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.

Cette thèse porte sur la description théorique de processus dynamiques ultra-rapides de molécules polyatomiques et de leur contrôle par impulsions laser. Nous avons d’abord étudié la photochimie de l’aniline à l’aide de calculs de structure électronique. Nous avons d´écrit plusieurs régions clé des surfaces d’énergie potentielle et analysé ces résultats en relation avec les données expérimentales existantes. La photochimie de la pyrazine a été étudiée par des calculs de dynamiques quantique basés sur un Hamiltonien modèle incluant les quatre états électroniques excités de plus basse énergie et seize modes de vibration. Nous montrons que l’état sombre Au(nπ∗) joue un rôle important dans la dynamique de la molécule après photo-excitation. Un modèle simplifié à deux états et quatre modes a été utilisé pour étudier le contrôle par laser de la dynamique de la pyrazine photo-excitée. Nous proposons un mécanisme visant à augmenter la durée de vie de l’état B2u(ππ∗) en utilisant l’effet Stark induit par une impulsion laser intense non-résonante. / The central subject of this thesis is the theoretical description of ultrafast dynamical processes in molecular systems of chemical interest and of their control by laser pulses. We first use electronic structure calculations to study the photochemistry of aniline. A umber of previously unknown features of the potential energy surfaces of the low-lying elec-tronic states are reported, and analyzed in relation with the experimental results available. We use quantum dynamics simulations, based on a model Hamiltonian including the four lowest excited electronic states and sixteen vibrational modes, to investigate the photochem-istry of pyrazine. We show that the dark Au(nπ∗) state plays an important role in the ultrafast dynamics of the molecule after photoexcitation. The laser control of the excited state dynamics of pyrazine is studied using a simplified two-state four-mode model Hamiltonian. We propose a control mechanism to enhance the lifetime of the bright B2u(ππ∗) state using the Stark effect induced by a strong non-resonant laser pulse. We finally focus on the laser control of the tunneling dynamics of the NHD2 molecule, using accurate full-dimensional potential energy and dipole moment surfaces. We use simple effective Hamiltonians to explore the effect of the laser parameters on the dynamics and design suitable laser fields to achieve the control. These laser fields are then used in MCTDH quantum dynamics simulations. Both enhancement and suppression of tunneling are achieved in our model.

Dans le cadre géographique de la basse Provence, cette thèse étudie la réalisation des grands aménagements hydrauliques entre la Renaissance et le XXe siècle. Ils concernent des amenées d’eau à usages multiples : canal de Craponne, projet Floquet, canal de Boisgelin, de Carpentras, de Marseille, du Verdon et de Provence, alors que les canaux de Van Ens concernent l’assainissement. En examinant ces réalisations de plusieurs points de vue : techniques, économiques et financiers et en les replaçant dans leur contexte politique, des caractéristiques communes apparaissent. Le mode de financement joue un rôle important dans le succès des opérations, avec des erreurs ou des illusions qui ont mis longtemps à se dissiper. Pour réussir, ces aménagements ont exigé un temps de maturation important et la présence d’un ou plusieurs hommes compétents qui en ont organisé la réalisation.En conclusion, la capacité de gestion de l’eau apparaît comme un indicateur du degré de civilisation atteint. / This thesis is a study of the big hydraulic facilities in the lower Provence region, between the Renaissance and the XXth century. It concerns mainly bringing water for various purposes : Floquet Project, Craponne, Boisgelin, Carpentras, Marseille, Verdon, and Provence canals (the Van Ens canal are devoted to drainage work).These constructions are studied under technical, economical, financial, and political point of view.Common characteristics appear; different financing methods (public or private), project gestation time, mistakes and different managements are studied to try to explain the reasons of success of these constructions.

Dans un contexte de densification des villes et de leurs réseaux de transport, les gens sont de plus en plus exposés au bruit. Ainsi, le résultat de l'étude d'impact vibro-acoustique joue un rôle primordial dans l'expansion du réseau ferroviaire. L'une des principales sources est le bruit de roulement : La rugosité de la surface de la roue et du rail produit un déplacement imposé sur ces derniers. Ce déplacement entraine une réponse vibratoire des roues et de la voie ferrée et leurs rayonnements acoustiques. Cette thèse propose une amélioration de la modélisation vibro-acoustique de la voie ferrée.Pour la réponse vibratoire, le coté infini de la voie et sa déformation dans les 3 dimensions rendent les modèles analytiques et les éléments finis non-optimales dans la gamme de fréquence de l’audible. La méthode élément fini semi-périodique (SAFEM) est utilisée dans cette thèse pour modéliser une voie à support continue. Elle est ensuite couplée au théorème de Floquet pour modéliser une voie à support périodique. Cependant, cette technique génère des problèmes numériques qui ont imposé un algorithme adapté. La méthode d'Arnoldi du second ordre (SOAR) est utilisée avant de résoudre l'équation SAFEM permet de résoudre ces problèmes ainsi qu’apporter la stabilité requise. Des comparaisons avec d’autres modèles et des données expérimentales permettent de valider la méthode.Pour le rayonnement acoustique, la simulation de grand domaine en haute fréquence rendent inadapté l'utilisation de techniques conventionnelles (FEM, BEM, ...). La méthode proposée ici : la théorie variationnelle du rayon complexe est particulièrement bien adaptée à ce cas. Les principales caractéristiques de l'approche VTCR sont l'utilisation d'une formulation faible du problème acoustique, qui permet de considérer automatiquement les conditions limites entre sous-domaines. Ensuite, l'utilisation d'une répartition intégrale des ondes planes dans toutes les directions permet de simuler le champ acoustique. Les inconnues du problème sont leurs amplitudes. Cette méthode qui a déjà montré son efficacité pour les domaines fermés a été étendue au domaine ouvert et couplée à la réponse vibratoire. Des comparaisons avec des solutions analytiques et des simulations FEM à basse fréquence permettent de valider la méthode. / In a context of urban and transport network densification, people are increasingly exposed to noise. Consequently, the result of vibro-acoustic impact assessment has a pivotal role in rail network expansion. One of the main sources is the rolling noise: Roughness on the wheel and rail surface produce an imposed displacement one the both. This last, generates vibrational response of wheels and the railway track and their acoustic radiation. This PhD thesis presents some improvements of the vibro-acoustic railway track modelling.Concerning vibrational response, the infinite dimension in the longitudinal direction of the track and its deformation in the 3 dimensions, make the analytical models and finite elements non-optimal. The Semi-analytical finite element method (SAFEM), used in this thesis, is particularly well adapted in this case. Firstly, it is used to model railway track on a continuous support. Then, it is coupled with Floquet theorem to model tracks with a periodic support. However, this technique suffers from numerical problems that imposed an adapted algorithm. The second-order Arnoldi method (SOAR) is used to tackle them. This reduction allows to eliminate critical values improving the robustness of the method. Comparison with existing techniques and experimental results validate this model.Concerning acoustic radiation, big domains simulations at high frequency are almost unfeasible when using conventional techniques (FEM, BEM,…). The method used in this thesis, the Variational theory of complex ray (VTCR) is particularly well adapted to these cases. The principal features of VTCR approach are the use of a weak formulation of the acoustic problem, which allows to consider automatically boundary conditions between sub-domains. Then, the use of an integral repartition of plane waves in all the direction allow to simulate the acoustic field. The unknowns of the problem are their amplitudes. This method well assessed for closed domain, has been extended to open domain and coupled to vibrational response of the rail. Comparison with analytic solution and FEM simulation at low frequency allow to validate the method.Coupling these both methods allowed to simulate complex real life vibro-acoustic scenarios. Result of different railway tracks are presented and validated

Robust states exist at the interfaces between topologically trivial and nontrivial phases of matter. These boundary states are expression of the nontrivial bulk properties through a connection dubbed the bulk-boundary correspondence. Whether the bulk is topological or not is determined by the value of a topological invariant. This quantity is defined with respect to symmetries and dimensionality of the system, such that it takes only quantized values. For static topological phases that are realized in ground-states of isolated, time-independent systems, the topological invariant is related to the properties of the Hamiltonian operator. In contrast, Floquet topological phases that are realized in open systems with periodical pumping of energy are topologically characterized with a unitary Floquet operator i.e., the time-evolution operator over the entire period. Topological phases of matter can be distinguished by the dimensionality of robust boundary states with respect to the protecting bulk. This dissertation concerns recently discovered higher-order topological phases where the difference between dimensionalities of bulk and boundary states is larger than one. Using analytical and numerical single-particle techniques, we focus on instances where static higher-order topology can be understood with insights from the mature field of Floquet topology. Namely, even though static systems do not admit a Floquet description, we find examples of higher-order systems to which certain unitary operators can be attributed. The understanding of topological characteristics of these systems is therefore conditioned by the knowledge on topological properties of unitary operators, among which the Floquet operator is well-known. The first half of this thesis concerns toy models of static higher-order topological phases that are topologically characterized in terms of unitary operators. We find that a class of these systems called quadrupole topological insulators exhibit a wider range of topological phases than known previously. In the second half of this dissertation, we study reflection matrices of higher-order topological phases and show that they can exhibit the same topological features as Floquet systems. Our findings suggest a new route to experimental realizations of Floquet systems, the one that avoids noise-induced decoherence inevitable in many other experimental setups.

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.:1. Introduction 2. System definition and equivalent systems 3. Analysis of nonlinear time delay systems 4. Analytical solution of linear time delay systems 5. Frequency domain approach 6. Hill-Floquet method 7. Applications 8. Concluding remarks A Appendix / 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.:1. Introduction 2. System definition and equivalent systems 3. Analysis of nonlinear time delay systems 4. Analytical solution of linear time delay systems 5. Frequency domain approach 6. Hill-Floquet method 7. Applications 8. Concluding remarks A Appendix

Cet ouvrage porte sur la réaction du clergé canadien face à l’invasion américaine de 1775-1776. Alors que l’historiographie considère généralement que les prêtres de la colonie restèrent fidèles au gouvernement britannique à cette occasion, trois curés se détachèrent au contraire de cette image de loyalisme : Eustache Chartier de Lotbinière (1716-1785), Pierre-René Floquet (1716-1782) ainsi que Pierre Huet de La Valinière (1732-1806). Soupçonnés par les autorités ecclésiastiques et coloniales d’entrenir des sympathies pour les révolutionnaires américains, ces hommes furent frappés par diverses sanctions, affectant durablement le déroulement de leur carrière. / This dissertation examines the reaction of Canadian clergy to the American invasion of 1775-1776. While historians have generally considered that the priests of the colony remained loyal to the British Government on this occasion, three priests stand in contrast to this image of loyalty: Eustache Chartier de Lotbinière (1716-1785), Pierre-René Floquet (1716 -1782), Joseph Huguet (1725-1783) and Pierre Huet de La Valinière (1732-1806). Suspected by church and colonial authorities to have shown sympathy to the American revolutionaries, these men were struck by various sanctions that permanently affected the development of their careers.

This work was supported by Fundação para a Ciência e a Tecnologia, through the scholarship SFRH/BD/35821/2007 / Tese de doutoramento. Engenharia Mecânica. Faculdade de Engenharia. Universidade do Porto. 2012

This thesis reports our work on transport related problems in mesoscopic physics using analytical as well as numerical techniques. Some of the problems we studied are: effect of interactions and static impurities on the conductance of a ballistic quantum wire[1], aspects of quantum charge pumping [2, 3, 4], DC and AC conductivity of a (dissipative) quantum Hall (edge) line junctions[5, 6], and junctions of three or more Luttinger liquid (LL)quantum wires[7]. This thesis begins with an introductory chapter which gives a brief glimpse of the underlying physical systems and the ideas and techniques used in our studies. In most of the problems we will look at the physical effects caused by e-e interactions and static scattering processes. In the second chapter we study the effects of a static impurity and interactions on the conductance of a 1D-quantum wire numerically. We use the non-equilibrium Green’s function (NEGF) formalism along with a self-consistent Hartree-Fock approximation to numerically study the effects of a single impurity and interactions between the electrons (with and without spin) on the conductance of a quantum wire [1]. We study the variation of the conductance with the wire length, temperature and the strength of the impurity and electron-electron interactions. We find our numerical results to be in agreement with the results obtained from the weak interaction RG analysis. We also discover that bound states produce large density deviations at short distances and have an appreciable effect on the conductance which is not captured by the renormalization group analysis. In the third chapter we use the equations of motion (EOM) for the density matrix and Floquet scattering theory to study different aspects of charge pumping of non-interacting electrons in a one-dimensional system. We study the effects of the pumping frequency, amplitude, band filling and finite bias on the charge pumped per cycle, and the spectra of the charge and energy currents in the leads[2]. The EOM method works for all values of parameters, and gives the complete time-dependences of the current and charge at any site of the system. In particular we study a system with oscillating impurities at several sites and our results agree with Floquet and adiabatic theory where these are applicable, and provides support for a mechanism proposed elsewhere for charge pumping by a traveling potential wave in such systems. For non-adiabatic and strong pumping, the charge and energy currents are found to have a marked asymmetry between the two leads, and pumping can work even against a substantial bias. We also study one-parameter charge pumping in a system where an oscillating potential is applied at one site while a static potential is applied in a different region [3]. Using Floquet scattering theory, we calculate the current up to second order in the oscillation amplitude and exactly in the oscillation frequency. For low frequency, the charge pumped per cycle is proportional to the frequency and therefore vanishes in the adiabatic limit. If the static potential has a bound state, we find that such a state has a significant effect on the pumped charge if the oscillating potential can excite the bound state into the continuum states or vice versa. In the fourth chapter we study the current produced in a Tomonaga-Luttinger liquid (TLL) by an applied bias and by weak, point-like impurity potentials which are oscillating in time[4]. We use bosonization to perturbatively calculate the current up to second order in the impurity potentials. In the regime of small bias and low pumping frequency, both the DC and AC components of the current have power law dependences on the bias and pumping frequencies with an exponent 2K−1 for spinless electrons, where Kis the interaction parameter. For K<1/2, the current grows large for special values of the bias. For non-interacting electrons with K= 1, our results agree with those obtained using Floquet scattering theory for Dirac fermions. We also discuss the cases of extended impurities and of spin-1/2 electrons. In chapter five, we present a microscopic model for a line junction formed by counter or co-propagating single mode quantum Halledges corresponding to different filling factors and calculate the DC [5] and AC[6] conductivity of the system in the diffusive transport regime. The ends of the line junction can be described by two possible current splitting matrices which are dictated by the conditions of both lack of dissipation and the existence of chiral commutation relations between the outgoing bosonic fields. Tunneling between the two edges of the line junction then leads to a microscopic understanding of a phenomenological description of line junctions introduced by Wen. The effect of density-density interactions between the two edges is considered exactly, and renormalization group (RG) ideas are used to study how the tunneling parameter changes with the length scale. The RG analysis leads to a power law variation of the conductance of the line junction with the temperature (or other energy scales) and the line junction may exhibit metallic or insulating phase depending on the strength of the interactions. Our results can be tested in bent quantum Hall systems fabricated recently. In chapter six, we study a junction of several Luttinger Liquid (LL) wires. We use bosonization with delayed evaluation of boundary conditions for our study. We first study the fixed points of the system and discuss RG flow of various fixed points under switching of different ‘tunneling’ operators at the junction. Then We study the DC conductivity, AC conductivity and noise due to tunneling operators at the junction (perturbative).We also study the tunneling density of states of a junction of three Tomonaga-Luttinger liquid quantum wires[7]. and find an anomalous enhancement in the TDOS for certain fixed points even with repulsive e-e interactions.

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.