Control of Patterns in Spatiotemporal Chaos in Optics

Lu, Weiping; Yu, Dejin; Harrison, Robert G.

doi:10.1103/physrevlett.76.3316

Cited by 93 publications

(51 citation statements)

References 14 publications

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“…Second, Lu et al [7] considered feedback constructed by combining comparisons of the current state of the system both to a version delayed by the temporal period of the target state and to versions shifted by the spatial period of the target state. Numerical integration showed that the control can direct the system towards, and stabilize, a pattern in a transversely extended laser model, but the method does not appear to be a good candidate for all-optical implementation.…”

Section: Introductionmentioning

confidence: 99%

Controlling extended systems with spatially filtered, time-delayed feedback

et al. 1997

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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.

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Section: Introductionmentioning

confidence: 99%

Controlling extended systems with spatially filtered, time-delayed feedback

et al. 1997

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“…Second, the only information required a priori is the period τ of the target UPO, rather than a detailed knowledge of the profile of the orbit, or even any knowledge of the form of the original ODEs, which may be useful in experimental setups. The method has been implemented successfully in a variety of laboratory situations [2,3,4,5,6,7,8], as well as analytically and numerically in spatially extended pattern-forming systems [9,10,11,12]; more examples can be found in a recent review by Pyragas [13].…”

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confidence: 99%

Stabilization of Long-Period Periodic Orbits Using Time-Delayed Feedback Control

Postlethwaite¹

2009

SIAM J. Appl. Dyn. Syst.

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Abstract. The Pyragas method of feedback control has attracted much interest as a method of stabilising unstable periodic orbits in a number of situations. We show that a time-delayed feedback control similar to the Pyragas method can be used to stabilise periodic orbits with arbitrarily large period, specifically those resulting from a resonant bifurcation of a heteroclinic cycle. Our analysis reduces the infinite-dimensional delay-equation governing the system with feedback to a three-dimensional map, by making certain assumptions about the form of the solutions. The stability of a fixed point in this map corresponds to the stability of the periodic orbit in the flow, and can be computed analytically. We compare the analytic results to a numerical example and find very good agreement.

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“…Some proposals of controlling spatially extended systems, 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 introduced for controlling unstable periodic patterns (UPP) in extended systems.…”

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confidence: 99%

“…We adjust the pump and delay parameters S and T of Eqs. (3) and (4) so that the system enters the chaotic region. This region, in fact, is split into two different regimes.…”

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Control of Defects and Spacelike Structures in Delayed Dynamical Systems

et al. 1997

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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. Some proposals of controlling spatially extended systems, 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 introduced 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 bywhere y y͑t͒ [ ‫ޒ‬ m , dot denotes temporal derivative, F is a nonlinear function, and y d ϵ y͑t 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 s 1 uT ,where 0 # s # T is a continuous spacelike variable and u [ ‫ގ‬ plays the role of a discrete temporal variable [5]. By such a representation the long range interactions introduced by the delay are reinterpreted as short range interactions along the u direction, since now y d ϵ y͑s, u 2 1͒. In this framework, the formation and propagation of space-time structures, as defects and/or spatiotemporal intermittency 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 suppress these defects, stabilizing the oscillations of a delayed system. The control restores regular patterns in two different chaotic regimes, namely, phase turbulence and amplitude 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: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 syst...

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Control of Patterns in Spatiotemporal Chaos in Optics

Cited by 93 publications

References 14 publications

Controlling extended systems with spatially filtered, time-delayed feedback

Controlling extended systems with spatially filtered, time-delayed feedback

Stabilization of Long-Period Periodic Orbits Using Time-Delayed Feedback Control

Control of Defects and Spacelike Structures in Delayed Dynamical Systems

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