Diffusion Parameter Control of Spatiotemporal Chaos

There have been recently a considerable experimental and theoretical effort to characterize Spatiotemporal Chaos (STC) [Cross & Hohenberg, 1994; Gollub, 1994]. Weak STC seems to be an ubiquitous phenomenon in large non-equilibrium systems. In some cases STC arise in the proximity of threshold and can be described within the context of weakly nonlinear theories. These theories are well developed in the form of so–called complex Ginzburg–Landau equations (CGLE) [Cross & Hohenberg, 1993]. The CGLE is a prototypical equation for a complex field A that exhibit STC [Chaté, 1995]. It accounts for the slow modulations, in space and time of the oscillatory state in a physical system which undergoes a Hopf bifurcation [van Saarloos & Hohenberg, 1992]. The control of spatiotemporal chaos is a complicated problem, and so, there is a wide variety ofmethods intended to control such chaotic behavior. There have been several attempts to achieve such control in the CGLE [Aranson et al., 1994; Bleich & Socolar, 1996; Mertens et al., 1994; Battogtokh & Mikhailov, 1996; Montagne & Colet, 1997]. The most common approach is adding time-delayed feedback terms to the CGLE. The feedback can be either local [Bleich & Socolar, 1996] (at each spatial point, the field at the same point at previous times is fed back) or global [Mertens et al., 1994; Battogtokh & Mikhailov, 1996] (at each spatial point a term proportional to the integral of the field over the spatial variable is fed back). Feedback has also been used for control in a nonlinear drift–wave equation driven by a sinusoidal wave [Gang, 1993] and, in conjunction with a spatial filter, it has been applied to stabilize rolls and hexagonal structures in a model for a transversally extended three level laser [Lu et al., 1996] and to control filamentation in a model for wide aperture semiconductor lasers based on the Swift-Hohenberg equation [Bleich et al., 1997].