Kuramoto global attractor convergence to the Ott–Antonsen manifold

Let $\Phi_\mathcal H$ be the graph parametrizing the two-dimensional global attractor $W^u(\frac{1}{2\pi})$, let $q_{OA}(z)$ denote the corresponding state on the Ott–Antonsen manifold, and let $\mathcal G_a$ be the Gevrey space used in the paper. Ott–Antonsen convergence conjecture. For any $\alpha\in[0,1[$ and any $z\in[0,\alpha]$,

$$\Phi_\mathcal H(z)\underset{K\to+\infty}{\longrightarrow}q_{OA}(z)\quad\text{in }\mathcal G_a\quad\forall\,1<a<\frac{1}{\alpha}.$$

Moreover, the two-dimensional global attractor $W^u(\frac{1}{2\pi})$ converges to the two-dimensional invariant manifold $\mathcal M_{OA}$ of the Kuramoto equation with no diffusion in analytic-function space, with speed $\frac{1}{K}$ as $K\to+\infty$. This is one of the paper's two conjectures, supported by numerical observations; the convergence and rate are not established in the supplied text.