Feedback control of subcritical Turing instability with zero mode.

A global feedback control of a system that exhibits a subcritical monotonic instability at a nonzero wave number (short-wave or Turing instability) in the presence of a zero mode is investigated using a Ginzburg-Landau equation coupled to an equation for the zero mode. This system is studied analytically and numerically. It is shown that feedback control, based on measuring the maximum of the pattern amplitude over the domain, can stabilize the system and lead to the formation of localized unipulse stationary states or traveling solitary waves. It is found that the unipulse traveling structures result from an instability of the stationary unipulse structures when one of the parameters characterizing the coupling between the periodic pattern and the zero mode exceeds a critical value that is determined by the zero mode damping coefficient.