Determining the maximum power system loadability, as well as preventing the system from being operated close to the stability limits is very important in power systems planning and operation. The application of optimization techniques to power systems security and electricity markets is a rather relevant research area in power engineering. The study of optimization models to determine critical operating conditions of a power system to obtain secure power dispatches in an electricity market has gained particular attention. This thesis studies and develops optimization models and techniques to detect or avoid voltage instability points in a power system in the context of a competitive electricity market. A thorough analysis of an optimization model to determine the maximum power loadability points is first presented, demonstrating that a solution of this model corresponds to either Saddle-node Bifurcation (SNB) or Limit-induced Bifurcation (LIB) points of a power flow model. The analysis consists of showing that the transversality conditions that characterize these bifurcations can be derived from the optimality conditions at the solution of the optimization model. The study also includes a numerical comparison between the optimization and a continuation power flow method to show that these techniques converge to the same maximum loading point. It is shown that the optimization method is a very versatile technique to determine the maximum loading point, since it can be readily implemented and solved. Furthermore, this model is very flexible, as it can be reformulated to optimize different system parameters so that the loading margin is maximized. The Optimal Power Flow (OPF) problem with voltage stability (VS) constraints is a highly nonlinear optimization problem which demands robust and efficient solution techniques. Furthermore, the proper formulation of the VS constraints plays a significant role not only from the practical point of view, but also from the market/system perspective. Thus, a novel and practical OPF-based auction model is proposed that includes a VS constraint based on the singular value decomposition (SVD) of the power flow Jacobian. The newly developed model is tested using realistic systems of up to 1211 buses to demonstrate its practical application. The results show that the proposed model better represents power system security in the OPF and yields better market signals. Furthermore, the corresponding solution technique outperforms previous approaches for the same problem. Other solution techniques for this OPF problem are also investigated. One makes use of a cutting planes (CP) technique to handle the VS constraint using a primal-dual Interior-point Method (IPM) scheme. Another tries to reformulate the OPF and VS constraint as a semidefinite programming (SDP) problem, since SDP has proven to work well for certain power system optimization problems; however, it is demonstrated that this technique cannot be used to solve this particular optimization problem.

Determining the maximum power system loadability, as well as preventing the system from being operated close to the stability limits is very important in power systems planning and operation. The application of optimization techniques to power systems security and electricity markets is a rather relevant research area in power engineering. The study of optimization models to determine critical operating conditions of a power system to obtain secure power dispatches in an electricity market has gained particular attention. This thesis studies and develops optimization models and techniques to detect or avoid voltage instability points in a power system in the context of a competitive electricity market. A thorough analysis of an optimization model to determine the maximum power loadability points is first presented, demonstrating that a solution of this model corresponds to either Saddle-node Bifurcation (SNB) or Limit-induced Bifurcation (LIB) points of a power flow model. The analysis consists of showing that the transversality conditions that characterize these bifurcations can be derived from the optimality conditions at the solution of the optimization model. The study also includes a numerical comparison between the optimization and a continuation power flow method to show that these techniques converge to the same maximum loading point. It is shown that the optimization method is a very versatile technique to determine the maximum loading point, since it can be readily implemented and solved. Furthermore, this model is very flexible, as it can be reformulated to optimize different system parameters so that the loading margin is maximized. The Optimal Power Flow (OPF) problem with voltage stability (VS) constraints is a highly nonlinear optimization problem which demands robust and efficient solution techniques. Furthermore, the proper formulation of the VS constraints plays a significant role not only from the practical point of view, but also from the market/system perspective. Thus, a novel and practical OPF-based auction model is proposed that includes a VS constraint based on the singular value decomposition (SVD) of the power flow Jacobian. The newly developed model is tested using realistic systems of up to 1211 buses to demonstrate its practical application. The results show that the proposed model better represents power system security in the OPF and yields better market signals. Furthermore, the corresponding solution technique outperforms previous approaches for the same problem. Other solution techniques for this OPF problem are also investigated. One makes use of a cutting planes (CP) technique to handle the VS constraint using a primal-dual Interior-point Method (IPM) scheme. Another tries to reformulate the OPF and VS constraint as a semidefinite programming (SDP) problem, since SDP has proven to work well for certain power system optimization problems; however, it is demonstrated that this technique cannot be used to solve this particular optimization problem.

Les systèmes écologiques sont des systèmes complexes qui ne peuvent pas être d´écrits par un unique modèle mathématique. De nombreux modèles peuvent être construits pour un même système, selon les internets du modélisateur et ses choix dans la construction du modèle. Quel est l’impact de ces choix dans la construction du modèle sur les prédictions de la dynamique des systèmes écologiques et les informations qu’elles fournissent sur la résilience de ces systèmes est la question générale qui guide le travail présente dans cette thèse. Cette thèses focalise sur un choix entre formulations de modèle basées sur des mécanismes biologiques et qui décrivent les données empiriques avec la même efficacité. Ces modèles sont proches l’un de l’autre, donc on s’attendrait `a ce que leurs prédictions soient similaires. Cependant, nous montrons avec un exemple générique de modèle prédateur-proie que des formulations similaires du processus de prédation peuvent prédire des dynamiques qualitativement différentes en terme de: (i) nombre et type d'états stables, et (ii) réponse et résilience du système face à une perturbation extérieure. Ces différences dans les prédictions du modèle sont expliquées par une analyse mathématique détaillée du modèle prédateur-proie. Ensuite, ce modèle est étendu à des réseaux trophiques compos´es de dizaines d’espèces. La complexité de ces réseaux (nombre d'espèces et d’interactions) explique leur persistance, alors que leur dynamique temporelle est fortement affectée par la fonction utilisée pour modéliser la prédation. Des méthodes sont ´également proposées pour quantifier la sensibilités d’un modèle. Finalement, nous montrons que si un minimum de détails biologiques sont pris en compte, des modèles prédateurs-proies sont moins sensibles `a la formulation de la prédation. Ceci nous donne des pistes pour gérer les incertitudes dans la construction d’un modèle, qui sont intrinsèques à la complexité des systèmes naturels. / Ecological systems are complex systems which cannot be described by a single mathematical model. Multiple modelsof a same system can be built, depending on modeller’s interests and on its choices during model construction. Howfar these choices in model construction can affect the predicted dynamics of ecological systems and the informationthey provide on their resilience? is the general question that leads the research presented in this thesis. This thesisfocuses on a choice between model formulations that are based on biological mechanisms and describe empiricaldata with the same accuracy. These models are close to each other, so they are expected to predict similar systemdynamics. However, we show through a generic example of predator-prey model that similar formulations of thepredation process can predict qualitatively different system dynamics in term of: (i) number and type of stablestates, and (ii) system response to external disturbance and its potential for recovery. These differences in modelpredictions are explained by a detailed mathematical analysis of the predator-prey model. Next, this model isextended to complex food webs made of tens of species. The complexity of these networks (number of species andinteractions) drives their persistence, whereas their temporal dynamics is strongly affected by the function used tomodel predation. Methods to quantify model sensitivity are also proposed. Finally, we show that if a minimumlevel of biological details is included, predator-prey models are less sensitive to predation formulation. This providea clue to deal with uncertainties in model construction, which are intrinsic to the complexity of natural systems.

Cette thèse est consacrée à l’étude du phénomène de Newhouse et des bifurcations en dynamique holomorphe à plusieurs variables. Elle comporte trois Théorèmes principaux. Le premier de ces trois résultats est un Gap Lemma complexe. En dynamique réelle, le Gap Lemma de Newhouse donne un critère sur le produit des épaisseurs de deux ensembles de Cantor dynamiques pour prouver que leur intersection est non vide. On en donne une généralisation partielle au cas des ensembles de Cantor dynamiques dans C. Plus précisément, on introduit une notion d’épaisseur pour un ensemble de Cantor dynamique planaire et on fournit un critère sur le produit de deux épaisseurs afin d’obtenir une intersection entre deux ensembles de Cantor dynamiques. On montre également que l’épaisseur est une quantité qui varie continûment, ce qui permet d’obtenir des intersections persistantes d’ensembles de Cantor dynamiques. Le second Théorème de cette thèse démontre l’existence du phénomène de Newhouse dans l’espace des automorphismes polynomiaux de degré d pour n’importe quel degré d ≥ 2 dans C^{3}. Au contraire de la situation dans C^{2}, le degré est ici connu et optimal. Le point clef de la preuve est l’introduction dans le domaine complexe d’un outil issu de la dynamique réelle : le blender de Bonatti et Diaz. On formalise le concept de blender complexe et on donne un automorphisme polynomial de C^{3} de degré 2 possédant un blender. Puis, on l’utilise afin de construire successivement des tangences persistantes et des sous-ensembles résiduels d’automorphismes ayant une infinité de puits. Enfin, le dernier résultat porte sur les bifurcations d’endomorphismes holomorphes de P^{2}(C) très particuliers, appelés exemples de Lattès, semi-conjugués à une application affine sur un tore. Dujardin a conjecturé que ces derniers étaient accumulés par des ouverts de bifurcations. On montre que tout exemple de Lattès de degré suffisamment élevé est accumulé par de telles bifurcations robustes. Ceci implique en particulier que tout exemple de Lattès possède un itéré dans l’adhérence de l’intérieur du lieu de bifurcation. La démonstration est basée sur l’obtention d’intersections persistantes entre l’ensemble postcritique et un ensemble hyperbolique répulsif contenu dans l’ensemble de Julia. La preuve est divisée en deux parties : on donne tout d’abord un toy-model qui permet d’obtenir des intersections persistantes entre l’ensemble limite d’un certain type d’IFS, appelé IFS correcteur, et une courbe. Ensuite, dans un second temps, on perturbe l’exemple de Lattès pour créer simultanément un IFS correcteur dans l’ensemble de Julia et une courbe bien orientée dans l’ensemble postcritique / In this PhD thesis, we study Newhouse’s phenomenon and bifurcations in the context of dynamics in several complex variables. We prove three main Theorems. The first one is a complex Gap Lemma. In real dynamics, Newhouse’s Gap Lemma gives a criterion on the product of the thicknesses of two dynamical Cantor sets K and L to show that K ∩ L is not empty. We show a partial generalization of this result for dynamical Cantor sets in C. A relevant notion of thickness in this case is defined and we give some criterion on the product of two thicknesses to show that two dynamical Cantor sets in C must intersect. We also show that the thickness varies continuously, which generates persistent intersections of dynamical Cantor sets. In the second Theorem, we show that there exists a polynomial automorphism f of C^{3} of degree 2 such that for every automorphism g sufficiently close to f, g admits a tangency between the stable and unstable laminations of some hyperbolic set. As a consequence, for each d ≥ 2, there exists an open set of polynomial automorphisms of degree at most d in which the automorphisms having infinitely many sinks are dense. In contrary to the case of C^{2}, the degree is known. To prove these results, we give a complex analogous to the notion of blender introduced by Bonatti and Diaz. In particular, we use a blender to produce robust tangencies. In the third and last result, we study the phenomenon of robust bifurcations in the space of holomorphic maps of P^{2}(C). We prove that any Lattès example of sufficiently high degree belongs to the closure of the interior of the bifurcation locus. This gives a partial answer to a conjecture of Dujardin. In particular, every Lattès map has an iterate with this property. To show this, we design a method creating robust intersections between the limit set of a particular type of iterated functions system in C^{2} with a well-oriented complex curve. Then we show that any Lattès map of sufficiently high degree can be perturbed so that the perturbed map exhibits this geometry

We study one-parameter families of quasi-periodically forced monotone interval maps and provide sufficient conditions for the existence of a parameter at which the respective system possesses a non-uniformly hyperbolic attractor. This is equivalent to the existence of a sink-source orbit, that is, an orbit with positive Lyapunov exponent both forwards and backwards in time. The attractor itself is a non-continuous invariant graph with negative Lyapunov exponent, often referred to as ‘SNA’. In contrast to former results in this direction, our conditions are C² -open in the fibre maps. By applying a general result about saddle-node bifurcations in skew-products, we obtain a conclusion on the occurrence of non-smooth bifurcations in the respective families. Explicit examples show the applicability of the derived statements.

This thesis examines how early warning signals perform when tested on climate systems thought to exhibit future tipping point behaviour. A tipping point in a dynamical system is a large and sudden change to the state of the system, usually caused by changes in external forcing. This is due to the state the system occupies becoming unstable, causing the system to settle to a new stable state. In many cases, there is a degree of irreversibility once the tipping point has been passed, preventing the system from reverting back to its original state without a large reversal in forcing. Passing tipping points in climate systems, such as the Amazon rainforest or the Atlantic Meridional Overturning Circulation, is particularly dangerous as the effects of this will be globally felt. Fortunately there is potential for early warning signals, designed to warn that the system is approaching a tipping point. Generally, these early warning signals are based on analysis of the time series of the system, such as searching for ‘critical slowing down’, usually estimated by an increasing lag-1 autocorrelation (AR(1)). The idea here is that as a system’s state becomes less stable, it will start to react more sluggishly to short term perturbations. While early warning signals have been tested extensively in simple models and on palaeoclimate data, there has been very little research into how these behave in complex models and observed data. Here, early warning signals are tested on climate systems that show tipping point behaviour in general circulation models. Furthermore, it examines why early warning signals might fail in certain cases and provides prospect for more ‘system specific indicators’ based on properties of individual tipping elements. The thesis also examines how slowing down in a system might affect ecosystems that are being driven by it.

Ce travail de thèse porte sur les instabilités dans les structures minces hyperélastiques. Nous analysons les mécanismes de sélection du motif de flambement dans un solide prismatique fortement pré-contraint. Pour ce système, le modèle de poutre classique d'Euler-Bernoulli n'est pas pertinent du fait de cette forte précontrainte et il est nécessaire de recourir à une description 3-d pour expliquer l'apparition de modes instables de petite longueur d'onde. Notre analyse, fondée sur la théorie de l'élasticité finie incrémentale, montre que la longueur typique du motif de flambement est sélectionnée par l'importance relative de la pré-contrainte et du rapport d'aspect du solide prismatique. Nous nous interrogeons ainsi sur les limites du modèle classique et proposons une piste de réflexion pour la construction de nouveaux modèles. Celle-ci repose sur un développement à deux échelles combiné aux équations d'équilibre du système formulées sous forme faible. Il est alors facile de résoudre les équations exactes à chaque ordre, ce qui donne accès à la cinématique complète du système. Nous mettons en œuvre cette méthode pour l'exemple simple d'un barreau homogène en compression, ce qui nous permet d'établir le modèle classique d'Euler-Bernoulli à partir des équations de l'élasticité 3-d. La dernière partie de ce travail de thèse porte sur l'étude du flambement de systèmes invariants d'échelle : l'étude expérimentale et numérique de la compression d'un prisme à base triangulaire met en lumière une transition inédite d'un mode de flambement étendu vers des motifs localisés. / This Ph.D. work deals with buckling instabilities arising in thin hyper-elastic structures. We focus on instabilities arising in a prismatic solid submitted to finite incompatible pre-strains. We observe that the traditional 1-d Euler-Bernoulli beam model is not applicable to such a system because of the finite inhomogeneous pre-stress. The latter triggers short-wavelength instabilities that are note described by the classical 1-d models. We rely on the 3-d elasticity theory and propose a quantitative criterion on the ratio between the pre-stress and the cross-sectional aspect-ratio of the prismatic solid, that predicts the typical wavelength of the buckling pattern. This work questions the validity of classical 1-d models and suggests that extensions of these models are possible. We propose a method for the systematic derivation of reduced models. It relies on asymptotic expansions of the variational formulation of the equilibrium equations, starting from the complete expression of the energy. In this framework, kinematics can be entirely determined by solving the exact equations, order by order. We successfully apply this method to a homogeneous and isotropic beam submitted to a homogeneous compression and recover the classical Euler-Bernoulli beam model. In a last part of the manuscript we investigate the transition from extended wrinkling to localized creasing in a scale-invariant system made of a prismatic solid with a triangular cross-section, both experimentally and numerically.

Aplica-se a Teoria Qualitativa de Equações Diferenciais aos problemas de sincronismo de fase, associando às diversas regiões do espaço de parâmetros os tipos de atratores esperados. <p style=\"margin: 11px 0px;\">Três casos básicos são estudados: <ol type=\"i\" style=\"padding: 0px 40px\"> Malha de Sincronismo de fase Autônoma de 2ª Ordem Modulação em Frequência Acidental em Malha de Sincronismo de Fase de 2ª Ordem Malha de Sincronismo de Fase Autônoma de 3ª Ordem <p style=\"margin: 11px 0px;\">No caso (i), usando resultados clássicos da teoria de sistemas dinâmicos, discute-se os pontos de equilíbrio e os ciclos limite. <p style=\"margin: 11px 0px;\">No caso (ii), usando o método de Melnikov propõem-se critérios para previsão de aparecimento de atratores caóticos.<p style=\"margin: 11px 0px;\">No caso (iii), usando o teorema de bifurcações de Hopf, a estabilidade dos pontos de equilíbrio e a formação dos ciclos limite são analisadas / The Qualitative Theory of Differential Equations is applied to the phaselock problems, and the several parameters space regions are associated to the expected attractors. <p style=\"margin: 11px 0px;\">Three basic cases are studied: <ol type=\"i\" style=\"padding: 0px 40px\"> Autonomous Second Order Phaselock Loop Accidental Frequency Modulation on Second Orer Phaselock Loop Autonomous Third Order Phaselock Loop <p style=\"margin: 11px 0px;\">In case i), using classical results of dynamical systems theory, the equilibrium points and limit cycles are analyses. <p style=\"margin: 11px 0px;\">In case ii), the Melnikov technique gives some criteria for chaotic attractors.<p style=\"margin: 11px 0px;\">In case iii), Hopf bifurcation theorem provides propositions about equilibrium points and limit cycles.

We propose in this thesis a way to study deformations of plane curves that take into consideration the geometry of the curves as well as their singularities. We deal in details with local phenomena that occur generically in two-parameter families of curves. We obtain information on the inflections and vertices appearing on the deformed curves. We also obtain the configurations of the evolutes of the curves and of their deformations, and apply our results to orthogonal projections of space curves. Finally, we consider the profile (outline, apparent contour) of a smooth surface in the Euclidian 3-space. This is the image of the singular set of an orthogonal projection of the surface. The profile is a plane curve and may have singularities. We study the changes in the geometry of the profile as the direction of projection changes locally in the unit sphere. / Propomos nesta tese um método para estudar deformações de curvas planas que leva em consideração a geometria delas, bem como as suas singularidades. Consideramos em detalhes os fenômenos locais que ocorrem genericamente em famílias de curvas com dois parâmetros. Obtemos informações sobre as inflexões e vértices que aparecem nas curvas deformadas. Obtemos também as configurações das evolutas das curvas e das suas deformações e aplicamos os nossos resultados nas projeções ortogonais de curvas espaciais. Finalmente, consideramos o perfil de uma superfície regular no espaço Euclidiano R3. O perfil é a imagem do conjunto singular de uma projeção ortogonal da superfície, esta é uma curva plana e pode ter singularidades. Estudamos as alterações na geometria do perfil quando a direção de projeção muda localmente na esfera unitária.

CoordenaÃÃo de AperfeiÃoamento de NÃvel Superior / Numa cama, numa greve, numa festa ou numa revoluÃÃo: era assim que Glauber Rocha imaginava que A Idade da Terra (1980), seu Ãltimo filme, poderia ser experimentado. A obra leva ao limite um processo de pesquisa que radicalizava o fazer cinema e inventava novos possÃveis para a experiÃncia estÃtica. à um trabalho que tento pensar junto a duas outras obras do realizador, Claro (1975), filmado durante o exÃlio em Roma, e Di Cavalcanti (1977), curta-metragem que ele produziu no impulso, ao saber da morte do amigo pintor. Cada um desses filmes desencadeia procedimentos singulares de ocupaÃÃo do mundo e de invenÃÃo de caminhos para o cinema. Eles sÃo entendidos aqui como bifurcaÃÃes em formas configuradas de sensibilidade, na medida em que embarcam na proliferaÃÃo de veredas para a fabricaÃÃo fÃlmica. Trata-se de uma preocupaÃÃo mais ampla em discutir as polÃticas dos filmes, maneiras singulares pelas quais o cinema pode inscrever no mundo uma operaÃÃo de rotura estÃtica que à jà uma polÃtica. Nesse sentido, cabe pensar com as obras algumas figuras que afirmam o cinema como campo de resistÃncias. Uma das questÃes que norteiam as discussÃes deste trabalho à a dimensÃo de um devir minoritÃrio, pensado a partir de Deleuze e Guattari, um movimento no qual o cinema pode se engajar para tensionar com um visÃvel e um audÃvel que se apresentam como fatos majoritÃrios. Nesse processo, à a prÃpria vida em comum que està em jogo. Essas operaÃÃes de invenÃÃo se dÃo em um emaranhado de temporalidades, o que leva a uma abordagem metodolÃgica que se coloca como assumidamente anacrÃnica, na medida em que tenta pensar o que uma obra faz agitar no tempo e como ela pode fazer esse tempo se abrir, numa articulaÃÃo que tento fazer com os pensamentos de Benjamin, Warburg, Didi-Huberman e Agamben. Dessa maneira, cabe investigar a dimensÃo de uma contemporaneidade desse cinema de Glauber, na medida em que as proliferaÃÃes buscadas pelas formas fÃlmicas nÃo estÃo circunscritas a um instante de uma cronologia de eventos sucessivos, mas se espalham rizomaticamente pelos tempos, em movimento proliferante e diferenciante. A polÃtica e a estÃtica seriam questÃes de como fazemos pontes nas temporalidades, de quais traÃados sÃo arrancados como possÃveis. Ou de como os engajamentos no tempo podem suscitar acontecimentos e instaurar resistÃncias em devir. / In a bed, a strike, a party or a revolution: that was how Glauber Rocha once thought that his last film, A Idade da Terra (1980), could be experienced. The work takes to the limit a research that turns making cinema a process more radical. It also invents news possibilities to aesthetic experience. Itâs a film that I try to think with two others works from Glauber: Claro (1975), shot when he was in exile in Rome, and Di Cavalcanti (1977), a short film that he produced on an impulse, when he heard that his painter friend had died. Each of these films triggers singular procedures on occupying the world and on inventing paths to the cinema. They are taken here as bifurcations on configured forms of sensibility, as they try to proliferate ways of making a film. This is a broader concern in discussing the politics of films, singular ways in which cinema can inscribe in the world an operation of aesthetic fracture that is already political. In this sense, it is thinking with the works that we can propose some figures who claim cinema as field of resistances. One of the questions that guide the discussions of this work is the concept of a becoming minority, thought from Deleuze and Guattari, a movement in which cinema can engage to create tension with a visible and an audible that present themselves as major facts. In this process, it is common life that is at stake. Those operations of invention are taken in a entanglement of temporalities, which leads to an approach frankly anachronistic, on thinking what a work can shake in time and how it can open this time, in which I try to articulate with the thoughts of Benjamin, Warburg, Didi-Huberman and Agamben. Thus, the idea is to investigate how we could consider a contemporaneity in this cinema of Glauber, in that the proliferations researched by filmic forms are not restricted to a chronology of successive events, rather they spread in a rhizomatic way through temporalities, a proliferative and differentiating movement. Politics and aesthetics would be questions of how we bridge the temporalities, of which tracings are turn into possibilities. Or how the engagements in time can raise events and create resistances in becoming.