Le but de cette thèse est d'étudier d'une façon expérimentale et théorique le comportement synchrone d'un groupe d'oscillateurs contrôlés par la lumière (LCOs). Ces LCOs sont très simples du point de vue électronique et ont la propriété d'imiter le comportement des lucioles puisqu'ils interagissent par des impulsions de lumière. En même temps, les LCOs sont une bonne approche pour étudier d'autres systèmes qui agissent comme des oscillateurs d'intégration et de tir car un LCO est un oscillateur de relaxation à deux échelles de temps : un long processus de charge alterné avec un très court processus de décharge. Une série d'expériences a été menée pour pouvoir comprendre le processus de synchronisation des LCOs. Nous avons trouvé que l'acquisition de la synchronisation est due aux effets de la perturbation à savoir: le raccourcissement de la charge et l'allongement de la décharge. Les mesures expérimentales ainsi que la physique liée aux LCOs nous ont permis de formuler un modèle qui a été utilisé pour trouver d'une façon analytique la courbe de réponse de phase (PRC) qui caractérise un LCO. Le modèle a ensuite été validé en comparant les résultats expérimentaux et théoriques. Le modèle reproduit même, le phénomène de bifurcation qui apparaît lorsque trois LCOs sont couplés et disposés en ligne : deux états stables différents apparaissent selon les conditions initiales. L'accord trouvé entre théorie et expérience nous permet d'utiliser le modèle pour étudier d'autres situations qui ne sont pas facilement abordables du point de vue expérimental. Nous avons étudié analytiquement deux LCOs identiques couplés. Même pour ce cas idéal, nous étions obligés de faire des simplifications pour pouvoir trouver des solutions exactes. On a trouvé pour ce système deux états possibles qui dépendent des conditions initiales, la synchronisation (stable) et l'anti-synchronisation (instable). Nous avons également montré que le temps de synchronisation augmente avec la distance entre LCOs. La construction des langues d'Arnold (régions de synchronisation) nous a permis de distinguer des régions de synchronisation pure d'ordre n:m et des régions de superposition synchronisation--modulation. Nous avons travaillé numériquement avec des systèmes de LCOs affectés de bruits uniforme et Gaussien. Le comportement synchrone de ce système a été caractérisé en utilisant des paramètres statistiques simples tels que la moyenne de la différence de phase linéaire et la variance de la différence de phase cyclique. Nous avons démontré que le bruit, bien qu'il puisse perturber la synchronisation, peut aussi la favoriser entre deux LCOs qui ne se synchroniseraient pas en conditions normales, surtout quand le bruit est Gaussien et que les variances du bruit ne sont pas égales. Nous avons étudié en termes statistiques la synchronisation de LCOs couplés localement et arrangés en ligne, en anneau et en réseau. Nous avons montré que la synchronisation totale se produit plus facilement pour des LCOs disposés en anneau. Concernant le temps de synchronisation, il est imprédictible. Les résultats analytiques et numériques suggèrent que la synchronisation totale est le phénomène le plus probable quand le nombre d'oscillateurs n'est pas très grand. Finalement, nous avons étudié des LCOs statiques et mobiles couplés globalement. Dans les deux cas, nous avons trouvé que la synchronisation est moins probable quand le nombre d'oscillateurs augmente. Pour la condition statique, en considérant un couplage du type champ moyen, nous avons observé que le temps de synchronisation diminue avec le nombre de LCOs. Cependant, pour la situation plus réaliste dans laquelle l'interaction entre LCOs dépend de la distance les séparant, le temps de synchronisation devient à nouveau imprédictible. Enfin, nous avons étudié l'influence de la mobilité sur la synchronisation, problème qui est important en biologie et en robotique. Notre système, de par ses caractéristiques et sa base expérimentale, est beaucoup plus proche de la réalité que ceux considérés d'habitude dans la littérature. Les résultats obtenus peuvent s'appliquer à des systèmes biologiques (lucioles, cellules cardiaques, neurones, …), mais également à la robotique, où la communication à longue portée par la lumière et l'émergence de patterns de synchronisation pourraient être très utiles dans le but d'effectuer des tâches spécifiques.

In the first part of this dissertation we explore the consequences of high frequency operation of Josephson junction arrays. At high frequencies these systems are no longer well modeled by Kirchhoffs laws, and new dynamical equations are derived directly from Maxwells equations. From these equations we derive a reduced set of averaged equations which greatly simplify the analysis of high frequency arrays. The averaged equations allow us to examine experimental strategies for obtaining higher power outputs from arrays. These strategies rely on resonant architectures that place the junctions near antinodes of a desired standing wave mode of the fluctuating current. Simple, heuristic rules are derived for the proper placement of junctions. The second part of the dissertation is devoted to stochastic resonance. A new theory is proposed to explain both two-state and excitable stochastic resonance. Previous theories explaining the two types of stochastic resonance yield similar results while using different analytic strategies. A constrained asymmetric rate model is derived that in one limit produces the proper result for the two-state system, while in another limit models the excitable system. The result that the constrained asymmetric rate model gives in the excitable limit is off by a factor of two, and this discrepancy is examined. Furthermore, we study the consequences of adding a colored noise source to the classic two-state model of stochastic resonance. We will find that when both white and colored noise sources are present, stochastic resonance will occur as a function of colored noise strength only if the correlation time of the colored noise source is small enough. Two theories are proposed to explain this phenomenon and both are examined in detail.

In this thesis, convergence of time averages near statistical attractors of continuous flows are investigated. A relation between statistical attractor and essential Ω-limit set is proved, and using this a general definition for statistical attractor is given. Sufficient conditions are given for an observable to admit a convergent time average along the orbits of the flow. The general results are applied to flows on a torus, and in particular to systems of coupled phase oscillators that admit attracting heteroclinic networks in their phase space. A particular heteroclinic network that we call heteroclinic ratchet is observed and analysed in detail. Heteroclinic ratchets give rise to a novel phenomenon, unidirectional desynchronization of oscillators (ratcheting). The results obtained about the convergence of time averages near statistical attractors implies that heteroclinic ratchets induce, besides its other interesting consequences, frequency synchronization without phase synchronization. Different coupling structures that can give rise to ratcheting of oscillators are also investigated.

This dissertation is dedicated to the rigorous study of discontinuous transitions in star graphs of coupled phase oscillators. A star graph consists of a central node, called hub, connected to peripheral nodes called leaves. We consider the setting where the frequency of the leaves is identical and the hub has a higher frequency when isolated. This captures the effect of positive correlation between the hub high number of connections and its high natural frequency. Hub higher frequency turns out to be the key feature for discontinuity in the transition from incoherent to synchronous behavior. This transition has been observed numerically and explained via a non-rigorous analytical treatment in the thermodynamic limit. Using Möbius group reduction and the theory of persistence of normally hyperbolic invariant manifold, we prove that this transition is indeed discontinuous for a certain set of initial conditions. / Esta dissertação dedica-se em estudar rigorosamente transições descontínuas de osciladores de fase acoplados em grafos estrelas. Um grafo estrela é composto de um nó central, chamado hub, conectado a nós periféricos chamados folhas. Consideramos a situação na qual a frequência das folhas é igual e o hub tem frequência mais elevada, o efeito de correlação positiva entre o grande número de conexões do hub e sua frequência. A elevada frequência do hub resulta por ser o aspecto crucial na descontinuidade da transição do comportamento incoerente para o síncrono. Esta transição foi observada numericamente e estudada por meio de tratamento analítico não rigoroso no limite termodinâmico. Usando técnica de redução a partir do grupo de Möbius e a teoria de variedades invariantes normalmente hiperbólicias, provamos que esta transição é de fato descontínua para certo conjunto de condições iniciais.

This dissertation is dedicated to the rigorous study of discontinuous transitions in star graphs of coupled phase oscillators. A star graph consists of a central node, called hub, connected to peripheral nodes called leaves. We consider the setting where the frequency of the leaves is identical and the hub has a higher frequency when isolated. This captures the effect of positive correlation between the hub high number of connections and its high natural frequency. Hub higher frequency turns out to be the key feature for discontinuity in the transition from incoherent to synchronous behavior. This transition has been observed numerically and explained via a non-rigorous analytical treatment in the thermodynamic limit. Using Möbius group reduction and the theory of persistence of normally hyperbolic invariant manifold, we prove that this transition is indeed discontinuous for a certain set of initial conditions. / Esta dissertação dedica-se em estudar rigorosamente transições descontínuas de osciladores de fase acoplados em grafos estrelas. Um grafo estrela é composto de um nó central, chamado hub, conectado a nós periféricos chamados folhas. Consideramos a situação na qual a frequência das folhas é igual e o hub tem frequência mais elevada, o efeito de correlação positiva entre o grande número de conexões do hub e sua frequência. A elevada frequência do hub resulta por ser o aspecto crucial na descontinuidade da transição do comportamento incoerente para o síncrono. Esta transição foi observada numericamente e estudada por meio de tratamento analítico não rigoroso no limite termodinâmico. Usando técnica de redução a partir do grupo de Möbius e a teoria de variedades invariantes normalmente hiperbólicias, provamos que esta transição é de fato descontínua para certo conjunto de condições iniciais.

This dissertation analyzes the fundamental limits for the determination of the network structure of loosely coupled oscillators based on observing the behavior of the network, specifically, node synchronization. The determination of the requisite characteristics and underlying behaviors necessary for the application of a theoretical mechanism for determining the underlying network topology in a network of loosely coupled natural oscillators are the desired outcome. To that end, this effort defines an analytical framework where key components of networks of coupled oscillators are isolated in order to determine the relationships between the various components. The relationship between the number of nodes in a network, the number of connections in the network, the number of connections of a given node, the distribution of the phases of the network, and the resolution of measurement of the components of the network, and system noise is investigated.

Spontaneous order happens in non-equilibrium systems composed of interacting elements. This phenomenon manifests in both the formation of space-time patterns in reaction-diffusion systems and collective rhythmic behaviors in coupled oscillators. In this thesis, we present the results of two studies: 1) The response of a multi-species predator-prey system to perturbation. 2) The features of a rich attractor space in a system of repulsively coupled Kuramoto oscillators. In the first part, we address this question: how does a complex coarsening system with non-trivial in-domain dynamics respond to perturbations? We choose a cyclic predator-prey model with six species each attacking three others. As a result of this interaction network, two competing domains form, while inside each domain three species play a rock-paper-scissors game which results in the formation of spirals inside the domains. We perturb the system by changing the interaction scheme which leads to a change of alliances and therefore a different spatial pattern. As expected, perturbing a complex space-time pattern results in a complex response. In the second part, we explore the attractor space of a system of repulsively coupled oscillators with non-homogeneous natural frequencies on a hexagonal lattice. Due to the negative coupling and the particular choice of geometry, some of the links between oscillators become frustrated. Coupled oscillators with frustration show similar features as frustrated magnetic systems. We use the parameters of the system like the coupling constant and the width of the frequency distribution to understand the system's attractor space. Further, we study the effects of external noise on the system. We also identify the breaking of time-translation invariance in the absence of external noise, in our system. / Doctor of Philosophy / Spontaneous ordering is a ubiquitous phenomenon observed in natural systems containing many interacting elements. In some systems the order is observed in the form of spatial patterns. It also can be seen in a population of coupled oscillators in the form of collective rhythmic behaviors. In this thesis, we present the results of two studies. For the first study, we choose a cyclic predator-prey system that shows a non-trivial space-time pattern. The system consists of six species each attacking three others, cyclically. By choosing such an interaction network, two competing domains form, while inside each domain three species play a rock-paper-scissors game. As a result of the inner competition, spirals form inside the domains. We study the response of the system to a perturbation. To perturb the system, we change the interaction scheme which leads to a change of alliances and therefore, a different spatial pattern. In the second study, we explore the patterns of clustering and synchronization in a system of repulsively coupled oscillators with non-homogeneous natural frequencies. Due to the negative coupling and the particular choice of geometry, some of the links between oscillators become frustrated. We use the parameters of the system such as the coupling constant and the width of the frequency distribution to understand the system’s attractor space. Further, we examine the effect of external noise on the system.

In this thesis, we present a dynamical systems analysis of models of movement coordination, namely the Haken-Kelso-Bunz (HKB) model and the Jirsa-Kelso excitator (JKE). The dynamical properties of the models that can describe various phenomena in discrete and rhythmic movements have been explored in the models' parameter space. The dynamics of amplitude-phase approximation of the single HKB oscillator has been investigated. Furthermore, an approximated version of the scaled JKE system has been proposed and analysed. The canard phenomena in the JKE system has been analysed. A combination of slow-fast analysis, projection onto the Poincare sphere and blow-up method has been suggested to explain the dynamical mechanisms organising the canard cycles in JKE system, which have been shown to have different properties comparing to the classical canards known for the equivalent FitzHugh-Nagumo (FHN) model. Different approaches to de fining the maximal canard periodic solution have been presented and compared. The model of two HKB oscillators coupled by a neurologically motivated function, involving the effect of time-delay and weighted self- and mutual-feedback, has been analysed. The periodic regimes of the model have been shown to capture well the frequency-induced drop of oscillation amplitude and loss of anti-phase stability that have been experimentally observed in many rhythmic movements and by which the development of the HKB model has been inspired. The model has also been demonstrated to support a dynamic regime of stationary bistability with the absence of periodic regimes that can be used to describe discrete movement behaviours.

Le thème central de la thèse est l'interférométrie atomique et la réduction du bruit de phase lié aux vibrations de l'environnement. Les interféromètres atomiques sont des instruments pouvant permettre des mesures fondamentales de grande précision et cette précision est fortement liée à la vitesse des atomes. L'interféromètre qui était utilisé à Toulouse travaillait alors avec des atomes de lithium aux énergies thermiques et le projet était de pouvoir utiliser des atomes fortement ralentis. Les vibrations du sol (bruit sismique) deviennent alors un inconvénient majeur et il est indispensable de s'en affranchir le plus possible. La première partie du travail fut de prévoir une isolation vis à vis du bruit sismique. Un premier filtre est réalisé en plaçant l'ensemble de la manipulation sur un pendule suspendu par 3 fils. Celui ci atténue les vibrations de fréquence supérieure à sa fréquence propre mais amplifie celles de fréquences voisines de celle-ci. Il faut donc un pendule asservi et cela implique un sismomètre sensible pouvant fonctionner sous ultravide. Nous avons développé une stabilisation des mouvements horizontaux en nous appuyant tout d'abord sur des sismomètres simples mais peu sensibles, puis nous avons cherché à améliorer les performances en réalisant un capteur de déplacement basé sur un interféromètre de Michelson à coins de cube. Nous avons suivi en cela les travaux de l'équipe de M.Zumberge qui utilise une détection de deux signaux en quadrature ce qui permet une mesure de déplacement avec une sensibilité meilleure que 4.10-13 m/vHz à 1 Hz et entrepris d'adapter cette technologie à un fonctionnement sous ultravide. Mais les difficultés rencontrées et l'abandon de l'interféromètre tel qu'il était pour en developper un nouveau ne nous ont pas permis d'atteindre complètement notre but et de pouvoir tester le sismomètre in-situ.Cependant, la mise au point et l'optimisation du sismomètre nous a amené à nous pencher sur la théorie des pendules à lame élastique, lesquels sont largement utilisés dans ce genre de capteurs. Il nous est apparu que cette théorie était très incomplète et nous avons entrepris une étude plus systématique de tels pendules ce qui a donné lieu à une publication et fait l'objet de la seconde partie de la thèse. [...] / The main theme of my thesis is atomic interferometry and in particular the reduction of the phase noise induced by vibrations. Atomic interferometers are good devices to achieve accurate and fundamental measuring. The sensibility of these devices is related to the flying time of the atoms inside the apparatus. At Toulouse, our interferometer worked with atoms at thermal velocity and to increase the sensibility we wanted slower atoms. However, this will at the same time increase the effect of vibrations, witch result in a larger phase noise and a jamming of the fringes. In order to reduce this effect, I've put the core of the interferometer on a 3 wires pendulum. A pendulum attenuates the vibrations of frequencies much higher than its resonant frequencies but amplifies those with frequencies close to its resonances. To avoid this phenomenom, we have to enslave the pendulum on the signal given by seismometers. With a first realisation, I was able to stabilize 2 horizontal movements with 2 low sensibility seismometers. To increase the performances, I needed high sensibility seismometer and the possibility to operate under ultra vacuum. I've made a deplacement sensor based on the Michelson interferometer with cube corners, following the works of Zumberge's team. By choosing cleverly the polarisation of the laser beam, we can detect 2 signals in quadrature and the sensibility achieved is better than 4x10-13 m/vHz at 1 Hz. The next step was to migrate this seismometer in ultra vacuum but the retirement of the interferometer using slow down lithium atoms at the benefit of an atomic fountain of rubidium stopped this project. However, this work on the seismometer led me to think about elastic blade pendulums, widely used in such sensors. I've complete the theory, showing the presence of 2 resonance frequencies and, as a test, I've build a such pendulum, for witch I've measured the caracteristics with some position and velocity sensors I've developped for this purpose. I was able to measure precisely the damping of the oscillations of the pendulum and study more precisely the different origins of the damping. Two of them have given some additionnal work : a)the coupling with the resonances of the frame witch support the pendulum may have an effect on the quality factor of the pendulum. [...]

<p>Mechanical oscillators form the primary structure of a wide variety of devices including energy harvesters and vibration absorbers, and also have parallel systems in electrical fields for signal processing. In the area of wave propagation, recent study in periodic chains have focused on active tuning methods to control bandgap regions, bands in the frequency response in which no propagating wave modes exist. In energy harvesting, several coupled systems have been proposed to enhance the peak power or bandwidth of a single harvester through arrays or dynamic magnification. Though there are applications in several fields, the work in this dissertation can all fit into the category of coupled non-linear oscillators. In each sub-field, this study demonstrates means to advance state of the art techniques by adding nonlinearity to a coupled system of linear oscillators, or by adding a coupled device to a nonlinear oscillator.</p><p>The first part of this dissertation develops the analytical methods for studying wave propagation in nonlinear systems. A framework for studying rotational systems is presented and used to design an testbed for wave propagation experiments using a chain of axially aligned pendulums. Standard analytical methods are also adapted to allow uncertainty analysis techniques to provide insight into the relative impact of variations in design parameters. Most analytical insight in these systems is derived from a linearlized model and assumes low amplitude oscillations. Additional study on the nonlinear system is performed to analyze the types of deviations from this behavior that would be expected as amplitudes increase and nonlinear effects become more prominent.</p><p>The second part of this dissertation describes and demonstrates the first means of passive control of bandgap regions in a periodic structure. By imposing an asymmetrical bistability to an oscillator in each unit cell, it is analytically shown that each potential well has different wave propagation behaviors. Experimental demonstrations are also provided to confirm the simulated results.</p><p>The final section performs analytical and numerical analysis of a new system design to improve the performance of a nonlinear energy harvester by adding an excited dynamic magnifier. It is shown that this addition results in higher peak power and wider bandwidth than the uncoupled harvester. Unlike standard dynamic magnifiers, this performance does not come at the expense of power efficiency, and unlike harvester arrays, does not require the added cost of multiple energy harvesters.</p> / Dissertation