Neste trabalho estudamos a dinâmica assintótica não linear de algumas equações parabólicas do tipo reação-difusão sob perturbações nos parâmetros e perturbações singulares no domínio do tipo dumbbell. Mais precisamente, trataremos dos atratores provenientes destes problemas, buscaremos compreender a dependência destes conjuntos assintóticos de estados em relação ao parâmetro, investigando a continuidade com taxa dos mesmos. O programa que executaremos para obtenção da taxa de continuidade dos atratores, bem como de toda a estrutura, mostra-nos fortes propriedades de dissipatividade exponencial de alguns semigrupos / In this work we study the asymptotic nonlinear dynamical of some reaction-diffusion parabolic equations under perturbations in parameter and singular perturbations in a dumbbell domain. More precisely, we treat of the attractors from these problems, we seek understand the dependence these asymptotic set of states in relationship the parameter, investigating continuity with rate. The program that we will follow to prove the continuity of the attractors with rate well as the entire structure, we show that these semigroups possess strong exponential dissipative properties

Nesse trabalho apresentamos o estudo do artigo [25] que analisa a existência de atratores globais para uma classe de equações da onda amortecida da forma \'épsilon u IND. tt\' + \'alpha\' (x) u IND. t\' + \'BETA\' (x)u - \'\\SIGMA SOBRE i, j\' \'\\PARTIAL ind. I\' (\'a IND. i j\' (x) \'PARTIAL IND. j u\') = f(x, u) , x PERTENCE A ÔMEGA\'\', t \'PERTENCE A\' [0,\'infinito\'), u (x, t) = 0, x \'PERTENCE A\' \'\\PARTIAL ÔMEGA\', t \'PERTENCE A\' [0, \'infinito\') definidas em um domínio arbitrário \'ÔMEGA\' / In this work we describe the results of the paper [25]. In [25] the authors prove existence of global attractors for the following semilinear damped wave equation \'\\épsilon u IND. t\'t + \'alpha\'(x)u IND. t\' + \'beta\' (x)u - \'\\SIGMA SOBRE i, j \'\\PARTIAL IND. i\' (\'a IND. i j\' (x) \'\\PARTIAL IND. j u\') = f (x, u), x \'IT BELONGS\' \'ÔMEGA\', t \'IT BELONGS\' [0, \'INFINITY\'), u(x,t), x \'IT BELONGS\' \'\\PARTIAL ÔMEGA\', t \'IT BELONGS\' [), \'INFINITY\'0 on an arbitrary domain \'OMEGA\'

Diante da crescente competição no mercado e as mudanças constantes que vêm ocorrendo, as empresas tiveram que elaborar novas estratégias e pensar a partir de novas lentes. A nova estruturação das organizações como a formação de redes e arranjos produtivos, é consequência dessas mudanças que as organizações vêm enfrentando devido ao mundo globalizado e a concorrência que aumenta a cada dia. Nesse contexto, o presente trabalho tem como objetivo investigar como os atratores sociais moldam as dinâmicas organizacionais em Arranjos Produtivos Locais. A pesquisa teve como objeto de estudo os APL\'s de Ibitinga, Juruaia e Grande ABC. A metodologia teve abordagem qualitativa com caráter exploratório, e quanto aos meios é classificada como pesquisa documental, pesquisa de campo e estudo de múltiplos casos. Os dados foram coletados por meio de entrevistas baseada na ténica de incidente crítico e coleta documental. Para a análise de dados foi utilizada a técnica de análise de conteúdo e análise documental. Foi identificado o sistema técnico como o principal atrator social, com particular destaque à importância das lideranças e dos sistemas sociais no desenvolvimento e aperfeiçoamento dos arranjos produtivos em tela. / In front of the market competition\'s increases and its constant changes that have been occurring, companies had to develop new strategies and think from new point of view. The new organization structuring like build networking and productive arrangements, is consequence of changes that those organizations are facing due the globalization and the competition that increases day by day. In this context, this paper has an objective investigate how the social attractors fits the organizational dynamics on Local Productive Arrangements. The research has Ibitinga, Juruaia and Grande ABC\'s LPAs as study object. The methodology has qualitative approach with an exploratory character, and as to means classified as documental research, field research and multiple cases. The data was collected through interview based on the critical incident technique and documentary collection technique. To the data analysis was used the technique of content analysis and documental analysis. Was identified the technical system as the main social attractor, with particular emphasis to the importance of the leadership and the social systems on development and improvement of the productive arrangements in the screen.

Redes neurais atratoras são redes de neurônios artificiais com realimentacão e sem estrutura de conexão pré-definida. Estes tipos de redes apresentam uma rica dinâmica dissipativa e são freqüentemente utilizadas como memórias associativas. Tais dispositivos tem a propriedade de recuperar uma memória previamente armazenada, mesmo quando expostos a informação parcial ou degradada daquela memória. Armazenar uma memória significa criar um atrator para ela na dinâmica da rede e isto e feito especificando-se adequadamente os pesos sinápticos. Nesta tese, nos concentramos basicamente em duas maneiras de se definir os pesos sinapticos, que dão origem ao modelo da pseudo-inversa e ao modelo dos pesos ótimos. Para redes neurais extremamente diluídas, onde a conectividade C e o número de neurônios N satisfazem à condição C&#171 In N obtivemos os diagramas de fase no espaço completo de parâmetros dos modelos da pseudo-inversa e dos pesos ótimos através da analise da dinâmica da correlação de recuperação dos padrões armazenados. Alem disso, investigamos as propriedades de recuperação de redes neurais completamente conectadas através de duas abordagens: a investigação analítica da vizinhança dos padrões armazenados e a enumeração exaustiva dos atratores por meio de simulações numéricas. Finalmente. estudamos analiticamente o problema da categorizarão no modelo da pseudo-inversa. A categorizar;ao em redes neurais atratoras e a capacidade da rede treinada com exemplos de um conceito desenvolver um atrator para este conceito. / Attractor neural networks are feedback neural networks with no pre-defined connection structure. These types of neural networks present a rich dissipative dynamics and, in general, are used as associative memory devices. Such devices have the capacity to retrieve a previously stored memory, even when exposed to partial or degraded information. To store a memory means to create an attractor for it in the network dynamics, and this is done by specifying the set of synaptic weighs. In this thesis, we concentrate on two classical ways of specifying the synaptics weighs: the pseudo-inverse and the optimal weighs models. For extremely diluted neural networks, for which the connectivity C and the number of neurons N satisfy the condition C &#171 In N, we obtain the phase diagrams in the complete space of the model parameters through the analytical study of the retrieval overlap dynamics. We also investigate the retrieval properties of fully connected neural networks using two approaches: the analytical study of the neighborhood of the stored patterns, and the exhaustive enumeration of the attractors via numerical simulations. Finally, we study analytically the problem of categorization in the pseudo-inverse model. Categorization in attractor neural networks is the capacity to create an attractor for a concept to which the network has had access only through a finite number of examples.

Neste trabalho, estudou-se a sincronização de uma classe de sistemas dinâmicos não-lineares acoplados. Do ponto de vista teórico, apresentam-se resultados que fornecem condições suficientes sobre o campo vetorial e estimativas dos parâmetros de acoplamentos que garantem sincronização de um conjunto de soluções de uma classe de sistemas não-lineares acoplados. Diferentemente da grande maioria dos resultados existentes na literatura de sincronização de sistemas não-lineares acoplados, os resultados propostos nesta tese podem ser aplicados para demonstrar sincronização em sistemas que não possuem atratores globais, incluindo casos instáveis, onde as soluções são não-limitadas. Quando o sistema não possui atrator global, foi utilizado um resultado, também proposto nesta tese, que fornece estimativas uniformes de atratores, para estimar conjuntos positivamente invariantes contidos na região de sincronização do sistema. Os resultados teóricos propostos foram empregados para demonstrar sincronização em um sistema formado por dois pêndulos acoplados e também por dois sistemas de Duffing acoplados. Do ponto de vista aplicado, estuda-se o problema de coerência de geradores em sistemas elétricos de potência. Valendo-se dos resultados teóricos desta tese, um índice foi proposto para detectar e identificar geradores fracamente coerentes, os chamados clusters. A metodologia de análise de coerência proposta nesta tese não requer grande esforço computacional e poderia ser utilizada em aplicações em tempo real. Os resultados mostraram que a análise deste índice fornece, a priori, sem a necessidade de simulações numéricas, informações importantes sobre a presença de acoplamento forte entre as máquinas, a localização dos pontos de equilíbrio instáveis de controle, assim como a existência de modos de instabilidade combinados. / Synchronization of a class of coupled non-linear systems is studied in this work. From the theoretical point of view, we present synchronization results that provide sufficient conditions on the vector field and estimates of the coupling parameters that guarantee synchronization. Differently from the existing approaches in the nonlinear systems literature, our results can be applied to demonstrate synchronization in systems that do not have global attractors, including even unstable cases, where the solutions are unbounded. When the system does not globally synchronize, a result that provides uniform estimates of attractors is used to present an estimate of a positively invariant set contained in the synchronization region. The theoretical results are applied to demonstrate synchronization between two nonlinear pendulums and two coupled Duffing\'s systems. From the applied point of view, we study the problem of coherency between generators in electrical power systems. Using the theoretical results of this thesis, an index is proposed to detect and identify groups of weakly-coherent generators, the so called clusters. The proposed coherency analysis methodology proposed in this text does not require a great computational effort and is suitable for online applications. Our results have shown that this index analysis provides important information about the strong coupling between the generators, the location of the controlling unstable equilibrium points and the existence of combined unstable modes.

Submitted by Maria Aparecida (cidazen@gmail.com) on 2017-07-17T12:51:37Z No. of bitstreams: 1 Pericles Carvalho.pdf: 1223900 bytes, checksum: f4ce61e56294f3bb3b3e192efa2993f2 (MD5) / Made available in DSpace on 2017-07-17T12:52:05Z (GMT). No. of bitstreams: 1 Pericles Carvalho.pdf: 1223900 bytes, checksum: f4ce61e56294f3bb3b3e192efa2993f2 (MD5) Previous issue date: 2017-05-02 / This paper begins with the definition of SRB measures, and the introduction of several concepts in ergodic theory necessary for the development of the presented results. We prove the existence and uniqueness of SRB measures for uniformly expanding transformations in compact connected manifolds whose Jacobian is H¨older continuous. Then, we present the definition of hyperbolic sets, hyperbolic attractors and their respective fundamental properties. As a main result, we prove the existence of SRB measures for hyperbolic attractors contained in compact manifolds, and its uniqueness if the hyperbolic attractor is transitive. First, it is shown the existence of invariant measures absolutely continuous along the unstable foliation. Then, we note that the restriction of this measure over certain subsets have the SRB property. Using the transitivity of the hyperbolic attractor, it is shown that there exists a unique subset such that this restriction is an SRB measure. We conclude that the system supports a unique SRB measure. / Este trabalho inicia-se com a definição de medida SRB, e a apresentação de diversos conceitos da teoria ergódica importantes para o desenvolvimento dos resultados apresentados. É demonstrada a existência e unicidade de medidas SRB para transformações expansoras em variedades compactas e conexas cujo jacobiano é Holder. Em seguida, apresenta-se a definição de conjuntos hiperbólicos, atratores hiperbólicos e suas respectivas propriedades fundamentais. Como resultado principal, é demonstrada a existência de medidas SRB para atratores hiperbólicos, bem como sua unicidade para o caso de atratores hiperbólicos transitivos, ambos dentro de variedades compactas. Primeiramente, é mostrado que existem medidas invariantes absolutamente contínuas ao longo da folheação instável. Em seguida, verifica-se que a restrição desta medida sob determinados conjuntos possuem a propriedade SRB. Utilizando a transitividade do atrator hiperbólico, mostra-se que existe um único conjunto tal que esta restrição seja uma medida SRB. Conclui-se que o sistema admite uma única medida SRB.

The inflationary paradigm is one which was designed to answer questions that arose from classical Hot Big Bang cosmology. The period of rapid expansion in the early Universe provides a mechanism to solve the flatness, horizon and relic problems. More importantly, since the theory was first introduced it has been realised that it also provides a mechanism to generate the initial perturbations from which structure in the Universe can grow. In the zoo of potential inflationary models there is a dominant class: slow-roll inflation. The idea that the energy density of the inflationary field is dominated by its potential highly simplifies the calculations required to predict observable quantities. This simplification relies on all the information required to know the subsequent dynamics of the field to be encoded in the space Φ-Φ̇; it must be an effective phase space. I show that Φ-Φ̇ can be considered to be such a space for the most general scalar-tensor theory which gives second-order equations of motion: Horndeski theory. There are theoretical issues associated with this reduction that are illuminated through specific examples in which they occur. A theoretical issue with inflation is that there is an overabundance of models, with some capable of predicting any value of the possible observables. The second block of work in this thesis looks at a particular set of models that make the same observational prediction. These 'attractor' models utilise a non-minimal coupling between the inflationary fields and gravity and are studied in depth, both in the case of one and several fields. Firstly, I examine the Universal Attractors, a single field subset of these models. I show, in detail, the observational prediction such a model makes in the case of a strong non-minimal coupling and then examine the constraints it would be possible to put on such a coupling if a confirmed detection of primordial gravitational waves was made. Despite the discussion existing in the literature there is a small deviation of the Universal Attractor models from the predictions of the Starobinsky model. Furthermore, the coupling, ξ is found to be constrained so that |ξ| < 1 in the case where there a level of detectable primordial tensor modes. While the attractor models have an effective one-field description in reality there are several other fields that are assumed to be fixed during the inflationary phase. This claim requires careful examination as the field-space of the models generally is not flat. This curvature can cause a destabilising effect with certain parameters and so I investigate how susceptible the α-attractors and related models are to the destabilisation. A key result of this chapter is to highlight how important it is to not rely on the slow-roll approximation when assessing the effect of the instability, as the region where the effect begins to become large corresponds with the region where slow-roll begins to break down. Assuming the slow-roll approximation is valid leads to an over-estimation of the effect that the instability mechanism has. Despite this, some of the models considered are seen to experience the instability for certain ranges of model parameters. Making the assumption that any occurrence of the instability will, at the very least, move the observational prediction of the model outside the currently constrained range allows a constraint on the model parameter in question which directly translates to a theoretical lower bound on the tensor-scalar ratio, r > 0.0005.

A simple asset pricing model with two types of adaptively learning traders, fundamentalists and technical analysts, is studied. Fractions of these trader types, which are both boundedly rational, change over time according to evolutionary learning, with technical analysts conditioning their forecasting rule upon deviations from a benchmark fundamental. Volatility clustering arises endogenously in this model. Two mechanisms are proposed as an explanation. The first is coexistence of a stable steady state and a stable limit cycle, which arise as a consequence of a so-called Chenciner bifurcation of the system. The second is intermittency and associated bifurcation routes to strange attractors. Both phenomena are persistent and occur generically in nonlinear multi-agent evolutionary systems. (author's abstract) / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

This dissertation can be essentially divided into two parts. The first, consisting of Chapters I, II, and III, studies the graph theoretic nature of complex systems. This includes the spectral properties of such systems and in particular their influence on the systems dynamics. In the second part of this dissertation, or Chapter IV, we consider a new class of one-dimensional dynamical systems or functions with an eventual negative Schwarzian derivative motivated by some maps arising in neuroscience. To aid in understanding the interplay between the graph structure of a network and its dynamics we first introduce the concept of an isospectral graph reduction in Chapter I. Mathematically, an isospectral graph transformation is a graph operation (equivalently matrix operation) that modifies the structure of a graph while preserving the eigenvalues of the graphs weighted adjacency matrix. Because of their properties such reductions can be used to study graphs (networks) modulo any specific graph structure e.g. cycles of length n, cliques of size k, nodes of minimal/maximal degree, centrality, betweenness, etc. The theory of isospectral graph reductions has also lead to improvements in the general theory of eigenvalue approximation. Specifically, such reductions can be used to improved the classical eigenvalue estimates of Gershgorin, Brauer, Brualdi, and Varga for a complex valued matrix. The details of these specific results are found in Chapter II. The theory of isospectral graph transformations is then used in Chapter III to study time-delayed dynamical systems and develop the notion of a dynamical network expansion and reduction which can be used to determine whether a network of interacting dynamical systems has a unique global attractor. In Chapter IV we consider one-dimensional dynamical systems of an interval. In the study of such systems it is often assumed that the functions involved have a negative Schwarzian derivative. Here we consider a generalization of this condition. Specifically, we consider the functions which have some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class of functions. This includes both systems with regular as well as chaotic dynamic properties.

We introduce numerical methods for the analysis of random dynamical systems. The subdivision and the continuation algorithm are powerful tools which will be demonstrated for a system from ship dynamics. With our software package we are able to show that the well known safe basin is a moving fractal set. We will also give a numerical approximation of the attracting invariant set (which contains a local attractor) and its evolution.