive power on a biological system, through the use of complementary mathematical methods.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2008-02-26Bitstream added on 2014-06-13T18:30:53Z : No. of bitstreams: 1 pena_is_me_sjrp.pdf: 488383 bytes, checksum: 40a97f3540caa6b8f6f2691c3a402579 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Sistemas dinâmicos híbridos se diferenciam por exibir simultaneamente variados tipos de comportamento dinâmico (contínuo, discreto, eventos discretos) em diferentes partes do sistema. Neste trabalho foram estudados resultados de estabilidade no sentido de Lyapunov para sistemas dinâmicos híbridos gerais, que utilizam uma noção de tempo generalizado, definido em um espaço métrico totalmente ordenado. Mostrou-se que estes sistemas podem ser imersos em sistemas dinâmicos descontínuos definidos em R+, de forma que sejam preservadas suas propriedades qualitativas. Como foco principal, estudou-se resultados de estabilidade para sistemas dinâmicos descontínuos modelados por semigrupos de operadores, em que os estados do sistema pertencem à espaços de Banach. Neste caso, de forma alternativa à teoria clássica de estabilidade, os resultados não utilizam as usuais funções de Lyapunov, sendo portanto mais fáceis de se aplicar, tendo em vista a dificuldade em se encontrar tais funções para muitos sistemas. Além disso, os resultados foram aplicados à uma classe de equações diferenciais com retardo. / Hybrid dynamical systems are characterized for showing simultaneously a variety of dynamic behaviors (continuous, discrete, discrete events) in different parts of the System. This work discusses stability results in the Lyapunov sense for general hybrid dynamical systems that use a generalized notion of time, defined in a completely ordered metric space. It has been shown that these systems may be immersed in discontinuous dynamical systems defined in R+, so that their quality properties are preserved. As the main focus, it is studied stability results for discontinuous dynamical systems modeled by semigroup operators, in which the states belong to Banach spaces. In this case, an alternative to the classical theory of stability, the results do not make use of the usual Lyapunov functions, and therefore are easier to apply, in view of the difficulty in finding such functions for many systems. Furthermore, the results were applied to a class of time-delay discontinuous differential equations.

Orientador: Masayoshi Tsuchida / Banca: José Manoel Balthazar / Banca: Waldemar Donizete Bastos / Resumo: Nesta dissertação vamos fazer uma análise do sistema pêndulo eletromecânico utilizando a teoria de perturbações através dos métodos da média e múltiplas escalas. Nosso objetivo é obter soluções analáticas aproximadas para o sistema e fazer análise dos casos de ressonâncias internas, alám de estudar a estabilidade dos estados estacionários. O sistema pêndulo eletromecânico tem uma dinâmica muito rica, pois apresenta curvas características dos efeitos de histerese, fenômenos de saltos nas amplitudes dos movimentos realizáveis, curvas com características mole e dura ("softening e hardening") e, além disso, diversas ressonâncias internas. Devido a complexidade das equações do sistema pêndulo eletromecânico, elas são difíceis de serem tratadas analíticamente, já que existe iteração ressonante entre as três partes (bloco, motor e pêndulo), e não podemos restringir o estudo das interações ressonantes à apenas duas partes e desprezar a outra parte. Neste trabalho analisamos o caso em que existe interação ressonante entre o bloco e o motor, mas sem interação ressonante com o pêndulo, mas, no entanto, sem desprezar os efeitos do movimento do pêndulo. Em seguida, discutimos a possibilidade de efeitos de saltos nas amplitudes dos movimentos realizáveis, apresentamos alguns pontos onde o sistema perde a estabilidade, já que a discuss~ao sobre comportamento geral do sistema érestrito a variedade central, e analisamos a estabilidade dos pontos fixos tomando como exemplo o estudo feito por Kononenko. A estabilidade dos pontos fixos do sistema é feita pela utilização do critério R-H, juntamente com a teoria da variedade central já que, no caso analisado, existe auto valor zero / Abstract: In this work we study the eletromechanical pendulum system with pertubation theory. We use the average method and the multiple scales method to get a approximate analytic solution for the problem, and analyse the various internal resonances and the stationary states stability. The eletromechanical pendulum is a very complex dynamical system and it shows very interesting e®ects such as histeresis, jump phenomenon, curves of hardening and softening type and, also, various kinds of internal resonances. The equations of the system are very complicated and so they are very hardy to study in an analytic way, because there is resonant interaction between the three components parts of the system and we can't restrict our study to interactions of just two parts of the system. In this work we analyse the case of resonant interaction between the block and the load without resonant interaction between the block and pendulum, but taking in account the pendulum e®ects. We treat the possibility of jump phenomenon, some points where the system loose stability are localized, and we analyse the stability of the stationary states as done by Kononenko. The analysis of stability of the stationary states is done by Routh-Hurwtiz criterion(R- H criterion) and center manifold, because the Jacobian matrix has an eigenvalue with zero real part / Mestre

Resumo: Sistemas dinâmicos híbridos se diferenciam por exibir simultaneamente variados tipos de comportamento dinâmico (contínuo, discreto, eventos discretos) em diferentes partes do sistema. Neste trabalho foram estudados resultados de estabilidade no sentido de Lyapunov para sistemas dinâmicos híbridos gerais, que utilizam uma noção de tempo generalizado, definido em um espaço métrico totalmente ordenado. Mostrou-se que estes sistemas podem ser imersos em sistemas dinâmicos descontínuos definidos em R+, de forma que sejam preservadas suas propriedades qualitativas. Como foco principal, estudou-se resultados de estabilidade para sistemas dinâmicos descontínuos modelados por semigrupos de operadores, em que os estados do sistema pertencem à espaços de Banach. Neste caso, de forma alternativa à teoria clássica de estabilidade, os resultados não utilizam as usuais funções de Lyapunov, sendo portanto mais fáceis de se aplicar, tendo em vista a dificuldade em se encontrar tais funções para muitos sistemas. Além disso, os resultados foram aplicados à uma classe de equações diferenciais com retardo. / Abstract: Hybrid dynamical systems are characterized for showing simultaneously a variety of dynamic behaviors (continuous, discrete, discrete events) in different parts of the System. This work discusses stability results in the Lyapunov sense for general hybrid dynamical systems that use a generalized notion of time, defined in a completely ordered metric space. It has been shown that these systems may be immersed in discontinuous dynamical systems defined in R+, so that their quality properties are preserved. As the main focus, it is studied stability results for discontinuous dynamical systems modeled by semigroup operators, in which the states belong to Banach spaces. In this case, an alternative to the classical theory of stability, the results do not make use of the usual Lyapunov functions, and therefore are easier to apply, in view of the difficulty in finding such functions for many systems. Furthermore, the results were applied to a class of time-delay discontinuous differential equations. / Orientador: Geraldo Nunes Silva / Coorientador: Luís Antônio Fernandes de Oliveira / Banca: Carlos Alberto Raposo da Cunha / Banca: Waldemar Donizete Bastos / Mestre

The use of Hilbert space theory became an important tool for ergodic theoreticians ever since John von Neumann proved the fundamental Mean Ergodic theorem in Hilbert space. Recurrence is one of the corner stones in the study of dynamical systems. In this dissertation some extended ideas besides those of the basic, well-known recurrence results are investigated. Hilbert space theory proves to be a very useful approach towards the solution of multiple recurrence problems in ergodic theory. Another very important use of Hilbert space theory became evident only relatively recently, when it was realized that non-commutative dynamical systems become accessible to the ergodic theorist through the important Gelfand-Naimark-Segal (GNS) representation of C*-algebras as Hilbert spaces. Through this construction we are enabled to invoke the rich catalogue of Hilbert space ergodic results to approach the more general, and usually more involved, non-commutative extensions of classical ergodic-theoretical results. In order to make this text self-contained, the basic, standard, ergodic-theoretical results are included in this text. In many instances Hilbert space counterparts of these basic results are also stated and proved. Chapters 1 and 2 are devoted to the introduction of these basic ergodic-theoretical results such as an introduction to the idea of measure-theoretic dynamical systems, citing some basic examples, Poincairé’s recurrence, the ergodic theorems of Von Neumann and Birkhoff, ergodicity, mixing and weakly mixing. In Chapter 2 several rudimentary results, which are the basic tools used in proofs, are also given. In Chapter 3 we show how a Hilbert space result, i.e. a variant of a result by Van der Corput for uniformly distributed sequences modulo 1, is used to simplify the proofs of some multiple recurrence problems. First we use it to simplify and clarify the proof of a multiple recurrence result by Furstenberg, and also to extend that result to a more general case, using the same Van der Corput lemma. This may be considered the main result of this thesis, since it supplies an original proof of this result. The Van der Corput lemma helps to simplify many of the tedious terms that are found in Furstenberg’s proof. In Chapter 4 we list and discuss a few important results where classical (commutative) ergodic results were extended to the non-commutative case. As stated before, these extensions are mainly due to the accessibility of Hilbert space theory through the GNS construction. The main result in this section is a result proved by Niculescu, Ströh and Zsidó, which is proved here using a similar Van der Corput lemma as in the commutative case. Although we prove a special case of the theorem by Niculescu, Ströh and Zsidó, the same method (Van der Corput) can be used to prove the generalized result. Copyright 2004, 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: Beters, FJC 2004, A Hilbert space approach to multiple recurrence in ergodic theory, MSc dissertation, University of Pretoria, Pretoria, viewed yymmdd < http://upetd.up.ac.za/thesis/available/etd-02222006-104936 / > / Dissertation (MSc (Applied Mathematics))--University of Pretoria, 2007. / Mathematics and Applied Mathematics / unrestricted

Le travail qui suit comporte quatre chapitres : le premier est centré autour de la propriété de mélange faible pour les échanges d'intervalles et flots de translation. On y présente des résultats obtenus avec Artur Avila qui renforcent des résultats précédents dus à Artur Avila et Giovanni Forni. Le deuxième chapitre est consacré à un travail en commun avec Zhiyuan Zhang et concerne les propriétés d'ergodicité et d'accessibilité stables pour des systèmes partiellement hyperboliques de dimension centrale au moins égale à deux. On montre que sous des hypothèses de cohérence dynamique, center bunching et pincement fort, la propriété d'accessibilité stable est dense en topologie C^r, r>1, et même prévalente au sens de Kolmogorov. Dans le troisième chapitre, on expose les résultats d'un travail réalisé en collaboration avec Julie Déserti, consacré à l'étude d'une famille à un paramètre d'automorphismes polynomiaux de C^3 ; on montre que de nouveaux phénomènes apparaissent par rapport à ce qui était connu dans le cas de la dimension deux. En particulier, on étudie les vitesses d'échappement à l'infini, en montrant qu'une transition s'opère pour une certaine valeur du paramètre. Le dernier chapitre est issu d'un travail en collaboration avec Jiangong You, Zhiyan Zhao et Qi Zhou ; on s'intéresse à des estimées asymptotiques sur la taille des trous spectraux des opérateurs de Schrödinger quasi-périodiques dans le cadre analytique. On obtient des bornes supérieures exponentielles dans le régime sous-critique, ce qui renforce un résultat précédent de Sana Ben Hadj Amor. Dans le cas particulier des opérateurs presque Mathieu, on montre également des bornes inférieures exponentielles, qui donnent des estimées quantitatives en lien avec le problème dit "des dix Martinis". Comme conséquences de nos résultats, on présente des applications à l'homogénéité du spectre de tels opérateurs ainsi qu'à la conjecture de Deift. / The following work contains four chapters: the first one is centered around the weak mixing property for interval exchange transformations and translation flows. It is based on the results obtained together with Artur Avila which strengthen previous results due to Artur Avila and Giovanni Forni. The second chapter is dedicated to a joint work with Zhiyuan Zhang, in which we study the properties of stable ergodicity and accessibility for partially hyperbolic systems with center dimension at least two. We show that for dynamically coherent partially hyperbolic diffeomorphisms and under certain assumptions of center bunching and strong pinching, the property of stable accessibility is dense in C^r topology, r>1, and even prevalent in the sense of Kolmogorov. In the third chapter, we explain the results obtained together with Julie Déserti on the properties of a one-parameter family of polynomial automorphisms of C^3; we show that new behaviours can be observed in comparison with the two-dimensional case. In particular, we study the escape speed of points to infinity and show that a transition exists for a certain value of the parameter. The last chapter is based on a joint work with Jiangong You, Zhiyan Zhao and Qi Zhou; we get asymptotic estimates on the size of spectral gaps for quasi-periodic Schrödinger operators in the analytic case. We obtain exponential upper bounds in the subcritical regime, which strengthens a previous result due to Sana Ben Hadj Amor. In the particular case of almost Mathieu operators, we also show exponential lower bounds, which provides quantitative estimates in connection with the so-called "Dry ten Martinis problem". As consequences of our results, we show applications to the homogeneity of the spectrum of such operators, and to Deift's conjecture.

In this paper we study local rigidity of actions of simply connected Lie groups. In particular, we apply the Nash-Moser inverse function theorem to give sufficient conditions for the action of a simply connected Lie group to be locally rigid. Let $G$ be a Lie group, $H &lt; G$ a simply connected subgroup and $\Gamma &lt; G$ a cocompact lattice. We apply the result for general actions of simply connected groups to obtain sufficient conditions for the action of $H$ on $\Gamma\backslash G$ by right translations to be locally rigid. We also discuss some possible applications of this sufficient condition / I den här texten så studerar vi lokal rigiditet av gruppverkan av enkelt sammanhängande Liegrupper. Mer specifikt, vi applicerar Nash-Mosers inversa funktionssats för att ge tillräckliga villkor för att en gruppverkan av en enkelt sammanhängande grupp ska vara lokalt rigid. Låt $G$ vara en Lie grupp, $H &lt; G$ en enkelt sammanhängande delgrupp och $\Gamma &lt; G$ ett kokompakt gitter. Vi applicerar resultatet för generella gruppverkan av enkelt sammanhängande grupper för att få tillräckliga villkor för att verkan av $H$ på $\Gamma\backslash G$ med translationer ska vara lokalt rigid. Vi diskuterar också några möjliga tillämpningar av det tillräckliga villkoret.

This dissertation presents a Bayesian generative modeling approach for complex dynamical systems for emotion-interaction patterns within multivariate data collected in social psychology studies. While dynamical models have been used by social psychologists to study complex psychological and behavior patterns in recent years, most of these studies have been limited by using regression methods to fit the model parameters from noisy observations. These regression methods mostly rely on the estimates of the derivatives from the noisy observation, thus easily result in overfitting and fail to predict future outcomes. A Bayesian generative model solves the problem by integrating the prior knowledge of where the data comes from with the observed data through posterior distributions. It allows the development of theoretical ideas and mathematical models to be independent of the inference concerns. Besides, Bayesian generative statistical modeling allows evaluation of the model based on its predictive power instead of the model residual error reduction in regression methods to prevent overfitting in social psychology data analysis. In the proposed Bayesian generative modeling approach, this dissertation uses the State Space Model (SSM) to model the dynamics of emotion interactions. Specifically, it tests the approach in a class of psychological models aimed at explaining the emotional dynamics of interacting couples in committed relationships. The latent states of the SSM are composed of continuous real numbers that represent the level of the true emotional states of both partners. One can obtain the latent states at all subsequent time points by evolving a differential equation (typically a coupled linear oscillator (CLO)) forward in time with some known initial state at the starting time. The multivariate observed states include self-reported emotional experiences and physiological measurements of both partners during the interactions. To test whether well-being factors, such as body weight, can help to predict emotion-interaction patterns, we construct functions that determine the prior distributions of the CLO parameters of individual couples based on existing emotion theories. Besides, we allow a single latent state to generate multivariate observations and learn the group-shared coefficients that specify the relationship between the latent states and the multivariate observations. Furthermore, we model the nonlinearity of the emotional interaction by allowing smooth changes (drift) in the model parameters. By restricting the stochasticity to the parameter level, the proposed approach models the dynamics in longer periods of social interactions assuming that the interaction dynamics slowly and smoothly vary over time. The proposed approach achieves this by applying Gaussian Process (GP) priors with smooth covariance functions to the CLO parameters. Also, we propose to model the emotion regulation patterns as clusters of the dynamical parameters. To infer the parameters of the proposed Bayesian generative model from noisy experimental data, we develop a Gibbs sampler to learn the parameters of the patterns using a set of training couples. To evaluate the fitted model, we develop a multi-level cross-validation procedure for learning the group-shared parameters and distributions from training data and testing the learned models on held-out testing data. During testing, we use the learned shared model parameters to fit the individual CLO parameters to the first 80% of the time points of the testing data by Monte Carlo sampling and then predict the states of the last 20% of the time points. By evaluating models with cross-validation, one can estimate whether complex models are overfitted to noisy observations and fail to generalize to unseen data. I test our approach on both synthetic data that was generated by the generative model and real data that was collected in multiple social psychology experiments. The proposed approach has the potential to model other complex behavior since the generative model is not restricted to the forms of the underlying dynamics.