VOLUME 79, NUMBER 26 PHYSICAL REVIEW LETTERS 29 DECEMBER 1997 Control of Defects and Spacelike Structures in Delayed Dynamical Systems S. Boccaletti, 1 D. Maza, 2 H. Mancini, 2 R. Genesio, 3 and F. T. Arecchi 1 1 Istituto Nazionale di Ottica, I50125 Florence, Italy and Department of Physics, University of Florence, I50125 Florence, Italy 2 Department of Physics and Applied Mathematics, Universidad de Navarra, Pamplona, Spain 3 Department of Sistemi e Informatica, University of Florence, I50139 Florence, Italy (Received 30 April 1997; revised manuscript received 18 September 1997) In many nonequilibrium dynamical situations delays are crucial in inducing chaotic scenarios. In particular, a delayed feedback in an oscillator can break the regular oscillation into trains mutually uncorrelated in phase, whereby the phase jumps are localized as defects in an extended system. We show that an adaptive control procedure is effective in suppressing these defects and stabilizing the regular oscillations. The analysis of the transient times for achieving control demonstrates that stabilization is obtained within an amplitude turbulent regime, analogous to what is present in spatially distributed systems. The control technique is robust against the presence of large amounts of noise. [S0031-9007(97)04933-8] PACS numbers: 05.45. + b, 47.52. + j, 47.54. + r Since the original idea of Ott, Grebogi, and Yorke [1], many different theoretical schemes [2] and experimental applications [3] have faced the problem of controlling unstable periodic orbits (UPO’s) in chaotic concentrated systems, i.e., in systems modeled by ordinary differential equations. Some proposals of controlling spatially extended sys- tems, i.e., systems ruled by partial differential equations whose order parameter y is a m dimensional vector (m $ 1) in phase space, with k components (k $ 1) in real space, have been put forward for the case k  2 [4]. However, experimentally implementable tools have not yet been in- troduced for controlling unstable periodic patterns (UPP) in extended systems. The essential problems arising in the passage from concentrated to extended systems are already present in delayed dynamical systems, i.e., systems ruled by  y  F  y, y d  , (1) where y  yt  [  m , dot denotes temporal derivative, F is a nonlinear function, and y d  yt 2 T , T being a time delay. Experimental evidence of the analogy between delayed and extended systems was provided for a CO 2 laser with delayed feedback [5] and supported by a theoretical model [6]. Most of the statistical indicators for delayed systems, such as the fractal dimensions, are extensive parameters proportional to T , which thus plays a role analogous to the size for the extended case [7]. The conversion from the former to the latter case is based on a two variable time representation, defined by t  s1uT , (2) where 0 #s# T is a continuous spacelike variable and u [  plays the role of a discrete temporal variable [5]. By such a representation the long range interactions intro- duced by the delay are reinterpreted as short range interac- tions along the u direction, since now y d  ys, u2 1. In this framework, the formation and propagation of space-time structures, as defects and/or spatiotemporal in- termittency can be identified [5,6]. When T is larger than the oscillating period of the system, the behavior of a delayed system is analogous to an extended one with k  1. In particular, it may display phase defects, i.e., points where the phase suddenly changes its value and the amplitude goes to zero. In this Letter we introduce a control technique to sup- press these defects, stabilizing the oscillations of a delayed system. The control restores regular patterns in two dif- ferent chaotic regimes, namely, phase turbulence and am- plitude turbulence, this last one implying the presence of a large number of defects. The control efficiency persists even in the presence of a large amount of noise. For the sake of exemplification, we make reference to the following delayed dynamics:  A  ´A 1b 1 A 2 t 2 T A 1b 2 A 4 t 2 T A , (3)  ´  m μ S 2 m 1 m ´2 kA 2 ∂ . (4) Here, all quantities are real. A is an order parameter, ´ is the time-dependent linear gain, b 1 , b 2 , m 1 , k are suitable fixed parameters, m is a measure of the ratio between the characteristic time scales for A and ´, and S is a measure of the power provided to the system. Equations (3) and (4) are rather general. For instance, when T  0, S , 0, b 1 . 0, b 2 , 0, m. 0, m 1 . 0, k . 0 they model an excitable system, producing the so called Leontovitch bifurcation, evidence of which has been shown experimentally on a CO 2 laser with intracavity satu- rable absorber [8]. For T fi 0, they are similar to the mod- els already used to describe self-sustained oscillations of confined jets [9], or memory induced low frequency oscil- lations in closed convection boxes [10], or even the pulsed dynamics of a fountain [11]. Equations (3) and (4) have been found also to be a good model for the temperature evolution in a well controlled time-dependent convection 5246 0031-9007 97 79(26) 5246(4)$10.00 © 1997 The American Physical Society