Adaptive control of chaos in a delayed high dimensional laser system

Summary form only given. We present an efficient method to control a delayed feedback system. These systems, for delays long with respect to the intrinsic decorrelation time corresponding to short-delay dynamics, display a high dimensional chaotic behavior. A suitable space-like representation provides an analogy between delayed systems and space extended systems. The analogy rests on the fact that the time variable /spl tau/ can be decomposed into a continuous spacelike variable /spl sigma/ (0/spl les//spl sigma//spl les/t) and a discrete timelike variable n/spl tau/ t/spl equiv//spl sigma/+n/spl tau/ where /spl tau/ is the delay time. In this framework, the long-range temporal interaction due to the delayed feedback can be considered as a local interaction from one to the next delay unit. Thus, the behavior of such a system can be studied by representing the original signal x(t) as a space-time plot of a one-dimensional spatial system of length L=/spl tau/ on a discrete-time lattice. Typically, when the delay /spl tau/ is larger than the characteristic oscillating period, it is possible to observe phase defects, i.e. points where the phase presents a discontinuity and the amplitude goes to zero. We refer to the pseudo-spatial structures produced by a CO/sub 2/ laser with delayed feedback. In particular we demonstrate the possibility of controlling irregular patterns by applying a real-time implementation of an adaptive algorithm. This method allows stabilization of the period and the amplitude of the chaotic oscillations by eliminating the zero amplitude points which can originate spatial defects. The adaptive control has been realized based on the combination of a washout filter with its time derivative in a negative feedback configuration.