Selected structures in a finite model of coupled oscillators approach the Ott-Antonsen reduced description at a rate that scales inversely with the number of oscillators, according to a mathematical analysis. The structures are unstable manifolds, and the same rate holds for their continuum counterpart after a separate step-function representation. The paper also establishes a mean-field limit that is uniform in time for trajectories lying on the finite unstable manifolds. Because the study analyzes equations, equilibria, invariant manifolds and trajectories rather than an empirical sample, the result comes from deterministic analysis rather than experimental measurement.
A model with strong assumptions
The model is the homogeneous, all-to-all Kuramoto system: every oscillator has the same natural frequency, and every oscillator is coupled to every other. The study analyzes equations rather than participants or a dataset. That scope matters because the claims are tied to this model structure, not to heterogeneous frequencies or non-all-to-all networks.
The geometry of the equilibrium
At the equilibrium the paper calls incoherent, the spectral analysis identifies one reported eigenvalue at zero and another at half the coupling strength K. The eigenspace for the zero value is orthogonal to the span of the cosine and sine vectors, while the eigenspace for the second value is spanned by those vectors. The construction also uses a conserved mean phase, the average of all oscillator phases, so trajectories remain in a corresponding hyperplane. For the family of incoherent equilibria, the paper gives an explicit two-dimensional unstable-manifold parametrization, with the main theorem stated for systems containing at least three oscillators.
Two measures of convergence
To compare the finite and population-level objects, the authors use Hausdorff distance, a measure of how far two sets are from one another. In one construction, finite states are lifted to empirical measures before comparison with the Ott-Antonsen manifold. In another, they are lifted to step functions before comparison with the continuum counterpart. In both constructions, the stated manifold-distance error is of order 1/N when the manifold parameter beta runs from zero to a cutoff strictly below one. The bounds come with constants that depend on the parameter range, and the same manifold-distance result is not supplied at the synchronized boundary.
From sets to trajectories
Convergence of sets is only part of the result. As the oscillator count tends to infinity, the dynamics on the finite unstable manifolds simplify to, and are identified with, the reduced dynamics on the Ott-Antonsen manifold. The paper also establishes a uniform-in-time mean-field limit for trajectories on those manifolds. In practical terms, the theorem controls the approximation across time for the specified initial-data sequences and parameter regimes, rather than asserting a result at just one selected time.
Synchronization, with a boundary case
The analysis also describes long-term synchronization under its stated conditions. For initial beta strictly between zero and one, Ott-Antonsen trajectories converge exponentially in p-Wasserstein distance, a way to measure separation between probability distributions, toward a synchronized state, with p at least one. Finite-model trajectories on the unstable manifolds converge to their synchronized phase state when the oscillator count is at least three. The result is conditional on the specified initial data and invariant manifolds.
The rate has a boundary
The main caveat concerns the boundary at beta equal to zero. When the limiting initial beta is zero, uniform-in-time convergence cannot generally be expected for an arbitrary sequence of finite initial beta values; the paper says only a long-time convergence result is then available. A stronger statement is available when both the limiting and finite initial beta values are exactly zero for every oscillator count: the paper obtains a uniform-in-time Wasserstein error bound proportional to a theorem constant divided by that count. The stated rate applies only under that exact-zero initialization.
What the result does not cover
The conclusions remain tied to the explicitly constructed setting. They cover the finite-dimensional homogeneous all-to-all model and trajectories lying on its finite unstable manifolds, not arbitrary network structures or arbitrary trajectories. The analysis contains no empirical sample, so it does not provide experimental evidence about real oscillator populations. Its geometric comparisons are conditional on the stated parameter ranges and initial-data sequences used by the theorems.
Paper data and sources
Original title: Unstable Manifolds for the Kuramoto Model: Convergence to the Ott-Antonsen Manifold
Authors: Christian Kuehn, Giacomo Landi
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
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