The main contribution of this thesis can be formulated in terms of a synchronization problem describing interactions between a system of harmonic oscillators (HO) and a given dynamical system, called HO synchronization. This thesis investigates the applications of HO synchronization in the area of controlling chaos. It proposes an innovative feedback stabilization technique, called HO feedback control, that utilizes an output of the synchronized system to stabilize periodic orbits of dynamical systems. A particularly attractive application of this stabilization technique is found in controlling chaotic systems, where the aim is to stabilize unstable periodic orbits embedded in the chaotic attractors. This thesis utilizes the same concept of synchronization to develop a novel method of detecting unstable periodic motions in chaotic time series, called HO time series analysis. To do so, the proposed method does not require any information about the underlying dynamics beyond a recorded time history. Therefore, it is appealing in experimental situations. The information obtained from the HO time series analysis can be used in various methods of controlling chaos, including the HO feedback control. In a different, but related application, this thesis proposes a novel transformation of a time delay system to a system of ordinary differential equations featuring the same concept of synchronization, called HO transformation. This transformation yields an efficient finite dimensional approximation to the original time delay system. It is practically important as it allows implementation of classical theories and conventional tools, developed for finite dimensional systems, to analyze time delay systems. This thesis utilizes the HO transformation to reveal the relation between the delayed feedback control and the HO feedback control. / February 2017

Un microsystème électromécanique (MEMS – Micro Electro-Mechanical Systems) est une structure qui intègre un ou plusieurs éléments mécaniques réalisant la fonction de capteur ou d’actionneur à l’échelle du micron. Jusqu’au milieu des années 1990, les MEMS étaient réservés à des applications spécifiques, notamment dans les domaines spatiales et aéronautiques. Depuis une décennie, ces microsystèmes connaissent un essor important dans de nombreux secteurs grand public comme l’automobile, l’informatique ou encore les télécommunications. La conception et l’évaluation des performances de ces microsystèmes nécessitent des outils de modélisation robustes et fiables. Ainsi, les présents travaux de recherche ont pour objectif la modélisation, déterministe et stochastique, de MEMS à actionnement électrostatique pour le calcul spécifique du « phénomène de collage électrostatique » (pull-in phenomenon).Les différentes approches de modélisation associant les deux physiques mises en jeu, à savoir les comportements électrostatique et élastique, sont investiguées. La présentation des résultats, sur deux cas types de poutres issues de MEMS électrostatiques, suit une approche didactique, allant du modèle le moins précis (analytique) vers le modèle le plus fin (éléments finis). Elle permet ainsi de mettre en évidence les effets prédominants, notamment la déformation des parties mobiles et les effets de bord. Enfin, afin de tenir compte des incertitudes sur la géométrie et les matériaux, une étude stochastique, à l’aide d’approches non intrusives de types Monte Carlo et décomposition en chaos polynomial, est également effectuée pour le calcul de la tension de collage d’un dispositif MEMS. / MEMS are micro-structures integrating one or several mechanical elements performing as transducers or actuators. Until the mid 90s, MEMS technology was exclusively dedicated to aerospace and aeronautical applications. In the last decade, these microsystems have known a real boom, spreading in specific consumer applications like automotive, IT and telecommunications. The design and performance analysis of MEMS requires reliable and robust simulation tools. Thereby, the presented works, aim at the deterministic and stochastic modelling of electrostatic MEMS, focusing on the well known pull-in phenomenon.The different modelling approaches describing the electrostatic, elastic and the coupled electromechanical behaviours are investigated. The results are illustrated on two beam-like typical structures, following a didactic approach, from the simplest model (analytical) towards the most precise (Finite Element), emphasizing the predominant effects, like the deformation of the active parts and fringing fields.Furthermore, in order to take into account the uncertainties on the geometry or the material properties, a stochastic analysis is also done, using non intrusive approaches, as Monte-Carlo method and the polynomial chaos decomposition approach, for the determination of the pull-in voltage of a MEMS device.

Pour certaines classes de matériaux de structure, la taille du Volume Elémentaire Représentatif peut être très supérieure à celle du domaine usuellement considéré pour une caractérisation expérimentale. Le tenseur d'èlasticité du milieu présente alors des fluctuations spatiales et statistiques qu'il convient de modéliser par un champ aléatoire. Le travail de thèse a consisté en la construction, l'identification expérimentale et la mise en œuvre d'un modèle probabiliste du champ aléatoire du tenseur d'élasticité à l'échelle mésoscopique. Pour ce faire, deux approches sont privilégiées La première est basée sur la construction d'un modèle probabiliste associé à la fraction volumique aléatoire mésoscopique, combiné à un schéma d'homogénéisation. Une analyse expérimentale par ultrasons est réalisée sur un matériau modèle et permet, à l'aide de la résolution numérique d'un problème inverse, d'obtenir les trajectoires du champ. L'identification des paramètres du modèle est ensuite effectuée en s'appuyant sur le Principe du Maximum de Vraisemblance. La seconde approche porte sur l'identification et la mise en œuvre d'un modéle probabiliste direct du champ aléatoire du tenseur d'élasticité, proposé dans la littérature. Les paramètres du modèle sont déterminés grâce à la caractérisation ultrasonore, via la résolution d'un problème d'optimisation. Les deux approches fournissent des estimations semblables pour les longueurs de corrélation spatiale du champ aléatoire et valident le choix de l'échelle d'analyse mésoscopique. Enfin, une analyse de convergence probabiliste permet de discuter de la taille du VER en fonction des longueurs de corrélation spatiale du champ mésoscopique. / For sorne classes of materials, the size of the Representative Volume Element can be much larger than the one of the domain typically used in experimental testing. The elasticity tensor of such media then exhibits both spatial and statistical fluctuations and has to be modelled as a random field. This thesis is dedicated to the construction, experimental identification and use of a probabilistic model of the random elasticity tensor at mesoscale. For this purpose, two kinds of approaches are considered. The first one is based on the construction of a probabilistic model for the mesoscopic volume fraction, combined to a homogenization scheme. An ultrasound experimental analysis is performed on a model material and allows the experimental trajectories of the random field to be identified by solving an inverse problem. The identification of the parameters is carried out by using the Maximum Likelihood Principle. The second approach is focused on the identification and use of a probabilistic model for the elasticity tensor random field that was recently proposed in the literature. The parameters of the model are computed by combining the ultrasound results with an optimization problem. Both approaches yield similar predictions of the spatial correlation lengths of the mesoscopic random field and validate the choice of the scale for the mesoscopic analysis. Finally, a probabilistic convergence analysis is performed and allows one to discuss the size of the RVE in terms of the correlation lengths of the mesoscopic random field.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, South Africa, in fulfilment of the requirements for the Degree of Doctor of Philosophy. / Quantum chaos is associated with the phenomenon of avoided level crossings on a large scale which leads to a statistical behaviour similar to that of a Gaussian Orthogonal Ensemble (GOE) of matrices. (Abbreviation abstract) / Andrew Chakane 2019

Nonlinear dynamical properties of semiconductor lasers have attracted considerable attention, and their rich behaviors enable many popular research topics. The research effort of this thesis has emphasized on two areas - one is photonic microwave generation based on period one dynamic of semiconductor lasers; the other is laser's chaotic dynamic. Microwave photonics has attracted considerable attention recently because of its practical applications in radio-over-fiber (RoF) communications links. A stable photonic microwave allows it to convey, in a cost-effective manner, wideband signals over optical fibers with low loss, large bandwidth and immunity of electromagnetic interference. Microwave photonics technologies consist of photonic microwave generation, processing, control and distribution. Many photonic microwave generation techniques have been proposed, which includes direct modulation, optical heterodyne technique, external modulation, mode-locked semiconductor lasers, optoelectronic oscillator (OEO) and period one (P1) dynamic of semiconductor lasers. Among these techniques, photonic microwave generation based on P1 oscillation dynamic has gained special attention due to its many advantages, such as: widely tunable oscillation frequency, and nearly single sideband (SSB) spectrum. The aim of this thesis in the photonic microwave generation area is to produce photonic microwaves based on P1 dynamic using low-cost vertical-cavity surface-emitting lasers (VCSELs). The technical contents in this area cover two parts. The first part is to generate broadly tunable photonic microwaves. Continuous tuning of the microwave frequency from 4GHz to up to an instrumentation limited 15GHz is experimental achieved through the adjustment of the injection power and the frequency detuning between the master laser and the VCSEL. Numerical simulations using a common spin flip model are also carried out, which agree qualitatively with the experimental results. The second part of the photonic microwave generation in this thesis is to explore effective approaches to not only reduce the linewidth but also improve the stability of the generated microwave. Due to spontaneous emission noise in the semiconductor laser, P1 dynamic inherently imposes phase noise, which increases the microwave linewidth of the generated microwave. This considerably affect the signal transmission performance of the modulated microwave signal in RoF applications. To address this challenge, single optical feedback and double optical feedback are applied in the experiments. The experimental results demonstrate that both single feedback and double feedback can reduce the linewidth of the generated microwave to about one tenth of linewidth without the optical feedback. However, single optical feedback may induce many side peaks due to external cavity frequency from the feedback cavity, the feedback phase needs to be carefully adjusted to suppress the side peaks. The side peaks can be suppressed by introducing the second optical feedback. The double optical feedback can also significantly enhance the stability of the generated microwave. The results of the numerical simulations are in good agreement with the experimental results. The other important dynamic of semiconductor lasers is chaos, which has attracted considerable research interest due to its many potential applications in secure communications, chaotic optical time-domain reflectors, chaotic lidars and physical random number generators. Optical feedback is the simplest method to generate chaos in semiconductor lasers, but a typical chaos generated by optical feedback has unwanted recurrence features termed time delay (TD) signature because of the optical round trip in the external cavity. The complexity, bandwidth and TD signature of chaos are the three main parameters for evaluating its applicability in abovementioned application scenarios. In order to find the correct operating parameters to achieve low TD signature and high complexity of chaos simultaneously, in this thesis, the influence of bias current and the feedback strength on the complexity and time-delay signature of chaotic signals in semiconductor lasers with optical feedback is investigated experimentally and theoretically. In the experiment, the effect of the data acquisition method on quantification of complexity is also examined. The experimental results show that the TD signature is approximately in an inverse relationship with the complexity of chaos when the semiconductor laser is subject to low or strong optical feedback. However, the inverse relationship disappears when the laser operates at higher bias currents with intermediate feedback strength. Numerical simulation based on Lang Kobayashi laser equations show qualitative agreements with the experimental results.

La quantification d’incertitudes est une démarche consistant à prendre en compte les incertitudes des coefficients caractéristiques (matériaux, géométries, sources ...) d’un modèle mathématique en vue d’estimer l’effet de ces méconnaissances sur les grandeurs physiques recherchées. Dans ce travail de thèse, nous nous sommes intéressés aux approches probabilistes de propagation d’incertitudes portées par les lois de comportement (perméabilités et conductivités) aux sein de modèles éléments finis de l’électromagnétisme quasi-statique de taille industrielle. Cette thèse vise à comparer les deux approches spectrales NISP et SSFEM qui sont basées sur une représentation fonctionnelle dans le chaos polynomial des grandeurs d’intérêt aléatoires. Cette étude de comparaison est effectuée en terme de précision numérique et de coût de calcul, et pour des grandeurs d’intérêt scalaires et vectorielles complexes. Les applications numériques nous ont montré que la SSFEM peut être assez compétitive par rapport à la NISP pour des problèmes probabilistes à grandes dimensions stochastiques. Il en résulte que celle-ci est la méthode de prédilection pour l’étude des systèmes électromagnétiques dont les lois de comportement des matériaux sont aléatoires. Enfin, les deux méthodes spectrales ont été appliquées sur un problème de détection de bouchage par la magnétite des plaques entretoises des générateurs de vapeur d’une centrale nucléaire. Dans cette étude probabiliste, nous nous sommes attelés à quantifier la contribution des incertitudes, subsistant dans les conductivités et perméabilités de la magnétite et de la plaque, à la variabilité des signaux et du ratio SAX. / The uncertainty quantification technique aims to quantify the effect of uncertainties of input parameters of numerical models, e.g. material, geometry, source terms, on the quantity of interest. In this thesis, we focus on probabilistic approaches in order to spread uncertainties of magnetic and electric behavior laws over large scale electromagnetic finite element models. The main objective of this work is to compare two spectral stochastic methods (Non Intrusive Spectral Projection (NISP) and Spectral Stochastic Finite Element Method (SSFEM)), which are based on chaos polynomial representation of the random quantities. The comparison between the NISP and the SSFEM is carried out by confronting the computational costs and the precision when scalar and vector complex quantities of interest are computed. The numerical applications show that the SSFEM method become as competitive as the NISP method in terms of computational cost when solving probabilistic problems with large number of random parameters. Thus, the SSFEM method is chosen as the best adapted to solve electromagnetic problems when the behavior laws are random. In fact, the NISP method is inappropriate to compute vector complex quantities when equipped with adaptive sparse grid procedures. Finally, the NISP and SSFEM methods are used to study the clogging of the Tube Support Plate (TSP) of steam generators of nuclear power plants. The effect of uncertainties of the permeability and the conductivity of the TSP and the magnetite (clogging product) on the control signal and the SAX ratio is investigated.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references (leaves 179-187). Available also in a digital version from Dissertation Abstracts.

Abstract In this paper, several three dimensional (3-D) four-wing smooth quadratic autonomous chaotic systems are analyzed. It is shown that these systems have a number of similar features. A new 3-D continuous autonomous system is proposed based on these features. The new system can generate a four-wing chaotic attractor with less terms in the system equations. Several basic properties of the new system is analyzed by means of Lyapunov exponents, bifurcation diagrams and Poincare maps. Phase diagrams show that the equilibria are related to the existence of multiple wings.

The finest resolution that can be achieved in any real chaotic system is limited by the presence of noise. This noise can be used to define neighborhoods of the deterministic periodic orbits using the local eigenfunctions of the Fokker-Planck operator and its adjoint. We extend the work of D. Lippolis to include hyperbolic periodic orbits. The dynamics along the stable and unstable directions are separated. The neighborhoods on the stable and unstable manifolds can be defined in the same way as the neighborhoods for entirely stable or entirely unstable orbits. The neighborhoods are then returned to the original coordinates. The Fokker-Planck evolution can be described as a finite Markov transition graph between these neighborhoods. Its spectral determinant is used to calculate the time averages of observables. We apply this technique to calculate long time observables of the Lozi map.

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 114-121). Available also in a digital version from Dissertation Abstracts.