Dans cette thèse nous étudions les propriétés d'ondes stationnaires dans des structures composées d'une couche diélectrique nonlinéaire de type Kerr et des couches métalliques et diélectriques linéaires. Nous élaborons différents modèles pour étudier les propriétées de plasmons-solitons dans deux types de structures : (i) une région diélectrique nonlinéaire semi-infinie, des couches de métal et de diélectrique linéaires et (ii) une couche de diélectrique nonlinéaire d'épaisseur finie entre deux régions métalliques (guide d'onde métallique à coeur nonlinéaire). Pour le premier type de structures, nous montrons qu'en utilisant une structure à quatre couches, il est possible d'obtenir des plasmons-solitons de basses puissance. Pour des guides d'onde métalliques à coeur nonlinéaire, nous trouvons de modes d'ordres supérieurs. Pour certains des modes symétriques, nous observons une bifurcation par brisure de symétrie donnant naissance à des modes asymétriques dans une structure symétrique. / In this PhD thesis, we study the properties of stationary transverse magnetic polarized waves in structures composed of a Kerr-type nonlinear dielectric layer, metal and linear dielectric layers. We develop several models to study the properties of plasmon-soliton waves in two types of structures: a semi-infinite nonlinear dielectric in contact with metal and linear dielectric layers and a finite-size nonlinear dielectric layer sandwiched between two metal regions (nonlinear slot waveguide). Our models allow us to compute the nonlinear dispersion relations and the corresponding field profiles. For the first type of structure, we prove that using the four-layer structures that we propose, it is possible to obtain plasmon-soliton waves at the power levels. For nonlinear slot waveguide structures, we discover the existence of new, higher order modes. For some of the symmetric modes, we observe a symmetry breaking bifurcation giving birth to asymmetric modes in symmetric structure.

Apesar do crescente desenvolvimento das metodologias de detecção de bifurcações de Hopf em sistemas de energia elétrica nas últimas décadas, alguns aspectos ainda carecem de progressos, especialmente frente à crescente necessidade de aumento da segurança na operação dos sistemas eletroenergéticos. Um destes aspectos diz respeito ao tempo de processamento destas metodologias para serem utilizadas em tempo real na operação do sistema. Este tempo computacional é pouco considerado para este fim pelas metodologias atualmente existentes. Assim sendo, o presente trabalho apresenta um novo desenvolvimento relativo a estimação da margem de estabilidade do sistema de potência referente à bifurcação de Hopf e considera o tempo computacional envolvido neste processo a fim de incluir a margem devido a Hopf na operação em tempo real do sistema. O desenvolvimento apresentado estende uma das metodologias que compõe o estado da arte através da flexibilização de um dos parâmetros de interesse em relação a margem de estabilidade devido a bifurcação de Hopf, a saber, a frequência do autovalor no ponto de birfurcação. Esta metodologia utiliza o método de Newton em um conjunto de equações, e neste trabalho ainda é proposta a utilização de um tratamento da esparsidade para este conjunto de equações, deixando o algoritmo mais rápido. De forma a apresentar a eficiência desta metodologia proposta, esta foi testada em dois sistemas, o sistema Kundur de duas áreas e o sistema IEEE 39 barras. Os resultados obtidos são comparados frente a resultados obtidos também para a metodologia clássica utilizada em centros de operação. Através destes resultados é possível mostrar a possibilidade de sua utilização em tempo real e elucidar as grandes melhorias obtidas através do desenvolvimento proposto. / Despite the increasing development of Hopf bifurcations detection methods for electric power systems in the last decades, some aspects still require to progress, especially with the increasing necessity for higher safety in the electrical energy systems operation. One of these aspects concerns to the processing time of these methodologies to be used in real-time system operation. The computational time is disregarded for this purpose by the methods currently available. Therefore, this paper presents a new development for the power system stability margin estimation due to Hopf bifurcation and considers the computational time involved in this process to include the margin due to Hopf in electrical energy real-time operation. The presented development extends a methodology that makes up the state-of-the-art through an interest parameter relaxation in the stability margin due to Hopf bifurcation, namely, the eigenvalue frequency at the bifurcation point. This method uses Newton\'s method on a set of equations, and this work also proposes the use of a sparsity treatment for this set of equations, speeding up the algorithm. In order to demonstrate the proposed methodology efficiency, it was tested in two systems, the two areas Kundur system and the IEEE 39 bus system. The results are compared against the results of the classic methodology used in operation centers. Through these results it is possible to show the possibility of their use in real time and elucidate the major improvements resulting from the proposed development.

Apesar do crescente desenvolvimento das metodologias de detecção de bifurcações de Hopf em sistemas de energia elétrica nas últimas décadas, alguns aspectos ainda carecem de progressos, especialmente frente à crescente necessidade de aumento da segurança na operação dos sistemas eletroenergéticos. Um destes aspectos diz respeito ao tempo de processamento destas metodologias para serem utilizadas em tempo real na operação do sistema. Este tempo computacional é pouco considerado para este fim pelas metodologias atualmente existentes. Assim sendo, o presente trabalho apresenta um novo desenvolvimento relativo a estimação da margem de estabilidade do sistema de potência referente à bifurcação de Hopf e considera o tempo computacional envolvido neste processo a fim de incluir a margem devido a Hopf na operação em tempo real do sistema. O desenvolvimento apresentado estende uma das metodologias que compõe o estado da arte através da flexibilização de um dos parâmetros de interesse em relação a margem de estabilidade devido a bifurcação de Hopf, a saber, a frequência do autovalor no ponto de birfurcação. Esta metodologia utiliza o método de Newton em um conjunto de equações, e neste trabalho ainda é proposta a utilização de um tratamento da esparsidade para este conjunto de equações, deixando o algoritmo mais rápido. De forma a apresentar a eficiência desta metodologia proposta, esta foi testada em dois sistemas, o sistema Kundur de duas áreas e o sistema IEEE 39 barras. Os resultados obtidos são comparados frente a resultados obtidos também para a metodologia clássica utilizada em centros de operação. Através destes resultados é possível mostrar a possibilidade de sua utilização em tempo real e elucidar as grandes melhorias obtidas através do desenvolvimento proposto. / Despite the increasing development of Hopf bifurcations detection methods for electric power systems in the last decades, some aspects still require to progress, especially with the increasing necessity for higher safety in the electrical energy systems operation. One of these aspects concerns to the processing time of these methodologies to be used in real-time system operation. The computational time is disregarded for this purpose by the methods currently available. Therefore, this paper presents a new development for the power system stability margin estimation due to Hopf bifurcation and considers the computational time involved in this process to include the margin due to Hopf in electrical energy real-time operation. The presented development extends a methodology that makes up the state-of-the-art through an interest parameter relaxation in the stability margin due to Hopf bifurcation, namely, the eigenvalue frequency at the bifurcation point. This method uses Newton\'s method on a set of equations, and this work also proposes the use of a sparsity treatment for this set of equations, speeding up the algorithm. In order to demonstrate the proposed methodology efficiency, it was tested in two systems, the two areas Kundur system and the IEEE 39 bus system. The results are compared against the results of the classic methodology used in operation centers. Through these results it is possible to show the possibility of their use in real time and elucidate the major improvements resulting from the proposed development.

Nous étudions mathématiquement et numériquement des problèmes de réaction-diffusion avec convection. Dans la première partie, nous montrons sous certaines conditions que les opérateurs considérés ont la propriété de Fredholm, sont propres, et nous construisons un degré topologique pour ces opérateurs. Nous utilisons le degré pour étudier les bifurcations pour un problème d'ondes progressives de réaction-diffusion-convection, et nous montrons l'existence de fronts de réaction modifiés par la convection naturelle. Nous nous intéressons également aux instabilités convectives pour ces solutions. Nous étudions dans la deuxième partie l'influence de la tension de surface sur la stabilité des fronts. Dans le cas de liquides non miscibles, nous montrons que l'interaction de la tension de surface et de la réaction chimique peut conduire à une instabilité nouvelle. Dans le cas de liquides miscibles, nous modélisons la tension transitoire par une contrainte supplémentaire dans les équations de Navier-Stokes. Nous montrons que le problème mathématique correspondant a une solution unique, et nous observons numériquement que les gradients de concentration peuvent engendrer des courants convectifs. Nous simulons l'évolution d'une goutte miscible sous l'influence de ces courants~: elle est comparable à celle d'une goutte non miscible sous l'action de la tension de surface, avec une tendance à s'arrondir ou à se scinder en gouttelettes. Nous montrons numé\-ri\-quement que la tension transitoire peut amplifier de petites déformations de fronts plans.

A large body of literature exists regarding linear and nonlinear dynamic absorbers, but the vast majority of it deals with linear primary structures. However, nonlinearity is a frequency occurrence in engineering applications. Therefore, the present thesis focuses on the mitigation of vibrations of nonlinear primary systems using nonlinear dynamic absorbers. Because most existing contributions about their design rely on optimization and sensitivity analysis procedures, which are computationally demanding, or on analytic methods, which may be limited to small-amplitude motions, this thesis sets the emphasis on a tuning procedure of nonlinear vibration absorbers that can be computationally tractable and treat strongly nonlinear regimes of motion. The proposed methodology is a two-step procedure relying on a frequency-energy based approach followed by a bifurcation analysis. The first step, carried out in the free vibration case, imposes the absorber to possess a qualitatively similar dependence on energy as the primary system. This gives rise to an optimal nonlinear functional form and an initial set of absorber parameters. Based upon these initial results, the second step, carried out in the forced vibration case, exploits the relevant information contained within the nonlinear frequency response functions, namely, the bifurcation points. Their tracking in parameter space enables the adjustment of the design parameter values to reach a suitable tuning of the absorber. The use of the resulting integrated tuning methodology on nonlinear vibration absorbers coupled to systems with nonlinear damping is then investigated. The objective lies in determining an appropriate functional form for the absorber so that the limit cycle oscillation suppression is maximized. Finally, the proposed tuning methodology of nonlinear vibration absorbers may impose the use of complicated nonlinear functional forms whose practical realization, using mechanical elements, may be difficult. In this context, an electro-mechanical nonlinear vibration absorber relying on piezoelectric shunting possesses attractive features as various functional forms for the absorber nonlinearity can be achieved through proper circuit design. The foundation of this new approach are laid down and the perspectives are discussed.

Cette thèse se compose de trois parties. Dans la première est étudiée la vitesse d'un front entre un état homogène et un état périodique lorsque l'on ajoute les termes non résonants à la forme normale. On voit que cette vitesse est nulle sur un intervalle fini autour du point de Maxwell.<br /><br />La seconde partie est consacrée à l'étude d'une goutte déposée adiabatiquement sur un solvant plus dense. La goutte tombe dans le solvant jusqu'à une hauteur minimale, ensuite la fragmentation a lieu et les gouttelettes secondaires remontent à la surface. On a developpé un modèle théorique qui inclut l'essentiel du phénomène et prédit les échelles correctes du temps de montée et de la hauteur minimale.<br /><br />La troisième partie concerne l'étude linéaire du modèle de la valve à cristaux liquides avec rétro-action optique. Elle permet de comprendre un nouveau type de structures localisées qui apparaissent comme des pics isolés sur une structure spatiale de plus faible amplitude.

Les écoulements vrillés, qui sont largement utilisés dans les turbines à gaz, sont connus pour être sujet à des bifurcations entre différentes topologies (grandes reconfigurations de l'écoulement) qui peuvent affecter les performances et la sécurité du moteur. Ce travail se concentre sur l'étude de ces bifurcations en utilisant la Simulation aux Grandes Echelles (SGE). Cette étude montre qu'un petit changement dans les conditions dynamique du fluide, induite par les différents modèles de sous-maille utilisés, peut provoquer une transition entre deux régimes d'écoulement distincts lorsque l'écoulement tourbillonnaire est proche des conditions critiques de transition. La sensibilité de la SGE aux modèles de sous-maille est également identifiée comme le résultat d'un manque de résolution à certains endroits critiques, un problème qui est analysé en utilisant une méthode d'adaptation de maillage. L’adaptation de maillage est testée sur des cas académiques et industriels. Ici, par ajustement de la résolution du maillage sur la base des caractéristiques de l'écoulement étudié (raffinement et grossissement de la grille en maintenant constant le coût numérique), des améliorations substantielles peuvent être obtenues, en terme de prédictions de la SGE. Ce travail peut être considéré comme une des premières étapes vers la mise en place d'une procédure standard (reproductible et indépendante de l’utilisateur) de maillage pour la SGE. / Swirling flows, which are widely employed in gas turbines, are known to undergo bifurcation between different topologies (large reconfigurations of the flow field) affecting the engine performance and safety. This work focuses on the study of such bifurcations using Large-Eddy Simulation (LES). It shows that a small change in the fluid dynamics conditions, induced by the different Sub-Grid Scale (SGS) models used in the simulations, can cause a transition between two, distinct, flow states when the swirling flow is close to transition conditions. The sensitivity of LES to SGS modeling is also identified as the result of a lack of mesh resolution at some critical locations, a problem which is analyzed using mesh adaptation. Mesh adaptation is tested on canonical and industrial flows. Here, by adjusting the mesh resolution based on the characteristics of the flow examined (refining and coarsening the grid keeping constant the numerical cost), substantial improvements of the LES predictions can be obtained. This work can be considered as the first step toward the establishment of a standard (repeatable and user independent) meshing procedure for LES.

Orientador: Claudio Aguinaldo Buzzi / Banca: João Carlos da Rocha Medrado / Banca: Luciana de Fátima Martins / Resumo: Nesse trabalho são apresentados alguns resultados importantes sobre bifurcações de codimensão dois de campos de vetores. O resultado principal dessa dissertação e o teorema que d a o diagrama de bifurcação e os retratos de fase da Bifurcação de Bogdanov-Takens. Para a demonstracão são usadas algumas técnicas basicas de Sistemas Dinâmicos e Teoria das Singularidades, tais como Integrais Abelianas, desdobramentos de Sistemas Hamiltonianos, desdobramentos versais, Teorema de Preparação de Malgrange, entre outros. Outra importante bifurcação clássica apresentada e a Bifurca cão do tipo Hopf-Zero, quando a matriz Jacobiana possui um autovalor simples nulo e um par de autovalores imagin arios puros. Foram usadas algumas hipóteses que garantem propriedades de simetria do sistema, dentre elas, assumiuse que o sistema era revers vel. Assim como na Bifurcação de Bogdanov-Takens, foram apresentados o diagrama de bifurcao e os retratos de fase da Bifurcação Hopf-zero bifurcação reversível. As técnicas usadas para esse estudo foram a forma normal de Belitskii e o método do Blow-up polar. / Abstract: In this work is presented some important results about codimension two bifurcations of vector elds. The main result of this work is the theorem that gives the local bifurcation diagram and the phase portraits of the Bogdanov-Takens bifurcation. In order to give the proof, some classic tools in Dynamical System and Singularities Theory are used, such as Abelian Integral, versal deformation, Hamiltonian Systems, Malgrange Preparation Theorem, etc. Another classic bifurcation phenomena, known as the Hopf-Zero bifurcation, when the Jacobian matrix has a simple zero and a pair of purely imaginary eigenvalues, is presented. In here, is added the hypothesis that the system is reversible, which gives some symmetry in the problem. Like in Bogdanov-Takens bifurcation, the bifurcation diagram and the local phase portraits of the reversible Hopf-zero bifurcation were presented. The main techniques used are the Belitskii theory to nd a normal forms and the polar Blow-up method. / Mestre

We investigate in this thesis the generic properties of curves in the Minkowski plane R2 1 and of smooth Lorentzian surfaces. The generic properties of curves in R2 1 are obtained by studying the contacts of curves in R2 1 with lines and pseudo-circles. These contacts are captured by the singularities of the families of height and distancesquared functions on the curves. On the other hand, the generic properties of smooth Lorentzian surfaces are obtained by studying certain Binary Differential Equations defined on the surfaces.

Nous présentons dans cette étude le développement de la convection à partir de différentes perturbations de l'état conductif d'une couche fluide confinée dans une cavité cylindrique, chauffée par le bas et avec une surface supérieure libre. La discrétisation spatiale du domaine repose sur la méthode des éléments spectraux et les itérations temporelles sont assurées par une méthode splitting.Au déclenchement de la convection, les structures convectives correspondent à des modes de Fourier, et les seuils critiques dépendent du rapport de forme de la cavité, et des nombres de Biotet de Marangoni qui caractérisent la surface libre. Les transitions d'écoulements au-delà du seuil primaire sont caractérisées quantitativement en fonction du nombre de Rayleigh pour différentes valeurs du nombre de Biot et Ma = 0. Les résultats présentés sont obtenus en résolvant l'ensemble des équations non-linéaires de conservation à travers une méthode de continuation. Lorsque la convection se déclenche sous la forme d'un mode axisymétrique m = 0, l'évolution non-linéaire montre la coexistence de différentes structures convectives, des structures axisymétriques avec écoulement montant ou descendant au centre de la cavité et des structures correspondant à des combinaisons de modes qui apparaissent sur des branches secondaires sous-critiques.L'action d'un champ magnétique constant est ensuite étudiée pour des fluides conducteurs dans une même configuration comprenant une surface supérieure libre. Nous montrons l'effet stabilisateur du champ magnétique sur les seuils primaires ainsi que son action sélective sur les différents modes de convection. Nous analysons l'évolution des structures convectives au delà de ces seuils et montrons comment le champ magnétique modifie les transitions entre ces structures.En soumettant le bain fondu à un champ magnétique tournant, le mouvement de rotation du fluide se superpose aux mouvements de convection thermique et on observe une diminution des fluctuations de température et un retard du déclenchement de l'instabilité de Rayleigh-Bénard(lorsque les deux parois haut/bas du bain sont rigides). La rotation influe sur ce déclenchement qui de stationnaire devient oscillatoire, à l'exception du mode m = 0 de Fourier, pour qui la transition reste stationnaire jusqu'à une certaine valeur critique du nombre de Taylor magnétique.La dynamique de l'écoulement axisymétrique de part et d'autre de cette valeur critique sera étudiée en détail. / The growth of thermal convection out of different perturbations of the conductive base state is investigated using a spectral element time-stepping code. The fluid is subject to a vertical heat transfer in a cylindrical cavity with an upper free surface corresponding to the so-called Rayleigh-Bénard-Marangoni situation and the heat exchange through the free surface is evaluated via the Biot number. The results of the stability diagrams show that the evolution of the primary thresholds are largely influenced by the Biot number, the Marangoni number, and the aspect ratio of the cavity. Flow transitions are elucidated in quantitative detail as a function of the Rayleigh number for different Biot numbers in the tension free limit Ma = 0. The results shown are obtained by solving the full nonlinear field equations numerically among a continuation method. When an axisymmetric m = 0 Fourier mode is obtained at onset, the non-linear evolution shows the coexistence of different convective structures, the axisymmetric structures with up-ow or down-ow at the center and mixed-mode structures which appear on secondary subcritical branches. The action of a constant magnetic field is then considered for melts in the same type of configuration with an upper free surface. We show the global stabilizing effect of the magnetic field on the primary bifurcation thresholds and the selective effect on the different instability modes. We also analyze the evolution of the convective structures above the thresholds and show how the magnetic field modifies the transitions between these structures. When applying a magnetic body forcing in the azimuthal direction (RMF), one can damp the unavoidable thermal fluctuations inside the melt and delay the transition to the Rayleigh-Bénard instability (for rigid-rigid circular plates at top and bottom). The rotation effect also changes the transitions from steady to oscillatory, except for the m = 0 Fourier mode where the transitionis first steady until a critical Taylor number and then becomes oscillatory. The dynamics of the transitions to the axisymmetric flow, below and above this value of critical magnetic Taylor number, is particularly interesting and will be described.