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SOE: Fachverband Physik sozio-ökonomischer Systeme

SOE 20: Networks: From Topology to Dynamics IV (with BP, DY)

SOE 20.5: Talk

Thursday, March 17, 2011, 16:15–16:30, GÖR 226

Adaptive time-delayed feedback control of unstable fixed points and cluster states in networks — •Judith Lehnert¹, Anton Selivanov², Peter Guzenko³, Philipp Hövel¹, Thomas Dahms¹, Alexander Fradkov^2,3, and Eckehard Schöll¹ — ¹Institut f. Theo. Physik, Technische Universität Berlin — ²Saint Petersburg State University, Russia — ³Institute for Problems of Mechanical Engineering, Russian Academy of Sciences

Time-delayed feedback as originally proposed by Pyragas is a simple and efficient scheme for the control of unstable periodic orbits and unstable fixed points. It is non-invasive since it employs the difference of an output signal s(t) and its delayed counterpart s(t−τ), multiplied by the feedback gain K. In the (K,τ)-plane this gives rise to domains of successful control where the largest Lyapunov exponent becomes negative. Here, we show that it is possible to obtain an optimal value of the feedback gain by applying adaptive control, namely the speed gradient method. Furthermore, we study the role of different initial conditions and show the robustness of our method with respect to noise. We apply our method to stabilize unstable fixed points as well as to reach different M-cluster states in networks. By the latter we refer to a state where the nodes of a network are zero-lag synchronized within M clusters with a constant phase difference between these clusters.