Traveling and standing patterns induced by delay feedback in uniform oscillatory reaction–diffusion system

The influence of delay feedback (DF) on pattern formation in reaction–diffusion system is investigated in the parametric domain where the undelayed system exhibits uniform oscillations. Depending on the feedback parameters, both traveling patterns and standing patterns are observed. The linear stability analysis implies the traveling patterns arise from an oscillatory instability at a finite wavenumber (wave instability), whereas the standing patterns are formed due to another type of instability – the instability of homogeneous limit cycle with respect to spatial perturbations. It is demonstrated that, rather than stabilizing the system, the DF can unusually induce two kinds of instability.