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Added by
Kestutis Pyragas Prof.
Publication date
2013
fields
Physics
and research's language is
English
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We present an algorithm for a time-delayed feedback control design to stabilize periodic orbits with an odd number of positive Floquet exponents in autonomous systems. Due to the so-called odd number theorem such orbits have been considered as uncontrollable by time-delayed feedback methods. However, this theorem has been refuted by a counterexample and recently a corrected version of the theorem has been proved. In our algorithm, the control matrix is designed using a relationship between Floquet multipliers of the systems controlled by time-delayed and proportional feedback. The efficacy of the algorithm is demonstrated with the Lorenz and Chua systems.
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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. 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.
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.
We present an analysis of time-delayed feedback control used to stabilize an unstable steady state of a neutral delay differential equation. Stability of the controlled system is addressed by studying the eigenvalue spectrum of a corresponding characteristic equation with two time delays. An analytic expression for the stabilizing control strength is derived in terms of original system parameters and the time delay of the control. Theoretical and numerical results show that the interplay between the control strength and two time delays provides a number of regions in the parameter space where the time-delayed feedback control can successfully stabilize an otherwise unstable steady state.
We investigate the possibility to suppress noise-induced intensity pulsations (relaxation oscillations) in semiconductor lasers by means of a time-delayed feedback control scheme. This idea is first studied in a generic normal form model, where we derive an analytic expression for the mean amplitude of the oscillations and demonstrate that it can be strongly modulated by varying the delay time. We then investigate the control scheme analytically and numerically in a laser model of Lang-Kobayashi type and show that relaxation oscillations excited by noise can be very efficiently suppressed via feedback from a Fabry-Perot resonator.
In this paper we study the stabilization of rotating waves using time delayed feedback control. It is our aim to put some recent results in a broader context by discussing two different methods to determine the stability of the target periodic orbit in the controlled system: 1) by directly studying the Floquet multipliers and 2) by use of the Hopf bifurcation theorem. We also propose an extension of the Pyragas control scheme for which the controlled system becomes a functional differential equation of neutral type. Using the observation that we are able to determine the direction of bifurcation by a relatively simple calculation of the root tendency, we find stability conditions for the periodic orbit as a solution of the neutral type equation.
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