Дисертації з теми "Partially Hyperbolic System"

In the last years, an extremely powerful method has been developed: the functional approach. It consists in the study of the spectral properties of the transfer operators on suitable Banach spaces. In this work we apply this approach to partially hyperbolic systems in two dimensions, establishing the germ of a general theory. To illustrate the scope of the theory, the results are used in the case of fast-slow partially hyperbolic systems, pointing out how to pursue the arguments for further progresses.

Let f : T3 → T3 be a C2 volume preserving partially hyperbolic diffeomorphism homotopic to a linear Anosov automorphism A : T3 → T3. We prove that if f is Kolmogorov, then f is Bernoulli. We study the characteristics of atomic disintegration of the volume measure whenever it occurs. We prove that if the volume measure m has atomic disintegration on the center leaves then the disintegration has one atom per center leaf. We give a condition, depending only on the center Lyapunov exponent of the diffeomorphism, that guarantees atomic disintegration of the volume measure on center leaves. We construct an open family of diffeomorphisms satisfying this condition which generates the first examples of foliations which are both measurable and minimal. In this same construction we give the first examples of partially hyperbolic diffeomorphisms with zero center Lyapunov exponent and homotopic to a linear Anosov.
Seja f : T3 → T3 um difeomorfismo C2 parcialmente hiperbólico, homotópico a um automorfismo de Anosov linear e preservando a medida de volume m. Provamos que se f é Kolmogorov então f é Bernoulli. Estudamos as características da desintegração atômica da medida de volume quando esta ocorre. Provamos que se a medida de volume m tem desintegração atômica nas folhas centrais então a desintegração tem um átomo por folha central. Apresentamos uma condição, a qual depende apenas do expoente de Lyapunov central do difeomorfismo, que garante desintegração atômica da medida de volume. Construímos uma família aberta de difeomorfismos satisfazendo esta condição, o que gerou os primeiros exemplos de folheações que são mensuráveis e ao mesmo tempo minimais. Nesta mesma construção damos os primeiros exemplos de difeomorfismos parcialmente hiperbólicos com expoente de Lyapunov central nulo e homotópico a um Anosov linear.

Neste trabalho estudamos relações entre expoente de Lyapunov e continuidade absoluta da folheação central para difeomorfismos parcialmente hiperbólicos conservativos de \'T POT. 3\'. Sobre tal tema, provamos que tipicamente (\'C POT. 1\' aberto e \'C POT. 2\' denso) os difeomorfismos parcialmente hiperbólicos, conservativos de classe \'C POT. 2\' , do toro \'T POT. 3\', apresentam folheação central não absolutamente contínua. Desta maneira, respondemos positivamente uma pergunta proposta em [20]. Também neste trabalho, estudamos entropia topológica para Sistema Iterado de Funções. Neste contexto, damos uma nova demonstração para uma conjectura proposta em [14] e provada primeiramente em [15]. Apresentamos um método geométrico que nos permite calcular entropia para transformações de \'S POT. 1\', como em [15]. Além de disso o método apresentado se verifica para casos mais gerais, como por exemplo: transformações não comutativas
In this work we study relations between Lyapunov exponents, absolute continuity of center foliation for conservative partially hyperbolic diffeomorphisms of \'T POT. 3\'. About this theme, (on a \'C POT. 1\' open and \'C POT. 2\'dense set) of conservative partially hyperbolic \'C POT. 2\' diffeomorphisms of the 3-torus presents non absolutely continuous center foliation. So, we answer positively a question proposed in [20]. Also in this work, we study topological entropy for Iterated Functions Systems. In this setting, we give a proof for a conjecture proposed in [14] and firstly proved in [15]. We present a geometrical method that allows us to calcule the entropy for transformations of \'S POT. 1\', like in [15]. Furthermore this method holds for more general cases, for example: non commutative transformations

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós-Graduação em Engenharia de Automação e Sistemas, Florianópolis, 2017.
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Sistemas com parâmetros distribuídos representam uma vasta gama de processos da engenharia. Neste caso, as variáveis do sistema irão conter termos dependentes do tempo assim como gradientes espaciais e, portanto, é natural representa-los por equações diferenciais parciais. Exemplos podem ser encontrados em diversas áreas: desde processos químicos e térmicos, sistemas de produção e distribuição de energia, e problemas relacionados ao transporte de fluidos e ciência médica. Esta tese trata dois tipos de problemas: estabilização de equações diferenciais parciais lineares hiperbólicas com variável de controle na condição de contorno e controle regulatório de sistemas descritos por equações diferenciais parciais quasi-lineares hiperbólicas com variável de controle no domínio. Com relação ao primeiro, estudaram-se duas metodologias de controle: (i) uma lei de controle estática que garante convergência do sistema para o ponto de equilíbrio desejado. A metodologia de controle utiliza uma função de Lyapunov para encontrar os valores dos parâmetros do controlador que garantem estabilidade exponencial em malha fechada. Resultados de simulação para o problema de supressão de golfadas em sistemas de produção de petróleo são apresentados para ilustrar a eficiência do método; (ii) uma lei de controle baseada nas ferramentas clássicas do domínio da frequência. Neste caso, aplicamos a transformada de Laplace na equação diferencial parcial para obter uma função de transferência irracional e então, ferramentas clássicas do domínio da frequência são usadas para projetar o controlador, de maneira similar aos sistemas de dimensão finita com função de transferência racional. Estes resultados foram aplicados experimentalmente no problema de controle de oscilações termoacústicas do tubo de Rijke, mostrando a efetividade do método. Para o segundo problema, utiliza-se o método das características combinado com a técnica de controle por modos deslizantes. O método das características é usado para transformar o sistema de equações diferenciais parciais em um conjunto de equações diferenciais ordinárias que descrevem o sistema original. O projeto de controle é então realizado a partir deste conjunto de equações diferenciais ordinárias através de resultados bem conhecidos da teoria de equações diferenciais ordinárias. Os resultados obtidos foram testados experimentalmente em dois sistemas de escala industrial: uma planta solar e um fotobiorreator tubular.

Abstract : Distributed parameter systems represent a wide range of engineeringprocesses. In this case, the system variables will contain temporally dependentterms as well spatial gradients and, therefore, it is natural to representthem by partial dierential equations. Examples can be found in manyelds: chemical and thermal processes, production and distribution energysystems, and problems related to uid transport and medical science.This thesis deals with two dierent problems: stabilization of linear hyperbolicpartial dierential equations with boundary control and regulatorycontrol of systems described by quasilinear hyperbolic partial dierentialequations with in domain control. Concerning the boundary control problem,we studied two control methodologies: (i) a static control law thatguarantees convergence of the system to the desired equilibrium point. Thiscontrol methodology uses a Lyapunov function to nd the values of thecontrol parameters that guarantee closed-loop exponential stability. Simulationresults for the slugging control problem in oil production facilities arepresented to illustrate the eciency of the methodology; (ii) a control lawbased on the frequency domain tools. In this case, we applied the Laplacetransform on the partial dierential equation to obtain an irrational transferfunction and then classical frequency domain tools are used to designthe control law. These results were applied experimentally to the controlproblem of thermoacoustic oscillations in the Rijke tube, showing the effectivenessof the method. Regarding the regulatory control problem, weuse the method of characteristics together with the sliding mode controlmethodology. The method of characteristics is used to transform the partialdierential equations into a system of ordinary dierential equations thatdescribes the original system without any kind of approximation. Then,the control design is performed on the ordinary dierential equations withwell-known results of the theory of lumped parameter systems. The resultswere validated experimentally in two industrial scale systems: a solar powerplant and a tubular photobioreactor.

Wir betrachten Optimalsteuerungsprobleme, die von partiellen Differentialgleichungen beziehungsweise Variationsungleichungen mit Differentialoperatoren erster Ordnung abhängen. Wir führen die Reformulierung eines Tagebauplanungsproblems, das auf stetigen Funktionen beruht, ein. Das Resultat ist ein Optimalsteuerungsproblem für Viskositätslösungen einer Eikonalgleichung. Die Existenz von Lösungen dieses und bestimmter Hilfsprobleme, die von semilinearen PDG‘s mit künstlicher Viskosität abhängen, wird bewiesen, Stationaritätsbedingungen hergeleitet und ein schwaches Konsistenzresultat für stationäre Punkte präsentiert. Des Weiteren betrachten wir Optimalsteuerungsprobleme, die von stationären Variationsungleichungen erster Art mit linearen Differentialoperatoren erster Ordnung abhängen. Wir diskutieren Lösbarkeit und Stationaritätskonzepte für diese Probleme. Für letzteres vergleichen wir Ergebnisse, die entweder durch die Anwendung von Penalisierungs- und Regularisierungsansätzen direkt auf Ebene von Differentialoperatoren erster Ordnung oder als Grenzwertprozess von Stationaritätssystemen für viskositätsregularisierte Optimalsteuerungsprobleme unter passenden Annahmen erhalten werden. Um die Konsistenz von ursprünglichem und regularisierten Problemen zu sichern, wird ein bekanntes Ergebnis für Lösungen von VU’s mit degeneriertem Differentialoperator erweitert. In beiden Fällen ist die erhaltene Stationarität schwächer als W-stationarität. Die theoretischen Ergebnisse werden anhand numerischer Beispiele verifiziert. Wir erweitern diese Ergebnisse auf Optimalsteuerungsprobleme bezüglich zeitabhängiger VU’s mit Differentialoperatoren erster Ordnung. Hierfür wird die Existenz von Lösungen bewiesen und erneut ein Stationaritätssystem mit Hilfe verschwindender Viskositäten unter bestimmten Beschränktheitsannahmen hergeleitet. Die erhaltenen Ergebnisse werden anhand von numerischen Beispielen verifiziert.
We consider problems of optimal control subject to partial differential equations and variational inequality problems with first order differential operators. We introduce a reformulation of an open pit mine planning problem that is based on continuous functions. The resulting formulation is a problem of optimal control subject to viscosity solutions of a partial differential equation of Eikonal Type. The existence of solutions to this problem and auxiliary problems of optimal control subject to regularized, semilinear PDE’s with artificial viscosity is proven. For the latter a first order optimality condition is established and a mild consistency result for the stationary points is proven. Further we study certain problems of optimal control subject to time-independent variational inequalities of the first kind with linear first order differential operators. We discuss solvability and stationarity concepts for such problems. In the latter case, we compare the results obtained by either utilizing penalization-regularization strategies directly on the first order level or considering the limit of systems for viscosity-regularized problems under suitable assumptions. To guarantee the consistency of the original and viscosity-regularized problems of optimal control, we extend known results for solutions to variational inequalities with degenerated differential operators. In both cases, the resulting stationarity concepts are weaker than W-stationarity. We validate the theoretical findings by numerical experiments for several examples. Finally, we extend the results from the time-independent to the case of problems of optimal control subject to VI’s with linear first order differential operators that are time-dependent. After establishing the existence of solutions to the problem of optimal control, a stationarity system is derived by a vanishing viscosity approach under certain boundedness assumptions and the theoretical findings are validated by numerical experiments.

Thesis (Ph.D. in Computational and Applied Mathematics)--S.M.U.
Title from PDF title page (viewed Mar. 16, 2009). Source: Dissertation Abstracts International, Volume: 69-08, Section: B Adviser: Peter K. Moore. Includes bibliographical references.

Ce travail est consacré à l’étude théorique et numérique de systèmes hyperboliques d’équations aux dérivées partielles et aux équations de transport, avec des termes de relaxation et des conditions aux bords. Dans la première partie, on étudie la stabilité raide d’approximations numériques par différences finies du problème mixte donnée initiale-donnée au bord pour l’équation des ondes amorties dans le quart de plan. Dans le cadre du schéma discret en espace, nous proposons deux méthodes de discrétisation de la condition de Dirichlet. La première est la technique de sommation par partie et la seconde est basée sur le concept de condition au bord transparente. Nous proposons également une comparaison numérique des deux méthodes, en particulier de leur domaine de stabilité. La deuxième partie traite de schémas numériques d’ordre élevé pour l’équation de transport avec une donnée entrante sur domaine borné. Nous construisons, implémentons et analysons la procédure de Lax-Wendroff inverse au bord entrant. Nous obtenons des taux de convergence optimaux en combinant des estimations de stabilité précises pour l’extrapolation des conditions au bord avec des développements de couche limite numérique. Dans la dernière partie, nous étudions la stabilité de solutions stationnaires pour des systèmes non conservatifs avec des termes géométrique et de relaxation. Nous démontrons que les solutions stationnaires sont stables parmi les solutions entropique processus, qui généralisent le concept de solutions entropiques faibles. Nous supposons essentiellement que le système est complété par une entropie partiellement convexe et que, selon la dissipation du terme de relaxation, la stabilité ou la stabilité asymptotique des solutions stationnaires est obtenue
The dissertation focuses on the study of the theoretical and numerical analysis of hyperbolic systems of partial differential equations and transport equations, with relaxation terms and boundary conditions. In the first part, we consider the stiff stability for numerical approximations by finite differences of the initial boundary value problem for the linear damped wave equation in a quarter plane. Within the framework of the difference scheme in space, we propose two methods of discretization of Dirichlet boundary condition. The first is the technique of summation by part and the second is based on the concept of transparent boundary conditions. We also provide a numerical comparison of the two numerical methods, in particular in terms of stability domain. The second part is about high order numerical schemes for transport equations with nonzero incoming boundary data on bounded domains. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at incoming boundary. We obtain optimal convergence rates by combining sharp stability estimate for extrapolation boundary conditions with numerical boundary layer expansions. In the last part, we study the stability of stationary solutions for non-conservative systems with geometric and relaxation source term. We prove that stationary solutions are stable among entropy process solution, which is a generalisation of the concept of entropy weak solutions. We mainly assume that the system is endowed with a partially convex entropy and, according to the entropy dissipation provided by the relaxation term, stability or asymptotic stability of stationary solutions is obtained

Depuis Ruelle, puis Rugh, Baladi, Tsujii, Liverani et d'autres, on sait que la fuite vers l'équilibre statistique dans de nombreux systèmes dynamiques chaotiques est gouvernée par le spectre de résonances de Ruelle de l'opérateur de transfert. A la suite de récents travaux de Faure, Sjöstrand et Roy, cette thèse propose une approche semiclassique de systèmes dynamiques chaotiques de type partiellement expansifs. Une partie du mémoire est consacrée aux extensions d'applications expansives vers des groupes de Lie compacts, en se reistreignant essentiellement aux extensions vers le groupe spécial unitaire SU(2). On se sert de la théorie des états cohérents pour les groupes de Lie, développée dans les années 70 par Perelomov et Gilmore, pour mettre en oeuvre les outils semiclassiques et la théorie des résonances de Helfer et Sjöstrand. On en déduira une estimation de Weyl et un gap spectral pour les résonances de Ruelle prouvant que la fuite vers l'équilibre statistique dans ces modèles est gouvernée par un opérateur de rang fini (en accord avec les résultats obtenus par Tsujii pour les semi-flots partiellement expansifs). On étend ensuite cette approche aux modèles "ouverts" pour lesquels la dynamique présente un ensemble captif de Cantor. On montrera l'existence d'un spectre discret de résonances de Ruelle et on prouve une loi de Weyl fractale, analogue classique du théorème de Lin-Guillopé-Zworski pour les résonances du laplacien hyperbolique sur les surfaces à courbure négative constante. On montre aussi un gap spectral asymptotique. On expliquera pourquoi ces modèles semblent être des objets d'étude adaptés pour approcher des questions importantes et difficiles du chaos classique ou quantique. On pense en particulier au problème de la minoration du nombre de résonances, étudié dans le contexte des applications quantiques par Nonnenmacher et Zworski
Since the work of Ruelle, then Rugh, Baladi, Tsujii, Liverani and others, it is kown that the convergence towards statistical equilibrium in many chaotic dynamical systems is gouverned by the Ruelle spectrum of resonances of the so-called transfer operator. Following recent works from Faure, Sjöstrand and Roy, this thesis gives a semiclassical approach for partially expanding chaotic dynamical systems. The first part of the thesis is devoted to compact Lie groups extenstions of expanding maps, essentially restricting to SU(2) extensions. Using Perlomov's coherent state theory for Lie groups, we apply the semiclassical theory of resonances of Helfer and Sjöstrand. We deduce Weyl type estimations and a spectral gap for the Ruelle resonances, showing that the convergence towards equilibrium is controled by a finite rank operator (as Tsujii already showed for partially expanding semi-flows). We then extend this approach to "open" models, for which the dynamics exhibits a fractal invariant reppeler. We show the existence of a discrete spectrum of resonances and we prove a fractal Weyl law, the classical analogue of Lin-Guillopé-Zworski's theorem on resonances of non-compact hyperbolic surfaces. We also show an asymptotic spectral gap. Finally we breifly explain why these models are interseting "toy models" to explore important questions of classical and quantum chaos. In particular, we have in mind the problem of proving lower bounds on the number of resonances, studied in the context of open quantum maps by Nonnenmacher and Zworski

O objetivo deste trabalho é tratar da simulação do fenômeno de propagação de ondas em uma haste heterogênea elástico, composta por dois materiais distintos (um linear e um não-linear), cada um deles com a sua própria velocidade de propagação da onda. Na interface entre estes materiais existe uma descontinuidade, um choque estacionário, devido ao salto das propriedades físicas. Empregando uma abordagem na configuração de referência, um sistema não-linear hiperbólico de equações diferenciais parciais, cujas incógnitas são a velocidade e a deformação, descrevendo a resposta dinâmica da haste heterogénea. A solução analítica completa do problema de Riemann associado são apresentados e discutidos.
The objective of this work is the simulation of the wave propagation phenomenon in a heterogeneous elastic rod, composed by two distinct materials (a linear and a non-linear one), each of them with its own wave propagation speed. At the interface between these materials there is a discontinuity, a stationary shock, due to the jump of the physical properties. Employing a reference configuration approach, a nonlinear hyperbolic system of partial differential equations, whose unknowns are the velocity and the strain, describing the dynamical response of the heterogeneous rod. The complete analytical solution of the associated Riemann problem is presented and discussed.

Nous étudions des systèmes hyperboliques non linéaires, avec des conditions initiales oscillantes. La première partie utilise la théorie de la compacité par compensation qui permet d'exprimer la limite faible de toute suite de fonction continue d'une solution du système considéré. On démontre notamment un résultat de convergence pour un cas de système linéairement dégénéré particulier de 3 équations qui est une généralisation du cas de 2 équations. Un système intégro-différentiel est obtenu lors de l'étude d'un système de 2 équations dont les champs caractéristiques ne sont pas de même type. Enfin, suite à une analyse précédemment établie (système linéairement dégénéré de 2 équations), une approche numérique est effectuée. La deuxième partie est consacrée d'une part, à des tests numériques portant sur le système des équations d'Euler et, d'autre part, à un système de 4 équations sur lequel, dans un cas particulier, on obtient des résultats sur le type de convergence

Le travail qui suit comporte quatre chapitres : le premier est centré autour de la propriété de mélange faible pour les échanges d'intervalles et flots de translation. On y présente des résultats obtenus avec Artur Avila qui renforcent des résultats précédents dus à Artur Avila et Giovanni Forni. Le deuxième chapitre est consacré à un travail en commun avec Zhiyuan Zhang et concerne les propriétés d'ergodicité et d'accessibilité stables pour des systèmes partiellement hyperboliques de dimension centrale au moins égale à deux. On montre que sous des hypothèses de cohérence dynamique, center bunching et pincement fort, la propriété d'accessibilité stable est dense en topologie C^r, r>1, et même prévalente au sens de Kolmogorov. Dans le troisième chapitre, on expose les résultats d'un travail réalisé en collaboration avec Julie Déserti, consacré à l'étude d'une famille à un paramètre d'automorphismes polynomiaux de C^3 ; on montre que de nouveaux phénomènes apparaissent par rapport à ce qui était connu dans le cas de la dimension deux. En particulier, on étudie les vitesses d'échappement à l'infini, en montrant qu'une transition s'opère pour une certaine valeur du paramètre. Le dernier chapitre est issu d'un travail en collaboration avec Jiangong You, Zhiyan Zhao et Qi Zhou ; on s'intéresse à des estimées asymptotiques sur la taille des trous spectraux des opérateurs de Schrödinger quasi-périodiques dans le cadre analytique. On obtient des bornes supérieures exponentielles dans le régime sous-critique, ce qui renforce un résultat précédent de Sana Ben Hadj Amor. Dans le cas particulier des opérateurs presque Mathieu, on montre également des bornes inférieures exponentielles, qui donnent des estimées quantitatives en lien avec le problème dit "des dix Martinis". Comme conséquences de nos résultats, on présente des applications à l'homogénéité du spectre de tels opérateurs ainsi qu'à la conjecture de Deift
The following work contains four chapters: the first one is centered around the weak mixing property for interval exchange transformations and translation flows. It is based on the results obtained together with Artur Avila which strengthen previous results due to Artur Avila and Giovanni Forni. The second chapter is dedicated to a joint work with Zhiyuan Zhang, in which we study the properties of stable ergodicity and accessibility for partially hyperbolic systems with center dimension at least two. We show that for dynamically coherent partially hyperbolic diffeomorphisms and under certain assumptions of center bunching and strong pinching, the property of stable accessibility is dense in C^r topology, r>1, and even prevalent in the sense of Kolmogorov. In the third chapter, we explain the results obtained together with Julie Déserti on the properties of a one-parameter family of polynomial automorphisms of C^3; we show that new behaviours can be observed in comparison with the two-dimensional case. In particular, we study the escape speed of points to infinity and show that a transition exists for a certain value of the parameter. The last chapter is based on a joint work with Jiangong You, Zhiyan Zhao and Qi Zhou; we get asymptotic estimates on the size of spectral gaps for quasi-periodic Schrödinger operators in the analytic case. We obtain exponential upper bounds in the subcritical regime, which strengthens a previous result due to Sana Ben Hadj Amor. In the particular case of almost Mathieu operators, we also show exponential lower bounds, which provides quantitative estimates in connection with the so-called "Dry ten Martinis problem". As consequences of our results, we show applications to the homogeneity of the spectrum of such operators, and to Deift's conjecture

Dans cette thèse, nous nous sommes principalement intéressés à l’étude théorique et numérique de quelques équations qui décrivent la dynamique des densités des dislocations. Les dislocations sont des défauts microscopiques qui se déplacent dans les matériaux sous l’effet des contraintes extérieures. Dans un premier travail, nous démontrons un résultat d’existence globale en temps des solutions discontinues pour un système hyperbolique diagonal qui n’est pas nécessairement strictement hyperbolique, dans un espace unidimensionnel. Ainsi dans un deuxième travail, nous élargissons notre portée en démontrant un résultat similaire pour un système d’équations de type eikonal non-linéaire qui est en fait une généralisation du système hyperbolique déjà étudié. En effet, nous prouvons aussi l’existence et l’unicité d’une solution continue pour le système eikonal. Ensuite, nous nous sommes intéressés à l’analyse numérique de ce système en proposant un schéma aux différences finies, par lequel nous montrons la convergence vers le problème continu et nous consolidons nos résultats avec quelques simulations numériques. Dans une autre direction, nous nous sommes intéressés à la théorie de contraction différentielle pour les équations d’évolutions. Après avoir introduit une nouvelle distance, nous construisons une nouvelle famille des solutions contractantes positives pour l’équation d’évolution p-Laplace
In this thesis, we are mainly interested in the theoretical and numerical study of certain equations that describe the dynamics of dislocation densities. Dislocations are microscopic defects in materials, which move under the effect of an external stress. As a first work, we prove a global in time existence result of a discontinuous solution to a diagonal hyperbolic system, which is not necessarily strictly hyperbolic, in one space dimension. Then in another work, we broaden our scope by proving a similar result to a non-linear eikonal system, which is in fact a generalization of the hyperbolic system studied first. We also prove the existence and uniqueness of a continuous solution to the eikonal system. After that, we study this system numerically in a third work through proposing a finite difference scheme approximating it, of which we prove the convergence to the continuous problem, strengthening our outcomes with some numerical simulations. On a different direction, we were enthused by the theory of differential contraction to evolutionary equations. By introducing a new distance, we create a new family of contracting positive solutions to the evolutionary p-Laplacian equation

Ce travail présente quelques résultats sur les systèmes de particules en interaction pour l'interprétation probabiliste des équations aux dérivées partielles, avec des applications à des questions de dynamique moléculaire et de chimie quantique. On présente notamment une méthode particulaire permettant d'analyser le processus de la force biaisante adaptative, utilisé en dynamique moléculaire pour le calcul de différences d'énergies libres. On étudie également la sensibilité de dynamiques stochastiques par rapport à un paramètre, en vue du calcul des forces dans l'approximation de Born-Oppenheimer pour rechercher l'état quantique fondamental de molécules. Enfin, on présente un schéma numérique basé sur un système de particules pour résoudre des lois de conservation scalaires, avec un terme de diffusion anormale se traduisant par une dynamique de sauts sur les particules

On démontre que les difféomorphismes partiellement hyperboliques admettant un feuilletage central invariant compact sont dynamiquement cohérent et en plus ils ont la propriété de pistage. En supposant une direction instable uni-dimensionnelle et orientée on peut prouver que le difféomorphisme projette en un automorphisme hyperbolique de tore dans l'espace des feuilles centrales.

According to the work of Dennis Sullivan, there exists a smooth flow on the 5-sphere all of whose orbits are periodic although there is no uniform bound on their periods. The question addressed in this thesis is whether such an example can occur in the partially hyperbolic context. That is, does there exist a partially hyperbolic diffeomorphism of a compact manifold such that all the leaves of its center foliation are compact although there is no uniform bound for their volumes. We will show that the answer to the previous question under the very mild hypothesis of dynamical coherence is no. The thesis is organized as follows. In the first chapter we give the necessary background and results in partially hyperbolic dynamics needed for the rest of the work, studying in particular the geometry of the center foliation. Chapter two is devoted to a general discussion of compact foliations. We give proof or sketches of all the relevant results used. Chapter three is the core of the thesis, where we establish the non existence of Sullivan's type of examples in the partially hyperbolic domain, and generalize to diffeomorphisms whose center foliation has arbitrary dimension. The last chapter is devoted to applications of the results of chapter three, where in particular it is proved that if the center foliation of a dynamically coherent partially hyperbolic diffeomorphism is compact and without holonomy, then it is plaque expansive.

碩士
國立中興大學
電機工程學系所
100
This thesis presents an analytic solution to the hyperbolic partial differential equation systems. Obvious, the partial differential equations are more difficult than the ordinary differential equations system for study. Therefore, this thesis proposed a sample-data formulation to analytic solution to the boundary hyperbolic partial differential equation systems. With boundary conditions satisfying the regular form of Strum-Liouville problem, and used the eigenfunctions expansion method to making hyperbolic partial differential equation into an infinite sequence of discrete-time control problems. The finite-dimensional approximation of the discrete-time system is obtained by the minimum square error theorem of Fourier series. And this thesis for verification to the proposed analysis derivation of the equation result. Therefore, to introduce two numerical analysis simulation methods, ones are direct to used finite difference approximate method to solve hyperbolic partial differential equation systems, and the other ones are use proposed sampled-data formulation of finite-dimensional discrete-time system control. In the end, to compare our proposed method with finite difference approximate method for to verification this proposed method feasibility.