Cheung Wing-chuen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 80-82). / Abstracts in English and Chinese. / Chapter 1 --- Introduction to CR manifolds --- p.6 / Chapter 1.1 --- Non-equivalence of real analytic hypersurfaces in C2 --- p.6 / Chapter 1.2 --- CR manifold and Levi form --- p.9 / Chapter 1.3 --- The real hyperquadrics --- p.16 / Chapter 2 --- Normal Forms --- p.22 / Chapter 2.1 --- Formal theory of normal forms --- p.22 / Chapter 2.2 --- Geometric theory of normal forms --- p.33 / Chapter 3 --- Connections and Curvatures --- p.45 / Chapter 3.1 --- Solution of the equivalence problem --- p.45 / Chapter 3.2 --- Geometric interpretation of the solution --- p.59 / Chapter 4 --- Chains --- p.65 / Chapter 4.1 --- Identification of the two definitions of chains --- p.65 / Chapter 4.2 --- Chain-preserving maps --- p.73 / Chapter 4.3 --- Some pathological behaviour of chains --- p.76 / Bibliography --- p.80

Cette thèse est un recueil de constructions et de résultats variés autour de problèmes de résonances et de stabilités. Premièrement, on s'intéresse à la conception et à l'analyse de méthodes numériques pour des problèmes académiques tels que le problème de Dirichlet sur un segment ou l'équation de transport associée à une rotation du plan. Ensuite, on étend l'analyse linéaire classique des équations de Vlasov-Poisson autour d'états d'équilibre homogènes pour décrire des phénomènes multidimensionnels et non linéaires. Enfin, une large partie est consacrée à l'étude d'équations de Schrödinger non linéaires en dimension 1. D'une part, on étudie l'impact d'une semi-discrétisation naturelle sur les ondes solitaires progressives et la croissance des normes de Sobolev. D'autre part, on développe une nouvelle famille de formes normales permettant de décrire la dynamique des petites solutions régulières pendant des temps très longs. / This manuscript deals with many problems about resonance and stability. First, we design and analyse numerical methods for academic problems like the Dirichlet problem on a segment line or the transport equation associated with a two dimensional rotation. Then, we extend the classical linear analysis of Vlasov-Poisson equations near homogeneous equilibria to describe nonlinear and multidimensional phenomena. Finally, a large part of this thesis is devoted to nonlinear Schrödinger equations in dimension 1. On the one hand, we study the impact of a natural semi-discretisation on the solitary traveling waves and on the growth of the high order Sobolev norms. On the other hand, we develop a new family of normal forms to describe the dynamic of small and smooth solutions for very long times.

Orientador: Marco Antonio Teixeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T12:10:27Z (GMT). No. of bitstreams: 1 Roberto_LuciAnyFrancisco_D.pdf: 627926 bytes, checksum: 0c8cb4e26df805282fa716847859d82f (MD5) Previous issue date: 2008 / Resumo: Neste trabalho estudamos sistemas dinâmicos possuindo estruturas Hamiltonianas e reversíveis( / Abstract: In this work we study dynamical systems possessing Hamiltonian and time-reversible structures. The reversibility concept is de¯ned in terms of an involution. Initially we discuss the dynamics of Hamiltonian vector ¯elds with 2 and 3 degrees of freedom around an elliptic equilibrium in the presence of an involution which preserves the symplectic structure. The main results discuss the existence of one-parameter families of reversible periodic solutions terminating at the equilibrium. The main techniques that are used in the proofs are Belitskii and Birkho® normal forms and the Liapunov-Schmidt Reduction. Next we consider a case of the 3-body restricted problem in rotating coordinates. In this case the two primaries are oving in an elliptic collision orbit. By the continuation method of Poincare we characterize that the periodic circular orbits and the symmetric periodic elliptic orbits from the Kepler problem which can be prolonged to pseudo periodic orbits of the planar restricted 3{body problem in rotating coordinates with the two primaries moving in an elliptic collision orbit / Doutorado / Topologia e Geometria / Doutor em Matemática

Orientador : Vivaldo Fernando da Costa / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-04T02:42:32Z (GMT). No. of bitstreams: 1 Domingues_AdrianaFavaro_D.pdf: 789245 bytes, checksum: 9ca2b124a01cd871aa7f4bbb4c63c57a (MD5) Previous issue date: 2005 / Resumo: Este trabalho aborda o problema das oscilações eletromecânicas de baixa freqüência de modo interárea em sistemas de energia elétrica através de duas metodologias: primeiramente, através do método convencional da análise modal linear e, posteriormente, através da aplicação do método da forma normal dos campos vetoriais como ferramenta para viabilizar o acréscimo dos termos não lineares de segunda ordem resultantes da expansão em série de Taylor. Neste caso, o método da forma normal é aplicado a um Modelo de Sensibilidade de Potência com representação de segunda ordem, para investigar os efeitos das interações não lineares entre os modos naturais de oscilação de sistemas de energia elétrica. São consideradas, em ambas as análises, a inclusão de dispositivos FACTS e da modelagem dinâmica das cargas. As simulações são realizadas para um sistema simétrico de duas áreas e para o Sistema Equivalente Sul-Sudeste Brasileiro. A metodologia de análise proposta mostra-se bastante satisfatória como alternativa à simulação não linear no domínio do tempo e à análise modal convencional / Abstract: In this work, the analysis of power systems inter-area mode oscillations is performed by the application of two different methodologies: first, the linear modal analysis, and then the analysis including second order nonlinear terms from a Taylor series expansion, with the application of the method of normal forms of vector fields. In this case, the method of normal forms is applied to a Power Sensitivity Model including second order nonlinear terms, in order to investigate the effects of nonlinear interactions between system modes. Both methodologies consider the inclusion of FACTS devices and dynamic load model. Simulations are performed for a symmetric two-area test power system and for the Equivalent South-Southeast Brazilian system. The results obtained show that the methodology proposed is very effective as an alternative to linear modal analysis and timedomain simulation in the performance of inter-area mode oscillations analysis / Doutorado / Energia Eletrica / Doutor em Engenharia Elétrica