A convex-geometric framework for fully phase-locked states in the finite Kuramoto model
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910134964322304 |
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| author | Garijo, Antonio Gómez, Sergio Arenas, Alex |
| author_facet | Garijo, Antonio Gómez, Sergio Arenas, Alex |
| contents | 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 $\hat{\mathbfω}/K$ lies inside that convex image. The critical coupling $K_{\ell}$ 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 $t\hat{\mathbfω}$ 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 $K_b \ge K_{\ell}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14772 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A convex-geometric framework for fully phase-locked states in the finite Kuramoto model Garijo, Antonio Gómez, Sergio Arenas, Alex Physics and Society Statistical Mechanics Adaptation and Self-Organizing Systems 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 $\hat{\mathbfω}/K$ lies inside that convex image. The critical coupling $K_{\ell}$ 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 $t\hat{\mathbfω}$ 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 $K_b \ge K_{\ell}$. 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. |
| title | A convex-geometric framework for fully phase-locked states in the finite Kuramoto model |
| topic | Physics and Society Statistical Mechanics Adaptation and Self-Organizing Systems |
| url | https://arxiv.org/abs/2604.14772 |