Adaptive control and synchronization in delay-coupled complex networks

Time delays arise naturally in many complex networks, for instance in neural networks, as delayed coupling or delayed feedback due to finite signal transmission and processing times. Such time delays can either induce instabilities, multistability, and complex bifurcations, or suppress instabilities and stabilize unstable states. Thus, they can be used to control the dynamics. We study synchronization in delay-coupled oscillator networks, using a master stability function approach, and show that for large coupling delays synchronizability relates in a simple way to the spectral properties of the network topology, allowing for a universal classification. As illustrative examples we consider coupled chaotic lasers and neural networks. Within a generic model of Stuart-Landau oscillators we demonstrate that by tuning the coupling parameters one can easily control the stability of different synchronous periodic states, i.e., in-phase, cluster, or splay states. We show that adaptive time-delayed feedback control based on the speed gradient method can be used to find the appropriate value of these parameters, and one can even self-adaptively adjust the network topology to realize a desired cluster state. Our results are robust even for slightly nonidentical elements of the network.