The size of the sync basin resolved

Abstract

Sparsely coupled Kuramoto oscillators offer a fertile playground for exploring high-dimensional basins of attraction due to their simple yet multistable dynamics. For $n$ identical Kuramoto oscillators on cycle graphs, it is well known that the only attractors are twisted states, whose phases wind around the circle with a constant gap between neighboring oscillators ($θ_j = 2πq j/n$). It was conjectured in 2006 that basin sizes of these twisted states scale as $e^{-kq^2}$ to the winding number $q$. Here, we provide new numerical and analytical evidence supporting the conjecture and uncover the dynamical mechanism behind the Gaussian scaling. The key idea is that, when starting with a random initial condition, the winding number of the solution stabilizes rapidly at $t \propto \log n$, before long-range correlation can develop among oscillators. This timescale separation allows us to calculate the winding number as a sum of weakly-dependent random variables, leading to a Central Limit Theorem derivation of the basin scaling.

Figures (4)

• Figure 1: Distribution of the winding number $q$ at different times $t$ when starting from random initial conditions at $t=0$. The distribution remains Gaussian for all $t$, with the variance gradually decreasing during the early stage of the evolution but quickly converges to its final shape. Here, we set the number of oscillators $n=1280$ and each distribution is estimated from $10^5$ samples.
• Figure 2: Time till the winding number stops changing ($t_s$) and time till the system enters the invariant region $\mathcal{I}$ ($t_e$) both scale as $\log n$. This will be shown more rigorously in Step 3. Each curve is averaged over $10^4$ trajectories starting from random initial conditions and the shaded bands represent standard deviations. We always have $t_s \leq t_e$ for any individual trajectory, which is the point of Step 1.
• Figure 3: No long-range correlation between oscillators can develop before the winding number stops changing. Here, for $n=1280$ oscillators, we calculate the Pearson correlation $r$ between two oscillators that are distance $d$ apart at $t=t_s$ (winding number stabilized) and $t=t_e$ (entering the invariant region $\mathcal{I}$). The shaded area mark the expected $|r|$ for two random vectors whose entries are chosen uniformly and independently between $-\pi$ and $\pi$, $\mathbb E(|r|) = \frac{\sqrt{2}}{\sqrt{\pi}\sqrt{n-1}}\approx 0.0223$. The oscillators are essentially uncorrelated unless they are very close to each other ($d \leq 6$). We show the lack of long-range correlation analytically in Step 2.
• Figure 4: Distribution of the times $t_e^{(i)}$ at which phase differences $\eta_i$ enter $(-\pi/2,\pi/2)$. Half of the $\eta_i$ are already inside $(-\pi/2,\pi/2)$ for random initial conditions (thus the spike at $t=0$), while the nonzero entering times follow an exponential distribution. The data are collected from $10^4$ independent simulations of $n=1280$ Kuramoto oscillators from random initial conditions. The exponential distribution of $t_e^{(i)}$ is a key ingredient for Step 3.