NETWORKS OF DELAY-COUPLED OSCILLATORS How delay and topology affect synchronisation and dynamical properties

Delayed dynamical systems arise in various scientific fields; delays occur in the reaction of a driver in traffic dynamics, the incubation time of a disease, in axonal transmission in neural networks, or in the propagation time of light between two coupled semiconductor lasers, to name only a few examples. Such time delays can have dramatic influences which, as we show here, can be common to different systems despite their very different nature. The field of delayed complex systems is nowadays a quickly emerging field in multidisciplinary research. In this thesis we study different aspects of the dynamics of delay-coupled networks, focusing on the interplay between delayed coupling and the onset of spatio-temporal patterns in networks. We approach this problem mainly analytically, complemented by numerical simulations. Starting point were the synchronisation properties of delay-coupled semiconductor lasers, but this thesis has addressed much more general questions: which networks synchronise? Why do some networks synchronise while others do not? Which is the role of a coupling delay, the role of the network topology? Which properties are specific for semiconductor lasers? To address these questions, we have studied a hierarchy of models with different levels of generality and complexity. In networks of Kuramoto oscillators, the simplest model, in which an oscillator is only described by its phase - we study the effect of a coupling delay on periodic orbits. In networks of Stuart-Landau oscillators, which in addition allow for a variable amplitude, we investigate the different symmetry transitions on the route to chaotic behaviour. Finally we compare the correlation properties of networks chaotic deterministic units to those of coupled linear stochastic units. Many of these results are shown to be universal for delayed systems, independent of the model. To complement the generic features, we also study some specific properties of delayed optical systems, such as the desynchronisation dynamics in coupled semiconductor lasers or the appearance of square waves in opto-electronic oscillators. These results are not only of fundamental importance, but also affect proposed applications based on chaos synchronisation.