Mirroring of synchronization in a bi-layer master–slave configuration of Kuramoto oscillators

Abstract

The phenomenon of mirroring of synchronization is investigated in dynamically dissimilar, unidirectionally coupled, bi-layer master–slave configuration of globally coupled Kuramoto oscillators. The dynamics of the master layer depends solely on the distribution of the natural frequencies of its oscillators. On the other hand, the slave layer dynamics depends not only on the distribution of the natural frequencies of its oscillators but also on the unidirectional coupling with the master layer. The standard Kuramoto order parameter is used to study synchronization in the individual layers and of the bi-layer network. A transition to a completely mirroring state is observed in the dynamics of the slave layer, as the mirroring coefficient in the unidirectional coupling is increased. We derive analytically and verify numerically the conditions for the slave layer to fully mimic the synchronization properties of the master layer. It is further shown that while the master and slave layers are individually synchronized, the bi-layer network exhibits a state of frustrated synchronization.