Synchronization of Heterogeneous Oscillators by Noninvasive Time-Delayed Cross Coupling

In the early 1990's, significant advances for the control of unstable states of a dynamical systems were achieved. In particular, the work of Pyragas became the foundation of an entire research community. He introduced a technique which is known as 'time-delayed autosynchronization', 'time-delayed feedback control' or simply 'chaos control'. The target of this method is an unstable orbit with period T. By taking the difference between an output signal s(t) and the signal delayed by one period of the orbit, s(t-T), and feeding it back into the dynamical system, it was shown that this orbit can be stabilized. Many extensions, generalizations and case studies both in theory and experiment have been presented since then.
In this talk, the development of the field of chaos control is reviewed and important works are highlighted. Moreover, we present our recent study on an extension of the control method to a setup of two coupled non-identical systems. In this work, we demonstrate that nonidentical systems, in particular, nonlinear oscillators with different time scales, can be synchronized by a mutual coupling via time-delayed control signals. Each oscillator settles on an unstable state, say a fixed point or an unstable periodic orbit, with an asymptotically vanishing coupling force. We present the underlying theoretical considerations and numerical simulations, and, moreover, demonstrate the concept experimentally in nonlinear electronic oscillators.