The delayed feedback control (DFC) method has been invented in 1992 (this year is a 20th anniversary). Following the original paper by Pyragas [Phys. Lett. A, 170, 421, 1992], more than 1500 papers devoted or related to the DFC have been published. Many different modifications of the algorithm have been proposed, and significant achievements are attained in the theory of the DFC. Although this theory is non-trivial, currently the mechanism of the DFC action is rather well understood, and the main limitations of the algorithm are established. The DFC has been successfully implemented in a number of experimental systems of different physical nature. The aim of this talk is to present a brief review of important modifications of the DFC algorithm, significant theoretical results and experimental implementations attained during the past twenty years. The recent results concerning adaptive modifications of the DFC and analytical achievements based on phase reduction of time-delay systems will be discussed as well. 1. Brief review of experimental and theoretical results The DFC algorithm [1] is a simple, robust, and efficient method to stabilize unstable periodic orbits (UPOs) in chaotic systems. Nowadays, it becomes one of the most popular methods in the chaos control research [2]. The method allows a noninvasive stabilization of UPOs of dynamical systems in the sense that the control force vanishes when the target state is reached. The DFC algorithm is reference-free and makes use of a control signal K∆s(t) obtained from the difference ∆s(t) = s(t) − s(t − τ) between the current state s(t) of the system and the state of the system s(t−τ) delayed by one-period τ of the target orbit. The UPO may become stable under the appropriate choice of feedback strength K. Note that only the stability properties of the orbit are changed, while the orbit itself and its period remain unaltered. The controlled system can be treated as a black box, since the method does not require any exact knowledge of either the form of the periodic orbit or the system’s equations. The method is particularly appealing for experimentalists, since one does not need to know anything about the target orbit beyond its period τ. The DFC algorithm is notably superior to other control methods in fast dynamical systems, since it does not require any realtime computer processing. Successful implementation of the DFC algorithm has been attained in diverse experimental systems, including electronic chaotic oscillators, mechanical pendulums, lasers, gas discharge systems, a current-driven ion acoustic instability, a chaotic Taylor-Couette flow, chemical systems, high-power ferromagnetic resonance, helicopter rotor blades, and a cardiac system. (cf. [3] for review up to 2006). An important practical application of the DFC algorithm has been recently demonstrated by Yamasue et al. [4]. The authors have successfully implemented the DFC method in an atomic force microscope and managed to stabilize cantilever oscillations. As a result, they remove artifacts on a surface image. Another interesting application of the DFC for the analysis of bifurcations of periodic states in experimental systems has been recently considered in Ref. [5]. A reach variety of modifications of the DFC has been suggested in order to improve its performance (cf. [3]). Here we mention only the most important modification known as an extended DFC (EDFC), which has been introduced in Ref. [6]. The authors improved an original DFC scheme by using an information from many previous states of the system. The EDFC scheme achieves stabilization of UPOs with a greater degree of instability [7, 8]. The theory of DFC is difficult because the delayed feedback induces an infinite number of degrees of freedom. Even linear analysis of such systems is complicated due to the infinite number of Floquet exponents characterizing the stability of controlled orbits. Nevertheless, some analytical approaches have been developed in vicinity to various bifurcations of periodic orbits, such as the period doubling bifurcation [9, 10], the subcritical Hopf bifurcation [11, 12, 13] and the Nejmark-Sacker (discrete Hopf) bifurcation [14]. In 1997 Nakajima [15, 9] proved the so-called odd number limitation, which states that any UPOs with an odd number of real Floquet multipliers greater than unity can never be stabilized by any DFC technique. This limitation has been commonly accepted and intensively discussed in the literature. However, in 2007, Fiedler at al [16] have shown by a simple example that this limitation does not hold in general for autonomous systems (note that for non-autonomous systems it remains valid in general). Recently, a modified (corrected) proof of the limitation for autonomous systems has been presented by Hooton and 2012 International Symposium on Nonlinear Theory and its Applications NOLTA2012, Palma, Majorca, Spain, October 22-26, 2012
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Jan 1, 2015
Dewey H Hodges 
Open Access