Control of unstable steady states by extended time-delayed feedback

Abstract

Time-delayed feedback methods can be used to control unstable periodic orbits as well as unstable steady states. We present an application of extended time delay autosynchronization introduced by Socolar et al. to an unstable focus. This system represents a generic model of an unstable steady state which can be found for instance in a Hopf bifurcation. In addition to the original controller design, we investigate effects of control loop latency and a bandpass filter on the domain of control. Furthermore, we consider coupling of the control force to the system via a rotational coupling matrix parametrized by a variable phase. We present an analysis of the domain of control and support our results by numerical calculations.

Delay stabilization of periodic orbits in coupled oscillator systems

“…Stabilization of unstable periodic orbits has thus been identified as a task of central importance, even in the context of chaotic dynamics (Ott et al 1990;Boccaletti et al 2000;Gauthier 2003;Schöll & Schuster 2008). Delayed-feedback control, as proposed by Pyragas (1992), has proven to be a powerful non-invasive method for the stabilization of unstable periodic orbits (Schöll & Pyragas 1993;Baba et al 2002;Beck et al 2002;von Loewenich et al 2004) and unstable steady states (Hövel & Schöll 2005;Schikora et al 2006;Dahms et al 2007Dahms et al , 2008 in dynamical systems. It has, for instance, been successfully applied in spatially extended systems (Unkelbach et al 2003;Stegemann et al 2006;Postlethwaite & Silber 2007;Dahlem et al 2008, in press;Schneider et al 2009;Kyrychko et al in press) and even in noisedriven systems (Janson et al 2004;Pomplun et al 2005;Pototsky & Janson 2007;Prager et al 2007;Hizanidis & Schöll 2008;Majer & Schöll 2009).…”

Delayed feedback control of unstable steady states with high-frequency modulation of the delay

“…In parallel to experimental realization, substantial work has been done to understand the control mechanism analytically [28][29][30][31]. A notable result has been reported recently in the context of refuting the alleged odd-number limitation [32,33], believed to be a severe limitation of the delayed feedback control technique for almost a decade [34].…”

Control of coherence resonance in semiconductor superlattices

“…This method proved to be very powerful and has been successfully applied to various physical systems since then [1]. Other variants have been elaborated, e. g. extended time-delay autosynchronization (ETDAS) [30], and have been applied not only to deterministic systems [31,32,33,34] including fixed points but to stochastic systems as well [35,36].…”