The study of nonlinear dynamical systems via the spectrum of the Koopman operator has emerged as a paradigm shift, from the Poincaré's geometric picture that centers the attention on the evolution of states, to the Koopman operator's picture that focuses on the evolution of observables. The Koopman operator-theoretic framework rests on the idea of lifting the states of a nonlinear dynamical system to a higher dimensional space; these lifted states are referred to as the Koopman eigenfunctions. To determine the Koopman eigenfunctions, one performs a nonlinear transformation of the states by relying on the so-called observables, that is, scalar-valued functions of the states. In other words, one executes a change of coordinates from the state space to another set of coordinates, which are denominated Koopman canonical coordinates. The variables defined on these intrinsic coordinates will evolve linearly in time, despite the underlying system being nonlinear. Since the Koopman operator is linear, it is natural to exploit its spectral properties. In fact, the theory surrounding the spectral properties of linear operators has well-known implications in electric power systems. Examples include small-signal stability analysis and direct methods for transient stability analysis based on the Lyapunov function. From the applications' standpoint, this framework based on the Koopman operator is attractive because it is capable of revealing linear and nonlinear modes by only applying well-established tools that have been developed for linear systems. With the challenges associated with the high-dimensionality and increasing uncertainties in the power systems models, researchers and practitioners are seeking alternative modeling approaches capable of incorporating information from measurements. This is fueled by an increasing amount of data made available by the wide-scale deployment of measuring devices such as phasor measurement units and smart meters. Along these lines, the Koopman operator theory is a promising framework for the integration of data analysis into our mathematical knowledge and is bringing an exciting perspective to the community. The present dissertation reports on the application of the Koopman operator for identification, estimation, and control of electric power systems. A dynamic state estimator based on the Koopman operator has been developed and compares favorably against model-based approaches, in particular for centralized dynamic state estimation. Also, a data-driven method to compute participation factors for nonlinear systems based on Koopman mode decomposition has been developed; it generalizes the original definition of participation factors under certain conditions. / PHD / Electric power systems are complex, large-scale, and given the bidirectional causality between economic growth and electricity consumption, they are constantly being expanded. In the U.S., some of the electric power grid facilities date back to the 1880s, and this aging system is operating at its capacity limits. In addition, the international pressure for sustainability is driving an unprecedented deployment of renewable energy sources into the grid. Unlike the case of other primary sources of electric energy such as coal and nuclear, the electricity generated from renewable energy sources is strongly influenced by the weather conditions, which are very challenging to forecast even for short periods of time. Within this context, the mathematical models that have aided engineers to design and operate electric power grids over the past decades are falling short when uncertainties are incorporated to the models of such high-dimensional systems. Consequently, researchers are investigating alternative data-driven approaches. This is not only motivated by the need to overcome the above challenges, but it is also fueled by the increasing amount of data produced by today’s powerful computational resources and experimental apparatus. In power systems, a massive amount of data will be available thanks to the deployment of measuring devices called phasor measurement units. Along these lines, the Koopman operator theory is a promising framework for the integration of data analysis into our mathematical knowledge, and is bringing an exciting perspective on the treatment of high-dimensional systems that lie in the forefront of science and technology. In the research work reported in this dissertation, the Koopman operator theory has been exploited to seek for solutions to some of the challenges that are threatening the safe, reliable, and efficient operation of electric power systems.

This study critically examines the (UHI) effect in urban and suburban neighbourhoods of Quito, Ecuador, over a 19-year period, focusing on the interplay between atmospheric pollution and urban/ suburban temperature. Utilizing Empirical Dynamic Modeling(EDM) and Convergent Cross-Mapping (CCM), this study dives into the nonlinear dynamics of environmental factors, a method that traditional linear models fail to address effectively.The results unveil a consistent and strong positive correlation across various neighbourhoods, with temperature fluctuations indicating a typical UHI effect. This is most noticeable in urbanized areas where the temperature is significantly higher due to dense infrastructure and reduced greenery, a pattern that diminishes as one moves towards the outskirts. Specifically, pollutants like PM2.5 exhibit a non-uniform positive correlation, suggesting their collective increase or decrease across different regions, whereas CO shows a very slight and inconsistent inverse relationship across locations. The causal analysis further substantiates a significant interaction between PM2.5 concentrations and temperature, with the data revealing a reciprocal predictive capacity between these variables. The CCM analysis, through its graphical representation of predictive skills, confirms the causal effect of PM2.5 on urban temperature, marking an essential contribution to understanding the UHI effect and its implications for urban environmental dynamics. This study provides a comprehensive overview of the UHI phenomenon, highlighting the intricate relationship between urbanization, atmospheric pollution, and climate. The findings emphasize the necessity for urban planning and policy to consider these complex interactions to mitigate the effects of climate change on urban environments.

This thesis focuses on analyzing the synchronization of time delays between mutually coupled phase-locked loops (PLLs) in a flat hierarchy. Mutual synchronization refers to decentralized synchronization where there is no primary or secondary unit or control source. Consequently, it is an inherently self-organizing system in which each unit has equal rights, making it a democratic system. In this research, a dynamic nonlinear time-domain model is used to describe mutually delayed coupled oscillators. The predictions of this model are evaluated against experimental measurements. The time-domain model is based on the Kuramoto model. The Kuramoto model describes a network of coupled oscillators. As a first impression, this Kuramoto model is first analyzed for understanding of the effects of time delays between oscillators. The time domain model is based on a conventional PLL architecture modified to allow mutual coupling. The modifications include a circuit section that sums and weights all incoming phase differences. Overall, the measured results of this research study are in good agreement with the theoretical predictions of the time-domain model. The analysis allows the identification of the transient dynamics and the mechanisms that lead to mutual coupling and the formation of synchronized states through self-organized synchronization. The results show that the mutual coupling can self-organize its dynamics to synchronize even at time delays where the phenomenon of multistability of synchronized states occurs. A critical time delay beyond which a stable synchronized state cannot be achieved has been identified. The work also analyzes the dynamics and noise of synchronized states and finds that the dynamics near a synchronized state are correlated due to mutual coupling, leading to a reduction in noise. The noise correlation is affected by the direction of coupling, the number of nodes in the network, and the network topology. An improvement in phase noise of up to 14.42 dBc/Hz at 100 kHz offset from the carrier and 49.47ns delay was achieved using all-to-all coupling with four nodes. Furthermore, the hybrid approach, the combination of hierarchical and self-organizing synchronization architectures, is investigated. The dissertation presents an experimental study to understand how this affects a network of mutually delayed delay-coupled oscillators and whether the network of mutually coupled nodes can be abstracted as a secondary oscillator. A range in which the mutually delay-coupled network can be successfully synchronized by an external reference oscillator, depending on the synchronized state, is identified. In summary, this thesis provides valuable insights into the properties of mutually delay-coupled PLLs and their synchronization in flat hierarchies, and contributes to the understanding, design, and optimization of more practical networks of mutually delayed PLLs.:Abstract/Zusammenfassung Symbols and Abbreviations Previous Publications 1 Introduction 1.1 Classifications of Synchronization 1.2 A Historical Perspective on Mutual Synchronization 1.3 Extending the Understanding of Mutual Synchronization 1.4 Definitions and Methodologies 2 Model of Networks of Mutually Coupled PLLs 2.1 Coupled Oscillators – Kuramoto Model 2.1.1 Consequences of a Time Delay between Oscillators 2.1.2 Arbitrary Time Delays between Oscillators 2.2 Time-Domain Model of Delay-Coupled PLLs 2.2.1 Phase Detection 2.2.2 Loop-Filter 2.2.3 Voltage Controlled Oscillator 2.3 Prediction and Stability Analysis of Synchronized States 2.3.1 Assessing the Linear Stability of Synchronized States 2.3.2 Stability Consideration for Two Identical PLL Nodes 2.3.3 The Notion of Mode Locking 2.3.4 Effects of Heterogeneity on Synchronized States 2.4 Dynamics and Noise in Synchronized States 2.4.1 Gain and Phase Margin of a PLL Node 2.4.2 Phase Noise 2.5 Key Findings of the Theoretical Model 3 Design of Phase-Locked-Loops for Mutual Synchronization 3.1 PLL Nodes Dedicated for Mutual Synchronization 3.1.1 Phase Detection Circuitry 3.1.2 Adder Chain 3.2 Additional Circuitry for Implementing a Time Delay 4 Experimental Analysis of Mutually Time-Delayed Coupled PLLs 4.1 Synchronized States Including Oscillator Nonlinearity 4.2 Stability of Multistable Synchronized States 4.3 Critical Time Delay Between Two Coupled Nodes 4.4 Combining Hierarchical and Flat Synchronization Concepts 4.4.1 Entrainment of a Chain Network Topology 4.4.2 Entrainment of a Ring Network Topology 4.5 Heterogeneous Time Delays between Coupled PLLs 4.6 Phase Noise Analysis of Time Delay Coupled PLLs 4.6.1 Phase Noise for Two Mutually Coupled Nodes 4.6.2 The Impact of Coupling Directionality 4.6.3 Long Term Frequency Stability 4.6.4 Effect of Time Delay on Phase Noise 4.6.5 Network Topology Dependency on Phase Noise 5 Conclusion and Future Prospects Bibliography Own Publications – Periodicals Own Publications – Conference Proceedings Co-Authored Publications Other References List of Figures List of Tables Acknowledgement Curriculum Vitae / Diese Arbeit befasst sich mit der Analyse der Auswirkungen von Zeitverzögerungen auf die Synchronisation von gegenseitig gekoppelten Phasenregelschleifen (engl. phase-locked loop (PLL)) in einer flachen Hierarchie. Gegenseitige Synchronisation bezieht sich auf eine dezentrale Synchronisation, bei der es keine primäre oder sekundäre Einheit oder Steuerquelle gibt. Folglich ist es ein inhärent selbstorganisierendes System, in dem jede Einheit gleichberechtigt ist, was es zu einem demokratischen System macht. Für die Untersuchung wird ein dynamisches nichtlineares Zeitbereichsmodell verwendet, um gegenseitig verzögert gekoppelte Oszillatoren zu modellieren und die Vorhersagen dieses Modells anhand experimenteller Messungen zu bewerten. Dieses Zeitbereichsmodell basiert auf dem sogenannten Kuramoto-Modell, das ein Netzwerk gekoppelter Oszillatoren beschreibt. Um einen ersten Eindruck zu erhalten, wird zunächst dieses Kuramoto-Modell analysiert, um die Auswirkungen von Zeitverzögerungen zwischen den Oszillatoren zu verstehen. Das Zeitbereichsmodell basiert auf einer konventionellen PLL-Architektur, die modifiziert wurde, um eine gegenseitige Kopplung zu ermöglichen. Die Modifikationen beinhalten einen Schaltungsteil, der alle eingehenden Phasendifferenzen summiert und gewichtet. Die gemessenen Ergebnisse dieser Untersuchung stimmen insgesamt gut mit den theoretischen Vorhersagen des Zeitbereichsmodells überein. Die Analyse erlaubt es, die transiente Dynamik und die Mechanismen zu identifizieren, die zur gegenseitigen Synchronisation und zur Bildung synchronisierter Zustände durch selbstorganisierte Synchronisation führen. Die Ergebnisse zeigen, dass selbst bei Zeitverzögerungen, bei denen das Phänomen der Multistabilität synchronisierter Zustände auftritt, die gegenseitige Kopplung ihre Dynamik selbst organisieren kann, um sich zu synchronisieren. Die Untersuchung identifizierte eine kritische Zeitverzögerung, bei der kein stabiler synchronisierter Zustand erreicht werden kann. Die Arbeit analysiert auch die Dynamik und das Rauschen von synchronisierten Zuständen und stellt fest, dass die Dynamik in der Nähe eines synchronisierten Zustands aufgrund der gegenseitigen Kopplung korreliert ist, was zu einer Reduktion des Rauschens führt. Die Richtung der Kopplung und die Anzahl der Knoten im Netzwerk sowie die Netzwerktopologie beeinflussen die Korrelation des Rauschens. Eine Verbesserung des Phasenrauschens von bis zu 14.42 dBc/Hz bei einem Versatz von 100 kHz zum Träger und einer Verzögerung von 49.47 ns wurde durch eine globalen oder All-to-All-Kopplung mit vier Knoten erreicht. Des Weiteren wird der hybride Ansatz, die Kombination von hierarchischen und selbstorganisierenden Synchronisationsarchitekturen, untersucht. Die Arbeit präsentiert eine experimentelle Studie, um zu verstehen, wie dies ein Netzwerk von gegenseitig verzögert gekoppelten Oszillatoren beeinflusst und ob das Netzwerk von gegenseitig gekoppelten Knoten als sekundärer Oszillator abstrahiert werden kann. Dabei wird eine vom synchronisierten Zustand abhängige Domäne identifiziert, in der das wechselseitig gekoppelte Netzwerk durch einen externen Referenzoszillator erfolgreich synchronisiert werden kann. Insgesamt liefert diese wissenschaftliche Arbeit wertvolle Erkenntnisse über die Eigenschaften von gegenseitig verzögerungsgekoppelten PLLs und deren Synchronisation in einer flachen Hierarchie und trägt zum Verständnis, zum Entwurf und zur Optimierung von praktisch realisierten Netzwerken gegenseitig verzögerungsgekoppelter PLLs bei.:Abstract/Zusammenfassung Symbols and Abbreviations Previous Publications 1 Introduction 1.1 Classifications of Synchronization 1.2 A Historical Perspective on Mutual Synchronization 1.3 Extending the Understanding of Mutual Synchronization 1.4 Definitions and Methodologies 2 Model of Networks of Mutually Coupled PLLs 2.1 Coupled Oscillators – Kuramoto Model 2.1.1 Consequences of a Time Delay between Oscillators 2.1.2 Arbitrary Time Delays between Oscillators 2.2 Time-Domain Model of Delay-Coupled PLLs 2.2.1 Phase Detection 2.2.2 Loop-Filter 2.2.3 Voltage Controlled Oscillator 2.3 Prediction and Stability Analysis of Synchronized States 2.3.1 Assessing the Linear Stability of Synchronized States 2.3.2 Stability Consideration for Two Identical PLL Nodes 2.3.3 The Notion of Mode Locking 2.3.4 Effects of Heterogeneity on Synchronized States 2.4 Dynamics and Noise in Synchronized States 2.4.1 Gain and Phase Margin of a PLL Node 2.4.2 Phase Noise 2.5 Key Findings of the Theoretical Model 3 Design of Phase-Locked-Loops for Mutual Synchronization 3.1 PLL Nodes Dedicated for Mutual Synchronization 3.1.1 Phase Detection Circuitry 3.1.2 Adder Chain 3.2 Additional Circuitry for Implementing a Time Delay 4 Experimental Analysis of Mutually Time-Delayed Coupled PLLs 4.1 Synchronized States Including Oscillator Nonlinearity 4.2 Stability of Multistable Synchronized States 4.3 Critical Time Delay Between Two Coupled Nodes 4.4 Combining Hierarchical and Flat Synchronization Concepts 4.4.1 Entrainment of a Chain Network Topology 4.4.2 Entrainment of a Ring Network Topology 4.5 Heterogeneous Time Delays between Coupled PLLs 4.6 Phase Noise Analysis of Time Delay Coupled PLLs 4.6.1 Phase Noise for Two Mutually Coupled Nodes 4.6.2 The Impact of Coupling Directionality 4.6.3 Long Term Frequency Stability 4.6.4 Effect of Time Delay on Phase Noise 4.6.5 Network Topology Dependency on Phase Noise 5 Conclusion and Future Prospects Bibliography Own Publications – Periodicals Own Publications – Conference Proceedings Co-Authored Publications Other References List of Figures List of Tables Acknowledgement Curriculum Vitae

The future fusion power plants that are based on magnetic confinement will deal with plasmas that inevitably contain energetic (non-thermal) particles. These particles come, for instance, from fusion reactions or from external heating of the plasma. Ensembles of energetic ions can excite eigenmodes in the Alfvén frequency range to such an extent that the resulting wave fields redistribute the energetic ions, and potentially eject them from the plasma. The redistribution of ions may cause a substantial reduction of heating efficiency. Understanding the dynamics of such instabilities is necessary to optimise the operation of fusion experiments and of future fusion power plants. Two models have been developed to simulate the interaction between energetic ions and Alfvén eigenmodes. One is a bump-on-tail model, of which two versions have been developed: one fully nonlinear and one quasilinear. The quasilinear version has a lower dimensionality of particle phase space than the nonlinear one. Unlike previous similar studies, the bump-on-tail model contains a decorrelation of the wave-particle phase in order to model stochasticity of the system. When the characteristic time scale for macroscopic phase decorrelation is similar to or shorter than the time scale of nonlinear wave-particle dynamics, the nonlinear and the quasilinear descriptions quantitatively agree. A finite phase decorrelation changes the growth rate and the saturation amplitude of the wave mode in systems with an inverted energy distribution around the wave-particle resonance. Analytical expressions for the correction of the growth rate and the saturation amplitude have been derived, which agree well with numerical simulations. A relatively weak phase decorrelation also diminishes frequency chirping events of the eigenmode. The second model is called FOXTAIL, and it has a wider regime of validity than the bump-on-tail model. FOXTAIL is able to simulate systems with multiple eigenmodes, and it includes effects of different individual particle orbits relative to the wave fields. Simulations with FOXTAIL and the nonlinear bump-on-tail model have been compared in order to determine the regimes of validity of the bump-on-tail model quantitatively. Studies of two-mode scenarios confirmed the expected consequences of a fulfillment of the Chirikov criterion for resonance overlap. The influence of ICRH on the eigenmode-energetic ion system has also been studied, showing qualitatively similar effects as seen by the presence of phase decorrelation. Another model, describing the efficiency of fast wave current drive, has been developed in order to study the influence of passive components close to the antenna, in which currents can be induced by the antenna generated wave field. It was found that the directivity of the launched wave, averaged over model parameters, was lowered by the presence of passive components in general, except for low values of the single pass damping of the wave, where the directivity was slightly increased, but reversed in the toroidal direction. / De framtida fusionskraftverken baserade på magnetisk inneslutning kommer att hantera plasmor som oundvikligen innehåller energetiska (icke-termiska) partiklar. Dessa partiklar kommer exempelvis från fusionsreaktioner eller från externa uppvärmningsmekanismer av plasmat. Ensembler av energetiska joner kan excitera egenmoder i Alfvén-frekvensområdet i en sådan utsträckning att de resulterande vågfälten omfördelar de energetiska jonerna i rummet, och potentiellt slungar ut jonerna ur plasmat. Omfördelningen av joner kan orsaka en väsentligen minskad uppvärmningseffekt. Det är nödvändigt att förstå dynamiken hos denna typ av instabilitet för att kunna optimera verkningsgraden hos experiment och hos framtida fusionskraftverk. Två modeller har utvecklats för att simulera interaktionen mellan energetiska joner och Alfvén-egenmoder. Den första är en bump-on-tail-modell, av vilken två versioner har utvecklats: en fullt icke-linjär och en kvasi-linjär. I den kvasi-linjära versionen har partiklarnas fasrum en lägre dimensionalitet än i den icke-linjära versionen. Till skillnad från tidigare liknande studier innehåller denna bump-on-tail-modell en dekorrelation av våg-partikelfasen för att modellera stokasticitet hos systemet. När den karakteristiska tidsskalan för makroskopisk fasdekorrelation är ungefär samma som eller kortare än tidsskalan för icke-linjär våg-partikeldynamik så stämmer den icke-linjära och den kvasi-linjära beskrivningen överens kvantitativt. En ändlig fasdekorrelation förändrar vågmodens tillväxthastighet och satureringsamplitud i system med en inverterad energifördelning omkring våg-partikelresonansen. Analytiska uttryck för korrektionen av tillväxthastigheten och satureringsamplituden har härletts, vilka stämmer väl överens med numeriska simuleringar. En relativt svag fasdekorrelation försvagar även "frequency chirping events" (snabba frekvensskiftningar i korttids-Fourier-transformen av egenmodens amplitudutveckling) hos egenmoden. Den andra modellen, kallad FOXTAIL, har ett mycket bredare giltighetsområde än bump-on-tail-modellen. FOXTAIL kan simulera system med flera egenmoder, och den inkluderar effekter av olika enskilda partikelbanor relativt vågfälten. Simuleringar med FOXTAIL och med bump-on-tail-modellen har jämförts för att kvantitativt bestämma bump-on-tail-modellens giltighetsområde. Studier av scenarier med två egenmoder bekräftar de förväntade effekterna av när Chirikov-kriteriet för resonansöverlapp uppfylls. Även inflytandet av ICRH på dynamiken mellan egenmoder och energetiska joner har studerats, vilket har visat kvalitativt liknande effekter som har observerats i närvaron av fasdekorrelation. En annan modell, vilken beskriver effektiviteten hos "fast wave current drive" (strömdrivning med snabba magnetosoniska vågor), har utvecklats för att studera inflytandet av passiva komponenter nära antennen, i vilka strömmar kan induceras av vågfälten som genereras av antennen. Det visades att den utskickade vågens direktivitet, medelvärdesbildat över modellparametrar, generellt sett minskade vid närvaron av passiva komponenter, förutom vid låg "sinlge pass damping" (dämpning av vågen vid propagering genom hela plasmat), då direktiviteten istället ökade något, men bytte tecken i toroidal riktning. / <p>QC 20160927</p>

Systeme mit Zeitverzögerung sind von großem Interesse in Nichtlinearer Dynamik und allgegenwärtig in den Naturwissenschaften. Gegenstand dieser Doktorarbeit ist die raumzeitliche Dynamik räumlich-ausgedehnter, nichtlinearer Systeme mit Zeitverzögerung, mit besonderem Augenmerk auf deren lokalisierte Lösungen. Die betrachteten Systeme werden beschrieben durch partielle Differentialgleichungen und gekoppelte Systeme von gewöhnlichen Differentialgleichungen mit verzögerter Rückkopplung. Hinsichtlich der partiellen Differentialgleichungen untersucht diese Arbeit die Existenz und Stabilität der ebenen Wellenlösungen ebenso, wie die Existenz und Stabilität der lokalisierten Lösungen der eindimensionalen, komplexen, kubischen und kubisch-quintischen Ginzburg-Landau Gleichung mit verzögerter, optischer Rückkopplung. Das erste Ergebnis dieser Arbeit ist die vollständige Beschreibung der Menge der ebenen Wellenlösungen und ihre Stabilität für lange Verzögerungszeiten. Aufgrund der Symmetrie der Ginzburg-Landau Gleichung bildet diese Menge eine eindimensionale Familie, die zum Auftreten einer „Tube“ in Parameter-Koordinaten führt. Das zweite, neuartige Ergebnis ist die Beschreibung der Modulationsinstabilität dieser lokalisierten Strukturen. Diese Instabilität kann zu einer periodischen und chaotischen Zickzackbewegung der Lösung führen. Das dritte Resultat ist die Charakterisierung gebundener Impulsfolgen in einem System von gekoppelten gewöhnlichen Differentialgleichungen mit Zeitverzögerung, das zur Beschreibung einer Anordnung von modengekoppelten Lasers herangezogen wird. In diesem Regime interagieren die modengekoppelten Impulse in verschiedenen Lasern lokal über die Balance von Abstoßung und Anziehung. Resultierend daraus entstehen Cluster von Impulsen, die in einzelnen modengekoppelten Lasern nicht möglich sind. Sämtliche genannte Phänomene wurden analytisch und numerisch behandelt. / Systems with time-delay are ubiquitous in nature and attract significant interest in the field of nonlinear dynamics. The scope of this Thesis is the spatiotemporal dynamics in spatially extended nonlinear systems with time-delay, with a focus on the dynamics of localized structures. The systems under consideration are described by partial differential equations with delayed feedback and coupled systems of delay differential equations. For the partial differential equations, the existence and stability of plane wave solutions as well as localized structures are investigated in one-dimensional complex cubic and cubic-quintic Ginzburg-Landau equation with delayed feedback. The first result of this Thesis is the complete description of the set of plane wave solutions and their stability in the limit of large delay time. Due to the symmetry of Ginzburg-Landau equation, this set forms a one-dimensional family which leads to the appearance of the “tube” in parameter coordinates which is filled densely with plane wave solutions with the increase of the delay time. The second novel result is the description of modulational instability of localized structures in spatially extended systems with time-delay which can lead to periodic and chaotic zigzagging movement of the solution. The third result is the description of bound pulse trains in coupled delay systems depicting an array of mode-locked lasers. In this regime mode-locked pulses in different lasers interact locally via the balance of their repulsion and attraction. As a result, clusters of pulses emerge which can not exist in a solitary mode-locked laser. All of the aforementioned phenomena were described analytically and the results are supported by path continuation methods as well as direct numerical simulations with a specially designed software tool.

Dans la première partie de la thèse, les schémas d’intégration conservatifs sont appliqués aux poutres co-rotationnelles 2D. Les cinématiques d'Euler-Bernoulli et de Timoshenko sont abordées. Ces formulations produisent des expressions de l'énergie interne et l'énergie cinétique complexe et fortement non-linéaires. L’idée centrale de l’algorithme consiste à définir, par intégration, le champ des déformations en fin de pas à partir du champ de vitesses de déformations et non à partir du champ des déplacements au travers de la relation déplacement-déformation. La même technique est appliquée aux termes d’inerties. Ensuite, une poutre co-rotationnelle plane avec rotules généralisées élasto-(visco)-plastiques aux extrémités est développée et comparée au modèle fibre avec le même comportement pour des problèmes d'impact. Des exemples numériques montrent que les effets de la vitesse de déformation influencent sensiblement la réponse de la structure. Dans la seconde partie de cette thèse, une théorie de poutre spatiale d’Euler-Bernoulli géométriquement exacte est développée. Le principal défi dans la construction d’une telle théorie réside dans le fait qu’il n’existe aucun moyen naturel de définir un trièdre orthonormé dans la configuration déformée. Une nouvelle méthodologie permettant de définir ce trièdre et par conséquent de développer une théorie de poutre spatiale en incorporant l'hypothèse d'Euler- Bernoulli est fournie. Cette approche utilise le processus d'orthogonalisation de Gram-Schmidt couplé avec un paramètre rotation qui complète la description cinématique et décrit la rotation associée à la torsion. Ce processus permet de surmonter le caractère non-unique de la procédure de Gram-Schmidt. La formulation est étendue au cas dynamique et un schéma intégration temporelle conservant l'énergie est également développé. De nombreux exemples démontrent l’efficacité de cette formulation. / In the first part of the thesis, energymomentum conserving algorithms are designed for planar co-rotational beams. Both Euler-Bernoulli and Timoshenko kinematics are addressed. These formulations provide us with highly complex nonlinear expressions for the internal energy as well as for the kinetic energy which involve second derivatives of the displacement field. The main idea of the algorithm is to circumvent the complexities of the geometric non-linearities by resorting to strain velocities to provide, by means of integration, the expressions for the strain measures themselves. Similarly, the same strategy is applied to the highly nonlinear inertia terms. Next, 2D elasto-(visco)-plastic fiber co-rotational beams element and a planar co-rotational beam with generalized elasto-(visco)-plastic hinges at beam ends have been developed and compared against each other for impact problems. In the second part of this thesis, a geometrically exact 3D Euler-Bernoulli beam theory is developed.The main challenge in defining a three-dimensional Euler-Bernoulli beam theory lies in the fact that there is no natural way of defining a base system at the deformed configuration. A novel methodology to do so leading to the development of a spatial rod formulation which incorporates the Euler-Bernoulli assumption is provided. The approach makes use of Gram-Schmidt orthogonalisation process coupled to a one-parametric rotation to complete the description of the torsional cross sectional rotation and overcomes the non-uniqueness of the Gram-Schmidt procedure. Furthermore, the formulation is extended to the dynamical case and a stable, energy conserving time-stepping algorithm is developed as well. Many examples confirm the power of the formulation and the integration method presented.

Dans ce travail, nous étudions deux types de problèmes à retard. Le premier traite des oscillateurs optoélectroniques (OOEs). Un OOE est un système bouclé permettant de délivrer une onde électromagnétique radio-fréquence de grande pureté spectrale et de faible bruit électronique. Le second problème traite du couplage retardé de neurones. Une nouvelle forme de synchronisation est observée où un régime oscillant est une alternative à un état stationnaire stable. Ces deux problèmes présentent des oscillations de type slow-fast. Une grande partie de ma thèse est dévouée à l’analyse de ces régimes. Etant donné qu’il s’agit d’équations nonlinéaires à retard, les techniques asymptotiques classiques ont dû être revues. En plus d’une étude théorique, des expériences ont été effectuées. Le travail sur les OOEs a été rendu possible grâce aux invitations respectives de L. Larger dans son laboratoire à l’Université de Franche-Comté et de D.J. Gauthier à Duke University. Le travail sur le couplage de neurones a bénéficié d’expériences réalisées par L. Keuninckx du groupe « Applied Physics » de la Vrije Universiteit Brussel.<p>Une contribution importante de cette thèse est à la fois l’analyse mathématique mais aussi l’observation expérimentale d’ondes carrées stables asymétriques présentant des longueurs de plateau différentes mais ayant la même période dans un OOE. Une bifurcation de Hopf primaire d’un état stationnaire est le mécanisme menant à ces régimes. Un deuxième phénomène qui a été à la fois observé pour l’OOE et pour les neurones couplés est la coexistence entre plusieurs ondes carrées ayant des périodes différentes. Pour l’OOE, ces oscillations peuvent être reliées à plusieurs bifurcations de Hopf primaires qui sont proches les unes des autres à cause du grand délai. Le mécanisme de stabilité est similaire à celui de "Eckhaus" pour les systèmes spatialement étendus. Pour le couplage de cellules excitables, nous avons étudié des équations couplées de type FitzHugh-Nagumo (FHN) linéaires par morceaux et obtenu des résultats analytiques. Nous montrons que le mécanisme menant à ces régimes périodiques correspond à un point limite d’un cycle-limite. La robustesse de ces régimes par rapport au bruit a ensuite été explorée expérimentalement en utilisant des circuits électroniques couplés et retardés. Ce système peut être modélisé mathématiquement par les mêmes équations de type FHN. Pour terminer, nous montrons que les équations pour l’OOE et le FHN possèdent des propriétés similaires. Ceci nous permet de généraliser nos principaux résultats à une plus grande variété d’équations différentielles à retard. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Un des buts de cette thèse est d’approfondir la compréhension de la dynamique non-linéaire, notamment celle des MEMS, en proposant de nouvelles méthodes d’analyse paramétrique et de calcul de modes normaux non-linéaires. Dans une première partie, les méthodes de détection, de localisation et de suivi de points de bifurcation selon un unique paramètre sont rappelées. Ensuite, une nouvelle méthode d’analyse multiparamétrique basée sur la continuation récursive d’extremums est présentée. Cette méthode est ensuite appliquée à un absorbeur de vibration non-linéaire afin de repousser l’apparition de solutions isolées. Deuxièmement, une méthode de calcul de modes normaux non-linéaires est présentée. Une condition de phase optimale et une régularisation de l’équation de mouvement sont proposées afin d’obtenir une méthode de continuation plus robuste au niveau des interactions modales. Ensuite, un problème quadratique aux valeurs propres modifié pour le calcul de stabilité et de points de bifurcation est présenté. Finalement, le calcul de modes normaux non-linéaires a été étendu aux systèmes non-conservatifs permettant la continuation des résonances d’énergie en déplacement et des résonances de phase. Troisièmement, la dynamique non-linéaire de réseaux de MEMS basé sur plusieurs micro-poutres résonantes est analysée à l’aide des méthodes proposées. Tout d'abord, un phénomène de synchronisation de points de bifurcations dû au couplage électrostatique dans les réseaux de MEMS est expliqué. Puis, la dynamique non-linéaire d'un réseau dissymétrisé par l'ajout d'une petite masse sur une micro-poutre est analysée. Enfin, des mécanismes de détection de masse exploitant ces phénomènes non-linéaires sont présentés. / One of the goals of this thesis is to enhance the comprehension of nonlinear dynamics, especially MEMS nonlinear dynamics, by proposing new methods for parametric analysis and for nonlinear normal modes computation. In a first part, methods for the detection, the localization and the tracking of bifurcation points with respect to a single parameter are recalled. Then, a new method for parametric analysis, based on recursive continuation of extremum, is presented. This method is then applied to a Nonlinear Tuned Vibration Absorber in order to push isolated solutions at higher amplitude of forcing. Secondly, a method is presented for the computation of nonlinear normal modes. An optimal phase condition and a relaxation of the equation of motion are proposed to obtain a continuation method able to handle modal interactions. Then, a quadratic eigenvalue problem is shifted to compute the stability and bifurcation points. Finally, nonlinear normal modes are extended to non-conservatives systems permitting the continuation of phase and energy resonances. Thirdly, the nonlinear dynamics of MEMS array, based on multiple resonant micro-beams, is analyzed with the help of the proposed methods. A frequency synchronization of bifurcation points due to the electrostatic coupling is discovered. Then, the nonlinear dynamics of a MEMS array after symmetry breaking event induced by the addition of a small mass onto one of the beam of the array is analyzed. Finally, mass detection mechanisms exploiting the discovered phenomena are presented.

The peripheral skin microcirculation reflects the overall health status of the cardiovascular system and can be examined non-invasively by laser methods to assess early cardiovascular disease (CVD) risk factors, i.e. oxidative stress and endothelial dysfunction. Examples of methods used for this task are the laser Doppler flowmetry (LDF) and laser fluorescence spectroscopy (LFS), which respectively allow tracing blood flow and the amounts of the coenzyme NAD(P)H (nicotamide adenine dinucleotide) that is involved in the cellular production of ATP (adenosine triphosphate) energy. In this work, these methods were combined with iontophoresis and PORH (post-occlusive reactive hyperaemia) reactive tests to assess skin microvascular function and oxidative stress in mice and human subjects. The main focus of the research was exploring the nonlinear dynamics of skin LDF and NAD(P)H time series by processing the signals with the wavelet transform analysis. The study of nonlinear fluctuations of the microcirculation and cell energy metabolism allows detecting dynamic oscillators reflecting the activity of microvascular factors (i.e. endothelial cells, smooth muscle cells, sympathetic nerves) and specific patterns of mitochondrial or glycolytic ATP production. Monitoring these dynamic factors is powerful for the prediction of general vascular/metabolic health conditions, and can help the study of the mechanisms at the basis of the rhythmic fluctuations of micro-vessels diameter (vasomotion). In this thesis, the microvascular and metabolic dynamic biomarkers were characterised <i>in-vivo</i> in a mouse model affected by oxidative stress and a human cohort of smokers. Data comparison, respectively, with results from control mice and non-smokers, revealed significant differences suggesting the eligibility of these markers as predictors of risk associated with oxidative stress and smoke. Moreover, a relevant link between microvascular and metabolic oscillators was observed during vasomotion induced by α-adrenergic (in mice) or PORH (in humans) stimulations, suggesting a possible role of cellular Ca²⁺oscillations of metabolic origin as drivers of vasomotion which is a theory poorly explored in literature. As future perspective, further exploration of these promising nonlinear biomarkers is required in the presence of risk factors different from smoke or oxidative stress and during vasomotion induced by stimuli different from PORH or α-adrenergic reactive challenges, to obtain a full picture on the use of these factors as predictors of risk and their role in the regulation of vasomotion.

La surveillance des systèmes industriels et/ou embarqués constitue une préoccupation majeure en raison de l’accroissement de leur complexité et des exigences sur le respect des profilsde mission. La détection d’anomalies tient une place centrale dans ce contexte. Fondamentalement,les procédures de détection à base de modèles consistent à comparer le fonctionnement réel dusystème avec un fonctionnement de référence établi à l’aide d’un modèle sans défaut. Cependant,les systèmes à surveiller présentent souvent des dynamiques non linéaires et difficiles à caractériserde manière exacte. L’approche retenue dans cette thèse consiste à englober leur influencepar des incertitudes bornées. La propagation de ces incertitudes permet l’évaluation de seuils dedécision visant à assurer le meilleur compromis possible entre sensibilité aux défauts et robustesseaux perturbations tout en préservant une complexité algorithmique raisonnable. Pour cela, unepart importante du travail porte sur l’extension des classes de modèles dynamiques à incertitudesbornées pour lesquels des observateurs intervalles peuvent être obtenus avec les preuves d’inclusionet de stabilité associées. En s’appuyant sur des changements de coordonnées variant dans letemps, des dynamiques LTI, LPV et LTV sont considérées graduellement pour déboucher sur desclasses de dynamiques Non Linéaires à Incertitudes Bornées continues (NL-IB). Une transformationdes modèles NL-IB en modèles LPV-IB a été utilisée. Une première étude sur les non-linéaritésd’une dynamique de vol longitudinal est présentée. Un axe de travail complémentaire porte surune caractérisation explicite de la variabilité (comportement aléatoire) du bruit de mesure dansun contexte à erreurs bornées. En combinant cette approche à base de données avec celle à basede modèle utilisant un prédicteur intervalle, une méthode prometteuse permettant la détection dedéfauts relatifs à la position d’une surface de contrôle d’un avion est proposée. Une étude portenotamment sur la détection du blocage et de l’embarquement d’une gouverne de profondeur. / The monitoring of industrial and/or embedded systems is a major concern accordingto their increasing complexity and requirements to respect the mission profiles. Detection of anomaliesplays a key role in this context. Fundamentally, model-based detection procedures consist incomparing the true operation of the system with a reference established using a fault-free model.However, the monitored systems often feature nonlinear dynamics which are difficult to be exactlycharacterized. The approach considered in this thesis is to enclose their influence through boundeduncertainties. The propagation of these uncertainties allows the evaluation of thresholds aimingat ensuring a good trade-off between sensitivity to faults and robustness with respect to disturbanceswhile maintaining a reasonable computational complexity. To that purpose, an importantpart of the work adresses the extension of classes of dynamic models with bounded uncertaintiesso that interval observers can be obtained with the related inclusion and stability proofs. Based ona time-varying change of coordinates, LTI, LPV and LTV dynamics are gradually considered tofinally deal with some classes classes of nonlinear continuous dynamics with bounded uncertainties.A transformation of such nonlinear models into LPV models with bounded uncertainties has beenused. A first study on nonlinearities involved in longitudinal flight dynamics is presented. A complementarywork deals with an explicit characterization of measurement noise variability (randombehavior of noise within measurement) in a bounded error context. Combining this data-drivenapproach with a model-driven one using an interval predictor, a promising method for the detectionof faults related to the position of aircraft control surfaces is proposed. In this context, specialattention has been paid to the detection of runaway and jamming of an elevator.