This is a summary report on some existing results and methods regarding the problem of determining the basins of attraction of dynamical systems (in particular, two-dimensional diffeomorphisms) when there is a coexistence of attractors. Based on the work of Helena Nusse and James Yorke, it presents existence and characterization results for a certain kind of basin boundaries (namely, the Wada boundaries). The key feature of their approach is to redefine the idea of a basin boundary by introducing the notion of a `basin cell', which bypasses the problem of exactly locating the attractor of a system, which is often either not well-defined or hard to locate in practice. Moreover, the basin cells and their boundaries are characterized by utilizing the stable and unstable manifolds of the system, which are easier to locate by numerical methods, and thus their method provides both numerically verifiable characteristics and algorithms for computation. / text

Computer dialogs communicate important security messages, but their excessive use has produced habituation: a strong tendency by computer users to ignore security dialogs. Unlike physical warnings, whose design and use is regulated by law and based on years of research, computer security dialogs are often designed in an arbitrary manner. We need scientific solutions to produce dialogs that users will heed and understand. Currently, we lack an understanding of the factors that drive users’ attention to security dialogs, and how to counteract habituation. Studying computer security behavior is difficult because a) users are more likely to expose themselves to risk in a lab experiment than in daily life, b) the size of observed effects is usually very small, which makes it necessary to collect many observations, and c) it is complex to balance research interests and the ethical duty not to harm. My thesis makes two contributions: a novel methodology to study behavioral responses to security dialogs in a realistic, ethical way with high levels of ecological validity, and a novel technique to increase and retain attention to security dialogs, even in the presence of habituation.

This thesis is structured as follows. Chapter 1 introduces fractal sets before recalling basic mathematical concepts from dynamical systems, measure theory, dimension theory and probability theory. In Chapter 2 we give an overview of both deterministic and stochastic sets obtained from iterated function systems. We summarise classical results and set most of the basic notation. This is followed by the introduction of random graph directed systems in Chapter 3, based on the single authored paper [T1] to be published in Journal of Fractal Geometry. We prove that these attractors have equal Hausdorff and upper box-counting dimension irrespective of overlaps. It follows that the same holds for the classical models introduced in Chapter 2. This chapter also contains results about the Assouad dimensions for these random sets. Chapter 4 is based on the single authored paper [T2] and establishes the box-counting dimension for random box-like self-affine sets using some of the results and the notation developed in Chapter 3. We give some examples to illustrate the results. In Chapter 5 we consider the Hausdorff and packing measure of random attractors and show that for reasonable random systems the Hausdorff measure is zero almost surely. We further establish bounds on the gauge functions necessary to obtain positive or finite Hausdorff measure for random homogeneous systems. Chapter 6 is based on a joint article with J. M. Fraser and J.-J. Miao [FMT] to appear in Ergodic Theory and Dynamical Systems. It is chronologically the first and contains results that were extended in the paper on which Chapter 3 is based. However, we will give some simpler, alternative proofs in this section and crucially also find the Assouad dimension of some random self-affine carpets and show that the Assouad dimension is always `maximal' in both measure theoretic and topological meanings.

Este trabalho contém um estudo sobre equações de ondas fracamente dissipativas definidas em domínios de fronteira móvel ∂2u/∂t2/ + η∂u/∂t - Δu + g(u) = f(x,t), (x,t) ∈ ^Dτ, onde ^Dτ = ∪t∈(τ,+ ∞) Ot X . Dizemos que domínio Dτ possui fronteira móvel se admitirmos que a fronteira Γt de de Ot varia em relação a t. Nossa contribuição é dividida em três etapas. 1 - Provamos que o problema munido da condição de fronteira de Dirichlet é bem posto no sentido de Hadamard (existência global, unicidade e dependência contínua dos dados) para soluções fortes e fracas. Nessa etapa utilizamos um método clássico que transforma o domínio dependente de t em um domínio fixo. Como consequência observamos que o sistema é essencialmente não autônomo. 2 - Buscamos uma teoria de sistemas dinâmicos não autônomos para estudar o operador solução do problema como um processo U(t; τ) : Xτ → Xτ, t≥ τ, definido em espaços de fase Xt = H01(Ot) × L2(Ot) que são dependentes do tempo t. 3 - No contexto da dinâmica de longo prazo encontramos hipóteses para garantir que o sistema dinâmico associado ao problema de ondas em domínios de fronteira móvel possui um atrator pullback. Basicamente admitimos que o domínio é crescente e \"time-like\". Salientamos que o nosso trabalho é o primeiro que estuda tais equações de ondas sob o ponto de vista de sistemas dinâmicos não-autônomos. Para equações parabólicas, resultados no mesmo contexto foram obtidos anteriormente por Kloeden, Marín-Rubio e Real [JDE 244 (2008) 2062-2090] e Kloeden, Real e Sun [JDE 246 (2009) 4702-4730]. Entretanto o nosso problema á hiperbólico e nã possui a regularidade das equações parabólicas. / In this work we study a weakly dissipative wave equation defined in domains with moving boundary ∂2u/∂t2/ + η∂u/∂t - Δu + g(u) = f(x,t), (x,t) ∈ Dτ, where D&tau> = ∪t∈(τ,+ ∞) Ot X . We says that a domain D&tau has moving boundary if the boundary &Gama;t of Ot varies with respect to t. Our contribution is threefold. 1 - We prove that the wave equation equipped with Dirichlet boundary condition is well-posed in the sense of Hadamard (global existence, uniqueness and continuous dependence with respect to data) for weak and strong solutions. This is done by using a classical argument that transforms the time dependent domain in a fixed domain. As a consequence we see that the problem is essentially non-autonomous. 2 -We find a theory of non-autonomous dynamical systems in order to study the solution operator as a process U(t; τ) : Xτ → Xsub>t, t≥τ, defined in time dependent phase spaces Xt = H01 (Ot) × L2.(Ot. 3 - In the context of long-time behavior of solutions we find suitable conditions to guarantee the existence of a pullback attractor. Roughly speaking, we assume the domain Q is expanding and time-like. We emphasize that our work is the first one that consider wave equations in noncylindrical domains as non-autonomous dynamical systems. With respect to parabolic equations, similar results were early obtained by Kloeden, Marín-Rubio and Real [JDE 244 (2008) 2062-2090] and Kloeden, Real and Sun [JDE 246 (2009) 4702-4730]. However our problem is hyperbolic and does not enjoy regularity properties as the parabolic ones.

The focus of this thesis is to theoretically and experimentally investigate two new schemes of synchronizing chaotic attractors using chaotically operating diode resonators. The first method, called synchronization using control, is shown for the first time to experimentally synchronize dynamical systems. This method is an economical scheme which can be viably applied to low dimensional dynamical systems. The other, unidirectional coupling, is a straightforward means of synchronization which can be implemented in fast dynamical systems where timing is critical. Techniques developed in this work are of fundamental importance for future problems regarding high dimensional chaotic dynamical systems or arrays of mutually linked chaotically operating elements.

Neste trabalho estudamos a continuidade da dinâmica assintótica relativamente a perturbações e, em particular, exploramos a obtenção de melhorias para as taxas de convergência de atratores globais através da introdução de fatores de correção, inspirados pelos resultados da teoria de homogeneização e nos trabalhos de (BABIN; VISHIK, 1992) e (CARVALHO; CHOLEWA, 2011), e através da introdução de mecanismos que melhoram a transferência da taxa de convergência de semigrupos para a taxa de convergência de atratores, inspirados pelos trabalhos (SANTAMARÍA, 2013) e (BABIN; VISHIK, 1992; CARVALHO; CHOLEWA, 2011). A proposta inicial está centrada na obtenção de melhores taxas de convergência de atratores globais através da obtenção de equiatração e da melhoria da taxa de convergência dos semigrupos. Para isto, buscamos melhorar a taxa de convergência do resolvente dos operadores setoriais envolvidos, por meio de uma perturbação singular do resolvente limite que ainda gere uma família de operadores setoriais com resolventes que aproximam o resolvente do problema limite e aproximam melhor os resolventes das perturbações iniciais. Feito isto, obtemos uma melhora imediata de convergência dos semigrupos lineares, depois dos não lineares (através da fórmula da variação das constantes). Motivados pelos resultados de (SANTAMARÍA, 2013), que oferecem uma menor perda na transferência das taxas de convergência dos semigrupos para as taxas de convergência dos atratores, buscamos melhor compreender a propriedade Lipschitz Shadowing, que é responsável direta pela obtenção da taxa de convergência dos atratores diretamente da taxa de convergência dos semigrupos. Isto nos levou a descobrir que podemos obter as propriedade Lipschitz Shadowing e estabilidade estrutural para perturbações Lipschitz de semigrupos Morse-Smale. / Here we compare the continuity of the asymptotic dynamics with respect to perturbations and, in particular, we explored to obtain improvement of rates of convergence of the global attractor through the introduction of correction factors, inspired by the results of homogenization theory and work of (BABIN; VISHIK, 1992) and (CARVALHO; CHOLEWA, 2011), and the introduction of mechanisms that improve the transference of the convergence rate of semigroups to the convergence rate of attractors, inspired by the work of (SANTAMARÍA, 2013) and (BABIN; VISHIK, 1992; CARVALHO; CHOLEWA, 2011). The initial proposal is focused on achieving best rates of convergence of the global attractors by obtaining equi-atraction and improving the convergence rate of semigroups. For this, we seek to improve the rate of convergence of the resolvents of sectorial operators, through a singular perturbation of the resolvent associated with the limit problem and generate a new family of sectorial operators whose resolvents both approximate the resolvent of the limit problem as they were closer to the resolvents the initial perturbation. Having done this, we obtain an immediate improvement of convergence of linear semigroups, after the non-linear (using the variation of constants formula). Motivated by the results of (SANTAMARÍA, 2013), which offer an improvement in obtaining convergence rates, we seek to study property better Lipschitz Shadowing, which is basically responsible for obtaining the distance of the attractors directly from the convergence rate of the semigroups. This has led us to discover that we can both preserve the Lipschitz Shadowing property under Lipschitz perturbations of Morse-Smale semigroups, and The geometric stability of the attractors.

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.

A geometric model of a reinjected cuspidal horseshoe is constructed, that resembles the standard horseshoe, but where the set of points that escape are now reinjected and contribute to richer dynamics. We show it is observed in the unfolding of a three-dimensional vector field possessing an inclination-flip homoclinic orbit with a resonant hyperbolic equilibrium. We use techniques from classical dynamical systems theory and rigorous computational symbolic dynamics with algebraic topology to show that for suitable parameters the flow contains a strange attractor. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2016. / FAU Electronic Theses and Dissertations Collection

In this work, we develop an extension of the generalized Fourier transform for exterior domains due to T. Ikebe and A. Ramm for all dimensions n>2 to study the Laplacian, and fractional Laplacian operators in such a domain. Using the harmonic extension approach due to L. Caffarelli and L. Silvestre, we can obtain a localized version of the operator, so that it is precisely the square root of the Laplacian as a self-adjoint operator in L2 with DIrichlet boundary conditions. In turn, this allowed us to obtain a maximum principle for solutions of the dissipative two-dimensional quasi-geostrophic equation the exterior domain, which we apply to prove decay results using an adaptation of the Fourier Splitting method of M.E. Schonbek. / by Leonardo Kosloff. / Thesis (Ph.D.)--Florida Atlantic University, 2012. / Includes bibliography. / Mode of access: World Wide Web. / System requirements: Adobe Reader.