Shifted feedback suppression of turbulent behavior in advection-diffusion systems C. Evain, C. Szwaj, and S. Bielawski Laboratoire de Physique des Lasers, Atomes et Mol´ ecules, UMR CNRS 8523, Centre d’ ´ Etudes et de Recherches Lasers et Applications, FR CNRS 2416, Universit´ e des Sciences et Technologies de Lille, F-59655 Villeneuve d’Ascq Cedex (France). M. Hosaka 1* , A. Mochihashi 1,2† , and M. Katoh 1,2 (1) UVSOR Facility, Institute for Molecular Science, National Institutes of Natural Sciences, Okazaki 444-8585, Japan. (2) School of Physical Sciences, The Graduate University for Advanced Studies (SOKENDAI), Okazaki 444-8585, Japan. M.-E. Couprie Synchrotron SOLEIL, Saint Aubin, BP 34, 91192 Gif-sur-Yvette, France. (Dated: March 5, 2008) In spatio-temporal systems with advection, suppression of noise-sustained structures involves questions that are outside of the framework of deterministic dynamical systems control (such as OGY type methods). Here we propose and test an alternate strategy where a non-local additive feedback is applied, with the objective to create a new deterministic solution that becomes robust to noise. As a remarkable fact –though the needed parameter perturbations required have essentially a finite size– they turn out to be extraordinarily small in principle: 10 -8 in the Free-Electron laser experiment presented here. PACS numbers: 47.27.Rc, 05.45.-a, 42.65.Sf, 41.60.Cr A range of systems display dynamically-induced hyper- sensitivity to noise, which results in full-scale erratic fluctu- ations. This typically happens for wave patterns subjected to advection [1], which imposes a permanent drift of the struc- ture. Experimental examples include turbulence in pulsed lasers [2, 3], optical systems with transverse patterns [4], plas- mas [5], and fluid dynamics [6]. In these systems, control or more generally suppression of erratic behaviors involves specific difficulties, linked to their high-dimensionality and to the non-perturbative effect of noise. In particular, stabilization of a steady state is not a sufficient criterion for effective system control (“turbulent” behaviors are even observed in systems with a globally sta- ble steady state [7–9]). This makes traditional methods for deterministic dynamical system control (Ott-Grebogi-Yorke- type methods [10]) not directly applicable. In this letter, we examine a feedback strategy requiring very small perturbations, taking advantage of the strong amplifi- cation properties (the so-called transient growth [8]) of the system, and we test its efficiency on a Ginzburg-Landau equa- tion with advection. The feedback introduces a small nonlocal coupling between each site and another site located at a finite distance. This presents pictural similarities with Pyragas-type schemes [11], though the dynamical processes involved are strongly different. We show that the stabilization process oc- curs via the creation of a new deterministic solution, and that –in the local saturation coupling case– the process can be un- derstood in terms of convective/absolute instabilities. After * Present address: Nagoya University Graduate School of Engineering, 464- 8603 Nagoya, Japan. † Japan Synchrotron Radiation Research Institute (JASRI), SPring-8, Sayo- cho, Hyogo 679-5198 Japan. the general study of the process on Ginzburg-Landau equa- tions, we will present experimental results on the specific sys- tem which motivated this work: the UVSOR-II Free-Electron Laser. To test the feedback strategy, we consider the following advection-diffusion equation, with finite size: e t (z,t)+ ve z (z,t) = e zz (z,t)+ Rg(z)e(z,t) - Se(z,t)+ √ ηξ (z,t) +αe(z + a, t), (1) where e(z,t) is complex (as it is typically associated with the complex amplitude of a pattern). t and z represent time and space, v the advection velocity (v is supposed > 0 in the fol- lowing), R the real gain term for pattern formation (R> 0 here). η is the noise amplitude, and ξ (z,t) is a delta-correlated white noise term. g represents the spatial variation of the gain. g(z) is supposed to vary slowly with z and is supercritical [g(z) > 0] in a finite region near the center.  is a small parameter with  -1 characterizing the system’s size. In this paper, we take g(z)=1 -  2 z 2 , and the saturation term S is taken local in the first part of the paper: S = |e(z,t)| 2 . (2) The feedback is applied through the term αe(z + a), with α and a the gain and spatial shift parameters respectively. Without feedback (α =0), it is known [8] that noisy “turbu- lent” behavior typically appears when the velocity is beyond the convective/absolute threshold: v>v c =2 √ R. Beyond this point, the only stable attractor of the deterministic system (with η =0) is the solution e(z,t)=0. However the system is subjected to transient growth, and small noise is strongly amplified when it is advected through the system (Fig. 1). To achieve “control” (or more precisely to suppress erratic behaviors) the idea is to create a “loop” in the system, in the