Dynamics of Turing Pattern under Time-Delayed Feedback

We investigate Turing patterns under time-delayed feedback control. The Lengyel-Epstein model is solved numerically in one dimension to find various dynamics under feedback such as the propagation of spatially periodic structures at a constant velocity with and without a source and sink of waves, more complex spatiotemporal patterns and uniform oscillation depending on the strength of the feedback and the magnitude of the delay time. To elucidate some of the behaviors, we carry out theoretical analysis based on the amplitude equation for Turing patterns assuming that the feedback control is sufficiently weak.