Extended time-delay auto-synchronization (ETDAS) is a promising technique for stabilizing unstable periodic orbits in low-dimensional dynamical systems. The technique involves continuous feedback of signals delayed by multiples of the orbit's period in a manner that is especially well-suited for fast systems and optical implementation. We show how to analyze the stability of a given implementation of ETDAS without explicit integration of time-delay equations. To illustrate the method and point out some nontrivial features of ETDAS, we obtain the domain of control for a period-one orbit of the driven, damped pendulum.

We suggest a spatially local feedback mechanism for stabilizing periodic orbits in spatially extended systems. Our method, which is based on a comparison between present and past states of the system, does not require the external generation of an ideal reference state and can suppress both absolute and convective instabilities.As an example, we analyze the complex Ginzburg-Landau equation in one dimension, showing how the time-delay feedback enlarges the stability domain for travelling waves.

We stabilize unstable periodic orbits of a fast diode resonator driven at 10.1 MHz (corresponding to a drive period under 100 ns) using extended time-delay autosynchronization. Stabilization is achieved by feedback of an error signal that is proportional to the difference between the value of a state variable and an infinite series of values of the state variable delayed in time by integral multiples of the period of the orbit. The technique is easy to implement electronically and it has an all-optical counterpart that may be useful for stabilizing the dynamics of fast chaotic lasers. We show that increasing the weights given to temporally distant states enlarges the domain of control and reduces the sensitivity of the domain of control on the propagation delays in the feedback loop. We determine the average time to obtain control as a function of the feedback gain and identify the mechanisms that destabilize the system at the boundaries of the domain of control. A theoretical stability analysis of a model of the diode resonator in the presence of time-delay feedback is in good agreement with the experimental results for the size and shape of the domain of control. (c) 1997 American Institute of Physics.

We investigate a control technique for spatially extended systems combining spatial filtering with a previously studied form of time-delay feedback. The scheme is naturally suited to real-time control of optical systems. We apply the control scheme to a model of a transversely extended semiconductor laser in which a desirable, coherent traveling wave state exists, but is a member of a nowhere stable family. Our scheme stabilizes this state, and directs the system towards it from realistic, distant and noisy initial conditions. As confirmed by numerical simulation, a linear stability analysis about the controlled state accurately predicts when the scheme is successful, and illustrates some key features of the control including the individual merit of, and interplay between, the spatial and temporal degrees of freedom in the control.

A simple numerical experiment on Fitzhugh-Nagumo equations demonstrates that time-delay feedback methods can stabilize regular behavior in a paced, excitable oscillator without using large applied pulses. The method is robust against slow variations in the pacing rate and can be used to track an orbit into a parameter regime where it is unstable. Moreover, the method requires no information about the desired orbit other than period, which is determined directly by the observation of the pacing signal. Results are presented for feedback stabilization of a nonspiking orbit in the rapid pacing regime.