Delays, Connection Topology, and Synchronization of Coupled Chaotic Maps

We consider networks of coupled maps where the connections between units involve time delays. We show that, similar to the undelayed case, the synchronization of the network depends on the connection topology, characterized by the spectrum of the graph Laplacian. Consequently, scale-free and random networks are capable of synchronizing despite the delayed flow of information, whereas regular networks with nearest-neighbor connections and their small-world variants exhibit poor synchronization. On the other hand, the connection delays can actually be conducive to synchronization, so that it is possible for the delayed system to synchronize where the undelayed system does not. Furthermore, the delays determine the synchronized dynamics, leading to the emergence of a wide range of new collective behavior that the individual units are incapable of producing in isolation. In this way, connection delays enable the overall system to develop and sustain new dynamics through the coordination of its constituent units.