Synchronization of chaos in coupled systems of ordinary differential equations is an area of mathematics which has attracted much attention in recent years, in particular for the potential technological applications such systems have in engineering and industry. The motivation for this research was to understand mathematically, synchronization observed in systems of two and three solid state lasers studied by collaborators at the Georgia Institute of Technology. The main objectives of this thesis are to understand more clearly some of the dynamical phenomena associated with the synchronization of chaos, and to develop new techniques for the analysis of dynamical systems with symmetry; with a view to applying these techniques to models of solid state laser systems and other applications. First we introduce the main ideas of chaotic synchronization and some useful tools for the analysis of dynamical systems with symmetry. We then introduce a model for a solid state Nd:YAG laser and examine the types of dynamics which may be exhibited. Subsequently we look at systems of two and three coupled solid state lasers and examine the onset of synchronization in such systems, both in a fully symmetric system and in the case of two coupled lasers, the case of broken symmetry. We then contrast these results with those of a modified Rossler system and observe similar results in both cases. We examine how chaotic systems may be used for communication purposes and develop a new scheme for the communication of a signal using the synchronization of chaos. Finally we introduce a new definition of attractor and using topological and measure theoretic properties of sets, we reexamine the concepts of basin riddling and are able in certain situations to determine the presence or otherwise of riddling.

L’actuelle méthode de clarification du vin de Champagne à l’issue de sa seconde fermentation est basée sur une manipulation mécanique des bouteilles, principalement au travers de machines de remuage telles que des Gyropalettes. Le principe consiste à imprimer à la bouteille un ensemble ordonné de rotations et d’inclinaisons pour permettre le regroupement et le transport d’un dépôt (issu des étapes précédentes de vinifications), initialement réparti sur une surface importante de la bouteille, en un agrégat concentré dans la bague de cette dernière. Cette manipulation a toujours été pratiquée de façon empirique, basée sur des résultats d’observations in situ. En conséquence, les élaborateurs exécutent cette opération de remuage à partir de ce constat, sans avoir la possibilité d’optimiser le processus de façon plus efficace. Or, cette opération est consommatrice d’espace, de moyens et surtout de temps et ce, dans un contexte d’augmentation des volumes traités. C’est pourquoi, à travers cette thèse, nous présenterons une nouvelle méthode de travail basée sur l’utilisation de vibrations. Pour ce faire et afin de juger l’influence des paramètres et l’efficacité de la solution technique choisie, un ensemble d’outils ont été créés et éprouvés pendant les essais. Ils s’appuient sur l’observation et la quantification des évolutions des phénomènes dans la bouteille. Cette voie est encourageante et permet de réduire la durée du processus en conservant une qualité identique à celle du remuage traditionnel. / The current method of Champagne wine clarification, at the end of its second fermentation, is based on a mechanical manipulation of bottles; and it is essentially executed by machines such as Gyropalettes (« remuage » or riddling process). The process consists on handling the bottle by an orderly set of rotations and tilts to group and move a sediment (outcome of the previous step of winemaking), initially present over an important surface inside the bottle, into a concentrated aggregate in the bottle’s neck. The manipulation has been elaborated empirically; it was based on observations results. Therefore the elaborators execute the riddling operation empirically without having the possibility of optimizing this process in a more efficient way. Also, and because of the increased handled volume nowadays, this operation needs space, time and working tools. That’s why, through this thesis, we shall present a new working method based on imposed vibrations. My study aims to present a new working method based on imposed vibrations. In order to study the parameters influencing the process and the efficiency of the chosen technical solution, a set of tools based on observation and quantification of the phenomena’s evolution inside the bottle, were created and tested. As a conclusion, on the one hand this method reduces the time needed, and on the other hand it leads to the same result in terms of quality.