On the relay coupling of three fractional-order oscillators with time-delay consideration: Global and cluster synchronizations

The relay coupling of three fractional-order two-stage oscillators in the presence of time delay has been explored theoretically, numerically and analogically. The global stabilization of the system in a finite time is proven through Hölder and Gronwall inequalities, as well as through inequality scaling skills. The Synchronization of the system is characterized in terms of its parameters (coupling strength and time delay) by using time series, two parameters phase diagrams and two parameters transverse Lyapunov exponent diagrams. It is found that for smaller delay values, the network exhibits global phase synchronization whereas for higher delay values, phase synchronization just occurs between the two indirectly connected units (cluster phase synchronization). Striking phenomena such as amplitudes' death and chaotic beats oscillations are also observed from this relay coupling of three fractional-order two-stage oscillators. Furthermore, PSpice simulation results of the analog electronic circuit are in perfect accordance with both theoretical and numerical results.