We discuss how to realize simple Smale Flows on 3-manifolds. We focus on three questions: (1) What are the topological conjugate classes of Lorenz Smale flows that can be realized on S3? (2) Which 3-manifolds can also admit a Lorenz Smale flow? (3) What are the topological conjugate classes of simple Smale flows whose saddle set can be modeled by &nu(0+,0+,0,0) can be realized on S3? This dissertation extends the work of M. Sullivan and B. Yu.

A Smale space is a chaotic dynamical system with canonical coordinates of contracting and expanding directions. The basic sets for Smale’s Axiom A systems are a key class of examples. R.F. Williams considered the special case where the basic set had a totally disconnected contracting set and a Euclidean expanding one. He provided a construction using inverse limits of such examples and also proved that (under appropriate hyptotheses) all such basic sets arose from this construction. We will be working in the metric setting of Smale spaces, but the goal is to extend Williams’ results by removing all hypotheses on the unstable sets. We give criteria on a stationary inverse limit which ensures the result is a Smale space. We also prove that any irreducible Smale space with totally disconnected local stable sets is obtained through this construction. / Graduate

O problema de n?corpos é um dos problemas mais importantes em Sistemas Dinâmicos. Nós estudamos o modelo do problema dos três corpos restrito introduzido por Sitnikov. Nesse modelo os corpos primários tem a mesma massa e o terceiro corpo é de massa muito pequena com respeito aos corpos primários. Usando os métodos de Alekseev, nós mostramos a existência de uma ?ferradura de Smale?como um subsistema da dinâmica do terceiro corpo e concluímos ricas conseqüências probabilísticas. Nós também estudamos o problema pelo método de Melnikov / The n?body problem is one of the most important problems in dynamical systems. We study the model introduced by Sitnikov of restricted three body problem. In this model the primaries are of equal mass and the third body is very small with respect to the primaries. Using methods of Alekseev, we show the existence of ?Smale horseshoe?as a subsystem of the dynamic of the third body and conclude rich probabilistic consequences. We also study the same problem by Melnikov?s method

Les points critiques d’une fonction scalaire (minima, points col et maxima) sont des caractéristiques importantes permettant de décrire de gros ensembles de données, comme par exemple les données topographiques. L’acquisition de ces données introduit souvent du bruit sur les valeurs. Un grand nombre de points critiques sont créés par le bruit, il est donc important de supprimer ces points critiques pour faire une bonne analyse de ces données. Le complexe de Morse-Smale est un objet mathématique qui est étudié dans le domaine de la Visualisation Scientifique car il permet de simplifier des fonctions scalaires tout en gardant les points critiques les plus importants de la fonction étudiée, ainsi que les liens entre ces points critiques. Nous proposons dans cette thèse une méthode permettant de construire une fonction qui correspond à un complexe de Morse-Smale d’une fonction définie sur R^2 après suppression de paires de points critiques dans celui-ci.Tout d’abord, nous proposons une méthode qui définit une surface interpolant des valeurs de fonction aux points d’une grille de façon monotone, c’est-à-dire en ne créant pas de point critique. Cette surface est composée d’un ensemble de patchs de Bézier triangulaires cubiques assemblés de telle sorte que la surface soit globalement C^1. Nous donnons des conditionssuffisantes sur les valeurs d fonction et les valeurs de dérivées partielles aux points de la grille afin que la surface soit croissante dans la direction (x+y). Il n’est pas évident de créer des valeurs de dérivées partielles en chaque point de la grille vérifiant ces conditions. C’est pourquoi nous introduisons deux algorithmes : le premier permet de modifier des valeurs de dérivées partielles données en entrée afin que celles-ci vérifient les conditions et le second calcule des valeurs de dérivées partielles à partir des valeurs de fonctions aux points de la grille.Ensuite, nous décrivons une méthode de reconstruction de champs scalaires à partir de complexes de Morse-Smale simplifiés. Pour cela, nous commençons par approximer les 1-cellules (les liens entre les points critiques dans le complexe de Morse-Smale, ceux-ci sont décrits par des polylignes) par des courbes composées de courbes de Bézier cubiques. Nous décrivons ensuite comment notre interpolation monotone de valeurs aux points d’une grille est utilisée pour construire des surfaces monotones interpolant les courbes construites précédemment. De plus, nous montrons que la fonction reconstruite contient tout les points critiques du complexe de Morse-Smale simplifié et n’en contient aucun autre. / Critical points of a scalar function (minima, saddle points and maxima) are important features to characterize large scalar datasets, like topographic data. But the acquisition of such datasets introduces noise in the values. Many critical points are caused by the noise, so there is a need to delete these extra critical points. The Morse-Smale complex is a mathematical object which is studied in the domain of Visualization because it allows to simplify scalar functions while keeping the most important critical points of the studied function and the links between them. We propose in this dissertation a method to construct a function which corresponds to a Morse-Smale complex defined on R^2 after the suppression of pairs of critical points.Firstly, we propose a method which defines a monotone surface (a surface without critical points).This surface interpolates function values at a grid points. Furthermore, it is composed of a set of triangular cubic Bézier patches which define a C^1 continuous surface. We give sufficient conditions on the function values at the grid points and on the partial derivatives at the grid points so that the surface is increasing in the (x+y) direction. It is not easy to compute partial derivatives values which respect these conditions. That’s why we introduce two algorithms : the first modifies the partial derivatives values on input such that they respect the conditions and the second computes these values from the function values at the grid points.Then, we describe a reconstruction method of scalar field from simplified Morse-Smale complexes. We begin by approximating the 1-cells of the complex (which are the links between the critical points, described by polylines) by curves composed of cubic Bézier curves. We then describe how our monotone interpolant of values at grid points is used to construct monotone surfaces which interpolate the curves we computed before. Furthermore, we show that the function we compute contains all the critical points of the simplified Morse-Smale complex and has no others.

O problema de n?corpos é um dos problemas mais importantes em Sistemas Dinâmicos. Nós estudamos o modelo do problema dos três corpos restrito introduzido por Sitnikov. Nesse modelo os corpos primários tem a mesma massa e o terceiro corpo é de massa muito pequena com respeito aos corpos primários. Usando os métodos de Alekseev, nós mostramos a existência de uma ?ferradura de Smale?como um subsistema da dinâmica do terceiro corpo e concluímos ricas conseqüências probabilísticas. Nós também estudamos o problema pelo método de Melnikov / The n?body problem is one of the most important problems in dynamical systems. We study the model introduced by Sitnikov of restricted three body problem. In this model the primaries are of equal mass and the third body is very small with respect to the primaries. Using methods of Alekseev, we show the existence of ?Smale horseshoe?as a subsystem of the dynamic of the third body and conclude rich probabilistic consequences. We also study the same problem by Melnikov?s method

In recent decades, scientific data has become available in increasing sizes and precision. Therefore techniques to analyze and summarize the ever increasing datasets are of vital importance. A common form of scientific data, resulting from simulations as well as observational sciences, is in the form of scalar-valued function on domains of interest. The Morse-Smale complex is a topological data-structure used to analyze and summarize the gradient behavior of such scalar functions. This thesis deals with efficient parallel algorithms to compute the Morse-Smale complex as well as its application to datasets arising from cosmological sciences as well as structural biology. The first part of the thesis discusses the contributions towards efficient computation of the Morse-Smale complex of scalar functions de ned on two and three dimensional datasets. In two dimensions, parallel computation is made possible via a paralleizable discrete gradient computation algorithm. This algorithm is extended to work e ciently in three dimensions also. We also describe e cient algorithms that synergistically leverage modern GPUs and multi-core CPUs to traverse the gradient field needed for determining the structure and geometry of the Morse-Smale complex. We conclude this part with theoretical contributions pertaining to Morse-Smale complex simplification. The second part of the thesis explores two applications of the Morse-Smale complex. The first is an application of the 3-dimensional hierarchical Morse-Smale complex to interactively explore the filamentary structure of the cosmic web. The second is an application of the Morse-Smale complex for analysis of shapes of molecular surfaces. Here, we employ the Morse-Smale complex to determine alignments between the surfaces of molecules having similar surface architecture.

O objetivo principal deste trabalho é o estudo da estabilidade estrutural dos atratores de semigrupos. Começamos este trabalho apresentando o conceito e propriedades básicas de semigrupos que possuem atratores globais. Estudamos, então, semigrupos gradientes e dinamicamente gradientes, mostrando que eles são equivalentes e que uma pequena perturbação autônoma de um semigrupo gradiente continua sendo gradiente. Estudamos as variedades estável e instável de um ponto de equilíbrio hiperbólico e o comportamento de soluções periódicas sob perturbação. Concluímos este trabalho com o estudo dos semigrupos Morse-Smale. / The main goal of this work is the study of structural stability of global attractors. We start this work by presenting the concept and basic properties of semigroups and global attractors. We then studied gradient and dinamically gradient semigroups, showing that these concepts are equivalent and that a small autonomous pertubation of a gradient semigroup remains a gradient semigroup. We studied the stable and unstable manifolds in the neighbourhood of a hyperbolic equilibrium point and the behavior of periodic solutions under perturbation. Finally, we studied the Morse-Smale semigroups.

O objetivo principal deste trabalho é o estudo da estabilidade estrutural dos atratores de semigrupos. Começamos este trabalho apresentando o conceito e propriedades básicas de semigrupos que possuem atratores globais. Estudamos, então, semigrupos gradientes e dinamicamente gradientes, mostrando que eles são equivalentes e que uma pequena perturbação autônoma de um semigrupo gradiente continua sendo gradiente. Estudamos as variedades estável e instável de um ponto de equilíbrio hiperbólico e o comportamento de soluções periódicas sob perturbação. Concluímos este trabalho com o estudo dos semigrupos Morse-Smale. / The main goal of this work is the study of structural stability of global attractors. We start this work by presenting the concept and basic properties of semigroups and global attractors. We then studied gradient and dinamically gradient semigroups, showing that these concepts are equivalent and that a small autonomous pertubation of a gradient semigroup remains a gradient semigroup. We studied the stable and unstable manifolds in the neighbourhood of a hyperbolic equilibrium point and the behavior of periodic solutions under perturbation. Finally, we studied the Morse-Smale semigroups.

Un réseau de cellules est un graphe orienté dont chaque sommet (aussi appelé cellule) représente un ensemble de variables et dont les arcs symbolisent les interactions entre ces variables. Les réseaux de cellules jouent un rôle important dans la modélisation de phénomènes neurologiques, de systèmes économiques ou biologiques, etc.. Soit G un graphe orienté possédant N sommets, on dit qu'une application f=(f_1,...,f_N) de X=X_1×...×X_N dans X (où X_j=R^dj) est admissible, si pour tout sommet j, f_j(x) dépend de x_i seulement si i->j est un arc de G. Dans cette thèse nous montrons que si G est fortement connecté et auto-dépendant, génériquement par rapport à f appartenant à l'ensemble des applications admissibles de classe C¹, le système dynamique engendré par l'équation différentielle x'(t)=f(x(t)) vérifie la propriété de Kupka-Smale, c'est-à-dire tous les éléments critiques (points d'équilibre et orbites périodiques) sont hyperboliques et les variétés stable et instable des éléments critiques s'intersectent transversalement. Ainsi, pour un ensemble dense d'applications admissibles, le système dynamique est au moins localement stable par perturbation (admissible ou non). Nous considérons également l'ensemble des applications « dissipatives » f de classe C¹ dont la différentielle Df(x) est une matrice de Jacobi cyclique positive en tout point x. De telles applications définissent un système coopératif. Nous montrons que le système dynamique engendré par l'équation x'(t)=f(x(t)) vérifie génériquement la propriété de Morse-Smale par rapport à de telles applications f, c'est-à-dire le système vérifie la propriété de Kupka-Smale, les éléments critiques sont en nombre fini et l'ensemble des points non-errants est égal à l'ensemble des éléments critiques. Cette propriété entraîne la stabilité structurelle du système dynamique. Finalement, dans cette thèse nous étudions aussi des réseaux de cellules satisfaisant des contraintes de symétrie locale. Pour de tels systèmes, nous montrons tout d'abord des résultats génériques d'observation à symétrie près, de synchronisation et de décalage de phase. Nous utilisons ces résultats pour montrer la généricité de l'hyperbolicité des points d'équilibre ainsi qu'un lemme d'injectivité pour les trajectoires. Les résultats de généricité de cette thèse sont obtenus à l'aide de théorèmes de transversalité de type Sard-Smale. / A coupled cell network consists in a directed graph, with each node (also called cell) representing a set of variables and with each arrow representing the interaction between these variables. Coupled cell networks play an important role in the modeling of phenomena in neurology, economics or biology, etc.. Let G be a directed graph with N nodes. A mapping f=(f_1,...,f_N) of X=X_1×...×X_N to X (where X_j=R^dj) is admissible, if for each node j, f_j(x) depends on x_i only if i->j is an arrow of G. In this thesis, we show that if the graph G is strongly connected and self-dependant, generically with respect to f in the class of admissible C¹-functions, the dynamical system generated by the differential equation x'(t)=f(x(t)) satisfies the Kupka-Smale property, that is all the critical elements (i.e. the equilibria and periodic orbits) are hyperbolic and the stable and unstable manifolds of these critical elements intersect transversally. As a consequence, for a dense set of admissible functions, the dynamical system is locally stable with respect of small perturbations (admissible or not). We also consider the set of "dissipative" mappings f of class C¹, the differential Df (x) of which is a positive cyclic Jacobi matrix at any point x. Such maps define a cooperative system. We show that the dynamical system generated by the equation x'(t)=f(x(t)) is generically Morse-Smale with respect to such mappings f, that is the system is Kupka-Smale, the critical elements are in finite number and the non-wandering set is equal to the set of critical elements. This property implies the structural stability of the dynamical system. Finally, in this thesis we also study coupled cell networks satisfying local symmetry constraints. For such systems, we first show generic results of observation, synchronization and phase shift. We use these properties to show the genericity of hyperbolicity of equilibrium points and an injectivity lemma for trajectories. In the proof of these genericity results, we use different Sard-Smale type theorems.

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.