Applications of distributed-power systems are on the rise. They are already used in telecommunication power supplies, aircraft and shipboard power-distribution systems, motor drives, plasma applications, and they are being considered for numerous other applications. The successful operation of these multi-converter systems relies heavily on a stable design. Conventional analyses of power converters are based on averaged models, which ignore the fast-scale instability and analyze the stability on a reduced-order manifold. As such, validity of the averaged models varies with the switching frequency even for the same topological structure. The prevalent procedure for analyzing the stability of switching converters is based on linearized smooth averaged (small-signal) models. Yet there are systems (in active use) that yield a non-smooth averaged model. Even for systems for which smooth averaged models are realizable, small-signal analyses of the nominal solution/orbit do not provide anything about three important characteristics: region of attraction of the nominal solution, dependence of the converter dynamics on the initial conditions of the states, and the post-instability dynamics. As such, converters designed based on small-signal analyses may be conservative. In addition, linear controllers based on such analysis may not be robust and optimal. Clearly, there is a need to analyze the stability of power converters from a different perspective and design nonlinear controllers for such hybrid systems. In this Dissertation, using bifurcation analysis and Lyapunov's method, we analyze the stability and dynamics of some of the building blocks of distributed-power systems, namely standalone, integrated, and parallel converters. Using analytical and experimental results, we show some of the differences between the conventional and new approaches for stability analyses of switching converters and demonstrate the shortcomings of some of the existing results. Furthermore, using nonlinear analyses we attempt to answer three fundamental questions: when does an instability occur, what is the mechanism of the instability, and what happens after the instability? Subsequently, we develop nonlinear controllers to stabilize parallel dc-dc and parallel multi-phase converters. The proposed controllers for parallel dc-dc converters combine the concepts of multiple-sliding-surface and integral-variable-structure control. They are easy to design, robust, and have good transient and steady-state performances. Furthermore, they achieve a constant switching frequency within the boundary layer and hence can be operated in interleaving or synchronicity modes. The controllers developed for parallel multi-phase converters retain many of the above features. In addition, they do not require any communication between the modules; as such, they have high redundancy. One of these control schemes combines space-vector modulation and variable-structure control. It achieves constant switching frequency within the boundary layer and a good compromise between the transient and steady-state performances. / Ph. D.

[pt] Excitação paramétrica se dá a partir de coeficientes variantes no tempo na dinâmica de um sistema. Este tipo de excitação tem sido um amplo tema de pesquisa desde os campos da mecânica e eletrônica até dinâmica de fluidos. Ela aparece em problemas envolvendo sistemas dinâmicos, por exemplo, como uma forma de controle de vibrações em sistemas auto excitados, tornando este assunto digno de mais investigações. Abordando estabilidade no sentido de Lyapunov, esta dissertação fornece uma base didática de estabilidade desde conceitos básicos, como pontos de equilíbrio e planos de fase, até conceitos mais avançados, como excitação paramétrica e teoria de Floquet. Os objetos de estudo aqui são sistemas lineares com parâmetros periódicos no tempo, o que permite usar a teoria de Floquet para fazer afirmações a respeito da estabilidade da solução trivial do sistema. Vários exemplos são discutidos fazendo uso de um procedimento numérico desenvolvido para construir mapas de estabilidade e planos de fase. Os exemplos apresentados abrangem sistemas mecânicos, eletromagnéticos e eletromecânicos. Fazendo uso de mapas de estabilidade, diversas características de análise de estabilidade são abordadas. Duas estratégias diferentes para avaliar a estabilidade da solução trivial são comparadas: multiplicadores de Floquet e valor máximo dos expoentes característicos de Lyapunov. / [en] Parametric excitation is a type of excitation that arises from timevarying coefficients in a system s dynamics. More specifically, this dissertation deals with time-periodic coefficients. This type of excitation has been an extended topic of research from the fields of mechanics and electronics to fluid dynamics. It appears in problems involving dynamical systems, for example, as a way of controlling vibrations in self-excited systems, making this subject worthy of more investigations. By approaching stability in the sense of Lyapunov, this dissertation provides a didactic stability background from basic concepts, such as equilibrium points and phase diagrams, to more advanced ones, like parametric excitation and Floquet theory. The objects of study here are linear systems with time-periodic parameters. Floquet theory is used to make stability statements about the system s trivial solution. Several examples are discussed by making use of a developed numerical procedure to construct stability maps and phase diagrams. The examples presented herein encompass mechanical, electromagnetic and electromechanical systems. By making use of stability maps, several features that can be discussed in stability analysis are approached. Two different strategies to evaluate the stability of the trivial solution are compared: Floquet multipliers and the maximum value of Lyapunov characteristic exponents.

La Méthode de la Trajectoire Adiabatique Contrainte est développée dans le but de résoudre globalementl’équation de Schrödinger. Cette méthode utilise le formalisme de Floquet et une décomposition de Fourier pourdécrire les dépendances temporelles. Elle transforme ainsi un problème dynamique en un problème aux valeurspropres partiel dans un espace de Hilbert étendu au temps. Cette manipulation requiert l’application decontraintes sur les conditions initiales de l’état propre de Floquet recherché. Les contraintes sont appliquées parl’intermédiaire d’un opérateur absorbant artificiel. Cet algorithme est adapté à la description de systèmes dirigéspar des hamiltoniens dépendant explicitement du temps. Il ne souffre pas de l’accumulation d’erreurs au cours dutemps puisqu’il fournit une solution globale ; les erreurs éventuelles proviennent de la non-complétude des basesfinies utilisées pour la description moléculaire ou temporelle et de l’imperfection du potentiel absorbant dépendantdu temps nécessaire pour fixer les conditions initiales. Une forme générale de potentiel absorbant a étédéveloppée pour être en mesure d’intégrer un problème avec une condition initiale quelconque. Des argumentsrelatifs au suivi adiabatique dans le cas de Hamiltoniens non-hermitiens sont également présentés. Nous insistonssur le rôle des facteurs de phase géométrique. Les méthodes développées sont appliquées à des systèmesatomiques ou moléculaires soumis à des impulsions laser intenses, en relation avec la problématique du contrôlemoléculaire. Nous considérons plusieurs exemples : modèles d’atomes à deux ou trois niveaux, ion moléculairehydrogène et molécules froides de sodium. / The Constrained Adiabatic Trajectory Method (CATM) allows us to compute global solutions of the time-dependent Schrödinger equation using the Floquet formalism and Fourier decomposition. The dynamical problem is thustransformed into a “static” problem, in the sense that the time will be included in an extended Hilbert space. Thisapproach requires that suitable constraints are applied to the initial conditions for the relevant Floquet eigenstate.The CATM is well suited to the description of systems driven by Hamiltonians with explicit and complicated timevariations. This method does not have cumulative errors and the only error sources are the non-completeness ofthe finite molecular and temporal basis sets used, and the imperfection of the time-dependent absorbing potentialwhich is essential to impose the correct initial conditions. A general form is derived for the absorbing potential,which can reproduce any dispersed boundary conditions. Arguments on adiabatic tracking in the case of nonhermitianHamiltonians are also presented. We insist on the role of geometric phase factors. The methods areapplied to atomic and molecular systems illuminated by intense laser pulses, in connection with molecular controlproblems. We study several examples : two or three-level atomic models, hydrogen molecular ion, cold sodiummolecules.

Cette thèse analyse certains aspects de la dynamique hors équilibre de systèmes quantiques unidimensionnels lorsqu’ils sont soumis à des champs externes dépendant du temps. Nous considérons plus particulièrement le cas des forçages périodiques, et le cas d’une variation temporelle lente d’un paramètre de l’Hamiltonien qui permet de traverser une transition de phase quantique. La première partie contient une présentation des notions, des modèles et des outils nécessaires pour comprendre la suite de la thèse, avec notamment des rappels sur les modèles quantiques critiques (en particulier sur les chaines de spin et sur le modèle de Bose-Hubbard), le mécanisme de Kibble-Zurek, et la théorie de Floquet. Ensuite, nous étudions la dynamique hors équilibre des gaz de Tonks-Girardeau dans un potentiel harmonique dépendant du temps par différentes techniques : développements perturbatifs, diagonalisation numérique exacte et solutions analytiques exactes basées sur la théorie des invariants dynamiques d’Ermakov-Lewis. Enfin, nous analysons la dynamique hors équilibre des systèmes quantiques ouverts markoviens soumis à des variations périodiques des paramètres du système et de l’environnement. Nous formulons une théorie de Floquet afin d’obtenir des solutions exactes des équations de Lindblad périodiques. Ce formalisme de Lindblad-Floquet est utilisé pour obtenir une caractérisation exacte du fonctionnement en temps fini des machines thermiques quantiques. / This thesis analyzes some aspects regarding the dynamics of one-dimensional quantum systems which are driven out-of-equilibrium by the presence of time- dependent external fields. Among the possible kinds of driven systems, our focus is dedicated to the slow variation of a Hamiltonian’s parameter across a quantum phase transition and to the case of a time-periodic forcing. To begin with, we prepare the background and the tools needed in the following. This includes a brief introduction to quantum critical models (in particular to the xy spin chain and to the Bose-Hubbard model), the Kibble-Zurek mechanism and Floquet theory. Next, we consider the non-equilibrium dynamics of Tonks-Girardeau gases in time-dependent harmonic trap potentials. The analysis is made with different techniques: perturbative expansions, numerical exact diagonalization and exact methods based on the theory of Ermakov-Lewis dynamical invariants. The last part of the thesis deals instead with the non-equilibrium dynamics of markovian open quantum systems subject to time-periodic perturbations of the system parameters and of the environment. This has led to an exact formulation of Floquet theory for a Lindblad dynamics. Moreover, within the Lindblad-Floquet framework it is possible to have an exact characterization ofthe finite-time operation of quantum heat-engines.

This thesis is concerned with nonlinear analyses of BWR stability behaviour, contributing to a deeper understanding in this field. Despite negative feedback-coefficients of a BWR, there are operational points (OP) at which oscillatory instabilities occur. So far, a comprehensive and an in-depth understanding of the nonlinear BWR stability behaviour are missing, even though the impact of the significant physical parameters is well known. In particular, this concerns parameter regions in which linear stability indicators, like the asymptotic decay ratio, lose their meaning. Nonlinear stability analyses are usually carried out using integral (system) codes, describing the dynamical system by a system of nonlinear partial differential equations (PDE). One aspect of nonlinear BWR stability analyses is to get an overview about different types of nonlinear stability behaviour and to examine the conditions of their occurrence. For these studies the application of system codes alone is inappropriate. Hence, in the context of this thesis, a novel approach to nonlinear BWR stability analyses, called RAM-ROM method, is developed. In the framework of this approach, system codes and reduced order models (ROM) are used as complementary tools to examine the stability characteristics of fixed points and periodic solutions of the system of nonlinear differential equations, describing the stability behaviour of a BWR loop. The main advantage of a ROM, which is a system of ordinary differential equations (ODE), is the possible coupling with specific methods of the nonlinear dynamics. This method reveals nonlinear phenomena in certain regions of system parameters without the need for solving the system of ROM equations. The stability properties of limit cycles generated in Hopf bifurcation points and the conditions of their occurrence are of particular interest. Finally, the nonlinear phenomena predicted by the ROM will be analysed in more details by the system code. Hence, the thesis is not focused on rendering more precisely linear stability indicators like DR. The objective of the ROM development is to develop a model as simple as possible from the mathematical and numerical point of view, while preserving the physics of the BWR stability behaviour. The ODEs of the ROM are deduced from the PDEs describing the dynamics of a BWR. The system of ODEs includes all spatial effects in an approximated (spatial averaged) manner, e.g. the space-time dependent neutron flux is expanded in terms of a complete set of orthogonal spatial neutron flux modes. In order to simulate the stability characteristics of the in-phase and out-of-phase oscillation mode, it is only necessary to take into account the fundamental mode and the first azimuthal mode. The ROM, originally developed at PSI in collaboration with the University of Illinois (PSI-Illinois-ROM), was upgraded in significant points: • Development and implementation of a new calculation methodology for the mode feedback reactivity coefficients (void and fuel temperature reactivity) • Development and implementation of a recirculation loop model; analysis and discussion of its impact on the in-phase and out-of-phase oscillation mode • Development of a novel physically justified approach for the calculation of the ROM input data • Discussion of the necessity of consideration of the effect of subcooled boiling in an approximate manner With the upgraded ROM, nonlinear BWR stability analyses are performed for three OPs (one for NPP Leibstadt (cycle7), one for NPP Ringhals (cycle14) and one for NPP Brunsbüttel (cycle16) for which measuring data of stability tests are available. In this thesis, the novel approach to nonlinear BWR stability analyses is extensively presented for NPP Leibstadt. In particular, the nonlinear analysis is carried out for an operational point (OP), in which an out-of-phase power oscillation has been observed in the scope of a stability test at the beginning of cycle 7 (KKLc7_rec4). The ROM predicts a saddle-node bifurcation of cycles, occurring in the linear stable region, close to the KKLc7_rec4-OP. This result allows a new interpretation of the stability behaviour around the KKLc7_rec4-OP. The results of this thesis confirm that the RAM-ROM methodology is qualified for nonlinear BWR stability analyses. / Die vorliegende Dissertation leistet einen Beitrag zum tieferen Verständnis des nichtlinearen Stabilitätsverhaltens von Siedewasserreaktoren (SWR). Trotz der Tatsache, dass in diesem technischen System nur negative innere Rückkopplungskoeffizienten auftreten, können in bestimmten Arbeitspunkten oszillatorische Instabilitäten auftreten. Obwohl relativ gute Kenntnisse über die signifikanten physikalischen Einflussgrößen vorliegen, fehlt bisher ein umfassendes Verständnis des SWR-Stabilitätsverhaltens. Das betrifft insbesondere die Bereiche der Systemparameter, in denen lineare Stabilitätsindikatoren, wie zum Beispiel das asymptotische Decay Ratio (DR), ihren Sinn verlieren. Die nichtlineare Stabilitätsanalyse wird im Allgemeinen mit Systemcodes (nichtlineare partielle Differentialgleichungen, PDG) durchgeführt. Jedoch kann mit Systemcodes kein oder nur ein sehr lückenhafter Überblick über die Typen von nichtlinearen Phänomenen, die in bestimmten System-Parameterbereichen auftreten, erhalten werden. Deshalb wurde im Rahmen der vorliegenden Arbeit eine neuartige Methode (RAM-ROM Methode) zur nichtlinearen SWR-Stabilitätsanalyse erprobt, bei der integrale Systemcodes und sog. vereinfachte SWR-Modelle (ROM) als sich gegenseitig ergänzende Methoden eingesetzt werden, um die Stabilitätseigenschaften von Fixpunkten und periodischen Lösungen (Grenzzyklen) des nichtlinearen Differentialgleichungssystems, welches das Stabilitätsverhalten des SWR beschreibt, zu bestimmen. Das ROM, in denen das dynamische System durch gewöhnliche Differentialgleichungen (GDG) beschrieben wird, kann relativ einfach mit leistungsfähigen Methoden aus der nichtlinearen Dynamik, wie zum Beispiel die semianalytische Bifurkationsanalyse, gekoppelt werden. Mit solchen Verfahren kann, ohne das DG-System explizit lösen zu müssen, ein Überblick über mögliche Typen von stabilen und instabilen oszillatorischen Verhalten des SWR erhalten werden. Insbesondere sind die Stabilitätseigenschaften von Grenzzyklen, die in Hopf-Bifurkationspunkten entstehen, und die Bedingungen, unter denen sie auftreten, von Interesse. Mit dem Systemcode (RAMONA5) werden dann die mit dem ROM vorhergesagten Phänomene in den entsprechenden Parameterbereichen detaillierter untersucht (Validierung des ROM). Die Methodik dient daher nicht der Verfeinerung der Berechnung linearer Stabilitätsindikatoren (wie das DR). Das ROM-Gleichungssystem entsteht aus den PDGs des Systemcodes durch geeignete (nichttriviale) räumliche Mittelung der PDG. Es wird davon ausgegangen, dass die Reduzierung der räumlichen Komplexität die Stabilitätseigenschaften des SWR nicht signifikant verfälschen, da durch geeignete Mittlungsverfahren, räumliche Effekte näherungsweise in den GDGs berücksichtig werden. Beispielsweise wird die raum- und zeitabhängige Neutronenflussdichte nach räumlichen Moden entwickelt, wobei für eine Simulation der Stabilitätseigenschaften der In-phase- und Out-of-Phase-Leistungsoszillationen nur der Fundamentalmode und der erste azimuthale Mode berücksichtigt werden muss. Das ROM, welches ursprünglich am Paul Scherrer Institut (PSI, Schweiz) in Zusammenarbeit mit der Universität Illinois (USA) entwickelt wurde, ist in zwei wesentlichen Punkten erweitert und verbessert worden: • Entwicklung und Implementierung einer neuen Methode zur Berechnung der Rückkopplungsreaktivitäten • Entwicklung und Implementierung eines Modells zur Beschreibung der Rezirkulationsschleife (insbesondere wurde der Einfluss der Rezirkulationsschleife auf den In-Phase-Oszillationszustand und auf den Out-of-Phase-Oszillationszustand untersucht) • Entwicklung einer physikalisch begründeten Methode zur Berechnung der ROM-Inputdaten • Abschätzung des Einflusses des unterkühlten Siedens im Rahmen der ROM-Näherungen Mit dem erweiterten ROM wurden nichtlineare Stabilitätsanalysen für drei Arbeitspunkte (KKW Leibstadt (Zyklus 7) KKW Ringhals (Zyklus 14) und KKW Brunsbüttel (Zyklus 16)), für die Messdaten vorliegen, durchgeführt. In der Dissertationsschrift wird die RAM-ROM Methode ausführlich am Beispiel eines Arbeitspunktes (OP) des KKW Leibstadt (KKLc7_rec4-OP), in dem eine aufklingende regionale Leistungsoszillation bei einem Stabilitätstest gemessen worden ist, demonstriert. Das ROM sagt die Existenz eines Umkehrpunktes (saddle-node bifurcation of cycles, fold-bifurcation) voraus, der sich im linear stabilen Gebiet nahe der Stabilitätsgrenze befindet. Mit diesem ROM-Ergebnis ist eine neue Interpretation der Stabilitätseigenschaften des KKLc7_rec4-OP möglich. Die Resultate der in der Dissertation durchgeführten RAM-ROM Analyse bestätigen, dass das weiterentwickelte ROM für die Analyse des Stabilitätsverhaltens realer Leistungsreaktoren qualifiziert wurde.

Μελετάµε χωρικά εντοπισµένες και χρονικά περιοδικές λύσεις σε διακριτά συστήµατα που εκτείνονται σε µία χωρική διάσταση. Αυτού του είδους οι λύσεις είναι γνωστές µε τον όρο discrete breathers (DB) ή intrinsic localized modes (ILM). Στην ελληνική ϐιϐλιογραϕία, έχουν ονοµαστεί ∆ιακριτές Πνοές. Απαραίτητα χαρακτηριστικά για την εµϕάνιση τέτοιων λύσεων είναι η ύπαρξη ενός άνω φράγµατος του γραµµικού φάσµατος καθώς και η µη γραµµικότητα των εξισώσεων κίνησης, χαρακτηριστικά που συναντάµε σε πολλά φυσικά συστήµατα. Συγκεκριμένα, ασχολούµαστε µε πλέγµατα τύπου Klein Gordon και παρουσιάσουµε μια αποδείξη ύπαρξης τέτοιων λύσεων καθώς και αριθµητικά αποτελέσµατα µελετώντας παράλληλα την ευστάθεια των περιοδικών αυτών λύσεων µέσω της ϑεωρίας Floquet. Πέραν του κλασικού µοντέλου, όπου έχουµε αλληλεπιδράσεις πλησιέστερων γειτόνων, εισάγουµε επίσης ένα νέο µοντέλο µε αλληλεπιδράσεις µακράς εµβέλειας η οποία ελέγχεται µέσω µιας παράµετρου α και µελετάµε τις επιπτώσεις που έχει η μεταβολή του εύρους αλληλεπίδρασης στον χωρικό εντοπισµό και την ευστάθεια ενός DB. / We study time-periodic and spatially localized solutions in discrete dynamical systems describing Hamiltonian lattices in one spatial dimension. These solutions are called discrete breathers (DBs) or intrinsic localized modes (ILM). Necessary conditions for their occurrence are the boundedness of the spectrum of linear oscillations of the system as well as the nonlinearity of the equations of motion. More specifically, we focus on a Klein Gordon lattice and present an existence proof for such solutions, as well as numerical results revealing the stability (or instability) of DBs using Floquet theory. Besides reporting on the classical Klein Gordon model with nearest neighbor interactions, we also introduce long range interactions in our model, which are controlled by a parameter α and study the effect of varying the range of interactions on the spatial localization and the stability of a DB.

The non-adiabatic quantum molecular dynamics (NA-QMD) method couples self-consistently classical nuclear motion with time-dependent density functional theory (TDDFT) in basis expansion for the electron dynamics. It has become a versatile approach to study the dynamics of atoms, molecules and clusters in a wide range of scenarios. This work presents applications of the NA-QMD method to important benchmark systems and its systematic extension to include quantum effects in the nuclear motion. Regarding the first objective, a complete study of the strong-field ionization and dissociation dynamics of nature’s simplest molecule H2+ is performed. By including all electronic and nuclear degrees of freedom and all reaction channels, molecular rotation is shown to play an important role in the ionization process. In addition, strong orientation effects in the energy deposition process of the Buckminster fullerene C60 in short intense laser pulses are surprisingly found in full dimensional calculations. Their consequences on the subsequent nuclear relaxation dynamics shed new light on available experimental data and future experiments are proposed to confirm the detailed predictions. Regarding the second objective, the NA-QMD formalism is basically extended to take electron-nuclear correlations into account. This extension is achieved by means of a trajectory surface hopping scheme in the adiabatic Kohn-Sham framework. First studied examples from collision physics and photochemistry illustrate the relevance and importance of quantum effects in the nuclear dynamics.

The non-adiabatic quantum molecular dynamics (NA-QMD) method couples self-consistently classical nuclear motion with time-dependent density functional theory (TDDFT) in basis expansion for the electron dynamics. It has become a versatile approach to study the dynamics of atoms, molecules and clusters in a wide range of scenarios. This work presents applications of the NA-QMD method to important benchmark systems and its systematic extension to include quantum effects in the nuclear motion. Regarding the first objective, a complete study of the strong-field ionization and dissociation dynamics of nature’s simplest molecule H2+ is performed. By including all electronic and nuclear degrees of freedom and all reaction channels, molecular rotation is shown to play an important role in the ionization process. In addition, strong orientation effects in the energy deposition process of the Buckminster fullerene C60 in short intense laser pulses are surprisingly found in full dimensional calculations. Their consequences on the subsequent nuclear relaxation dynamics shed new light on available experimental data and future experiments are proposed to confirm the detailed predictions. Regarding the second objective, the NA-QMD formalism is basically extended to take electron-nuclear correlations into account. This extension is achieved by means of a trajectory surface hopping scheme in the adiabatic Kohn-Sham framework. First studied examples from collision physics and photochemistry illustrate the relevance and importance of quantum effects in the nuclear dynamics.

Cette thèse présente un travail théorique effectué dans le domaine de la physique de la matière condensée, et plus particulièrement la physique des solides. Ce domaine de la physique décrit le comportement des électrons dans les cristaux à très basses températures dans le but d'observer des effets quantiques à l'échelle mésoscopique.Cette thèse se situe à l'interface entre deux types de matériaux : le graphène et les isolants topologiques. Le graphène est une couche d’épaisseur monoatomique d’atomes de carbone arrangés en réseau nid d’abeilles, qui présente de nombreuses propriétés impressionnantes en optique, en mécanique et en électronique. Les isolants topologiques sont des matériaux qui sont isolants en volume et conduisent l'électricité sur les bords. Cette caractéristique découle d'une propriété topologique des électrons dans le volume. La topologie est une branche des mathématiques qui décrit des objets dans leur globalité en ne retenant que les caractéristiques invariantes par certaines déformations continues. Les états de bords des isolants topologiques sont robustes à certaines perturbations comme le désordre créé par des impuretés dans le matériau. Le lien entre ces deux sujets est double. D’une part les premiers modèles d’isolants topologiques de bande ont été formulés pour le graphène, par Haldane en 1988 et Kane et Mele en 2005, ouvrant ainsi la voie à la découverte des isolants topologiques à 2D et 3D dans des matériaux à fort spin-orbite. D’autre part, il a été prédit que le graphène, même sans spin-orbite, devient un isolant topologique lorsqu'il est irradié par une onde électromagnétique. Dans cette thèse, nous suivons deux directions en parallèle : décrire les caractéristiques topologiques d’une part et les propriétés de transport électronique d’autre part.En premier lieu, nous passons en revue le modèle des liaisons fortes pour le graphène, puis le modèle effectif qui décrit les électrons de basse énergie comme des fermions de Dirac sans masse. Nous introduisons ensuite le modèle de Haldane, un modèle simple défini sur le réseau en nid d’abeille et qui présente des bandes non triviales caractérisées par un invariant topologique, le nombre de Chern, non nul. Du fait de cette propriété topologique, ce modèle possède un état de bord chiral se propageant au bord de l’échantillon et une conductance de Hall quantifiée. Lorsque le graphène est irradié par un laser ayant une fréquence plus large que la largeur de bande du graphène, il acquiert un gap dynamique similaire au gap topologique du modèle de Haldane. Lorsque la fréquence est réduite, nous montrons que des transitions topologiques se produisent et l'apparition d'états de bords.Le travail principal de cette thèse est l'étude du transport électronique dans le graphène irradié dans un régime de paramètres réalisables expérimentalement. Une feuille de graphène est connectée à deux électrodes avec une différence de potentiel qui génère un courant. Nous calculons la conductance différentielle de l'échantillon selon le formalisme de Landauer-Büttiker étendu aux systèmes soumis à une modulation périodique. Il nous est possible d'obtenir la conductance en fonction de la géométrie de l’échantillon et de différents paramètres tels que le potentiel chimique, la fréquence et l'intensité de l’onde.Un autre type d'isolant topologique est l’isolant d’effet Hall quantique de spin. Ce type de phase possède deux états de bords dans lesquels les spins opposés se propagent dans des directions opposées. Le second travail de cette thèse concerne le transport électronique à travers cet état de bord irradié. Nous observons l'apparition d'un courant pompé en l'absence de différence de potentiel. Nous distinguons deux régimes : un pompage adiabatique quantifié à basse fréquence, et un régime de réponse linéaire non quantifiée à hautes fréquences. Par rapport aux études précédentes existantes, nous montrons un effet important de la présence des électrodes de mesure. / This thesis presents a theoretical work done in the field of condensed matter physics, and in particular solid state physics. This field of physics aims at describing the behaviour of electrons in crystalline materials at very low temperature to observe effects characteristic of quantum physics at the mesoscopic scale.This thesis lies at the interface between two types of materials : graphene and topological insulators. Graphene is a monoatomic layer of carbon atoms arranged in a honeycomb lattice that presents a wide range of striking properties in optics, mechanics and electronics. Topological insulators are materials that are insulators in the bulk and conduct electricity at the edges. This characteristic originates from a topological property of the electrons in the bulk. Topology is a branch of mathematics that aims to describe objects globally retaining only characteristics invariant under smooth deformations. The edge states of topological insulators are robust to certain king of perturbations such as disorder created by impurities in the bulk. The link between these two topics is two-fold. On one hand, the first models of band topological insulators were formulated for graphene, by Haldane in 1988 and Kane and Mele in 2005, opening the way to the discovery of 2D and 3D topological insulators in materials with strong spin-orbit coupling. On the other hand, it was predicted that graphene, even without spin-orbit coupling, turns to a topological insulator under irradiation by an electromagnetic wave. In this thesis, we follow two directions in parallel : describe the topological properties on one hand, and the electronic transport properties on the other hand.First, we review the tight-binding model of graphene, and the effective model that describes low-energy electrons as massless Dirac fermions. We then introduce the Haldane model, a simple model defined on the honeycomb lattice that presents non-trivial bands characterised by a topological invariant, the Chern number. Due to this topological property, this model possesses a chiral edge state that propagates around the sample and a quantized Hall conductance. When graphene is irradiated by a laser with a frequency larger than the graphene bandwidth, it acquires a dynamical gap similar to the topological gap of the Haldane model. When the frequency is lowered, we show that topological transitions happens and that different edge states appear.The main work of this thesis is the study of electronic transport in irradiated graphene in a regime of experimentally achievable parameters. A graphene sheet is connected to two electrodes with a potential difference that generates a current. We compute the differential conductance of the sample according to Landauer-Büttiker formalism extended to periodically driven systems. Using this simple formalism, we are able to obtain the conductance as a function of the geometry of the sample and of several parameters such as the chemical potential, the frequency and the intensity of the electromagnetic wave.Another kind of topological insulator is the quantum spin Hall insulator. This type of phase possesses two edge states in which opposite spins propagate in opposite directions. The second work of this thesis concerns electronic transport through this irradiated edge state. We observe the apparition of a pumped current in the absence of a potential difference. We observe two regimes : a quantized adiabatic at low frequency, and a non-quantized linear response regime at high frequency. Compared to previous studies, we show an important effect originating from the presence of electrodes.

In this thesis a new frequency domain approach for the analysis of time delay systems is presented. After linearization of a nonlinear delay differential equation (DDE) with constant distributed delay around a constant or periodic reference solution the so-called Hill-Floquet method can be used for the analysis of the resulting linear DDE. In addition, systems with fast or slowly time-varying delays, systems with variable transport delays originating from a transport with variable velocity, and the corresponding spatially extended systems are presented, which can be also analyzed with the presented method. The newly introduced Hill-Floquet method is based on the Hill’s infinite determinant method and enables the transformation of a system with periodic coefficients to an autonomous system with constant coefficients. This makes the usage of a variety of existing methods for autonomous systems available for the analysis of periodic systems, which implies that the typical calculation of the monodromy matrix for the time evolution of the solution over the principle period is no longer required. In this thesis, the Chebyshev collocation method is used for the analysis of the autonomous systems. Specifically, in this case the periodic part of the solution is expanded in a Fourier series and the exponential behavior of the solution is approximated by the discrete values of the Fourier coefficients at the Chebyshev nodes, whereas in classical spectral or pseudo-spectral methods for the analysis of linear periodic DDEs the complete solution is expanded in terms of basis functions. In the last part of this thesis, new results for three applications with time delay effects are presented, which were analyzed with the presented methods. On the one hand, the occurrence of diffusion-driven instabilities in reaction-diffusion systems with delay is investigated. It is shown that wave instabilities are possible already for single-species reaction diffusion systems with distributed or time-varying delay. On the other hand, the stability of metal cutting vibrations at machine tools is analyzed. In particular, parallel orthogonal turning processes with multiple discrete delays and turning processes with a time-varying delay due to a spindle speed variation are studied. Finally, the stability of the synchronized solution in networks with heterogeneous coupling delays is studied. In particular, the eigenmode expansion for synchronized periodic orbits is derived, which includes an extension of the classical master stability function to networks with heterogeneous coupling delays. Numerical results are shown for a network of Hodgkin-Huxley neurons with two delays in the coupling. / In dieser Dissertation wird ein neues Verfahren zur Analyse von Systemen mit Totzeiten im Frequenzraum vorgestellt. Nach Linearisierung einer nichtlinearen retardierten Differentialgleichung (DDE) mit konstanter verteilter Totzeit um eine konstante oder periodische Referenzlösung kann die sogenannte Hill-Floquet Methode für die Analyse der resultierende linearen DDE angewendet werden. Darüber hinaus werden Systeme mit schnell oder langsam variierender Totzeit, Systeme mit einer variablen Totzeit, resultierend aus einem Transport mit variabler Geschwindigkeit, und entsprechende räumlich ausgedehnte Systeme vorgestellt, welche ebenfalls mit der vorgestellten Methode analysiert werden können. Die neu eingeführte Hill-Floquet Methode basiert auf der Hillschen unendlichen Determinante und ermöglicht die Transformation eines Systems mit periodischen Koeffizienten auf ein autonomes System mit konstanten Koeffizienten. Dadurch können zur Analyse periodischer Systeme auch eine Vielzahl existierender Methoden für autonome Systeme genutzt werden und die Berechnung der Monodromie-Matrix für die Lösung des Systems über eine Periode entfällt. In dieser Arbeit wird zur Analyse des autonomen Systems die Tschebyscheff-Kollokationsmethode verwendet. Im Speziellen wird bei diesem Verfahren der periodische Teil der Lösung in einer Fourierreihe entwickelt und das exponentielle Verhalten durch die Werte der Fourierkoeffizienten an den Tschebyscheff Knoten approximiert, wohingegen bei klassischen spektralen Verfahren die komplette Lösung in bestimmten Basisfunktionen entwickelt wird. Im Anwendungsteil der Arbeit werden neue Ergebnisse für drei Beispielsysteme präsentiert, welche mit den vorgestellten Methoden analysiert wurden. Es wird gezeigt, dass Welleninstabilitäten schon bei Einkomponenten-Reaktionsdiffusionsgleichungen mit verteilter oder variabler Totzeit auftreten können. In einem zweiten Beispiel werden Schwingungen an Werkzeugmaschinen betrachtet, wobei speziell simultane Drehbearbeitungsprozesse und Prozesse mit Drehzahlvariationen genauer untersucht werden. Am Ende wird die Synchronisation in Netzwerken mit heterogenen Totzeiten in den Kopplungstermen untersucht, wobei die Zerlegung in Netzwerk-Eigenmoden für synchrone periodische Orbits hergeleitet wird und konkrete numerische Ergebnisse für ein Netzwerk aus Hodgkin-Huxley Neuronen gezeigt werden.