Localized structures in subcritically unstable, globally controlled pattern-forming systems

We investigate the nonlinear dynamics of subcritical instabilities in the presence of a feedback control. The control is based on a feedback loop between the linear growth rate and the maximum amplitude of the emerging pattern.

In the case of a subcritical monotonic instability, the globally controlled Sivashinsky equation is considered. It is shown that global control can prevent the blow-up, and spatially localized structures are formed. The subcritical oscillatory instability is studied in the framework of a globally controlled complex Ginzburg-Landau equation. In the latter case, the global control results in the formation of spatially localized pulses. In the one-dimensional case, depending on the values of the linear and nonlinear dispersion coefficients, several types of pulse dynamics are possible in which the computational domain contains: (i) a single stationary pulse, (ii) several co-existing stationary pulses, (iii) competing pulses that appear one after another at random locations so that at each moment of time there is only one pulse in the domain; (iv) temporal intermittency between cases (ii) and (iii); (v) spatio-temporally chaotic system of short pulses. In the two-dimensional case, alternating or chaotic pulses are found.