Max Planck Institute for Dynamics and Self-Organization -- Department for Nonlinear Dynamics and Network Dynamics Group
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MPI Kolloquium

Wednesday, 11.05.2011 16 c.t.

Controlling dynamics of complex networks with delayed coupling

by Prof. Dr. Eckehard Schöll
from TU Berlin Theoretische Physik/Nichtlineare Dynamik und Kontrolle, Berlin

Contact person: Jan Nagler

Location

Ludwig-Prandtl Hörsaal, Am Faßberg 11, AI-Gebäude

Abstract

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 [1]. 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 [2]. We study synchronization in delay-coupled oscillator networks, using a master stability function approach [3,4]. For general networks with large delay synchronizability relates in a simple way to the network topology, which allows for a universal classification. 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 in-phase synchronization or desynchronization for general network topologies, or between in-phase, cluster, or splay states in unidirectional rings. Our results are robust even for slightly nonidentical elements of the network. We also discuss applications to neural networks, in particular small-world networks with inhibitory couplings, and to chaotic laser networks. [1} W. Just, A. Pelster, M. Schanz, and E. Schoell (eds.): Theme Issue on Delayed Complex Systems, Phil. Trans. R. Soc. A 368, pp.301-513 (2010). [2] E. Schoell and H. G. Schuster (eds.): Handbook of Chaos Control (Wiley-VCH, Weinheim, 2008). [3] C.-U. Choe, T. Dahms, P. Hoevel, and E. Schoell: Controlling synchrony by delay coupling in networks: from in-phase to splay and cluster states, Phys. Rev. E 81, 025205(R) (2010). [4] V. Flunkert, S. Yanchuk, T. Dahms, and E. Schoell: Synchronizing distant nodes:a universal clssification of networks, Phys. Rev. Lett. 105, 254101 (2010).

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