Enhanced synchronization with proportional coupling in Kuramoto oscillator networks

Abstract

We introduce a novel coupling scheme for maximizing the synchronization of Kuramoto oscillator networks under a fixed coupling budget. We show that by scaling the interaction strength between oscillators according to their frequency detuning, synchronization is enhanced. The coupling scheme induces a change in criticality, driving the system from a continuous phase transition to an explosive transition by changing a single parameter. Our work offers a general route to efficient synchronization in engineered networks and provides insight into the critical behavior of the Kuramoto model.

Figures (9)

• Figure 1: Synchronization with proportional and uniform coupling for a network of $N=512$ Kuramoto oscillators with normally distributed natural frequencies. (a): Average order parameter $\bar{r}$ vs. normalized coupling strength $K/K_c$ with proportional coupling (yellow circles) compared to that with uniform all-to-all coupling (blue squares). Dashed black line shows the exact solution of Eq. \ref{['kuramoto_eqn']} with uniform coupling for $N\rightarrow \infty$RevModPhys.77.137. Inset: Histogram of the order parameter $r$ for $K/K_c=1.09$ showing a bimodal distribution in the case of proportional coupling in contrast to a unimodal distribution in the case of uniform coupling. (b-c): Hysteresis curves for uniform (b) and proportional (c) coupling. Red circles (blue crosses) correspond to the forward (backward) direction of adiabatically changing $K/K_c$.
• Figure 2: Effect of network size $N$ on synchronization with proportional coupling. (a) Synchronization order parameter $\bar{r}$ as a function of $N$ for $N = 16 \text{(pink)}$, $64\text{(blue)}$, $256\text{(orange)}$, $1024\text{(green)}$, $4096\text{(teal)}$.(b) Transition point $K(\bar{r}=0.5)/K_c$ (marked in a black dotted line in (a)) as function of $N$. (c) Inverse slope at the transition point as a function of $N$.
• Figure 3: Synchronization order parameter $\bar{r}$ as a function of the normalized coupling strength $K/K_c$ with partially correlated proportional coupling. The correlation between the coupling terms and the detuning differences is quantified by $\varepsilon=\text{Corr}(K_{ij},\abs{\Omega_i-\Omega_j})$. Uniform coupling (dashed line) is very similar to the random coupling case.
• Figure 4: Synchronization of a network of $N = 512$ oscillators for $K_{ij}\propto\abs{\Omega_i-\Omega_j}^p$. A sharp transition can be seen for $1<p<3$.
• Figure 5: Enhanced synchronization with proportional coupling. (a)$p_{opt}$, the optimal value of $p$ that generates the maximal $\bar{r}$ for each $K/K_c$ for proportional coupling. (b)$\bar{r}_{max}$ obtained for each $p_{opt}$ of (a) (blue circles) is higher than for uniform coupling (black dots) for all values of $K/K_c$.