2026/04/30 by Antonio Garijo, Sergio Gómez, Alex Arenas
Physics and Astronomy · #physics.soc-ph #cond-mat.stat-mech #nlin.AO
paper · pdf · doi:10.1088/2632-072x/ae8d6f
published as Journal of Physics: Complexity 7 (2026) 035009 · 18 pages, 4 figures
arxiv created 2026/07/15 · arxiv updated 2026/08/04
We study the finite-size Kuramoto model of all-to-all coupled phase oscillators with heterogeneous natural frequencies and characterize the minimal coupling strength required for the existence of a fully phase-locked equilibrium (in a co-rotating frame). To remove the degeneracy due to uniform phase shifts, we move to a reduced co-rotating frame and assess stability through the Jacobian of the reduced system: a fully phase-locked state is stable when this Jacobian is negative definite. This defines a stability region in the phase space. The Kuramoto vector field maps this region to a convex set in frequency space, so a fully-locked state at coupling K exists exactly when the rescaled frequency vector \mathbfω/K lies inside that convex image. The critical coupling Kℓ is defined as the smallest coupling strength for which a fully phase-locked equilibrium exists; geometrically, it corresponds to the first intersection of the ray t\mathbfω with the boundary of this convex set. Building on this convex-geometric structure, we construct an explicit polytope from analytically computable boundary points of the stability region, providing a closed-form upper bound Kb ≥ Kℓ. The bound is exact for frequencies aligned with polytope vertices and offers a fully explicit outer approximation for general frequency vectors. While not uniformly sharp in a quantitative sense, this construction exposes the underlying geometry of stable fully phase-locking solutions. These results provide a practical use the convex-geometric structure underlying stable fully-locked states in the Kuramoto model.