Abstract
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.