Regensburg 2007 – scientific programme

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DY: Fachverband Dynamik und Statistische Physik

DY 2: Time-delayed feedback and neural networks

DY 2.1: Talk

Monday, March 26, 2007, 10:30–10:45, H3

Refuting the odd number limitation of time-delayed feedback control — Bernold Fiedler², Valentin Flunkert¹, Marc Georgi², Philipp Hövel¹, and •Eckehard Schöll¹ — ¹Institut für Theoretische Physik, Technische Universität Ber lin, Hardenbergstr. 36, 10623 Berlin — ²Institut für Mathematik I, Freie Universität Berlin, Arnimalle 2-6, 14195 Berlin

We refute an often invoked theorem which claims that a periodic orbit with an odd number of real Floquet multipliers greater than unity can never be stabilized by time-delayed feedback control in the form proposed by Pyragas [1]. Using a generic normal form, we demonstrate that the unstable periodic orbit generated by a subcritical Hopf bifurcation, which has a single real unstable Floquet multiplier, can in fact be stabilized. We derive explicit analytical conditions for the control matrix in terms of the amplitude and the phase of the feedback control gain, and present a numerical example. Our results are of relevance for a wide range of systems in physics, chemistry, technology, and life sciences, where subcritical Hopf bifurcations occur.

[1] B. Fiedler, V. Flunkert, M. Georgi, P. Hövel, and E. Schöll: Refuting the odd number limitation of time-delayed feedback control, Phys. Rev. Lett. (2006), submitted http://arxiv.org/abs/nlin.CD/0609056