Dynamics in complex networks with time-delayed coupling
Time delays arise naturally in many complex networks, for instance in neural networks, as delayed coupling or delayed feedback due to nite signal transmission and processing times . We study synchronization in delay-coupled networks, using a master stability function approach . Using dierent models for the nodes' dynamics, we study patterns of synchronization for oscillatory, excitatory, and chaotic dynamics described by Stuart-Landau oscillators, FitzHugh-Nagumo neurons, and Lang-Kobayashi semiconductor lasers, respectively.
Within a generic model of Stuart-Landau oscillators (normal form of supercritical Hopf bifurcation) we derive analytical stability conditions and demonstrate that by tuning the coupling phase one can easily control the stability of synchronous periodic states. We propose the coupling phase as a crucial control parameter to switch between dierent states of synchronization and show that an adaptive time-delayed feedback control algorithm can be used to nd the appropriate value of the coupling phase . These results are robust even for slightly nonidentical elements of the network.
We discuss applications to neural networks, where our model operates in the excitable regime, i.e., it exhibits no self-sustained, but coupling-induced oscillations. The dynamics on regular networks is highly multistable exhibiting isochronous synchronization as well as cluster dynamics. Beyond regular networks, we study a small-world topology with inhibitory couplings, where a phase transition to desynchronization takes place as the number of inhibitory links approaches a critical value.
Finally, there is much interest in chaotic synchronization recently, especially in coupled semiconductor lasers. We show, on one hand, the emergence of isochronous and cluster synchronization, where several subgroups in the networks act isochronously. On the other hand, introducing a few dierent delays into the couplings enables two distant nodes to synchronize, while all other nodes are seemingly uncorrelated and do not lock into this dynamics.
 W. Just, A. Pelster, M. Schanz, and E. Schöll. Delayed complex systems. Theme Issue of Phil. Trans. R. Soc. A, 368 (2010). pp.301-513.
 C. U. Choe, T. Dahms, P. Hövel, and E. Schöll. Controlling synchrony by delay coupling in networks: from in-phase to splay and cluster states. Phys. Rev. E, 81, 025205(R) (2010).
 Anton Selivanov, J. Lehnert, T. Dahms, P. Hövel, A. L. Fradkov, and E. Schöll. Adaptive synchronization for delay-coupled networks of Stuart-Landau oscillators. Submitted (2011).