Made available in DSpace on 2016-12-12T20:15:53Z (GMT). No. of bitstreams: 1 resumo.pdf: 30947 bytes, checksum: f3ad8c83ce0b37705512036923795354 (MD5) Previous issue date: 2009-02-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we investigate the dynamical behavior of two dynamical systems: (i) a symmetric linear coupling of three quadratic maps, and (ii) the generalized Lorenz equations obtained by Stenflo. For the discrete-time system represented by the coupling of three quadratic maps, we study the emergence of quasiperiodic states arising from Naimark-Sacker bifurcations of stable periodic orbits pertaining to 1×2n cascade, in particular period-1 and period-2 orbits. We also study the change in the structure of the basin of attraction of the chaotic attractors, in the neighborhood of chaos-hyperchaos transition. For the continuous-time system represented by the Lorenz-Stenflo equations, we analytically investigate, by using Routh-Hurwitz Test, the stability of three fixed points, although without explicit solution of the eigenvalue equation. We determine the precise location where pitchfork and Hopf bifurcations of the fixed points occur, as a function of the parameters of the system. Lyapunov exponents, parameter-space and phase-space portraits, and bifurcation diagrams were used to numerically characterize periodic and chaotic attractors in both systems. / Neste trabalho investigamos o comportamento de dois sistemas dinâmicos (i) Um acoplamento linear simetrico de três mapas quadraticos, e (ii) as equações eneralizadas de Lorenz, obtidas por Stenfio Para o sistema discreto, representado elo acoplamento linear dos três mapas quadraticos, estudamos a emergência de tados quase-periodicos, surgindo da bifurcação de Naimark-Sacker, a partir de uma bita estavel pertencendo a cascata 1 x 2n em particular orbitas de periodo um e eriodo dois Tambem estudamos a mudança na estrutura das bacias de atração do rator caotico, na vizinhança da transição caos-hipercaos Para o sistema de tempo ntínuo representado pelas equações de Lorenz-Stenflo, investigamos analiticamente elo método de Routh-Hurwitz, a estabilidade dos três pontos de equilíbrio, mas sem solução explicita da equação de autovalores. Determinamos a localização precisa de as bifurcações do tipo forquilha e Hopf acontecem, a partir dos pontos de uilíbrio, como uma função dos parâmetros do sistema. Expoentes de Lyapunov, agramas no espaço de parâmetros e espaço de fase e diagramas de bifurcação foram ilizados para caracterizar numericamente os atratores periódicos e caóticos em ibos os sistemas

Orientador: Edson Denis Leonel / Coorientador: Peter Vaughan Elsmere McClintock / Banca: Roberto E. Lagos Monaco / Banca: Roberto Venegeroles Nascimento / Resumo: Nesta dissertação, investigamos propriedades estatísticas de alguns sistemas dinâmicos descritos por mapeamentos discretos nas proximidades de duas transições: (i) integrabilidade para não integrabilidade e; (ii) crescimento limitado de energia para crescimento ilimitado de energia (aceleração de Fermi). O foco principal está na descrição do comportamento da distribuição de probabilidade da velocidade/energia das partículas em dinâmica caótica. A quebra de simetria da distribuição de probabilidade leva a uma escala adicional àquelas já conhecidas na literatura e, com este estudo, acreditamos que a quebra de simetria também possa explicar um fenômeno que já vem sendo observado em mapeamentos discretos. Fenômeno este, até então descrito apenas fenomenologicamente, teve sua primeira observação na publicação seminal de investigação de leis de escala em mapeamentos discretos no periódico Phys. Rev. Let. 93, 014101 (2004), de Edson D. Leonel, Peter V. E. McClintock e Jafferson K. L. Silva. Nossa contribuição para o problema está no desenvolvimento de descrições analíticas e verificações numéricas, baseadas em um estudo sistemático do comportamento difusivo das trajetórias caóticas no espaço de fases dos sistemas dinâmicos de interesse / Abstract: In this dissertation, we investigate statistical properties of some dynamical systems described by discrete mappings near two types of transitions: (i) integrability to non-integrability; (ii) limited to unlimited diffusion in energy (Fermi acceleration). The main goal is to describe the behaviour of the probability density of the velocity/energy for a set of particles moving in a chaotic dynamics. The break of symmetry in the probability distribution leads to an additional scaling to those are already known in the literature and, with this study, we believe that the symmetry break might also explain a well-known phenomenon observed for discrete mappings. This phenomenon, it has been reported so far phenomenologically. A first observation in an area-preserving mapping was in a letter published in Phys. Rev. Let. 93, 014101 (2004), authored by Edson D. Leonel, Peter V. E. McClintock and Jafferson K. L. Silva. Our contribution to the problem is on the development of an analytical approach and numerical verifications, based essentially on a systematic study of the diffusive behaviour of chaotic trajectories on the phase space of dynamical systems of interest / Mestre

abstract: In this thesis, I investigate the C*-algebras and related constructions that arise from combinatorial structures such as directed graphs and their generalizations. I give a complete characterization of the C*-correspondences associated to directed graphs as well as results about obstructions to a similar characterization of these objects for generalizations of directed graphs. Viewing the higher-dimensional analogues of directed graphs through the lens of product systems, I give a rigorous proof that topological k-graphs are essentially product systems over N^k of topological graphs. I introduce a "compactly aligned" condition for such product systems of graphs and show that this coincides with the similarly-named conditions for topological k-graphs and for the associated product systems over N^k of C*-correspondences. Finally I consider the constructions arising from topological dynamical systems consisting of a locally compact Hausdorff space and k commuting local homeomorphisms. I show that in this case, the associated topological k-graph correspondence is isomorphic to the product system over N^k of C*-correspondences arising from a related Exel-Larsen system. Moreover, I show that the topological k-graph C*-algebra has a crossed product structure in the sense of Larsen. / Dissertation/Thesis / Ph.D. Mathematics 2011

abstract: In this thesis I introduce a new direction to computing using nonlinear chaotic dynamics. The main idea is rich dynamics of a chaotic system enables us to (1) build better computers that have a flexible instruction set, and (2) carry out computation that conventional computers are not good at it. Here I start from the theory, explaining how one can build a computing logic block using a chaotic system, and then I introduce a new theoretical analysis for chaos computing. Specifically, I demonstrate how unstable periodic orbits and a model based on them explains and predicts how and how well a chaotic system can do computation. Furthermore, since unstable periodic orbits and their stability measures in terms of eigenvalues are extractable from experimental times series, I develop a time series technique for modeling and predicting chaos computing from a given time series of a chaotic system. After building a theoretical framework for chaos computing I proceed to architecture of these chaos-computing blocks to build a sophisticated computing system out of them. I describe how one can arrange and organize these chaos-based blocks to build a computer. I propose a brand new computer architecture using chaos computing, which shifts the limits of conventional computers by introducing flexible instruction set. Our new chaos based computer has a flexible instruction set, meaning that the user can load its desired instruction set to the computer to reconfigure the computer to be an implementation for the desired instruction set. Apart from direct application of chaos theory in generic computation, the application of chaos theory to speech processing is explained and a novel application for chaos theory in speech coding and synthesizing is introduced. More specifically it is demonstrated how a chaotic system can model the natural turbulent flow of the air in the human speech production system and how chaotic orbits can be used to excite a vocal tract model. Also as another approach to build computing system based on nonlinear system, the idea of Logical Stochastic Resonance is studied and adapted to an autoregulatory gene network in the bacteriophage λ. / Dissertation/Thesis / Ph.D. Electrical Engineering 2011

This thesis is concerned with the formulation of a suitable notion of monotonicity of discrete and continuous-time dynamical systems on Lie groups and homogeneous spaces. In a linear space, monotonicity refers to the property of a system that preserves an ordering of the elements of the space. Monotone systems have been studied in detail and are of great interest for their numerous applications, as well as their close connections to many physical and biological systems. In a linear space, a powerful local characterisation of monotonicity is provided by differential positivity with respect to a constant cone field, which combines positivity theory with a local analysis of nonlinear systems. Since many dynamical systems are naturally defined on nonlinear spaces, it is important to seek a suitable adaptation of monotonicity on such spaces. However, the question of how one can develop a suitable notion of monotonicity on a nonlinear manifold is complicated by the general absence of a clear and well-defined notion of order on such a space. Fortunately, for Lie groups and important examples of homogeneous spaces that are ubiquitous in many problems of engineering and applied mathematics, symmetry provides a way forward. Specifically, the existence of a notion of geometric invariance on such spaces allows for the generation of invariant cone fields, which in turn induce notions of conal orders. We propose differential positivity with respect to invariant cone fields as a natural and powerful generalisation of monotonicity to nonlinear spaces and develop the theory in this thesis. We illustrate the ideas with numerous examples and apply the theory to a number of areas, including the theory of consensus on Lie groups and order theory on the set of positive definite matrices.

abstract: In response to the recent publication and media coverage of several books that support educating boys and girls separately, more public schools in the United States are beginning to offer same-sex schooling options. Indeed, students may be more comfortable interacting solely with same-sex peers, as boys and girls often have difficulty in their interactions with each other; however, given that boys and girls often interact beyond the classroom, researchers must discover why boys and girls suffer difficult other-sex interactions and determine what can be done to improve them. We present two studies aimed at examining such processes. Both studies were conducted from a dynamical systems perspective that highlights the role of variability in dyadic social interactions to capture temporal changes in interpersonal coordination. The first focused on the utility of applying dynamics to the study of same- and mixed-sex interactions and examined the relation of the quality of those interactions to participants' perceptions of their interaction partners. The second study was an extension of the first, examining how dynamical dyadic coordination affected students' self-perceived abilities and beliefs in science, with the intention of examining social predictors of girls' and women's under-representation in science, technology, engineering, and mathematics. / Dissertation/Thesis / Ph.D. Family and Human Development 2012

Submitted by Nilton Cesar Moretti Junior null (niiilton@hotmail.com) on 2018-07-31T04:58:47Z No. of bitstreams: 1 Dissertação-Final.pdf: 1149444 bytes, checksum: 6c44dc0b9f2462ee08c23da4a240fa0a (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2018-07-31T18:13:18Z (GMT) No. of bitstreams: 1 morettijunior_nc_me_sjrp.pdf: 1461434 bytes, checksum: 7a6418b1192346448fc927ec6c6650dc (MD5) / Made available in DSpace on 2018-07-31T18:13:18Z (GMT). No. of bitstreams: 1 morettijunior_nc_me_sjrp.pdf: 1461434 bytes, checksum: 7a6418b1192346448fc927ec6c6650dc (MD5) Previous issue date: 2017-08-30 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho estudamos vários resultados relacionados com sistemas dinâmicos, teoria dos números e combinatória. Em particular, provamos os teoremas de Van Der Waerden, Szemeredi, Koksma e Weyl. / In this work we study several results connected with dynamical systems, number thoery and combinatorics. In particular, we prove Van Der Waerden, Szemer edi, Koksma and Weyl’s theorems.

Cette thèse se situe à l’interface entre théorie des nœuds et théorie des systèmes dynamiques. Le thème central consiste, étant donné un champ de vecteurs dans une variété de dimension 3, à considérer ses orbites périodiques, et à s’interroger sur les informations qu’elles donnent sur le champ de vecteurs et la variété initiaux.La première partie est consacrée au flot géodésique défini sur le fibré unitaire tangentd’une surface, ou d’une orbiface, à courbure constante. L’observation de certains exemples (sphère, tore, surface modulaire) suggère la conjecture suivante, due à Étienne Ghys : l’enlacement entre deux familles homologiquement nulles quelconques d’orbites périodiques est toujours négatif. En d’autres termes, le flot géodésique serait lévogyre. Quand la courbure est négative, par les travaux de David Fried sur les flots d’Anosov, cette conjecture implique une propriété étonnante et très particulière : n’importe quelle collection homologiquement nulle d’orbites périodiques borde une section de Birkhoff pour le flot géodésique, et est par conséquent la reliure d’un livre ouvert. En ce sens, cette conjecture propose une généralisation de la construction de Norbert A’Campo de livres ouverts sur les fibrés unitaires tangents. Nous proposons la démonstration de cette conjecture dans les cas du tore, des orbifolds de type (2, q, infini), et de l’orbifold de type (2, 3, 7). La seconde partie est consacrée au comportement asymptotique des invariants des nœuds formés par les orbites périodiques d’un champ de vecteur, quand la longueur de l’orbite tend vers l’infini. Le but est de définir des invariants de champs de vecteurs stables par difféomorphisme. Dans le cas particulier des nœuds de Lorenz, nous montrons que les racines du polynôme d’Alexander admettent un comportement particulier : elles s’accumulent au voisinage du cercle-unité. / This thesis deals with interactions between knot theory and dynamical systems. Givena vector field on a 3-manifold, the main idea is to study its periodic orbits from the knottheoretical point of view, and to deduce informations about the vector field and the initial manifold. The first part is devoted to the study of the geodesic flow defined on the unit tangent bundle of a surface, or an orbiface, with constant curvature. Simple examples (sphere, torus, modular surface) suggest the following conjecture, due to Ghys : the linking number of two homologically zero collections of periodic orbits is always negative. In other words, the geodesic flow on any orbiface with constant curvature is left-handed. In the negatively curved case, the work of Fried imply another surprising property : any homologically trivial collection of periodic orbits bound a Birkhoff section for the geodesic flow, and is therefore the binding of an open book decomposition. In this setting, the conjecture is a generalization of A’Campo’s construction of open book decompositions on unit tangent bundles. In our work, we prove the conjectre for the torus, for the orbifolds of type (2, q, oo), and for the orbifold of type (2, 3, 7). The second part is devoted to the asymptotic behaviour of invariants of the knots made by the periodic orbits of a vector field, when the length of the orbits tend to infinity. The goal is to define invariants of the vector field under diffeomorphism. In the case of Lorenz knots, we show that the roots of the Alexander polynomial admit an asymptotic behaviour, namely that they accumulate on the unit circle.

Orientador: Edson Denis Leonel / Banca: Denis Gouvea Ladeira / Banca: Juliano Antônio de Oliveira / Resumo: Neste projeto aplicamos o formalismo de escala com o objetivo de explorar a evolução em direção ao equilíbrio perto de uma bifurcação tangente no mapa logístico. No ponto de bifurcação a órbita segue o caminho descrito por uma função homogênea com expoentes críticos bem definidos. Perto da bifurcação, a convergência para o equilíbrio é exponencial, cujo tempo de relaxação é marcado por uma lei de potência. Para obtermos os expoentes utilizamos dois procedimentos distintos: (1) o primeiro, fenomenológico, envolvendo hipóteses de escala, com o qual determinamos uma lei de escala entre os 3 expoentes críticos; (2) o segundo transforma uma equação de diferenças em uma equação diferencial, sendo resolvida com condições iniciais convenientes. Os resultados analíticos confirmam bem os resultados encontrados numericamente / Abstract: In this project we apply the scaling formalism to understand and describe the evolution towards the equilibrium at and near at a tangent bifurcation into logistic map. At the bifurcation the convergence to the steady state is described by a homogeneous function with well de ned critical exponents. Near the bifurcation, the evolution to the equilibrium is described by an exponential function whose relaxation time is described by a power law. We use two di erent approaches to obtain the critical exponents: (1) a phenomenological investigation based on three scaling hypotheses leading to a scaling law relating three critical exponents and; (2) a procedure transforming the di erence equation into a di erential equation which is solved under appropriate conditions. The numerical results give support for the theoretical approach / Mestre

Orientador: Juliano Antônio de Oliveira / Banca: Rene Orlando Metrano Torricos / Banca: Priscilla Andressa de Souza Silva / Resumo: Neste trabalho consideramos o mapa padrão descrito nas variáveis momento e ângulo, a partir do movimento de um rotor pulsado. Uma vez definido o modelo para o caso conservativo, construímos o espaço de fase para analisar a dinâmica do sistema. Observamos um mar caótico ao redor de ilhas periódicas e limitado por um conjunto de curvas invariantes spannig. Para caracterizar o caos, usamos os expoentes de Lyapunov. Estendemos os nossos estudos introduzindo dissipação no sistema. Dada a escolha dos parâmetros de controle, observamos que a estrutura mista observada no sistema conservativo decai exponencialmente para atratores caóticos. Os expoentes de Lyapunov foram usados para caracterizar os atratores caóticos. Introduzimos uma função de descontinuidade no sistema para investigar a raiz quadrada da variável ação quadrática média ao longo dos atratores caóticos. Uma lei de escala foi estabelecida e os expoentes de escala são encontrados numericamente. Finalmente, discutimos uma abordagem analítica para a variável ação quadrática média no mapeamento padrão dissipativo descontínuo / Abstract: In this work we consider the standard map described in the momentum and angle variables from the movement of a kicked rotor. Once the model for the conservative case is defined, we build the phase space to analyze the dynamics of the conservative system. We observe a chaotic sea surrounding periodic islands and limited by a set of invariant spannig curves. To characterize chaos we use the Lyapunov exponents. We extend our studies introducing dissipation in the system. Given the chose of the control parameters we obseve that the mixed structure observed in the conservative case decay exponentially for large chaotic attactors. The Lyapunov exponents were used to characterize the chaotic attactors. We introduce a discontinuity function in the system to investigate the root mean square of the quadratic action variable along of the chaotic attractors. A scaling law was established and the scaling exponents are found numerically. Finally a analytical approach for the quadratic mean action variable in the dissipative discontinuous standard mapping is discussed / Mestre