Single-point convergence for analytic gradient flows

The analytic potential $U$ is smooth by [F1], and its negative Riemannian gradient is a smooth vector field because $g$ is smooth and positive definite. Apply the completeness corollary on compact $M$ under countable choice: the maximal trajectory through any initial state at time $0$ is defined for every real time, in particular for all $t\ge 0$. Its forward image lies in its connected component $C$, since a connected time interval has connected image. The component is open by local connected coordinate balls and closed by the connected-component property, hence compact as a closed subset of $M$. Restricting the atlas, metric and potential to $C$ gives the hypotheses of the manifold point-limit lemma. That lemma supplies finite forward length and convergence to one critical point. Since $C$ is open, this is also a critical point on $M$. Empty $M$ has no trajectories; a zero-dimensional component is a singleton and the conclusions hold for its constant trajectory.

For the homogeneous Kuramoto system use the finite phase torus with its flat metric. Its energy is the analytic potential from [F2], and the rotating-frame gradient supplier identifies precisely the downward flow. The periodic convergence proposition therefore supplies one limiting critical point, equivalently an equilibrium. The general statement in step 1.1 also applies; the torus proposition is a separate direct route to this special case.

In the original frame the common added rotation cancels in every difference. The map taking torus phases to their difference in $\mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$ is continuous, so convergence of the rotating-frame phases makes every difference converge to a constant circle phase. This is the phase-locked long-term behavior in the declared definition. No convergence of the common original-frame rotation itself is asserted.