Stationary and switching synchronization states in an ensemble of four nonidentical phase oscillators with repulsive...

This study investigates the collective dynamics (phase synchronization, instantaneous frequencies synchronization, and mean frequencies synchronization) in an ensemble of four nonidentical phase oscillators with repulsive coupling. We use the Kuramoto-Sakaguchi system of ordinary differential equations as our mathematical model. Depending on the coupling strength in the presence of a small mismatch of the natural frequencies, all possible states of frequency synchronization were found: 4:0 (global), 3:1, 2:2, and 2:1:1 (cluster). It is shown that these states can be classified into two main types depending on the evolution of the instantaneous frequencies: stationary (4:0 and 2:2), characterized by constancy of phase mismatch and instantaneous frequencies, and switching (3:1 and 2:1:1), in which metastable processes with periodic switching of synchronous states are observed: For different time intervals, different types of locking of instantaneous frequencies of oscillator pairs and different types of phase mismatch were observed. A novel effect of collective dynamics-self-organized emergence of slow-fast behavior in the ensemble of elements which, if decoupled, exhibit uniform temporal evolution-was demonstrated. The rather long time intervals during which the distances between some oscillators stay nearly constant alternate with short epochs of rapid evolution, involving the break up of clusters and their subsequent reassembly. For the 4:0 and 2:2 states, analytical expressions for the synchronization frequencies were derived. The presence of bistability has been revealed depending on the initial conditions of different synchronization states at the same parameters: sets of individual frequencies and value of coupling strength.