Stationary and switching synchronization regimes in an ensemble of four nonidentical phase oscillators with repulsive couplings

TL;DR

This paper explores various synchronization regimes in a small ensemble of nonidentical phase oscillators with repulsive coupling, identifying stationary and switching behaviors, and deriving analytical expressions for certain regimes.

Contribution

It classifies synchronization modes into stationary and switching types and provides analytical formulas for some regimes, revealing bistability in the system.

Findings

Identified four synchronization modes: 4:0, 3:1, 2:2, 2:1:1.

Discovered bistability depending on initial conditions.

Derived analytical expressions for 4:0 and 2:2 regimes.

Abstract

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 modes of frequency synchronization were found: 4:0 (global), 3:1, 2:2, 2:1:1 (cluster). It is shown that these regimes 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 ratios 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 ratios were observed. For the 4:0 and 2:2 regimes, analytical expressions for the synchronization frequencies were derived. The presence of bistability has been revealed depending on the initial conditions of different synchronous regimes at the same parameters: sets of individual frequencies and value of coupling strength.