No description available.

La théorie classique des formes normales a pour but de simplifier des problèmes compliqués grâce à des changements de coordonnées réguliers pour ne conserver que les caractéristiques dynamiques du système. Plus précisément, on considère un système dynamique que l'on dit "élémentaire", comme par exemple la partie linéaire d'un champ de vecteurs au voisinage d'un point singulier, et on se donne une perturbation de ce système élémentaire. Les formes normales sont alors l'ensemble des représentants de ces perturbations à la conjugaison près d'une transformation régulière. Elles ne sont constituées que des termes qui caractérisent la dynamique du système perturbé et que l'on appelle "résonances". Dans la première partie de la thèse on cherche à comprendre la dynamique locale d'équations différentielles implicites de la forme F(x,y,y')=0, où F est un germe de fonction holomorphe au voisinage d'un point singulier. Pour cela on utilise la relation intime entre les systèmes implicites et les champs liouvilliens. La classification par transformation de contact des équations implicites provient de la classification symplectique des champs liouvilliens. On utilise alors toute la théorie des formes normales pour les champs de vecteurs, dans le cas holomorphe (Brjuno, Siegel, Stolovitch) et dans le cas réel (Sternberg), que l'on adapte pour les champs liouviliens avec des transformations symplectiques. On établit alors des résultats de classification des équations implicites en fonction des invariants dynamiques, ainsi que des conditions d'existence de solutions locales via les formes normales. / The aim of the classical theory of normal forms is to simplify complicated problems by using regular changes of coordinates, in order to keep the dynamical characteristics of the system. More precisely, we consider a dynamic system said to be "elementary", like a linear part of a vector field in the neighborhood of a singular point, and we focus on a perturbation of this elementary system. Normal forms are the set of all representatives of those perturbations under the action of the group of regular transformation. They are composed of terms which caracterise the dynamics of the perturbed system, and which are called "resonances". In the first part, we try to understand the local dynamic of implicit equations of the form $F(x,y,y')=0$, where $F$ is a germ of holomorphic function in a neighborhood of a singular point. To this end we use the relation between implicit systems and liouvillian vector fields. The classification by contact transformations of implicit equations come from the symplectic classification of liouvillian vector fields. We use all normal forms theory for vector fields, in complex case (Bjruno, Siegel, Stolovitch), and in real case (Sternberg), adapted to liouvillian fields with symplectic transformations. We establish classification results for implicit equations according to the dynamical invariants, and existence conditions of local solutions using normal forms. In the second part, we undertake the normalization of an analytic vector field in a neighborhood of the torus. Brjuno enunciates a theorem of normalization, under conditions of control of small divisors and integrability of the normal forms ; however he doesn't give any proof of that theorem.

Submitted by Franciele Moreira (francielemoreyra@gmail.com) on 2018-03-28T11:58:56Z No. of bitstreams: 2 Dissertação - Jeferson Arley Poveda Contreras - 2018.pdf: 763599 bytes, checksum: 6800571168e0aa9de85d151e4c912725 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-03-29T11:29:24Z (GMT) No. of bitstreams: 2 Dissertação - Jeferson Arley Poveda Contreras - 2018.pdf: 763599 bytes, checksum: 6800571168e0aa9de85d151e4c912725 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-03-29T11:29:24Z (GMT). No. of bitstreams: 2 Dissertação - Jeferson Arley Poveda Contreras - 2018.pdf: 763599 bytes, checksum: 6800571168e0aa9de85d151e4c912725 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-03-07 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / The goal of this work is study limit cycles of piecewise smooth vector fields. First, we present the basic theory, passing through the areas of analysis, qualitative theory of differential equations and algebra. We also present basic concepts of Filippov fields, which are indispensable for the study of piecewise smooth fields. In chapter one, was the main topic, a general method for finding limit cycles will be described; in the second chapter limit cycles are found in a piecewise smooth vector field with non-degenerate center being perturbed by a piecewise polynomial vector field. In the fourth chapter, we study limit cycles in piecewise smooth Hamiltonian fields. / O objetivo deste trabalho é estudar ciclos limite de campos de vetores suaves por parte. Primeiro apresentaremos a teoria básica, passando pelas áreas de análise, teoria qualitativa das equações diferenciais e álgebra. Apresentamos também conceitos básicos de campos de Filippov, os quais são imprescindíveis para o estudo dos campos suaves por partes. No capítulo dos, como tópico principal, será descrito um método geral para encontrar ciclos limite; no segundo três são encontrados ciclos limites em um campo de vetores suave por partes com um centro não degenerado sendo perturbado por um polinômio. No quarto capitulo estudaremos os ciclos limites de campos de vetores Hamiltonianos por parte.