Um dos principais problemas na teoria qualitativa das equações diferenciais em dimensão dois é apresentar, para uma dada família de sistemas diferenciais, uma classificação topológica dos retratos de fase de todos os sistemas dessa família. A proposta deste trabalho é estudar a técnica utilizada na classificação dos retratos de fase globais de sistemas diferenciais polinomiais da forma \'dx SUP dt\' = P(x,y) \'dy SUP dt = Q(x,y) onde P e Q são polinômios nas variáveis x e y e o máximo entre os graus de P e Q é 2. Para esse fim optamos pelo estudo da referência de Cairó e Llibre [5]. Na presente referência os autores obtém a classificação de todos os retratos de fase globais dos sistemas diferenciais polinomiais que possuem uma integral primeira racional, H, de grau 2. Esse estudo foi dividido em duas etapas. Na primeira, caracterizamos a função H através de seus coeficientes. Na segunda, encontramos todos os retratos de fase globais no disco de Poincaré. Para tais sistemas, existem exatamente 18 retratos de fase no disco de Poincaré, exceto pela reversão do sentido de todas as órbitas ou equivalência topológica / One of the main problems in the qualitative theory of 2-dimensional differential equations is, for a concrete family of differential systems, to describe a topological classification of the phase portraits for all the systems in this family. The purpose of this work is to study a technique used in the classification of global phase portraits of the planar polynomial diferential systems or simply quadratic systems of the form \'dx SUP. dt\' = P(x,y) \'dy SUP. dt\' = Q(x,y) where P and Q are real polynomials in x and y the maximum degree of P and Q is 2. Our basic reference is the paper of Cairó and Llibre [5]. In that work the authors give the classification of all global phase portraits of the planar quadratic differential systems having a rational first integral H of degree 2. Our work is divided in two parts. In the first part, we characterize the first integral H through its coeficients. In the second one, we describe all global phase portraits in the Poincaré disk. For such systems, there are exactly 18 different phase portraits in the Poincaré disk, up to a reversal of sense of all orbits or topological equivalence

Um dos principais problemas na teoria qualitativa das equações diferenciais em dimensão dois é apresentar, para uma dada família de sistemas diferenciais, uma classificação topológica dos retratos de fase de todos os sistemas dessa família. A proposta deste trabalho é estudar a técnica utilizada na classificação dos retratos de fase globais de sistemas diferenciais polinomiais da forma \'dx SUP dt\' = P(x,y) \'dy SUP dt = Q(x,y) onde P e Q são polinômios nas variáveis x e y e o máximo entre os graus de P e Q é 2. Para esse fim optamos pelo estudo da referência de Cairó e Llibre [5]. Na presente referência os autores obtém a classificação de todos os retratos de fase globais dos sistemas diferenciais polinomiais que possuem uma integral primeira racional, H, de grau 2. Esse estudo foi dividido em duas etapas. Na primeira, caracterizamos a função H através de seus coeficientes. Na segunda, encontramos todos os retratos de fase globais no disco de Poincaré. Para tais sistemas, existem exatamente 18 retratos de fase no disco de Poincaré, exceto pela reversão do sentido de todas as órbitas ou equivalência topológica / One of the main problems in the qualitative theory of 2-dimensional differential equations is, for a concrete family of differential systems, to describe a topological classification of the phase portraits for all the systems in this family. The purpose of this work is to study a technique used in the classification of global phase portraits of the planar polynomial diferential systems or simply quadratic systems of the form \'dx SUP. dt\' = P(x,y) \'dy SUP. dt\' = Q(x,y) where P and Q are real polynomials in x and y the maximum degree of P and Q is 2. Our basic reference is the paper of Cairó and Llibre [5]. In that work the authors give the classification of all global phase portraits of the planar quadratic differential systems having a rational first integral H of degree 2. Our work is divided in two parts. In the first part, we characterize the first integral H through its coeficients. In the second one, we describe all global phase portraits in the Poincaré disk. For such systems, there are exactly 18 different phase portraits in the Poincaré disk, up to a reversal of sense of all orbits or topological equivalence

Submitted by Erika Demachki (erikademachki@gmail.com) on 2017-06-23T19:07:29Z No. of bitstreams: 2 Dissertação - Yovani Adolfo Villanueva Herrera - 2017.pdf: 1975075 bytes, checksum: 4226bb99e3c16dbc26f70e7f7208e2c1 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Rejected by Erika Demachki (erikademachki@gmail.com), reason: on 2017-06-29T18:25:12Z (GMT) / Submitted by Erika Demachki (erikademachki@gmail.com) on 2017-06-29T18:25:47Z No. of bitstreams: 2 Dissertação - Yovani Adolfo Villanueva Herrera - 2017.pdf: 2011176 bytes, checksum: 4107b5d00ad34d3b4127265319a99868 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Erika Demachki (erikademachki@gmail.com) on 2017-06-29T18:28:16Z (GMT) No. of bitstreams: 2 Dissertação - Yovani Adolfo Villanueva Herrera - 2017.pdf: 2011176 bytes, checksum: 4107b5d00ad34d3b4127265319a99868 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-06-29T18:28:16Z (GMT). No. of bitstreams: 2 Dissertação - Yovani Adolfo Villanueva Herrera - 2017.pdf: 2011176 bytes, checksum: 4107b5d00ad34d3b4127265319a99868 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-05-30 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / The subject of this work is the theory of normal forms of smooth vector fields of constrained systems (systems of non-linear differential-algebraic equations). In this study we introduce the qualitative theory of ordinary differential equations, with topics such as stability, structural stability, bifurcations, limit cycles and catastrophes of differential equations, and the functional singularity theory. The goal of this work is classify and normalize constrained systems, first of all from the local point of view, we'll show an idea of the global one and our final objective will be extend this theory to differenciable manifolds of dimension $n \geq 2$. / O tema deste trabalho é a teoria das formas normais de campos vetoriais suaves de sistemas forçados (sistemas de equações diferenciais-algébricas não lineares). Neste estudo entram a teoria qualitativa de equações diferenciais ordinárias, com tópicos como estabilidade, estabilidade estrutural, bifurcações, ciclos limite e catástrofes de equações diferenciais e a teoria das singularidades de funções. O objetivo do trabalho é a classificação e normalização dos sistemas forçados, primeiramente do ponto de vista local, mostraremos uma ideia da análise global e será nossa finalidade estender esta teoria para variedades diferenciáveis de dimensão $n \geq 2$.

Les avancées technologiques ont permis le stockage de grandes masses de données en termes de taille (nombre d’observations) et de dimensions (nombre de variables).Ces données nécessitent de nouvelles méthodes, notamment en modélisation prédictive (data science ou science des données), de traitement statistique adaptées à leurs caractéristiques. Dans le cadre de cette thèse, nous nous intéressons plus particulièrement aux données dont le nombre de variables est élevé comparé au nombre d’observations.Pour ces données, une réduction du nombre de variables initiales, donc de dimensions, par la sélection d’un sous-ensemble optimal, s’avère nécessaire, voire indispensable.Elle permet de réduire la complexité, de comprendre la structure des données et d’améliorer l’interprétation des résultats et les performances du modèle de prédiction ou de classement en éliminant les variables bruit et/ou redondantes.Nous nous intéressons plus précisément à la sélection de variables dans le cadre de l’apprentissage supervisé et plus spécifiquement de la discrimination à catégories multiples dite multi-classes. L’objectif est de proposer de nouvelles méthodes de sélection de variables pour les modèles de discrimination multi-classes appelés Machines à Vecteurs Support Multiclasses (MSVM).Deux approches sont proposées dans ce travail. La première, présentée dans un contexte classique, consiste à sélectionner le sous-ensemble optimal de variables en utilisant le critère de "la borne rayon marge" majorante du risque de généralisation des MSVM. Quant à la deuxième approche, elle s’inscrit dans un contexte topologique et utilise la notion de graphes de voisinage et le critère de degré d’équivalence topologique en discrimination pour identifier les variables pertinentes qui constituent le sous-ensemble optimal du modèle MSVM.L’évaluation de ces deux approches sur des données simulées et d’autres réelles montre qu’elles permettent de sélectionner, à partir d’un grand nombre de variables initiales, un nombre réduit de variables explicatives avec des performances similaires ou encore meilleures que celles obtenues par des méthodes concurrentes. / The technological progress has allowed the storage of large amounts of data in terms of size (number of observations) and dimensions (number of variables). These data require new methods, especially for predictive modeling (data science), of statistical processing adapted to their characteristics. In this thesis, we are particularly interested in the data with large numberof variables compared to the number of observations.For these data, reducing the number of initial variables, hence dimensions, by selecting an optimal subset is necessary, even imperative. It reduces the complexity, helps to understand the data structure, improves the interpretation of the results and especially enhances the performance of the forecasting model by eliminating redundant and / or noise variables.More precisely, we are interested in the selection of variables in the context of supervised learning, specifically of multiclass discrimination. The objective is to propose some new methods of variable selection for multiclass discriminant models called Multiclass Support Vector Machines (MSVM).Two approaches are proposed in this work. The first one, presented in a classical context, consist in selecting the optimal subset of variables using the radius margin upper bound of the generalization error of MSVM. The second one, proposed in a topological context, uses the concepts of neighborhood graphs and the degree of topological equivalence in discriminationto identify the relevant variables and to select the optimal subset for an MSVM model.The evaluation of these two approaches on simulated and real data shows that they can select from a large number of initial variables, a reduced number providing equal or better performance than those obtained by competing methods.