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 (${\theta }_j=2\pi qj/n$). It was conjectured in 2006 that basin sizes of the twisted states scale as $e^{-k{q}^2}$ with the winding number $q$. Here, we provide 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 $t\propto logn$, 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.

Close

PrevNext

View figure in article

• Coupled oscillators
• Synchronization
• Dynamical systems