<p>In this thesis I present novel mechanisms for certain computational capabilities of the cerebral cortex, building on the established notion of attractor memory. A sparse binary coding network for generating efficient representation of sensory input is presented. It is demonstrated that this network model well reproduces receptive field shapes seen in primary visual cortex and that its representations are efficient with respect to storage in associative memory. I show how an autoassociative memory, augmented with dynamical synapses, can function as a general sequence learning network. I demonstrate how an abstract attractor memory system may be realized on the microcircuit level -- and how it may be analyzed using similar tools as used experimentally. I demonstrate some predictions from the hypothesis that the macroscopic connectivity of the cortex is optimized for attractor memory function. I also discuss methodological aspects of modelling in computational neuroscience.</p>

In the presented thesis, we study an application of nonstandard analysis to dynamical systems, in particular to ω-limit set, stability and global attractor. We recall the definition and properties of elementary embedding, in detail ex- plore the introduction of infinitesimals to the real line and study metric spaces using nonstandard methods, in particular continuity and compactness which are closely related to the theory of dynamical systems. Last we attend to dynamical systems and present nonstandard characterizations of some of its properties such as asymptotic compactness and dissipativity and using these characterizations we prove one of the basic results of this theory - existence of a global attractor. 1

Neste trabalho estudamos medidas invariantes para aplicações unimodais. Estamos especialmente interessados em detectar as situações que levam uma aplicação unimodal a não possuir uma medida piac, ou seja, uma medida de probabilidade invariante e absolutamente contínua em relação à medida de Lebesgue. Mostramos que a ordem do ponto crítico e a sua capacidade de recorrência são os fatores mais relevantes nesta questão. Os valores das derivadas da aplicação nos pontos periódicos tem uma infuência menor, mas suficiente para garantir que numa mesma classe de conjuga ção topológica podem existir duas aplicações unimodais com ponto crítico de mesma ordem, sendo que uma delas possui medida piac e a outra não possui. A capacidade de recorrência do ponto crítico, talvez o principal fator nesta questão, depende de aspectos combinatórios bem sofisticados. As ferramentas principais para analisar estes aspectos envolvem os conceitos de tempos de corte e de aplicações kneading. A existência ou não de medidas piac é uma propriedade de natureza métrica, e por isto, é necessário que tenhamos controle de como os iterados da aplicação unimodal distorcem a medida de Lebesgue. Então precisamos usar ferramentas de controle de distorção que incluem principalmente os Princípios de Koebe. Um ponto culminante deste trabalho diz respeito a relação entre existência de mediada piac e existência de atratores selvagens, isto é: atratores métricos que não são atratores topológicos e vice versa. Usamos aqui um argumento probabilístico de rara beleza. / In this work we study invariant measures for unimodal maps. We are especially interested in detecting situations that cause a unimodal map not to have a piac measure, i.e., a measure that is Probability Invariant and Absolutely Continuous with respect to Lebesgue measure. We show that the order of the critical point and its capacity for recurrence are the most relevant factors in this matter. The values of the derivatives of the map at periodic points have a small inuence, but enough to ensure that within a single class of topological conjugacy, there can be two unimodal maps with critical points of the same order, one of which has a piac measure and the other does not. The recurrence capacity of the critical point depends on very sophisticated combinatorial aspects and is probably the main factor in this issue. The main tools to analyze these aspects involve the concepts of cutting times and kneading maps. The existence of piac measures is a property of metric nature, and for this reason we need to have control of how iterations of the unimodal map distort the Lebesgue measure. We therefore need to use distortion control tools, including especially the Principles of Koebe. A culmination of this work concerns the relationship between existence of piac measures and the existence of wild attractors, i.e., metric attractors that not are topological attractors. Here we use a probabilistic argument of rare beauty.

Este trabalho se propõe a estudar o comportamento assintótico dos sistemas dinâmicos autônomos respaldado na Teoria das Dimensões. Mais precisamente, vamos compreender de que maneira nos é útil limitar a dimensão fractal do atrator global de um semigrupo a fim de estudar a dinâmica em dimensão finita, sem que se perca informações sobre a dinâmica ao fazê-lo. Para tanto, o Teorema de Mañé tem um papel decisivo junto às propriedades da dimensão de Hausdorff e a da dimensão fractal; nos permitindo encontrar uma projeção cuja restrição ao atrator é injetora sobre um espaço de dimensão finita. Constatamos ainda que esta abordagem por projeções se aplica largamente a semigrupos originados de equações diferenciais em espaços de Banach de dimensão infinita. / In this work, we study the asymptotic behavior of autonomous dynamical systems supported on the Dimension Theory. More precisely, we understand how fractal dimension finiteness of the global attractor of a semigroup can be used to study the dynamics in finite dimension, without losing information on the dynamics in doing so. For this purpose, the Mañés Theorem plays a decisive role considering the Hausdorff dimension properties and the fractal dimension; thanks to which we managed to find a projection whose restriction to the attractor is an injective application over a finite dimensional space. Besides, we also acknowledge that this projections approach is largely applied to semigroups arrising from differential equations in infinite dimensional Banach spaces.

Neste trabalho estudamos medidas invariantes para aplicações unimodais. Estamos especialmente interessados em detectar as situações que levam uma aplicação unimodal a não possuir uma medida piac, ou seja, uma medida de probabilidade invariante e absolutamente contínua em relação à medida de Lebesgue. Mostramos que a ordem do ponto crítico e a sua capacidade de recorrência são os fatores mais relevantes nesta questão. Os valores das derivadas da aplicação nos pontos periódicos tem uma infuência menor, mas suficiente para garantir que numa mesma classe de conjuga ção topológica podem existir duas aplicações unimodais com ponto crítico de mesma ordem, sendo que uma delas possui medida piac e a outra não possui. A capacidade de recorrência do ponto crítico, talvez o principal fator nesta questão, depende de aspectos combinatórios bem sofisticados. As ferramentas principais para analisar estes aspectos envolvem os conceitos de tempos de corte e de aplicações kneading. A existência ou não de medidas piac é uma propriedade de natureza métrica, e por isto, é necessário que tenhamos controle de como os iterados da aplicação unimodal distorcem a medida de Lebesgue. Então precisamos usar ferramentas de controle de distorção que incluem principalmente os Princípios de Koebe. Um ponto culminante deste trabalho diz respeito a relação entre existência de mediada piac e existência de atratores selvagens, isto é: atratores métricos que não são atratores topológicos e vice versa. Usamos aqui um argumento probabilístico de rara beleza. / In this work we study invariant measures for unimodal maps. We are especially interested in detecting situations that cause a unimodal map not to have a piac measure, i.e., a measure that is Probability Invariant and Absolutely Continuous with respect to Lebesgue measure. We show that the order of the critical point and its capacity for recurrence are the most relevant factors in this matter. The values of the derivatives of the map at periodic points have a small inuence, but enough to ensure that within a single class of topological conjugacy, there can be two unimodal maps with critical points of the same order, one of which has a piac measure and the other does not. The recurrence capacity of the critical point depends on very sophisticated combinatorial aspects and is probably the main factor in this issue. The main tools to analyze these aspects involve the concepts of cutting times and kneading maps. The existence of piac measures is a property of metric nature, and for this reason we need to have control of how iterations of the unimodal map distort the Lebesgue measure. We therefore need to use distortion control tools, including especially the Principles of Koebe. A culmination of this work concerns the relationship between existence of piac measures and the existence of wild attractors, i.e., metric attractors that not are topological attractors. Here we use a probabilistic argument of rare beauty.

O propósito deste trabalho é estudar a dinâmica assintótica de problemas parabólicos em domínios tipo dumbbell. Para isto primeiro estudaremos a semi-continuidade superior de atratores para problemas parabólicos com condição de fronteira do tipo Neumann homogênea e depois estudaremos a existência de equilíbrios estáveis não-constantes para problemas de reação-difusão com condições de fronteira tipo Neumann não-lineares. / The aim of this work is to study the asymptotic dynamics of parabolic problems in dumbbell type domains. To that end firstly, we study upper semicontinuity of attractors for parabolic problems with homogeneous Neumann boundary conditions and afterwards we study the existence of stable nonconstant equilibria for reaction-diffusion problems with nonlinear Neumann boundary conditions.

Submitted by (edna.saturno@ufrpe.br) on 2017-02-22T13:35:42Z No. of bitstreams: 1 Tatiane do Nascimento Medeiros.pdf: 1280315 bytes, checksum: 40c2ed259272c7c9980c97d49c5773b1 (MD5) / Made available in DSpace on 2017-02-22T13:35:42Z (GMT). No. of bitstreams: 1 Tatiane do Nascimento Medeiros.pdf: 1280315 bytes, checksum: 40c2ed259272c7c9980c97d49c5773b1 (MD5) Previous issue date: 2005-02-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Studies on fish eggs and larvae are relevant for a thorough knowledge about fish biology and systematics, and the identification and delimitation of spawning areas. Fish larvae and many other planktonic organisms present photopositive behaviour, which makes possible their attraction and capture by using artificial lights. The objective of this study was to comparatively analyse the use of a light-attractor model and a conycal-cylindrical plankton net for collecting fish larvae, in the estuary of Formoso river, in the state of Pernambuco. Samples were taken weekly, along the different moon phases, in October-November 2003 (cycle 1), March-April 2004 (cycle 2), July-August 2004 (cycle 3) and November 2004 (cycle 4), at a fixed station in the estuary located in front of the Cruzeiro do Reduto, between the Ariquindá river estuary and Pedra beach. The light-attractor was used during different randomly ordered light periods of 0, 1, 2, 5, 10 and 20 minutes; at 10 minutes intervals between each period, while plankton towing was conducted at the beginning and end of the attractor’s use. With the light-attractor, a total of 455 fish larvae were collected, distributed in sixteen families, 21.76% of which belonged to Engraulidae, 21.10% of Clupeidae, 14.06% of Gobiidae, and 12.97% of Gerreidae, the most abundant taxa. With he plankton net, 427 larvae were sampled, within twenty families, the most abundant being Engraulidae (29.7%), Clupeidae (9.37%), Gobiidae (23.4%), Gerreidae (4.21%), Eleotridae (4.21%) and Atherinopsidae (4.45%), as well as Pleuronectiformes (4.45%). Higher larvae occurrence was recorded for the light-attractor under first quarter (27.91%) and last quarter moon (38.46%), and new moon (33,28%), for the plankton net. The light-attractor was more efficient under five minute light exposure time, when a greater abundance of larvae was recorded. The individuals sampled with plankton net were smaller than those with light attraction, what may be attributed to their escape from the net during towing. The use of light-attractor was efficient, since the individuals sampled presented a better physical fitness than those from net towing, favouring their identification. Moreover, it allowed collecting live specimens which may be cultured in the laboratory, used for growth and physiology studies, as well as obtaining ontogenic series, of extreme value for the morphological characterization and identification of early life stages of fish development. / Os estudos de ovos e larvas são relevantes para o conhecimento global da biologia e sistemática das espécies de peixes e para a identificação e delimitação de áreas de desova. As larvas de peixes e muitos outros organismos planctônicos apresentam características de comportamento fotopositivo, o que possibilita a utilização de luzes artificiais para atraí-los e capturá-los. O objetivo deste trabalho foi analisar comparativamente o uso de um modelo de atrator luminoso e de rede de plâncton cônico-cilíndrica, para a coleta de larvas de peixes, no estuário do rio Formoso, em Pernambuco. As coletas foram semanais, ao longo das quatro fases da lua, nos meses de outubro-novembro/2003 (ciclo 1), março-abril/2004 (ciclo 2), julho-agosto/2004 (ciclo 3) e novembro/2004 (ciclo 4), em um ponto fixo de amostragem no estuário, localizado em frente ao Cruzeiro do Reduto, entre a foz do rio Ariquindá e a praia da Pedra. O atrator luminoso foi utilizado com diferentes tempos de exposição à luz: 0, 1, 2, 5, 10 e 20 minutos; com intervalos de 10 minutos entre cada exposição, ordenadas de modo aleatório a cada coleta, enquanto os arrastos foram realizados no início a ao término do uso do atrator luminoso. Com o atrator luminoso, foi capturado um total de 455 larvas de peixes, distribuídas entre dezesseis famílias, das quais Engraulidae (21,76%), Clupeidae (21,10%), Gobiidae (14,06%) e Gerreidae (12,97%) foram as mais abundantes. Com a rede de arrasto, foram capturadas 427 larvas, contemplando vinte famílias, sendo as mais abundantes Engraulidae (29,7%), Clupeidae (9,37%), Gobiidae (23,4%), Gerreidae (4,21%), Eleotridae (4,21%) e Atherinopsidae (4,45%), além da ordem Pleuronectiformes (4,45%). Maiores ocorrências de larvas foram observadas nas luas crescente (27,91%) e minguante (38,46%), para o atrator luminoso, e na lua nova (33,28%), para a rede de arrasto. O atrator luminoso foi mais eficiente com tempos de exposição à luz acima de 5 minutos, nos quais foi evidenciada maior abundância de larvas. Os indivíduos capturados com rede de arrasto apresentaram tamanhos menores que com o atrator luminoso, o que pode ser atribuído a um maior escape da rede durante os arrastos. O uso do atrator luminoso foi eficaz, uma vez que os exemplares capturados apresentam melhores características corpóreas que aqueles capturados com arrasto, favorecendo sua identificação. Permitiu, ainda, a coleta de exemplares vivos, que permite seu cultivo em laboratório, uso em estudos de crescimento e fisiologia, bem como na obtenção de séries ontogênicas, imprescindíveis à caracterização morfológica e identificação das fases iniciais de desenvolvimento de peixes.

In this paper we establish the regularity, exponential stability of global (weak) solutions and existence of uniform compact attractors of semiprocesses, which are generated by the global solutions, of a two-parameter family of operators for the nonlinear 1-d non-autonomous viscoelasticity. We employ the properties of the analytic semigroup to show the compactness for the semiprocess generated by the global solutions.

Chaos theory is well known as one of three revolutions in physical sciences in 20th-century, as one physicist called it: Relativity eliminated the Newtonian illusion of absolute space and time; quantum theory eliminated the Newtonian dream of a controllable measurable process; and chaos eliminates the Laplacian fantasy of deterministic predictability". Specially, when chaos synchronization was found in 1991, chaos theory becomes more and more attractive. Chaos has been widely applied to many scientific disciplines: mathematics, programming, microbiology, biology, computer science, economics, engineering, finance, philosophy, physics, politics, population dynamics, psychology, and robotics. One of most important engineering applications is secure communication because of the properties of random behaviours and sensitivity to initial conditions of chaos systems. Noise-like dynamical behaviours can be used to mask the original information in symmetric cryptography. Sensitivity to initial conditions and unpredictability make chaotic systems very suitable to construct one-way function in public-key cryptography. In chaos-based secure communication schemes, information signals are masked or modulated (encrypted) by chaotic signals at the transmitter and the resulting encrypted signals are sent to the corresponding receiver across a public channel (unsafe channel). Perfect chaos synchronization is usually expected to recover the original information signals. In other words, the recovery of the information signals requires the receiver's own copy of the chaotic signals which are synchronized with the transmitter ones. Thus, chaos synchronization is the key technique throughout this whole process. Due to the difficulties of generating and synchronizing chaotic systems and the limit of digital computer precision, there exist many challenges in chaos-based secure communication. In this thesis, we try to solve chaos generation and chaos synchronization problems. Starting from designing chaotic and hyperchaotic system by first-order delay differential equation, we present a family of novel cell attractors with multiple positive Lyapunov exponents. Compared with previously reported hyperchaos systems with complex mathematic structure (more than 3 dimensions), our system is relatively simple while its dynamical behaviours are very complicated. We present a systemic parameter control method to adjust the number of positive Lyapunov exponents, which is an index of chaos degree. Furthermore, we develop a delay feedback controller and apply it to Chen system to generate multi-scroll attractors. It can be generalized to Chua system, Lorenz system, Jerk equation, etc. Since chaos synchronization is the critical technique in chaos-based secure communication, we present corresponding impulsive synchronization criteria to guarantee that the receiver can generate the same chaotic signals at the receiver when time delay and uncertainty emerge in the transmission process. Aiming at the weakness of general impulsive synchronization scheme, i.e., there always exists an upper boundary to limit impulsive intervals during the synchronization process, we design a novel synchronization scheme, intermittent impulsive synchronization scheme (IISS). IISS can not only be flexibly applied to the scenario where the control window is restricted but also improve the security of chaos-based secure communication via reducing the control window width and decreasing the redundancy of synchronization signals. Finally, we propose chaos-based public-key cryptography algorithms which can be used to encrypt synchronization signals and guarantee their security across the public channel.

This dissertation is a contribution to the the study of the longtime dynamics of evolutionary equations in unbounded domains. It is of particular interest to prove the existence of global attractors for solutions of such equations. Th this end one need in general some type of asymtotical compactness. In the case the evolutionary PDE is defined on a bounded domain, asymptotical compactness follows from the regularity estimates and the compactnes of Sobolev embeddings and therefore the existence of attractors has been established for most of the disipative equations of mathematocal physics in a bounded domain. The problem is more challenging when the domain is unbounded since the Sobolev embeddings are no longer comapct, so that the usual regularity estimates may not be sufficient.To overcome this obstacle of compactness, A.V. Babin and M.I. Vishik introduced some weighted Sobolev spaces. In their pioneering paper, Proc. Roy. Soc. Edinb.