Systems of globally coupled phase oscillators can have robust attractors that are heteroclinic networks. Such a heteroclinic network is generated, where the phases cluster into three groups, within a specific regime of parameters when the phase oscillators are globally coupled using the function $g(\varphi) = -\sin(\varphi + \alpha) + r \sin(2\varphi + \beta)$. The resulting network switches between 30 partially synchronised states for a system of $N=5$ oscillators. Considering the states that are visited and the time spent at those states a spatio-temporal code can be generated for a given navigation around the network. We explore this phenomenon further by investigating the effect that noise has on the system, how this system can be used to generate a spatio-temporal code derived from specific inputs and how observation of a spatio-temporal code can be used to determine the inputs that were presented to the system to generate a given coding. We show that it is possible to find chaotic attractors for certain parameters and that it is possible to detail a genetic algorithm that can find the parameters required to generate a specific spatio-temporal code, even in the presence of noise. In closing we briefly explore the dynamics where $N>5$ and discuss this work in relation to winnerless competition.

L'analyse temps réel de la masse de données générée par les mécanismes de gestion de la vision dans les applications interactives est un problème toujours ouvert, promettant des avancées importantes dans des domaines aussi variés que la robotique, l’apprentissage à distance ou les nouvelles formes d’interactions avec l’utilisateur, sans clavier ni souris. Dans le cadre général de la vision, les algorithmes d’analyse de scène doivent trouver un compromis entre d'une part la qualité des résultats recherchés et d'autre part la quantité de ressources allouable aux différents tâches. Classiquement, ce choix est effectué à la conception du système (sous la forme de paramètres et d’algorithmes prédéfinis), mais cette solution limite le champ d’application de celui-ci. Une solution plus flexible consiste à utiliser un système de vision adaptatif qui pourra modifier sa stratégie d’analyse en fonction des informations disponibles concernant son contexte d’exécution. En conséquence, ce système doit posséder un mécanisme permettant de guider rapidement et efficacement l’exploration de la scène afin d’obtenir ces informations. Chez l’homme, les mécanismes de l’évolution ont mis en place le système d’attention visuelle. Ce système sélectionne les informations importantes afin de réduire la charge cognitive et les ambiguïtés d’interprétation de la scène. Nous proposons, dans cette thèse, un système d'attention visuelle, dont nous définissons l’architecture et les principes de fonctionnement. Ce dernier devra permettre l’interaction avec un système de vision afin qu’il adapte ses traitements en fonction de l’intérêt de chacun des éléments de la scène, i.e. ce que nous appelons saillance. A la croisée des chemins entre les modèles centralisés et hiérarchiques (ex : [Koch1985], puis [Itti1998]), et les modèles distribués et compétitifs (ex : [Desimone1995], puis [Deco2004, Rolls2006]), nous proposons un modèle hiérarchique, compétitif et non centralisé. Cette approche originale permet de générer un point de focalisation attentionnel à chaque pas de temps sans utiliser de carte de saillance ni de mécanisme explicite d’inhibition de retour. Ce nouveau modèle computationnel d'attention visuelle temps réel est basé sur un système d'équations proies / prédateurs, qui est bien adapté pour l'arbitrage entre un comportement attentionnel non déterministe et des propriétés de stabilité, reproductibilité, et réactivité. L'analyse des expérimentations menées est positive : malgré le comportement non-déterministe des équations proies / prédateurs, ce système possède des propriétés intéressantes de stabilité, reproductibilité, et réactivité, tout en permettant une exploration rapide et efficace de la scène. Ces propriétés ouvrent la possibilité d’aborder différents types d’applications allant de l’évaluation de la complexité d’images et de vidéos à la détection et au suivi d’objets. Enfin, bien qu’il soit destiné à la vision par ordinateur, nous comparons notre modèle au système attentionnel humain et montrons que celui-ci présente un comportement aussi plausible (voire plus en fonction du comportement défini) que les modèles classiques existants. / Providing real time analysis of the huge amount of data generated by computer vision algorithms in interactive applications is still an open problem. It promises great advances across a wide variety of fields : robotics, distance education, or new mouse-less and keyboard-less human computer interaction.When using scene analysis algorithms for computer vision, a trade-off must be found between the quality of the results expected, and the amount of computer resources allocated for each task. It is usually a design time decision, implemented through the choice of pre-defined algorithms and parameters. However, this way of doing limits the generality of the system. Using an adaptive vision system provides a more flexible solution as its analysis strategy can be changed according to the information available concerning the execution context. As a consequence, such a system requires some kind of guiding mechanism to explore the scene faster and more efficiently.In human, the mechanisms of evolution have generated the visual attention system which selects the most important information in order to reduce both cognitive load and scene understanding ambiguity.In this thesis, we propose a visual attention system tailored for interacting with a vision system (whose theoretical architecture is given) so that it adapts its processing according to the interest (or salience) of each element of the scene.Somewhere in between hierarchical salience based (ex: [Koch1985], then [Itti1998]) and competitive distributed (ex: [Desimone1995], then [Deco2004, Rolls2006]) models, we propose a hierarchical yet competitive and non salience based model. Our original approach allows the generation of attentional focus points without the need of neither saliency map nor explicit inhibition of return mechanism. This new real-time computational model is based on a preys / predators system. The use of this kind of dynamical system is justified by an adjustable trade-off between nondeterministic attentional behavior and properties of stability, reproducibility and reactiveness.Our experiments shows that despite the non deterministic behavior of preys / predators equations, the system exhibits interesting properties of stability, reproducibility and reactiveness while allowing a fast and efficient exploration of the scene. These properties are useful for addressing different kinds of applications, ranging from image complexity evaluation, to object detection and tracking. Finally, while it is designed for computer vision, we compare our model to human visual attention. We show that it is equally as plausible as existing models (or better, depending on its configuration).

This thesis is concerned with investigating new methods for the analysis of large-scale dynamical systems using convex optimization. The proposed methodology is based on composite Lyapunov theory and is computationally implemented using polynomial programming techniques. The main result of this work is the development of a system decomposition framework that makes it possible to analyze systems that are of such a scale that traditional methods cannot cope with. We begin by addressing the problem of model invalidation. A barrier certificate method for invalidating models in the presence of uncertain data is presented for both continuous and discrete time models. It is shown how a re-parameterization of the time dependent variables can improve the numerical conditioning of the underlying optimization problem. The main contribution of this thesis is the development of an automated dynamical system decomposition framework that permits us to verify the stability of systems that typically have a state dimension large enough to render traditional computational methods intractable. The underlying idea is to decompose a system into a set of lower order subsystems connected in feedback in such a manner that composite methods for stability verification may be employed. What is unique about the algorithm presented is that it takes into account both dynamics and the topology of the interconnection graph. In the first instance we illustrate the methodology with an ecological network and primal Internet congestion control scheme. The versatility of the decomposition framework is also highlighted when it is shown that when applied to a model of the EGF-MAPK signaling pathway it is capable of identifying biologically relevant subsystems in addition to stability verification. Finally we introduce stability metrics for interconnected dynamical systems based on the theory of dissipativity. We conclude by outlining a clustering based decomposition algorithm that explicitly takes into account the input and output dynamics when determining the system decomposition.

It is well known that some driven systems undergo transitions when a system parameter is changed adiabatically around a critical value. This transition can be the result of a fundamental change in the structure of the phase space, called a bifurcation. Most of these transitions are well classified in the theory of bifurcations. Among the driven systems, spatiotemporally periodic (STP) potentials are noteworthy due to the intimate coupling between their time and spatial components. A paradigmatic example of such a system is the Kapitza pendulum, which is a pendulum with an oscillating suspension point. The Kapitza pendulum has the strange property that it will stand stably in the inverted position for certain driving frequencies and amplitudes. A particularly interesting and useful STP system is an array of parallel electrodes driven with an AC electrical potential such that adjacent electrodes are 180 degrees out of phase. Such an electrode array embedded in a surface is called an Electric Curtain (EC). As we will show, by using two ECs and a quadrupole trap it is posible to produce an electric potential simular in form to that of the Kapitza pendulum. Here I will present the results of four related pieces of work, each focused on understanding the behaviors STP systems, long-range interacting particles, and long-range interacting particles in STP systems. I will begin with a discussion on the experimental results of the EC as applied to the cleaning of solar panels in extraterrestrial environments, and as a way to produce a novel one-dimensional multiparticle STP potential. Then I will present a numerical investigation and dynamical systems analysis of the dynamics that may be possible in an EC. Moving to a simpler model in order to explore the rudimentary physics of coulomb interactions in a STP potential, I will show that the tools of statistical mechanics may be important to the study of such systems to understand transitions that fall outside of bifurcation theory. Though the Coulomb and, similarly, gravitational interactions of particles are prevalent in nature, these long-range interactions are not well understood from a statistical mechanics perspective because they are not extensive or additive. Finally, I will present a simple model for understanding long-range interacting pendula, finding interesting non-equilibrium behavior of the pendula angles. Namely, that a quasistationary clustered state can exist when the angles are initially ordered by their index.

Title: A traffic flow with a bottelneck Author: Adam Kovařík Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Vladimír Janovský, DrSc. Supervisor's e-mail address: janovsky@karlin.mff.cuni.cz Abstract: In this paper we study a microscopic follow-the-leader traffic model on a circu- lar road with a bottleneck. We assume that all drivers are identical and overtaking is not permitted. We sketch a small part of the rich dynamics of the model including Hopf and Neimark-Sacker bifurcations. We introduce so called POM and quasi-POM solutions and an algorithm how to search them. The main goal of this work is to investigate how the optimal velocity model with a bottleneck deals with so called aggressive behavior of dri- vers. The effect of variable reaction time and a combination of both named factors is also tested. Using numerical simulations we'll find out that aggressiveness and faster reactions have positive effect on traffic flow. In the end we discuss models with two bottlenecks and with one extraordinary driver. Keywords: dynamical systems, ODEs, traffic flow, bottleneck, aggressiveness. 1

L'objectif de cette thèse est d'étudier les comportements typiques d'une famille de transformations du cylindre discret Z×T (où T=R/Z est le cercle de longueur un). Appliquez une rotation à chaque cercle du cylindre puis coupez tous les cercles en deux et déplacez une moitié de chaque cercle d'un niveau vers le bas et une moitié d'un niveau vers le haut. Nous utilisons pour cela des résultats existants en théorie des échanges d'intervalles et en théorie des surfaces de translation compactes. Tout d'abord, nous avons prouvé que pour presque toute suite bi-infinie de rotations, le système obtenu est conservatif (c'est à dire il n'y a pas d'ensemble errant de mesure strictement positive). Ensuite, nous avons prouvé que pour un ensemble Gδ-dense de paramètres, le système est en même temps conservatif, minimal et ergodique. Ce système a un rapport heuristique avec une famille de billards planaires, ainsi qu'une traduction dans des flots sur des surfaces de translation de genre infini. Cela est expliqué dans la thèse. / The main objective of this thesis is the study of the typical dynamical behaviour of a family of transformations on the discrete cylinder Z×T (where T=R/Z is the length one circle). Apply a rotation to every single circle of the cylinder then cut every circle in two and move half of each circle one level down and the other half one level up. To achieve this goal, we use existing results about interval exchange transformations and about compact translation surfaces. First, we proved that for almost every bi-infinite sequence of rotations, the obtained system is conservative (i.e. there is not wandering set of positive measure). Next, we proved that for a Gδ-dense set of parameters, the described system is ergodic, minimal and conservative. This system is heuristically related to a family of planar billiards, it has also a translation into flows on infinite genus translation surfaces.

Investigaremos algumas propriedades dinâmicas e de transporte para um conjunto de partículas clássicas não interagentes em diversos sistemas físicos. Os sistemas descritos aqui, em sua maioria, apresentam estrutura mista no espaço de fase no sentido de que curvas invariantes do tipo spanning, mares de caos e ilhas periódicas estão presentes. A descrição de cada sistema será feita utilizando mapeamentos discretos não lineares. Detalharemos a forma de obter os mapeamentos assim como discutiremos algumas de suas propriedades dinâmicas. Expoentes de Lyapunov serão utilizados para caracterizar a região de caos nos sistemas. Hipóteses de escala são usadas para provar que certos observáveis, por exemplo a energia média ao longo de mares de caos, são invariantes de escala. Consideraremos também que quando uma partícula, ou de forma equivalente um conjunto delas atinge uma determinada altura no espaço de fases, ela pode escapar. Ao estudar o escape de partículas, vemos que o histograma do número de partículas que atingem uma certa altura (ou energia) h no espaço de fases em uma dada iterada n, ao qual observamos ser invariante de escala, cresce rapidamente até atingir um máximo e então tende à zero para n grande. Quando a altura h varia proporcionalmente a posição da primeira curva invariante spanning, podemos confirmar uma invariância de escala do histograma de frequências. O mesmo ocorre para a probabilidade de sobrevivência da partícula à dinâmica. Neste contexto, abordaremos os seguintes problemas: (1) Um guia de ondas senoidalmente corrugado; (2) Uma família de mapas Hamiltonianos bidimensionais que recupera diversos modelos; (3) Partículas confinadas em uma caixa com potenciais infinitos nas bordas e contendo em seu interior um poço de potencial dependente periodicamente do tempo; (4) Analisaremos um bilhar ovóide com dependência temporal introduzida através de giro, onde para certas condições observamos que este não apresenta um aparente crescimento ilimitado de energia (aceleração de Fermi), desta forma sendo um possível contra-exemplo da conjectura LRA. Esta tese é um resumo de 8 artigos que foram publicados em revistas internacionais. / We investigate some dynamical and transport properties for a set of non-interacting classical particles. The systems here described, for the most part, present mixed structure in the phase space in the sense that invariant spanning curves, chaotic seas and periodic islands are present. The dynamics of each model is described by using non-linear mappings. We show all the details to construct the mappings and discuss some of their dynamical properties including fixed points stability among others. Lyapunov exponents will be obtained to characterize the chaotic dynamics observed in the phase space. Moreover some scaling hypotheses are used to prove that certain observables, including the average energy, are scaling invariant. We consider also that when a particle or an ensemble of them reach a certain portion of the phase space, they can escape. When studying the escape, we see that the histogram for the number of particles that reach certain height (or energy) h in the phase space for the iteration n, for which we observe to be scaling invariant, grows quickly until reaching a maximum and then goes towards zero for large enough n. When changing the height h proportionally to the position of the first invariant spanning curve, we can confirm the scaling invariance. The same happens for the survival probability for a particle in the chaotic dynamics. In this way, we will discuss the following problems: (1) A corrugated waveguide; (2) A family of two-dimensional Hamiltonian mappings which can reproduce different scaling exponents; (3) Particles confined to bounce in the interior of a time-dependent potential well; (4) We will analyse a rotating oval billiard, where for certain conditions we observed that this system does not present the unbounded energy growth (Fermi acceleration), in this way it is a possible counterexample of the LRA conjecture. This thesis is as summary of eight papers already published.

A precisão de uma ação motora depende de vários fatores, como: 1) grau de variabilidade da força gerada por cada músculo envolvido, 2) velocidade de geração da força, 3) coordenação das ativações dos músculos. A geração e o controle da força muscular possuem mecanismos que ainda precisam ser mais bem estabelecidos, tanto para o aprimoramento das teorias de controle motor, quanto para o desenvolvimento de técnicas que permitam a prevenção ou a compensação de certas deficiências. A perda de desempenho motor pode ser decorrente de doenças que afetam o sistema neuromuscular ou de alterações associadas ao envelhecimento. Sabe-se, por exemplo, que idosos podem possuir maior variabilidade e menor velocidade de desenvolvimento da força, quando comparados com jovens. Uma das formas de se entender os mecanismos responsáveis pelos fenômenos observados em experimentos neurofisiológicos, em indivíduos saudáveis, em pacientes ou em idosos, é por meio de uma representação adequada de tais mecanismos em modelos matemáticos. Tais modelos podem, pela escolha de um conjunto de parâmetros e de sinais de entrada, ser simulados, explorando-se toda gama de cenários plausíveis para a geração de um determinado fenômeno, tendo como referência os dados obtidos experimentalmente. Resumidamente, o presente trabalho trata do estudo do sistema neuromuscular por modelagem matemática e simulação computacional, com particular interesse nos músculos do tríceps sural e no primeiro interósseo dorsal (um músculo intrínseco da mão), sendo estes músculos amplamente utilizados em estudos experimentais e de modelagem. Maior enfoque é dado em contrações isométricas (i.e., ângulo articular mantido fixo), avaliando-se a organização do núcleo motor, em termos anatômicos e fisiológicos, recebendo como entrada a atividade sináptica das vias pré-motoneuronais, e estudando como diferentes arranjos das propriedades neurais podem resultar em características encontradas experimentalmente para a força muscular. Inicialmente foi feita uma ampla expansão de um simulador existente (ReMoto), tanto em aspectos de modelagem quanto de interface. Em seguida, este modelo expandido foi empregado para um estudo da influência do grau de rigidez muscular nas respostas reflexas do tornozelo. Posteriormente, um novo modelo de pool de motoneurônios, com ampla representação de características biofísicas, foi desenvolvido e, por fim, este modelo foi utilizado, em conjunto com modelos de tratos descendentes e da geração de força muscular, para representar a geração de forças isométricas em jovens e idosos. / The precision of a motor action depends on several factors such as: 1) the level of force variability of each involved muscle, 2) the rate of force development, 3) the coordination of the activations of the muscles involved. Several mechanisms underlying the force generation in a muscle and its control by the nervous system remain to be fully comprehended. An appropriate description of these mechanisms would allow an improvement in motor control theories and could contribute to the development of techniques for the prevention or compensation of some disabilities. Losses in motor performance may be caused by diseases affecting the neuromuscular system or due to aging processes. For instance, old adults may exhibit higher force variability and lower velocity of force development than young adults. Proper representations of such mechanisms in mathematical models constitute a promising way to test hypotheses raised by neurophysiological experiments to explain an observed phenomenon. These models can be used to investigate aspects of health/disease or young/old subjects and, by varying their parameter sets, it is possible to explore a broad range of plausible scenarios under which the experimentally observed phenomena are replicated. This project deals with the study of the neuromuscular system by mathematical modeling and computer simulations, applied to the triceps surae and the first dorsal interosseus (two of the most experimentally and theoretically studied muscles). The principal focus is on isometric contractions (i.e., fixed joint angle) and the study of the organization of the motor nucleus (anatomical and physiological aspects) receiving inputs from premotoneuronal pathways. The study analyzes how different patterns of organization result in experimentally observed aspects of muscle force. Initially, an existing simulator of the neuromuscular system (ReMoto) was broadly extended to include new models and a friendly interface. The extended model was used to investigate the influence of muscle stiffness on the reflex responses in the ankle joint. Next, a new motoneuron pool mathematical model was developed based on known biophysics. Finally, this model was integrated with models of pre-motoneuronal neurons estabilishing synapses with motoneurons and of muscle force generation in order to represent isometric force generation in young and old adults.

The theory of impulsive dynamical systems is an important tool to describe the evolution of systems where the continuous development of a process is interrupted by abrupt changes of state. This phenomenon is called impulse. In many natural phenomena, the real deterministic models are often described by systems which involve impulses. The aim of this work is to investigate topological properties of impulsive semidynamical systems. We establish necessary and sufficient conditions to obtain uniform and orbital stability via Lyapunov functions. We solve a problem of Jake Hale for impulsive systems where we obtain the existence of a maximal compact invariant set. Also, we obtain results about almost periodic motions and asymptotically almost periodic motions in the context of impulsive systems. Some asymptotic properties for impulsive systems and for their associated discrete systems are investigated. The new results presented in this text are in the papers [11], [15] and [16]. / A teoria de sistemas dinâmicos com impulsos é apropriada para descrever processos de evolução que sofrem variações de estado de curta duração e que podem ser consideradas instantâneas. Este fenômeno é chamado impulso. Para muitos fenômenos naturais, os modelos determinísticos mais realistas são frequentemente descritos por sistemas que envolvem impulsos. O objetivo deste trabalho é estudar propriedades topológicas para sistemas semidinâmicos impulsivos. Estabelecemos condições necessárias e suficientes para obtermos estabilidade uniforme e estabilidade orbital utilizando funções do tipo Lyapunov. Resolvemos um problema de Jack Hale para os sistemas impulsivos, onde obtemos a existência de um conjunto invariante compacto maximal. Além disso, obtemos resultados de movimentos quase periódicos e movimentos assintoticamente quase periódicos para sistemas impulsivos. Algumas propriedades assintóticas são estabelecidas para um sistema impulsivo e para seu sistema discreto associado. Os resultados novos apresentados neste trabalho estão presentes nos artigos [11], [15] e [16].

A importância do planejamento de sistemas hidrotérmicos é largamente reconhecida. A questão constitui-se em um complicado problema de otimização que envolve aspectos não-lineares e estocásticos, e que se torna mais complexo quando várias usinas são consideradas em conjunto, em virtude da ocorrência de interações. Apesar de se conhecerem formalizações teóricas do problema e da solução, a complexidade acarreta sérios entraves computacionais e muitas vezes consideram-se apenas cenários simplificados, como o determinístico ou estocástico com um modelo de reservatório equivalente. Este trabalho tem como objetivo conciliar a facilidade numérica de problemas determinísticos com a consideração indireta da natureza estocástica do problema. Com esta finalidade, propõe-se o uso de um funcional de custo que reflete o risco de vertimento a cada período. Outra linha estudada neste trabalho foi a de rastreamento de alvo utilizando a técnica de controle Linear Quadrático (LQ). A eficácia das estratégias propostas é avaliada através de diversos casos de estudo, incluindo comparação com o planejamento determinístico usual / The scheduling of hydrothermal systems is an important and challenging problem. In fact, the problem is complex since it involves the optimization of a cost functional related to a nonlinear and stochastic system. Although mathematical formulations of the problem are available, there are computational issues (as the course of dimensionality in dynamical programming) that make difficult to obtain the solution. In this situation, some simplifyied equivalent models or deterministic formulations were considered in the literature, which are simpler to deal with. The aim of this work is to obtain a solution thet presents the ease of computaion of deterministic models and, simultaneously, it reflects the stochastic nature of the problem, at some extension. The strategy consists of taking into account a cost functional that is related to the risk of spillage at each period. Another strategy considered here is the linear quadratic control with a target. Several case estudies are presented, which employ different models for the stochastic inflow, allwing us to evaluate the effectiveness of the strategies and to compare them with the usual deterministic control