Nesta tese foi estudado o transporte de partículas na borda do plasma confinado magneticamente em tokamaks a partir de um modelo para ondas de deriva proveniente de flutuaçõoes eletrostáticas geradas pela não uniformidade do plasma. Para investigar esse problema, consideramos o modelo com duas ondas de deriva, que possui uma complexa dinâmica não linear onde podemos encontrar tanto transporte anômalo quanto transporte difusivo. Para a encontras no plano de fases as Estruturas Lagrangianas Coerentes (ELCs) e os jatos, foram confeccionados mapas de Poincaré, diagramas de expoente de Lyapunov a tempo finito, diagramas de deslocamento quadrático, diagramas de autocorrelação da velocidade e o diagrama de retorno. Para avaliar o impacto dessas ELCs no transporte de partículas foram analisados a série temporal do desvio padrão médio, da dispersão relativa e dos saltos dentro do mapa de Poincar´e e também foram confeccionados histogramas com a distribuição desses saltos. Foi encontrado que, com duas ondas de deriva e para uma determinada combinação de parâmetros, surgem correntes de jato, que persistem por longos períodos, imersas na região caótica. Verificamos que, assim como nas ilhas, a região interna às correntes de jato são inacessíveis às ELCs. Também foi encontrado que, quando existe uma corrente de jato, o transporte observado na região caótica não é simétrico com uma pequena deriva na direção contraria ao jato. Esse fenômeno observado ocorre em contrapartida ao caso típico de sistemas com mistura em que as ELCs tem acesso a todo o plano de fase e o transporte é difusivo. / In this thesis we studied the particle transport in the edge of magnetically confined plasma in tokamaks using a model of drift waves due to electrostatic fluctuations generated by the non-uniformity of the plasma. To investigate this issue, we consider the model with two drift waves, which has a complex nonlinear dynamics where we can find both anomalous and diffusive transport. To find the Lagrangian Coherent Structures (LCSs) and the jets, we used Poincaré maps, Finite time Lyapunov exponent diagrams, quadratic displacement diagrams, autocorrelation velocity diagrams and return displacement diagram. To evaluate the impact of LCSs in the transport of particles, we analyzed the time series of both average standard deviation and relative dispertion and also histograms of the distribution of these jumps. It was found that, with two drift waves and for a given combination of parameters, a jet streams appear in the phase space and persist for long periods of time immersed in the chaotic region. We found that, as well as on the islands, the inner region of the jet streams are inaccessible to LCSs. It was also found that when there is a jet stream, the transport observed in the chaotic region is not symmetrical and have a small drift in the opposite direction to the jet. This phenomenon is observed in contrast to the typical case of systems with mixing in wich the LCSs have access to all the phase space and the trasnport is diffusive.

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: Paulo Regis Caron Ruffino / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-13T23:01:19Z (GMT). No. of bitstreams: 1 Costa_PauloHenriquePereirada_M.pdf: 875708 bytes, checksum: 8862a1813f1bb85b5d0269462a80501e (MD5) Previous issue date: 2009 / Resumo: O objetivo desse trabalho é estudar as equações de Hamilton no contexto estocástico. Sendo necessário para tal um pouco de conhecimento a cerca dos seguintes assuntos: cálculo estocástico, geometria de segunda ordem, estruturas simpléticas e de Poisson. Abordamos importantes resultados, dentre eles o teorema de Darboux (coordenadas locais) em variedades simpléticas, teorema de Lie-Weinstein que de certa forma generaliza o teorema de Darboux em variedades de Poisson. Veremos que apesar de o ambiente natural para se estudar sistemas hamiltonianos ser variedades simpléticas, no caso estocástico esses sistemas se adaptam bem em variedades de Poisson. Além disso, para atingir a nossa meta, estudaremos equações diferenciais estocásticas em variedades de dimensão finita usando o operador de Stratonovich / Abstract: This dissertation deals with transfering Hamilton's equations in stochastic context. This requires some knowledge about the following: stochastic calculus, second order geometry and Poisson and simplectic structures. Important results that will be discussed in this theory are Darboux's theorem (local coordinates) for simplectic manifolds, and Lie-Weintein's theorem that is in a certain way of Darboux's theorem on Poisson manifolds. We will see that although the natural environment for studying hamiltonian systems is symplectic manifolds, if we have a Poisson structure we will still be able to study them. Moreover, to achieve our goal, we will study stochastic differential equations on finite dimensional manifolds using the Stratonovich operator / Mestrado / Geometria Estocastica / Mestre em Matemática

Um dos problemas enfrentados pelos cientistas para o confinamento de plasma em Tokamaks, para se obter fusão termonuclear controlada, é o transporte radial de partículas pela borda do plasma. Nessa dissertação, estudamos o transporte através de um modelo que relaciona as flutuações eletrostáticas na borda do plasma às ondas de deriva. Essas ondas criam no plasma regiões de fluxo convectivo, formando ilhas que são, eventualmente, separadas por barreiras. Para apenas uma onda, o sistema é integrável e todas as trajetórias do plano de fase são curvas invariantes que, se não existirem barreiras, estão em ilhas divididas por separatrizes. Foi verificado que, quando uma segunda onda com velocidade de fase diferente da primeira é utilizada, o sistema não é mais integrável e a região anteriormente ocupada pelas separatrizes torna-se caótica. Com a quebra de separatrizes ocorre o transporte caótico de partículas. Quando uma separatriz é quebrada, surge em seu lugar uma estrutura que ainda preserva algumas características da separatriz, mas se modifica no espaço de fases ao longo do tempo. Essa estrutura é conhecida como Estrutura Lagrangiana Coerente (ELC). Nessa dissertação verificamos que as ELCs, por um lado, funcionam como barreiras de transporte, pois nenhuma trajetória consegue atravessa-la e, por outro lado, criam regiões no espaço de fases onde o transporte é alto, pois trajetórias próximas a elas tendem a ser aceleradas. Uma das principais contribuições obtidas ao se estudar ELCs no problema de duas ondas de deriva, aplicado ao confinamento de plasmas em Tokamaks, é a possibilidade de se prever a existência de ilhas, que funcionem como barreiras de transporte, no plano de fases que, por sua vez, são um importante mecanismo de aprisionamento de partículas. / One of the problems facing scientists in the confinement of plasma in tokamaks, to obtain controlled thermonuclear fusion, is the radial transport of particles at the plasma edge. In this dissertation, we study particle transport through a model that relates the electrostatic fluctuations at the edge of the plasma with drift waves. These waves create regions inside the plasma with convective flow, forming islands that are eventually separated by barriers. For one wave, the system is integrable and all the trajectories of phase space are invariant curves that are divided by separatrices. It was found that when a second wave with phase velocity different from the first is used, the system is no longer integrable and the region previously occupied by the separatrix becomes chaotic. With the destruction of the separatrix the transport of particles is chaotic. When a separatrix is broken, appears in its place a structure that preserves some features of the separatrix, but it is changing in phase space over time. This structure is known as Lagrangian Coherent Structure (LCS). In this dissertation we found that the LCSs, on the one hand, act as transport barriers, since no trajectory can cross it and, moreover, creates regions in phase space where particle transport is high, because trajectories close to them tend to be accelerated. One of the main contributions obtained by studying LCSs in the problem of two drift waves, applied to the confinement of plasma in tokamaks, is the ability to predict the existence of islands, which act as transport barriers, which are an important mechanism of trapping particles.

Un aéronef à voilure tournante est un système physique dynamique complexe. Le développement de ce type de système nécessite méthodes d’analyse (structurelle et comportementale) et de commande afin de maîtriser ses comportements. L’approche énergétique (bond graph et formalisme hamiltonien à port) permet une représentation multi-physique non linéaire, modulaire (acausale) et à différents niveaux de granularité. Parmi ses organes, les commandes de vol de l’aéronef permettent la transmission du pilotage aux rotors : canaliser la puissance motrice (2 MW) à partir d’une commande manuelle est impossible sans organes actifs d’assistance. Afin de représenter les cheminements et traitements des informations nécessaires aux organes actifs, la représentation multi-physique est complétée par une représentation informationnelle causale (schéma bloc).Les travaux exposés dans ce mémoire visent à ajouter le niveau de granularité intermédiaire et nécessaire entre la représentation multi-physique pure et une représentation combinée physique et informationnelle. Basée sur la démarche du PMBC (Physical Model Based Control), ils proposent une méthode originale permettant de représenter les organes d’assistance et leur commande par un modèle physique équivalent. La méthode est ici enrichie dans une démarche de conception des Systèmes d’Assistance à Opérateur : nous déterminons où doivent agir les organes actifs, selon quelles mesures et suivant quelles lois de commande. La méthode est illustrée sur un cas d’étude industriel : nous obtenons deux représentations de l’espace des solutions (les représentations physico-informationnelle détaillée et globale de son comportement) incluant la solution industrielle actuelle. / A rotorcraft is a complex dynamic physical system. The development of this kind of systems requires methods to analyze its structure and its behavior and to control this latter. The energetic framework (bond graph and Hamiltonian formulation) allows a multiphysical nonlinear representation, modular and with several levels of granularity. Among its components, flight controls transmit the orders from the pilot to rotors. Leading the motive power (about 2MW) directly from a handling control is almost impossible without active devices for assistance. In order to represent the flow of the control information and its processing, a cyberphysical representation combines a multiphysical representation with an informational representation (bloc diagram).This thesis work aims at proposing an intermediate granularity level between purely multiphysical representations and cyberphysical representations. Based on PMBC (Physical Model Based Control) approach, a new method to represent the assistance parts is proposed, by means of a physical equivalent model. The method is then enriched by a genuine design procedure of an Operator Assisting System: we determine where actuators must operate, according to which control laws and from which measurements. The method is applied to an industrial case: two representations of the possible design solutions set are obtained, a detailed cyberphysical representation and a global representation of its behavior. The actual industrial solution belongs to the defined set of possible solutions.

Dans cette thèse nous présentons les résultats sur l'emploi des systèmes Hamiltoniens à port et des systèmes de contact commandés pour la modélisation et la commande de systèmes issus de la Thermodynamique Irréversible. Premièrement nous avons défini une classe de pseudo-systèmes Hamiltoniens à port, appelée systèmes Hamiltoniens à port irréversibles, qui permet de représenter simultanément le premier et le second principe de la Thermodynamique et inclut des modèles d'échangeurs thermiques ou de réacteurs chimiques. Ces systèmes ont été relevés sur l'espace des phases thermodynamiques muni d’une forme de contact, définissant ainsi une classe de systèmes de contact commandés, c'est-à-dire des systèmes commandés non-linéaires définis par des champs de contacts stricts. Deuxièmement, nous avons montré que seul un retour d'état constant préserve la forme de contact et avons alors résolu le problème d'assignation d'une forme de contact en boucle fermée. Ceci a mené à la définition de systèmes de contact entrée-sortie et l'analyse de leur équivalence par retour d'état. Troisièmement, nous avons montré que les champs de contact n'étaient en général pas stables en leur zéros et avons alors traité du problème de la stabilisation sur une sous-variété de Legendre en boucle fermée. / This doctoral thesis presents results on the use of port Hamiltonian systems (PHS) and controlled contact systems for modeling and control of irreversible thermodynamic processes. Firstly, Irreversible PHS (IPHS) has been defined as a class of pseudo-port Hamiltonian system that expresses the first and second principle of Thermodynamics and encompasses models of heat exchangers and chemical reactors. These IPHS have been lifted to the complete Thermodynamic Phase Space endowed with a natural contact structure, thereby defining a class of controlled contact systems, i.e. nonlinear control systems defined by strict contact vector fields. Secondly, it has been shown that only a constant control preserves the canonical contact structure, hence a structure preserving feedback necessarily shapes the closed-loop contact form. The conditions for state feedbacks shaping the contact form have been characterized and have lead to the definition of input-output contact systems. Thirdly, it has been shown that strict contact vector fields are in general unstable at their zeros, hence the condition for the the stability in closed-loop has been characterized as stabilization on some closed-loop invariant Legendre submanifolds

Cette thèse est constituée de deux parties indépendantes. Dans le premier chapitre, nous introduisons une méthode numérique permettant d'intégrer des équations aux dérivées partielles représentant la dynamique Hamiltonienne de théories des champs. Cette méthode est un intégrateur multi-symplectique qui préserve localement le tenseur énergie-impulsion sur de très longues périodes de temps et avec précision. Son principal avantage est d'être extrêmement simple tout en restant bien définie localement. Nous la mettons à l'épreuve sur le cas particulier du modèle phi^4 en 1+1 dimensions; nous expliquons également comment l'implémenter en dimensions supérieures. De plus, nous faisons une présentation géométrique de la structure multi-symplectique et nous introduisons une construction permettant de résoudre le problème de dégénérescence pouvant l'affecter.Le second chapitre traite d'aspects hors équilibre dans les systèmes statistiques: nous nous intéressons en particulier à la question de l'impact d'un taux de refroidissement fini lors d'une trempe à travers une transition de phase du second ordre. Pour décrire plus fidèlement le régime hors équilibre qui se produit avant la transition de phase, nous étendons le mécanisme dit de Kibble-Zurek. Nous décrivons comment la taille caractéristique des objets géométriques présents dans le système dépend du temps et du taux de refroidissement; ceci, avant et une fois le point critique atteint. Ces prédictions théoriques sont mises à l'épreuve sur l'exemple du modèle d'Ising ferromagnétique. Nous décrivons également les propriétés géométriques des domaines qui apparaissent dans le système au cours de la dynamique de refroidissement. / This thesis is made up of two independent parts. In the first chapter, we introduce a novel numerical method to integrate partial differential equations representing the Hamiltonian dynamics of field theories. It is a multi-symplectic integrator that locally conserves the stress-energy tensor with an excellent precision over very long periods. Its major advantage is that it is extremely simple (it is basically a centered box scheme) while remaining locally well defined. We put it to the test in the case of the non-linear wave equation (with quartic potential) in one spatial dimension, and we explain how to implement it in higher dimensions. A formal geometric presentation of the multi-symplectic structure is also given as well as a technical trick allowing to solve the degeneracy problem that potentially accompanies the multi-symplectic structure. In the second chapter, we address the issue of the influence of a finite cooling rate while performing a quench across a second order phase transition. We extend the Kibble-Zurek mechanism to describe in a more faithfully way the out-of-equilibrium regime of the dynamics before crossing the transition. We describe the time and cooling rate dependence of the typical growing size of the geometric objects, before and when reaching the critical point. These theoretical predictions are demonstrated through a numerical study of the emblematic kinetic ferromagnetic Ising model on the square lattice. A description of the geometric properties of the domains present in the system in the course of the annealing and when reaching the transition is also given.

Notre travail se portera sur l’étude de la trajectoire exacte d’une particule chargée dans un champ magnétique de type tokamak. Nous considérerons des particules tests plongées dans un champ magnétique indépendant du temps. Par définition, elles n’interagissent pas entre-elles et n’ont aucun effet sur le champ magnétique. Le but de notre étude sera alors de mettre en évidence des différences de comportement majeures entre d’une part les trajectoires des particules et d’autre part les lignes de champ magnétique. Nous commencerons dans une première partie par expliciter nos motivations. Les techniques employées aujourd’hui pour simuler la dynamique d’un plasma de fusion reposent sur des modèles magnétohydrodynamiques ou gyrocinétiques dont les hypothèses sont parfois contestables. Nous exposerons ensuite l’approche hamiltonienne choisie pour simuler la trajectoire exacte d’une particule et introduirons les différents outils dont nous aurons besoin. Dans la seconde partie, nous présenterons brièvement l’outil numérique développé qu mettra en évidence des différences de comportements importantes entre ligne de champ et trajectoire de particule. Dans un premier temps, nous exhiberons du chaos de trajectoire en présence de ligne de champ intégrable en ajoutant au champ magnétique idéal une perturbation. Un même résultat sera démontré dans le cas d’un champ magnétique idéal axisymétrique. Ce résultat entrainera des questions importantes autour de l’invariance du moment magnétique μ. Enfin, la dernière partie de ce manuscrit portera sur le comportement des particules tests en présence d’un champ magnétique potentiellement chaotique. / We will work on the study of the exact trajectory of a charged particle within a magnetic field of the tokamak type. We will consider test particles immerged into a timeindependent magnetic field. They don’t interact with each other and have no effect on the magnetic field. The objective of our study will be to prove major behaviour differences between the particles’ trajectories and the magnetic field lines. We will start in part one by making our motivations explicit and especially see that the technics used nowadays to simulate the dynamic of fusion plasma are based on magnetohydrodynamic or gyrokinetic models which hypotheses are sometimes debatable. We will then present the Hamiltonian approach chosen to simulate the exact trajectory of a particle, and will introduce the various tools we will later need, including the characterizationof chaotic phenomena. we will briefly the developed digital tool which will highlight behaviour differences between field line and particle trajectory. Initially, we will exhibit trajectory chaos in the presence of integrable field lines by adding a perturbation to the ideal magnetic field. In a second phase, we will show that a similar result can be proven in the case of an asymmetrical ideal magnetic field. This result will lead to substantial questions regarding the invariance of the μ. The last part of this manuscript will finally focus on the behaviour of the test particles in the presence of a potentially chaotic magnetic field.

Orientador: Kyoko Furuya / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin / Made available in DSpace on 2018-08-04T01:53:26Z (GMT). No. of bitstreams: 1 SousaJunior_DelcidesFlaviode_M.pdf: 32680218 bytes, checksum: aa259799e554166260b37c235e19a803 (MD5) Previous issue date: 1998 / Resumo:Apresentamos nesta dissertação um estudo da conexão entre a Mecânica Clássica e a Mecânica Quântica através dos diagramas de energia vs. período para as principais famílias de órbitas periódicas de um dado sistema dinâmico. O diagrama quântico é definido através do espectro do sistema quântico correspondente, que mostra cicatrizes dessas famílias no regime semiclássico. Dois sistemas hamiltonianos, com dois graus de liberdade e apresentando comportamento misto ( caótico e regular ) , são estudados. O primeiro é o pêndulo elástico, usado como paradigma de caos clássico. Aspectos essenciais da sua dinâmica são estudados e o diagrama clássico de energia vs. período com as principais famílias de órbitas periódicas é construido. O segundo sistema é o Hamiltoniano Spin-Bóson, um sistema quântico para o qual trabalhos anteriores definiram um análogo clássico, para o qual estudou-se o comportamento caótico e famílias de órbitas periódicas. Uma versão quântica deste diagrama de energia vs. período é mostrada para este modelo. As duas versões são comparadas no regime de caos misto e o ajuste no limite semiclássico discutido. Uma concordância qualitativa é obtida, com indicações de que as cicatrizes são mais acentuadas nas regiões onde ocorrem bifurcações de órbitas periódicas / Abstract:We study the connection between Classical and Quantum Mechanics using the plots of Energy VS. Period for the main families of periodic orbits of certain dynamical system .The quantum E-t plot is defined through the spectrum of the corresponding quantum system, which shows scars of the classical families in the semiclagsical regime. Two Hamiltonian systems with two degrees of freedom both displaying mixed (chaotic and regular) behaviour are analized. The first one is the elastic pendulum, its behaviour ususally presented as a paradigm of classical chaos. Essential aspects of its dinamics are studied to some extent and the classical (E, t ) plot is shown. The second system is the Spin- Boson Hamiltonian, a quantum system for which previous works have defined a classical analogue with chaotic behaviour and compiled the main families of periodic orbits. A quantum version of the (E, t ) plot for this model is shown, and the classical and quantum plots are compared in the regime of soft chaos. The fitting in the semiclassical limit is discussed with a qualitative agreement that indicates enhancements of the scars in the regions where bifurcations of period orbits occur / Mestrado / Física / Mestre em Física

L'objectif de cette thèse est d'investiguer les relations entre les propriétés topologiques d'une variété de contact et la dynamique des flots de Reeb dans la variété de contact en question. Dans la première partie de la thèse, nous établissons une relation entre la croissance de l’homologie de contact cylindrique d'une variété de contact et l'entropie topologique des flots de Reeb dans cette variété de contact. Nous utilisons ce résultat dans les chapitres 8 et 9 pour montrer l'existence d'un grand nombre des nouvelles variétés de contact de dimension 3 dans lesquelles tous les flots de Reeb ont entropie topologique positive. Dans le chapitre 10, nous prouvons un résultat obtenu en collaboration avec Chris Wendl qui donne une obstruction dynamique pour qu'une variété de contact de dimension 3 soit planaire. Cette obstruction est utilisée pour montrer que, si une variété de contact de dimension 3 possède un flot de Reeb qui est uniformément hyperbolique (Anosov) avec variétés invariantes traversalement orientables, alors cette variété de contact n'est pas planaire. Dans le chapitre 11, nous étudions l'entropie topologique des flots de Reeb dans les fibrés unitaires des surfaces de genre plus grand que 1. Nous montrons que la restriction de chaque flot de Reeb en au ensemble limite de presque toute fibre unitaire a une entropie topologique positive. / In this thesis we study the relations between the contact topological properties of contact manifolds and the dynamics of Reeb flows. On the first part of the thesis, we establish a relation between the growth of the cylindrical contact homology of a contact manifold and the topological entropy of Reeb flows on this manifold. We build on this to show in Chapter 6 that if a contact manifold M admits a hypertight contact form A for which the cylindrical contact homology has exponential homotopical growth rate, then the Reeb flow of every contact form on M has positive topological entropy. Using this result, we exhibit in Chapter 8 and 9 numerous new examples of contact 3-manifolds on which every Reeb flow has positive topological entropy. On Chapter 10 we present a joint result with Chris Wendl that gives a dynamical obstruction for contact 3-manifold to be planar. We then use the obstruction to show that a contact 3-manifold that possesses a Reeb flow that is a transversely orientable Anosov flow, cannot be planar. On Chapter 11 we study the topological entropy for Reeb flows on spherizations. The result we obtain is a refinement of a result of Macarini and Schlenk, that states that every Reeb flow on the unit tangent bundle U of a high genus surface S has positive topological entropy. We show that for any Reeb flow on U, the omega-limit of almost every Legendrian fiber is a compact invariant set on which the dynamics has positive topological entropy.