Synchronization engineering: tuning the phase relationship between dissimilar oscillators using nonlinear feedback.

A mild, nonlinear, time-delayed feedback signal was applied to two heterogeneous oscillators in order to synchronize their frequencies with an arbitrary and controllable phase difference. The feedback was designed using phase models constructed from experimental measurements of the intrinsic dynamical properties of the oscillators. The feedback signal produced an interaction function that corresponds to the desired collective behaviour. The synchronized phase difference between the elements can be tuned to any value on the interval 0 and 2pi by shifting the phase of the interaction function using the feedback delay. Numerical simulations were conducted and experiments carried out with electrochemical oscillators.