This thesis addresses the problem of inference in factor graphs, especially the LDPC codes, almost solved by message-passing algorithms. In particular, the Belief Propagation algorithm (BP) is investigated as a particular message-passing algorithm whose suboptimality is discussed in the case where the factor graph has a loop-like topology. From the equivalence between the BP and the Bethe approximation in statistical physics that is generalized to the region-based approximation, is detailed the Generalized Belief Propagation algorithm (GBP), a message-passing algorithm between clusters of the factor graph. It is experimentally shown to surpass the BP in the cases where the clustering deals with the harmful topological structures that prevents the BP from rightly decoding any LDPC code, namely the trapping sets. We do not only confront the BP and the GBP algorithms according to their performance from the point of view of the channel coding with the error-rate, but also according to their dynamical behaviors for non-trivial error-events for which both algorithms can exhibit chaotic beahviors. By means of classical and original dynamical quantifiers, it is shown that the GBP algorithm can overcome the BP algorithm.

Main reason of this study is to set forth the internal paradox of the basic approach of the artificial intelligence in the medical field to by discussing on the theoretical and application levels and to suggest solutions in theory and practice against that. In order to rule out the internal paradox in the medical decision support systematic, a new medical model is suggested and based on this, concepts such as disease, health, etiology, diagnosis and treatment are questioned. Meanwhile, with the current scientific data, a simple application sample based on how a decision making system which was set up by fuzzy logic and which is based on the perception of human as a complex adaptive system has been explained. Finally, results of the research about accuracy and validity of this application, current improvements based on the current model and the location on the artificial intelligence theory is discussed.

Many seemingly simple systems can display extraordinarily complex dynamics which has been studied and uncovered through nonlinear dynamical theory. The leitmotif of this thesis is changing phase-space structures and their (linear or nonlinear) stabilities by adding control functions (which act on the system as external perturbations) to the relevant Hamiltonians. These phase-space structures may be periodic orbits, invariant tori or their stable and unstable manifolds. One-electron systems and diatomic molecules are fundamental and important staging ground for new discoveries in nonlinear dynamics. In past years, increasing emphasis and effort has been put on the control or manipulation of these systems. Recent developments of nonlinear dynamical tools can provide efficient ways of doing so. In the first subtopic of the thesis, we are adding a control function to restore tori at prescribed locations in phase space. In the remainder of the thesis, a control function with parameters is used to change the linear stability of the periodic orbits which govern the processes in question. In this thesis, we report our theoretical analyses on multiphoton ionization of Rydberg atoms exposed to strong microwave fields and the dissociation of diatomic molecules exposed to bichromatic lasers using nonlinear dynamical tools. This thesis is composed of three subtopics. In the first subtopic, we employ local control theory to reduce the stochastic ionization of hydrogen atom in a strong microwave field by adding a relatively small control term to the original Hamiltonian. In the second subtopic, we perform periodic orbit analysis to investigate multiphoton ionization driven by a bichromatic microwave field. Our results show quantitative and qualitative agreement with previous studies, and hence identify the mechanism through which short periodic orbits organize the dynamics in multiphoton ionization. In addition, we achieve substantial time savings with this approach. In the third subtopic we extend our periodic orbit analysis to the dissociation of diatomic molecules driven by a bichromatic laser. In this problem, our results based on periodic orbit analysis again show good agreement with previous work, and hence promise more potential applications of this approach in molecular physics.

Recently there has been growing interest to characterize and reduce uncertainty in stochastic dynamical systems. This drive arises out of need to manage uncertainty in complex, high dimensional physical systems. Traditional techniques of uncertainty quantification (UQ) use local linearization of dynamics and assumes Gaussian probability evolution. But several difficulties arise when these UQ models are applied to real world problems, which, generally are nonlinear in nature. Hence, to improve performance, robust algorithms, which can work efficiently in a nonlinear non-Gaussian setting are desired. The main focus of this dissertation is to develop UQ algorithms for nonlinear systems, where uncertainty evolves in a non-Gaussian manner. The algorithms developed are then applied to state estimation of real-world systems. The first part of the dissertation focuses on using polynomial chaos (PC) for uncertainty propagation, and then achieving the estimation task by the use of higher order moment updates and Bayes rule. The second part mainly deals with Frobenius-Perron (FP) operator theory, how it can be used to propagate uncertainty in dynamical systems, and then using it to estimate states by the use of Bayesian update. Finally, a method to represent the process noise in a stochastic dynamical system using a nite term Karhunen-Loeve (KL) expansion is proposed. The uncertainty in the resulting approximated system is propagated using FP operator. The performance of the PC based estimation algorithms were compared with extended Kalman filter (EKF) and unscented Kalman filter (UKF), and the FP operator based techniques were compared with particle filters, when applied to a duffing oscillator system and hypersonic reentry of a vehicle in the atmosphere of Mars. It was found that the accuracy of the PC based estimators is higher than EKF or UKF and the FP operator based estimators were computationally superior to the particle filtering algorithms.

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No. of bitstreams: 1 outa_r_dr_ilha.pdf: 8826970 bytes, checksum: 2d87ce6b0e50b1304328308f0df4e3bb (MD5) Previous issue date: 2017-10-27 / O objetivo deste trabalho é a elaboração de um controlador linear ótimo, que possa reduzir um sinal caótico do rotor dinâmico, em um sinal controlado. Para se obter o resultado esperado foi necessário desenvolver atividades ligadas à caracterização do experimento; análise de estabilidade pelo método de Lyapunov; aplicação da função de Lyapunov; análise da sensibilidade das condições iniciais utilizando o expoente de Lyapunov; desenvolvimento do projeto do controle ótimo linear. O resultado final mostra o desempenho da aplicação do controle linear ótimo no sinal caótico, cujo sinal foi reduzido para um comportamento estável e controlado. / The aim of this work is the elaboration of an optimal linear controller that can reduce a chaotic dynamic rotor signal in a controlled signal. To obtain the expected result, it was necessary to develop activities related to the characterization of the experiment; stability analysis by the Lyapunov method; application of the Lyapunov function; sensitivity analysis of the initial conditions by the Lyapunov exponent; development of linear optimum control. The result shows the performance of the optimal linear control in the chaotic signal, whose signal was reduced to a stable and controlled behavior.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O caos determinístico é um dos aspectos mais interessantes no que diz respeito à teoria moderna dos sistemas dinâmicos, e está intrinsecamente associado a pequenas variações nas condições iniciais de um dado modelo. Neste trabalho, é feito um estudo acerca do comportamento caótico em dois casos específicos. Primeiramente, estudam-se modelos préinflacionários não-compactos de Friedmann-Robertson-Walker com campo escalar minimamente acoplado e, em seguida, modelos anisotrópicos de Bianchi IX. Em ambos os casos, o componente material é um fluido perfeito. Tais modelos possuem constante cosmológica e podem ser estudados através de uma descrição unificada, a partir de transformações de variáveis convenientes. Estes sistemas possuem estruturas similares no espaço de fases, denominadas centros-sela, que fazem com que as soluções estejam contidas em hipersuperfícies cuja topologia é cilíndrica. Estas estruturas dominam a relação entre colapso e escape para a inflação, que podem ser tratadas como bacias cuja fronteira pode ser fractal, e que podem ser associadas a uma estrutura denominada repulsor estranho. Utilizando o método de contagem de caixas, são calculadas as dimensões características das fronteiras nos modelos, o que envolve técnicas e algoritmos de computação numérica, e tal método permite estudar o escape caótico para a inflação. / Deterministic chaos is the most interesting aspect with regard to the modern theory of dynamical systems, and is intrinsically associated with small changes in initial conditions of a given model. This paper is a study about the chaotic behavior in two specific cases. First, we study non compact pre-inflationary FRW models with a minimally coupled scalar field, and then anisotropic models of Bianchi IX. In both cases the material component is a perfect fluid. Such models have a cosmological constant and can be studied via a unified description using suitable transformations of variables. These systems have similar structures in phase space, called saddle-centers, which make the solutions to be contained in hypersurfaces whose topology is cylindrical. These structures dominate the relationship between collapse and escape to inflation, which can be treated as basins whose boundary can be fractal, and can be associated with a structure called a strange repeller. Using the boxcounting method, which involves methods and algorithms for numerical computation, we calculate the characteristic dimension of their sets. This method allows to study the chaotic escape to inflation.

Orientador: Fábio Roberto Chavarette / Resumo: O objetivo deste trabalho é a elaboração de um controlador linear ótimo, que possa reduzir um sinal caótico do rotor dinâmico, em um sinal controlado. Para se obter o resultado esperado foi necessário desenvolver atividades ligadas à caracterização do experimento; análise de estabilidade pelo método de Lyapunov; aplicação da função de Lyapunov; análise da sensibilidade das condições iniciais utilizando o expoente de Lyapunov; desenvolvimento do projeto do controle ótimo linear. O resultado final mostra o desempenho da aplicação do controle linear ótimo no sinal caótico, cujo sinal foi reduzido para um comportamento estável e controlado. / Abstract: The aim of this work is the elaboration of an optimal linear controller that can reduce a chaotic dynamic rotor signal in a controlled signal. To obtain the expected result, it was necessary to develop activities related to the characterization of the experiment; stability analysis by the Lyapunov method; application of the Lyapunov function; sensitivity analysis of the initial conditions by the Lyapunov exponent; development of linear optimum control. The result shows the performance of the optimal linear control in the chaotic signal, whose signal was reduced to a stable and controlled behavior. / Doutor

TEIXEIRA, Rivania Maria do Nascimento. Leis de escala em mapeamentos discretos. 2016. 85 f. Tese (Doutorado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2016. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2016-07-18T18:20:56Z No. of bitstreams: 1 2016_tese_rmnteixeira.pdf: 5826571 bytes, checksum: 6cde4ae78436e469a71b8b9608331776 (MD5) / Approved for entry into archive by Edvander Pires (edvanderpires@gmail.com) on 2016-07-18T18:22:35Z (GMT) No. of bitstreams: 1 2016_tese_rmnteixeira.pdf: 5826571 bytes, checksum: 6cde4ae78436e469a71b8b9608331776 (MD5) / Made available in DSpace on 2016-07-18T18:22:35Z (GMT). No. of bitstreams: 1 2016_tese_rmnteixeira.pdf: 5826571 bytes, checksum: 6cde4ae78436e469a71b8b9608331776 (MD5) Previous issue date: 2016 / In this work we are going to investigate the scale formalism in discret mappings. In 1D mappings, we explore the asymptotic decays to the steady state with focus in three types of bifurcation: transcriptical, pitchfork and period-doubling. We identify this behavior through a well defined generalized homogeneous function with critical exponents. Next to the bifurcation point, the decay to the fix point occurs by an exponential function, which is given by a power law that is independent of the non-linearity mapping. The numerical results obtained agree with the analytical results. We also apply the scale formalism in conservatives and dissipatives bidimensional mappings. In the conservative case, our goal was analyze the behavior of the chaotics orbits next to the phase transition from the integrable to the non-integrable. Next to that transition, we describe the dynamical system using a generalized homogeneous function for which we found a power law that describe the behavior of the criticality. Through a phenomenological discussion, we found critical exponents in agree with the analytical description. In the dissipative case, our main goal was to investigate the influence of a dissipative term in the dynamics, causing a phase transition - suppression of unlimited difusion of the action variable. Following a phenomenological approach with an analytical description, we were able to determine the critical exponents using a generalized homogeneous function. / Neste trabalho investigamos algumas aplicações do formalismo de escala em mapeamentos discretos. Exploramos os decaimentos assintóticos ao estado estacionário com foco em três tipos de bifurcações em mapeamentos unidimensionais: bifurcação transcrítica, bifurcação supercrítica de forquilha e bifurcação de duplicação de período. Caracterizamos este comportamento através de uma função homogênea generalizada com expoentes críticos bem definidos. Próximo ao ponto de bifurcação o decaimento ao ponto fixo ocorre através de uma função exponencial cujo o tempo de relaxação é caracterizado por uma lei de potência que independe da não linearidade do mapa. Os resultados obtidos numericamente harmonizam com os resultados analíticos. Aplicamos também o formalismo de escala em mapeamentos bidimensionais conservativos e dissipativos. No caso conservativo, nosso objetivo foi analisar o comportamento de órbitas caóticas próximas à transição de fase de integrável para não integrável. Próximo à esta transição, descrevemos o sistema dinâmico utilizando uma função homogênea generalizada para a qual encontramos um lei de escala que descreve o comportamento da ação quadrática média próximo à transição. Através de uma discussão fenomenológica, encontramos expoentes críticos que corroboram com a descrição analítica. No caso dissipativo, nosso principal objetivo foi investigar a influência na dinâmica ao ser introduzido um termo dissipativo, causando a supressão da difusão ilimitada da variável ação quadrática média. Seguimos uma descrição fenomenológica acompanhada de uma descrição analítica e assim, determinamos os expoentes críticos usando uma função homogênea generalizada.

Orientador: Prof. Dr. Roberto Venegeroles / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Física, 2015. / Discutiremos uma classe de sistemas dinâmicos intermitentes na qual o espaço de fase é composto de duas regiões distintas: uma região laminar, onde a partícula desenvolve uma dinâmica lenta e quase regular até saltar para uma região turbulenta, onde a mesma desenvolve uma dinâmica caó- tica de curta duração até ser reinjetada de volta para a fase laminar, num processo que assim se repete. A fase laminar é causada pela existência de medida invariante innita, e a composição regularidade-caos resulta numa modalidade de caos fraco na qual a separação de trajetórias inicialmente muito próximas torna-se subexponencial (em sistemas caóticos usuais, essa separação é exponencial). Por conta desse tipo intermitência, o mapa apresentará um comportamento ergódico diferente daquele observado pelo teorema de Birkho, de modo que a distribuição de médias temporais de observáveis (com fatores próprios de normalização) é descrita essencialmente por uma estatística Mittag-Leer, ao invés de distribuições com limites assintóticos para delta de Dirac. Apresentaremos a lei responsável por tal comportamento, o teorema de Aaronson-Darling-Kac, que nos permitirá estender adequadamente certos observáveis, tais como o expoente de Lyapunov e a entropia de Kolmogorov- Sinai, de modo a inferir precisamente a existência desse tipo de instabilidade. Após um estudo das principais características ergódicas de tais sistemas, investigamos o número de primeiras-passagens da fase laminar para a turbulenta, e como obter informações-chave por meio dessa quantidade. Mostraremos também que a teoria de processos de renovação, usualmente empregada na literatura para esse m, é insuciente para descrever precisamente esse tipo de intermitência. Historicamente, esse tipo de sistema surgiu do estudo de mapas de primeiro retorno de certas seções do atrator de Lorenz, realizado nos anos 80 por Pomeau e Manneville. Atualmente, cadeias de tais mapas são empregadas no estudo de difusão anômala e passeios aleatórios com tempos de espera. / We discuss a class of intermittent dynamical systems in which the phase space is made up of two distinct regions: a laminar region, where the particle develops a slow and almost regular dynamic until it jumps to a turbulent region, where it develops a chaotic dynamic of short duration that is reinjected back into the laminating step, in a repeating process. The laminar phase is caused by the existence of innite invariant measure, and the regularity-chaos composition results in a weak mode in which the trajectories separation of two initial nearly points becomes subexponential (in usual chaotic systems, this separation is exponential). Because of this type of intermittency, the map will present a dierent behavior from that observed by ergodic Birkho's theorem, so that the distribution of average observable time (with its own normalization factors) is described essentially by a Mittag-Leer statistics, rather than distributions with asymptotic limit to the Dirac delta. We will present the law responsible for such behavior, the theorem Aaronson-Darling-Kac, which will allow us to extend properly certain observables, such as the Lyapunov exponent and entropy Kolmogorov-Sinai, in order to infer precisely the existence of such instability . After a study of the main ergodic characteristics of such systems, we investigate the number of rst-passages of the laminar stage for the turbulence, and how to get key information by that amount. We will also show that the theory of renewal processes, usually used in the literature for this purpose, is insucient to accurately describe this type of intermittency. Historically, this type of system emerged from the study of rst return maps of certain sections of the Lorenz attractor was accomplished in the 80's by Pomeau and Manneville. Currently chains of these maps are used in the study of anomalous diusion and random walks with waiting times.

Orientador: Rafael Ribeiro Dias Vilela de Oliveira / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , 2016. / No espaço de fases de sistemas dinâmicos, podem existir diferentes regiões as quais correspondem a diferentes comportamentos futuros do sistema: as bacias de atração. Separando essas regiões, há um conjunto de pontos, o qual chamamos de fronteira, que pode possuir uma geometria regular ou fractal, essa última caracterizada por uma dimensão D não inteira. A principal consequência de um sistema dinâmico possuir uma fronteira fractal em seu espaço de fases está na dificuldade em se determinar o comportamento futuro do sistema. De fato, dado que a precisão com a qual conseguimos medir um ponto é finita, existe uma área no espaço de fases em que não sabemos ao certo a qual bacia de atração o ponto pertence. Em especial, caso a fronteira seja fractal, essa área é proporcional a N..D, onde é o erro de medição e N é a dimensão do sistema. Dessa forma, percebemos a importância de conseguirmos calcular a dimensão D da fronteira fractal. Nesse trabalho, primeiro apresentamos os principais conceitos de sistemas dinâmicos e geometria fractal, relacionando essas estruturas geométricas ao comportamento dinâmico caótico. Em seguida, definimos as fronteiras e estendemos a elas o conceito de geometria fractal. Por último, apresentamos os métodos vigentes para o cálculo numérico da dimensão de fronteiras fractais, a saber, o método da incerteza e o método da avaliação da função de saída e, baseados no primeiro método, desenvolvemos um método alternativo: o método da incerteza condicional. Observamos que o método desenvolvido nesse trabalho é válido como um novo método para o cálculo da dimensão de fronteiras fractais, podendo ser utilizado tanto em sistemas de tempo contínuo quanto discreto. / In the phase space of dynamical systems there may exist different regions which correspond to different final states: the basins of attraction. Between different basins of attraction, there is a set of points which we call basin boundary. Basin boundaries can be either smooth or fractal, the latter being characterized by a non-integer dimension D. The main consequence of fractal basin boundaries in the phase space of a dynamical system is the difficulty of determining the system¿s final state. Indeed, knowing that we can only measure a point with a finite precision, there is a phase space region where we cannot know in which of the basins of attraction the point really is by looking at the system¿s final state alone. In particular, for a fractal basin boundary, the area of the phase space where we cannot predict the final state with certainty is proportional to N..D, with being the measurement error and N the system¿s dimension. Therefore, it is important to know the dimension D of the fractal basin boundary. In this work, we first present the main concepts of dynamical systems and fractal geometry, linking these geometric structures to chaotic behavior in the system. Later, we define basin boundaries, both regular and fractal. At last, we present the two methods currently available to calculate the dimension of fractal basin boundaries in dynamical systems, namely the uncertainty method and the output function evaluation method. We propose a new method that is based on the former one called the conditional uncertainty method and we show that this method can calculate fractal dimensions of basin boundaries to a good accuracy either on continuous or discrete-time dynamics.