Phase-flip in relay oscillators via linear augmentation

In time-delay coupled relay system of three limit cycle of oscillators, linear augmentation control provides an effective strategy to induce phase-flip from relative phase zero to π between spatially separated oscillators. A coexisting dynamical regime of fixed point and anti-phase synchronization state is harnessed in relay system. Targeting phase-flip transition through linear augmentation is illustrated by using Stuart-Landau oscillators. However, such dynamical transition does not occur in the same system with conjugate coupling through dissimilar variables. Inducing relative phase difference without resorting to variation of internal system parameters and time-delay is important from the view of system control.