Microscopic correlations in the finite-size Kuramoto model of coupled oscillators

Supercritical Kuramoto oscillators with distributed frequencies can be separated into two disjoint groups: an ordered one locked to the mean field, and a disordered one consisting of effectively decoupled oscillators—at least so in the thermodynamic limit. In finite ensembles, in contrast, such clear separation fails: The mean field fluctuates due to finite-size effects and thereby induces order in the disordered group. This publication demonstrates this effect, similar to noise-induced synchronization, in a purely deterministic system. We start by modeling the situation as a stationary mean field with additional white noise acting on a pair of unlocked Kuramoto oscillators. An analytical expression shows that the cross-correlation between the two increases with decreasing ratio of natural frequency difference and noise intensity. In a deterministic finite Kuramoto model, the strength of the mean-field fluctuations is inextricably linked to the typical natural frequency difference. Therefore, we let a fluctuating mean field, generated by a finite ensemble of active oscillators, act on pairs of passive oscillators with a microscopic natural frequency difference between which we then measure the cross-correlation, at both super- and subcritical coupling.

Figure 1

Snapshot of passive oscillators (black dots) and active oscillators (large red dots) phases plotted against natural frequencies for a finite ensemble of 50 active oscillators (not all shown). The mean field is created by the active oscillators with natural frequencies drawn randomly from a Gaussian distribution (same sample as in Fig. 2) at slightly supercritical coupling $\varepsilon =1.85$ (same in Fig. 2). Three zoom levels of factor 10 are marked by shaded areas. Arrows in panel (c) point at ${\mathrm{\Omega }}_k\approx 1.7145$ and ${\mathrm{\Omega }}_k\approx 1.72$ for a new and an old phase slip, respectively. See the Supplemental Material [28] for an animated version of this figure. One second in the video equals one time unit.

Figure 2

Observed cross-correlation as defined by Eq. (3) of passive oscillators coupled to the mean field $Z$ of 50 active oscillators against their natural frequencies (mean $\langle R\rangle =0.58$, variance ${\sigma }_R^2=3.5\times {10}^{-3}$), shown for various levels of frequency mismatch $\mathrm{\Delta }$ between passive oscillators. (a) Absolute value of the (time-averaged) first harmonic of individual passive oscillators before ($z^{\tilde{\phi }}$, blue circles) and after ($z^{\psi }$, red squares) the Möbius transform. $z^{\psi }$ drops to near zero after the transformation, which shows that $\psi$ is now uniformly rotating. (b) Observed vs natural frequencies of passive (line) and active oscillators (dots). Gray vertical lines in panels (b) and (c) mark the natural frequencies of the active oscillators. (c) Synchronization index $\gamma$ between pairs of passive oscillators with natural frequencies ${\mathrm{\Omega }}_k\pm \mathrm{\Delta }/2$.

Figure 3

The cross-correlations of passive oscillators coupled to a mean field of 50 active ones at $\varepsilon =1$ (same natural frequency sample as Fig. 2). The mean field $Z$ fluctuates around zero; see Ref. [10]. (a) Cross-correlations in the linear scale; (b) the same in logarithmic scale to resolve the region $\gamma \lesssim 1$. Markers and colors as in Fig. 2.

Figure 4

Comparison of the observed cross-correlation $\gamma$ in sub- and supercritical ensembles ($\varepsilon =1$ and 1.85 with open and filled symbols, respectively) of different sizes $N_{\mathrm{osc}}=25,50,100,200$ with the theoretical expression (6) (solid line). The data points were generated in experiments as in Figs. 2 and 3. The median of all $\gamma$ for which $\gamma \left({10}^{-2}\right)<0.5$ (thereby excluding locked passive oscillators) is represented with error bars that mark the 25th and 75th percentiles, respectively. The value $d=\mathrm{\Delta }/{\sigma }^2$ is determined from the average over ${\sigma }^2=2{\varepsilon }^{2}D\{1+(|{\mathrm{\Omega }}_k-\overline{\omega }{|/\mathrm{\Delta })}^2\}$ with diffusion coefficient $D$, which was integrated from the autocorrelation function of $R$. For each ensemble size, only one set of natural frequencies is represented.