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

TL;DR

The paper analyzes synchronization in a quartet of nonidentical phase oscillators with repulsive coupling using the Kuramoto-Sakaguchi model to map all possible mean-frequency synchronization patterns: 4:0, 3:1, 2:2, and 2:1:1. It distinguishes stationary versus switching regimes, derives analytical expressions for the 4:0 synchronization frequency, and demonstrates that stronger coupling $d$ increases synchronization frequencies; it also reveals bistability under fixed parameters due to different initial conditions. The switching regimes (3:1 and 2:1:1) exhibit metastable dynamics with periodic switching of instantaneous-frequency locking and phase ratios, indicating rich cluster dynamics beyond global synchronization. These findings advance understanding of repulsively coupled oscillator networks and may inform design principles for engineered or natural systems where repulsion and heterogeneity coexist.

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.

Figures (7)

• Figure 1: Dependence of mean frequencies on the coupling strength. Panel (a) corresponds to the case of pairwise practically equal natural frequencies: $\gamma_1=3.00268, \gamma_2=3.00821, \gamma_3=3.00233, \gamma_4=3.00847$. The superscripts $A$ and $B$ denote two experiments with different initial conditions. In experiment $A$ at a coupling strength of $d\cong0.33$, one of the 2:2 regimes makes a smooth goes (with increasing $d$, a gradual reduction of the difference between the mean frequencies within the clusters occurs) to the global synchronization regime $\text{(4:0)}_1$ (see Fig. \ref{['fig:Fig_2']}(a)). In experiment $B$, for all values of the coupling parameter shown in the figure, a different 2:2 regime exists. Panel (b) corresponds to the case of non-close (compared to the previous case) natural frequencies: $\gamma_1=2.99987, \gamma_2=3.00158, \gamma_3=3.0053,\gamma_4= 3.00453$. In the ensemble, the global synchronization regime $\text{(4:0)}_2$ is realized (Fig. \ref{['fig:Fig_2']}(b)). In the inset of Fig. \ref{['fig:Fig_1']}(b), the evolution of the mean frequencies at large values of the coupling strength is presented.
• Figure 2: Phase representation of the 4:0 and 2:2 regimes for large values of the coupling parameter $d$. In the case of the 4:0 regimes ((a) -- splay state $\text{(4:0)}_1$, (b) -- in-phase--anti-phase synchronization regime $\text{(4:0)}_2$), all oscillators rotate with constant frequency $\Omega^s$. In the case of the 2:2 regime (c), being in anti-phase, the first and third oscillators rotate with frequency $\Omega_{13}^s$, while the second and fourth oscillators, also being in anti-phase, rotate with frequency $\Omega_{24}^s > \Omega_{13}^s$.
• Figure 3: Evolution of the mean $\Omega_j$ and instantaneous $\omega_j$ frequencies for the 2:2 synchronization regime. Parameters: natural frequencies $\gamma_1=2.99086, \gamma_2=3.00322, \gamma_3=2.99092, \gamma_4=3.00842$, coupling parameter $d=20$. The inset shows the interval of the instantaneous frequencies change.
• Figure 4: Evolution of the mean $\Omega_j$ and instantaneous $\omega_j$ frequencies for the 3:1 synchronization regime. Parameters: intrinsic frequencies $\gamma_1=3.00385, \gamma_2 =3.00571, \gamma_3=3.00958, \gamma_4=3.00627$; coupling strength $d=10$. The insets (a), (b) and (c) show magnified views of the corresponding time intervals where the ratios between the instantaneous frequencies change. For a detailed description, see the main text.
• Figure 5: Evolution of the sine of the half-phase difference $s_{jk}$, for the 3:1 regime. Parameters are the same as those in Fig. \ref{['fig:Fig_4']}. The lines marked with symbols correspond to the equality of the phase differences for two pairs of oscillators. For a detailed description, see the main text.