Synchronization in networks of nonlinear oscillators with coupling delays

We consider the synchronization problem of an arbitrary number of coupled nonlinear oscillators with delays in the interconnections. The emphasis is on coupled Lorenz systems. The network topology is described by a directed graph. Unlike the conventional approach of deriving directly sufficient synchronization conditions, our approach starts from a linearized stability analysis in a (gain, delay) parameter space of a synchronized equilibrium and extracts insights from an analysis of its bifurcations and from the corresponding emerging behavior. Instrumental to this analysis a factorization of the characteristic equation is employed that not only facilitates the analysis and reduces computational cost, but also allows to determine the precise role of the individual agents and the topology of the network in the (in)stability mechanisms. The study reveals fundamental limitations to synchronization and it explains under which conditions on the topology of the network and on the characteristics of the coupling, the systems are expected to synchronize. Furthermore, the main result shows that for sufficiently large coupling gains, coupled Lorenz systems exhibit a generic behavior that does not depend on the number of systems and the topology of the network, as long as some basic assumptions are satisfied, including the strong connectivity of the graph. The results are illustrated with networks of Lorenz systems with several topologies.