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.

Τα πρωτότυπα αποτελέσματα της διατριβής περιέχονται στα κεφ. 2, 3 και 4. Στο κεφ. 2 μελετούμε με αριθμητικές μεθόδους ένα 3-διάστατο διακριτό μοντέλο μακροοικονομίας με σταθερές ισοτιμίες. Χρησιμοποιώντας μια μέθοδο πλέγματος, βρίσκουμε την περιοχή ευστάθειας στον παραμετρικό χώρο, προσδιορίζουμε την καμπύλη διακλαδώσεων Hopf-Neimark και θεωρούμε σύντομα την εμφάνιση “γλωσσών” Arnold. Υπολογίζονται διαγράμματα διακλαδώσεων και εκθέτη Lyapunov που δίνουν πληροφορίες για τους επιχειρηματικούς κύκλους και την πολύπλοκη δυναμική του μοντέλου και. παρουσιάζουμε παραδείγματα κυκλικών και χαοτικών ελκυστών. Στο κεφ. 3 μελετούμε με τις ίδιες μεθόδους ένα διακριτό μοντέλο αλληλεπίδρασης περιοχών με σταθερές ισοτιμίες, επέκταση του προηγούμενου μοντέλου σε 5 διαστάσεις. Στόχος ήταν να δείξουμε πόσο εφικτή και αποτελεσματική είναι μία αριθμητική μελέτη για ηπίως πολυδιάστατα διακριτά δυναμικά συστήματα με πολλές παραμέτρους. Βρήκαμε ότι η κίνηση κεφαλαίων δεν αρκεί για τη δημιουργία κύκλων όταν είναι χαμηλή η εμπορική αλληλεπίδραση. Το κατώφλι εμπορίου προβλέπεται περίπου στο 15% των εμπορικών συναλλαγών. Αντίθετα, το μοντέλο δεν προβλέπει αναγκαίο ελάχιστο επίπεδο κίνησης κεφαλαίων για την εμφάνιση των κύκλων. Δίνουμε παραδείγματα διαγραμμάτων διακλάδωσης και εκθέτη Lyapunov που δείχνουν την εμφάνιση κύκλων ή ακολουθίας διπλασιασμού περιόδου, και παραδείγματα της ανάπτυξής τους. Το κεφ. 4 περιέχει σύντομη περιγραφή συμπληρωματικών αποτελεσμάτων στα παραπάνω μοντέλα, και στα αντίστοιχα μοντέλα μεταβλητής ισοτιμίας συναλλάγματος, καθώς και κατευθύνσεις μελλοντικής έρευνας. Στο κεφ. 5 περιγράφονται σύντομα οι υπολογιστικές τεχνικές που χρησιμοποιήσαμε. Η διατριβή δείχνει την αποτελεσματικότητα της αριθμητικής προσέγγισης για πολυδιάστατα διακριτά μοντέλα. / The original results of the dissertation are contained in ch. 2, 3 and 4, and concern mainly the problem of business cycles. In ch. 2 we explore numerically a 3D discrete Kaldorian macrodynamic model of open economy with fixed exchange rates. Using a grid search method we determine the stability region in parameter space, and the Hopf-Neimark bifurcation curve, and discuss briefly the occurrence of Arnold tongues. Bifurcation and Lyapunov exponent diagrams are computed providing information on the business cycles and illustrating the complex dynamics involved. Examples of cycles and chaotic attractors are presented. In ch. 3 we explore a 5D extension of the previous model using the same methods. The aim was to demonstrate the feasibility and effectiveness of the numerical approach for discrete dynamical systems of moderately high dimensionality and several parameters. We found that capital movement is not sufficient to generate interregional business cycles when trade interaction is low. The trade threshold is predicted at about 15% of trade transactions. By contrast, no minimum level of capital mobility exists as a requirement for the emergence of business cycles. Examples of bifurcation and Lyapunov exponent diagrams illustrating the occurrence of cycles or period doubling, and examples of their development, are given. Ch. 4 contains a short description of complementary results on the above models, and on two other models which extend the previous models to the case of flexible exchange rates, as well as some lines of future research. In ch. 5, the computational techniques employed in the present study are briefly described. The dissertation indicates the effectiveness of the numerical approach for high dimensional discrete models.

The aim of this thesis is to develop the dimension theory of self-affine carpets in several directions. Self-affine carpets are an important class of planar self-affine sets which have received a great deal of attention in the literature on fractal geometry over the last 30 years. These constructions are important for several reasons. In particular, they provide a bridge between the relatively well-understood world of self-similar sets and the far from understood world of general self-affine sets. These carpets are designed in such a way as to facilitate the computation of their dimensions, and they display many interesting and surprising features which the simpler self-similar constructions do not have. For example, they can have distinct Hausdorff and packing dimensions and the Hausdorff and packing measures are typically infinite in the critical dimensions. Furthermore, they often provide exceptions to the seminal result of Falconer from 1988 which gives the `generic' dimensions of self-affine sets in a natural setting. The work in this thesis will be based on five research papers I wrote during my time as a PhD student. The first contribution of this thesis will be to introduce a new class of self-affine carpets, which we call box-like self-affine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on self-affine carpets, but also helps to reconcile the `exceptional constructions' with Falconer's singular value function approach in the generic case. This will appear in Chapter 2 and is based on a paper which appeared in 'Nonlinearity' in 2012. In Chapter 3 we continue studying the dimension theory of self-affine sets by computing the Assouad and lower dimensions of certain classes. The Assouad and lower dimensions have not received much attention in the literature on fractals to date and their importance has been more related to quasi-conformal maps and embeddability problems. This appears to be changing, however, and so our results constitute a timely and important contribution to a growing body of literature on the subject. The material in this Chapter will be based on a paper which has been accepted for publication in 'Transactions of the American Mathematical Society'. In Chapters 4-6 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1-variable attractors, with the aim of developing the dimension theory of self-affine carpets in these directions. In order to put our work into context, in Chapter 4 we consider inhomogeneous self-similar sets and significantly generalise the results on box dimensions obtained by Olsen and Snigireva, answering several questions posed in the literature in the process. We then move to the self-affine setting and, in Chapter 5, investigate the dimensions of inhomogeneous self-affine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of self-similar sets. The material in Chapter 4 will be based on a paper which appeared in 'Studia Mathematica' in 2012, and the material in Chapter 5 is based on a paper, which is in preparation. Finally, in Chapter 6 we consider random self-affine sets. The traditional approach to random iterated function systems is probabilistic, but here we allow the randomness in the construction to be provided by the topological structure of the sample space, employing ideas from Baire category. We are able to obtain very general results in this setting, relaxing the conditions on the maps from `affine' to `bi-Lipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random self-similar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random self-affine carpets. The material in this Chapter will be based on a paper which has been accepted for publication by 'Ergodic Theory and Dynamical Systems'.

This thesis concerns an active research area within fractal geometry. In the first part, in Chapters 2 and 3, for directed graph iterated function systems (IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known. We give a constructive algorithm for calculating the set of gap lengths of any attractor as a finite union of cosets of finitely generated semigroups of positive real numbers. The generators of these semigroups are contracting similarity ratios of simple cycles in the directed graph. The algorithm works for any IFS defined on ℝ with no limit on the number of vertices in the directed graph, provided a separation condition holds. The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We obtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth packing moment of μ[subscript(u)], the self-similar measure at a vertex u, for the non-lattice case, with a corresponding limit for the lattice case. We do this (i) for any q ∈ ℝ if the strong separation condition holds, (ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negative matrix associated with the system is irreducible. In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum of μ[subscript(u)] to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper multifractal q box-dimension with respect to μ[subscript(u)], of the set consisting of all the intersections of the components of F[subscript(u)], is strictly less than the multifractal q Hausdorff dimension with respect to μ[subscript(u)] of F[subscript(u)].

Pas de résumé en français / It is generally maintained that one of cortex’ functions is the storage of a large number of memories. In this picture, the physical substrate of memories is thought to be realised in pattern and strengths of synaptic connections among cortical neurons. Memory recall is associated with neuronal activity that is shaped by this connectivity. In this framework, active memories are represented by attractors in the space of neural activity. Electrical activity in cortical neurones in vivo exhibits prominent temporal irregularity. A standard way to account for this phenomenon is to postulate that recurrent synaptic excitation and inhibition as well as external inputs are balanced. In the common view, however, these balanced networks do not easily support the coexistence of multiple attractors. This is problematic in view of memory function. Recently, theoretical studies showed that balanced networks with synapses that exhibit short-term plasticity (STP) are able to maintain multiple stable states. In order to investigate whether experimentally obtained synaptic parameters are consistent with model predictions, we developed a new methodology that is capable to quantify both response variability and STP at the same synapse in an integrated and statistically-principled way. This approach yields higher parameter precision than standard procedures and allows for the use of more efficient stimulation protocols. However, the findings with respect to STP parameters do not allow to make conclusive statements about the validity of synaptic theories of balanced working memory. In the second part of this thesis an alternative theory of cortical memory storage is developed. The theory is based on the assumptions that memories are stored in attractor networks, and that memories are not represented by network states differing in their average activity levels, but by micro-states sharing the same global statistics. Different memories differ with respect to their spatial distributions of firing rates. From this the main result is derived: the balanced state is a necessary condition for extensive memory storage. Furthermore, we analytically calculate memory storage capacities of rate neurone networks. Remarkably, it can be shown that crucial properties of neuronal activity and physiology that are consistent with experimental observations are directly predicted by the theory if optimal memory storage capacity is required.

Neste trabalho estudamos problemas de reação difusão em domínios unidimensionais que surgem de materiais compostos e obtemos resultados comparando os fluxos do problema original e do problema limite quando a difusão fica muito grande em partes do domínio. Provamos que os autovalores e autofunções do operador linear ilimitado associado à equação limite têm a propriedade de Sturm Liouville e provamos que as soluções do problema de reação difusão têm a propriedade do decrescimento do número de zeros ao longo do tempo. Estes resultados são usados para provar que as variedades instável e estável de pontos de equilíbrios são genericamente transversais e que o fluxo no atrator para o problema de reação difusão é genericamente estruturalmente estável. Estes fatos permitem obter a equivalência topológica dos fluxos restritos aos atratores dos problemas original e seu problema limite. / In this work we study some reaction-difusion problems in one dimensional domains that arise from composite materials. We obtain some results comparing the flux of the original problem and the flux of the limit problem when the difusion becomes large on parts of the physical domain. We prove that the eigenvalues and eigenfunctions of the linear unbounded operator associated with the equation have the Sturm Liouville property and also that the solutions of the reaction difusion equation have the property that the zeros do not increase with time. These results are used to obtain that the stable and unstable manifolds of equilibrium points are generically transversal and that the flux on the attractor for the reaction difusion problem is generically structurally stable. Using this we are able to prove the topological equivalence of the fluxs restricted to the attractors of the original and the limit problem.

Este trabalho é dedicado ao estudo do comportamento a longo prazo de uma equação de placas extensíveis acoplada a uma equação de calor do tipo hiperbólico. O problema corresponde a um modelo de termo-elasticidade baseado em teorias de calor do tipo não-Fourier. Considerando que efeitos de inércia de rotação estão presentes no modelo, mostramos que o efeito dissipativo do calor e suficiente para estabilizar exponencialmente o sistema, sem dissipações adicionais. Além disso, provamos que o sistema possui um atrator global de dimensão fractal finita e também atratores exponenciais. Nossos resultados generalizam e complementam diversos trabalhos existentes / This work is concerned with long-time dynamics of solutions of extensible plate equations with thermal memory. It corresponds to a model of thermoelasticity based on a theory of non-Fourier heat flux. By considering the case where rotational inertia is present we show that the thermal dissipation is sufficient to stabilize the system exponentially and guarantee the existence of a finite-dimensional global attractor. In addition the existence of an exponential attractor and some further properties are also considered. Our results complements several existing results

Neste trabalho estudamos um problema parabólico e um problema hiperbólico que não admitem unicidade de solução. Após garantir a existência de solução para cada um desses problemas, analisamos o comportamento assintótico de suas soluções por meio da teoria do atrator de trajetórias. Nossos resultados principais demonstram, sob hipóteses apropriadas, a semicontinuidade superior das famílias de atratores de trajetórias quando o coeficiente de difusão é grande. / In this work we study a parabolic problem and a hyperbolic problem that not admit uniqueness of solution. After to ensure existence of solution for each of these problems, we analyze the asymptotic behavior of their solutions by means of the theory of trajectory attractors. Our main results demonstrate, under appropriate assumptions, the upper semicontinuity of families of trajectory attractors when the diffusion coefficient is large.

Este trabalho é dedicado ao estudo do comportamento a longo prazo de uma classe de equações viscoelásticas não lineares com memória, da forma |\'upsilon IND. t\'| POT. ho\' \'upsilon IND. tt\' - DELTA \'upsilon\' - \'DELTA upsilon IND. tt\' + \'INT. SUP. t INF. \\tau\' upsilon (t- s) \'DELTA epsilon\' (s) ds = h, \'\\tau\' > 0, definida num domínio limitado de \'R POT. N\'. Tal classe de problemas foi estudada por diversos autores desde 2001, com \'\\tau = 0. Os resultados existentes são principalmente devotados à existência de soluções globais, decaimento da energia, com ou sem dissipações adicionais, existência com dados pequenos, entre outros. Entretanto, a questão da unicidade de soluções e existência de atratores globais não foram discutidas em trabalhos anteriores. No presente trabalho, apresentamos resultados de unicidade e existência de atratores globais para essa classe de problemas num contexto mais geral, incluindo o caso em que \'\\tau\' = -\'INFINITO\'. Além disso, incluímos um problema complementar, de quarta ordem onde estudamos a existência de atratores exponenciais / This work is concerned with the long-time behaviour of a class nonlinear viscoelastic equations of the form |\'upsilon IND. t\'| POT. ho\' \'upsilon IND. tt\' - DELTA \'upsilon\' - \'DELTA upsilon IND. tt\' + \'INT. SUP. t INF. \\tau\' upsilon (t- s) \'DELTA epsilon\' (s) ds = h, \'ho\' > 0, defined in a bounded domain of \'R POT. N\'. Such class of problems was studied by several authors since 2001, with \'\\tau\' = 0. Existing results are mainly devoted to global existence, energy decay, with or without additional dampings, existence with small data, among others. However, uniqueness and existence of global attractors were not considered previously. In the present work, we establish some results on the uniqueness of solutions and existence of global attractors in a more general setting, including \'\\tau\' = - \'INFINITY\'. In addition, we have added a second problem concerned with a fourth order equation where we study the existence of exponential attractors

Este trabalho investiga a influência da quantização em sistemas de controle em rede. São tratados problemas de estabilidade e estabilização de sistemas lineares de tempo discreto envolvendo quantização finita nas entradas da planta controlada, considerando dois tipos de quantizadores: os uniformes e os logarítmicos. Como consequência da quantização finita, ocorrem também efeitos de saturação e zonamorta dos sinais de entrada. Tais comportamentos não-lineares são considerados explicitamente na análise. Para plantas instáveis, o objetivo é estimar a região onde os estados estarão confinados em regime permanente. Esta região, denominada atrator dos estados, é estimada por meio de um conjunto elipsoidal. Ao mesmo tempo, determina-se um conjunto elipsoidal de condições iniciais admissíveis, para o qual se garante a convergência das trajetórias para o atrator em tempo finito. Primeiramente, esses conjuntos são determinados para o caso de um controlador dado e, posteriormente, sintetiza-se um controlador que minimiza o atrator. Em se tratando de plantas estáveis, investiga-se como o desempenho dinâmico é afetado pela quantização. Para tanto, utiliza-se como critério o coeficiente de decaimento exponencial que é garantido para o sistema. Nesta parte, excluem-se os comportamentos na região de saturação e na região da zona-morta. Primeiramente, o coeficiente de decaimento garantido é estimado para um sistema com controlador dado. Neste caso, faz-se uma análise de degradação de desempenho induzida pela quantização com relação ao comportamento do sistema em malha fechada sem quantização. Posteriormente, sintetiza-se um controlador que minimiza este coeficiente na presença da quantização. Na obtenção dos resultados, utilizam-se condições de setor respeitadas pelas não linearidades e formulam-se os problemas na forma de inequações matriciais que podem ser resolvidas a partir de problemas de otimização baseados em LMIs. / This work investigates the in uence of quantization over networked control systems. At rst, we tackle stability and stabilization problems of discrete-time linear systems involving nite quantization on the input of the controlled plant, considering two kinds of quantizers: uniform and logarithmic. As a consequence of the nite quantization, saturation and dead-zone e ects on the input signals are also present. These non-linear behaviors are explictly considered in the analysis. For unstable plants, the objective is to estimate the region where the states will be ultimately bounded. This region, which we call the attractor of the states, is estimated through an ellipsoidal set. Simultaneously, we determine an ellipsoidal set of admissible initial conditions, for which the trajectories will converge to the attractor in nite time. At rst, the sets are determined for the case where the controller is given and, in the sequel, a controller that minimizes the attractor is designed. When dealing with stable plants, we investigate how the dynamic performance is a ected by the quantization. To do that, we use as criterion the exponential decay rate which is guaranteed for the system. At this point, we exclude the behaviour in the saturation and deadzone regions. At rst, the guaranteed decay rate is estimated for a system where the controller is given. In this case, we analyze the deterioration of the performance in uenced by the quantization, compared to the behavior of the closed-loop system without quantization. In the sequel, a controller that minimizes that rate in the presence of quantization is designed. To obtain the results, we use sector conditions which are respected by the nonlinearities and we state the problems as matrix inequalities which can be solved using LMI-based optimization problems.