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arxiv: 2604.14772 · v1 · submitted 2026-04-16 · ⚛️ physics.soc-ph · cond-mat.stat-mech· nlin.AO

A convex-geometric framework for fully phase-locked states in the finite Kuramoto model

Antonio Garijo , Sergio G\'omez , Alex Arenas This is my paper

Pith reviewed 2026-05-10 09:21 UTC · model grok-4.3

classification ⚛️ physics.soc-ph cond-mat.stat-mechnlin.AO
keywords Kuramoto modelphase-locked statesconvex geometrysynchronizationcritical couplingstability regionfrequency vectorspolytope bound
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The pith

The Kuramoto vector field maps the stability region of phase configurations to a convex set in frequency space, so a fully phase-locked state at coupling K exists exactly when the rescaled frequencies lie inside that image.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper characterizes the minimal coupling strength needed for a stable fully phase-locked state in the finite all-to-all Kuramoto model with heterogeneous frequencies. It reduces the system to a co-rotating frame, identifies the region of phase space where the Jacobian is negative definite as the stability region, and shows that the vector field maps this region to a convex set in frequency space. A fully locked equilibrium therefore exists at coupling K precisely when the rescaled frequency vector lies inside the image of that region. The critical coupling is the smallest K for which the ray through the frequency vector first touches the boundary of the convex image. From this geometry the authors construct an explicit polytope that supplies a closed-form upper bound on the critical coupling.

Core claim

The Kuramoto vector field maps the stability region (phase configurations with negative-definite reduced Jacobian) to a convex set in frequency space; consequently a fully phase-locked equilibrium at coupling strength K exists if and only if the rescaled frequency vector lies inside this convex image, and the critical coupling K_ℓ is the first intersection of the ray tω̂ with the boundary of the image. An explicit polytope built from analytically computable boundary points of the stability region yields a closed-form upper bound K_b ≥ K_ℓ that is exact whenever the frequency vector aligns with a vertex of the polytope.

What carries the argument

the convex image of the stability region (negative-definite reduced Jacobian) under the Kuramoto vector field

Load-bearing premise

The Kuramoto vector field maps the stability region to a convex set in frequency space.

What would settle it

For any small fixed N and chosen frequency vector, numerically trace the boundary of the stability region, compute its image under the vector field, and verify whether the predicted ray-intersection K matches the smallest coupling at which a negative-definite Jacobian equilibrium actually appears.

Figures

Figures reproduced from arXiv: 2604.14772 by Alex Arenas, Antonio Garijo, Sergio G\'omez.

Figure 1
Figure 1. Figure 1: Dependence of the Kuramoto order parameter Rˆ on the coupling strength K for a fixed nonzero frequency vector ωˆ with n + 1 = 100. The frequencies are randomly drawn from a normal N(0, 1) distribution, setting ωn+1 = 0, adding an outlier at ω1 (equal to three times the largest of the absolute values of the other frequencies) to increase the heterogeneity, and substracting their mean so as the sum of all ωˆ… view at source ↗
Figure 2
Figure 2. Figure 2: Convex-geometric construction used to bound the critical coupling. Left: the stability region Ω ⊂ T n (shaded) defined by Spec(DF(θˆ)) ⊂ (−∞, 0), with marked boundary points p ± i ∈ ∂Ω. Right: its convex image F(Ω) ⊂ R n (shaded) and the polytope Pn = conv{F(p ± i )} (red), which provides an explicit outer approximation for locating the first intersection of the ray tωˆ and hence an upper bound Kb ≥ Kℓ. di… view at source ↗
Figure 3
Figure 3. Figure 3: Stability region Ω ⊂ T 2 (left) and its convex image F(Ω) ⊂ R 2 (right) for n = 2. The polytope P2 is marked with a dotted black line, together with the sets S1 and S2, their reference points F(p ± 1 ) and F(p ± 2 ), and the diagonal points F(d ±) (right). Each face of the Polytope Pn is contained in a hyperplane of R n . We can assume, without loss of generality, that the equation of a hyperplane, not con… view at source ↗
Figure 4
Figure 4. Figure 4: Zoom of the phase diagram of the Kuramoto model for n + 1 = 100 and with a vector of frequencies laying at one of the vertices of the polytope, namely, ωˆ = −F(p + 1 ) = (n, −1, . . . , −1). Similarly to [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
read the original abstract

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 $\hat{\mathbf{\omega}}/K$ 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 $t\hat{\mathbf{\omega}}$ 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops a convex-geometric framework for fully phase-locked equilibria in the finite all-to-all Kuramoto model. After reducing to a co-rotating frame to remove the uniform phase-shift degeneracy, it defines a stability region in phase space via negative-definiteness of the Jacobian of the reduced vector field. The central claim is that the Kuramoto map (whose components are sums of sine differences) sends this region to a convex set in frequency space; consequently a fully locked state exists at coupling K precisely when the rescaled frequency vector lies inside that convex image, with the critical coupling K_ℓ given by the first intersection of the ray tω̂ with the boundary. From analytically computable boundary points the authors construct an explicit polytope that furnishes a closed-form upper bound K_b ≥ K_ℓ, exact when ω̂ aligns with a vertex.

Significance. If the convexity assertion holds, the work supplies a geometrically transparent existence criterion and an explicit, computable outer bound for the synchronization threshold in finite heterogeneous Kuramoto networks. The polytope construction is parameter-free once the boundary points are fixed and yields exact results along certain rays, which is a concrete strength. The framework could be useful for both theoretical analysis and numerical verification of locking thresholds, provided the convexity property is rigorously established.

major comments (2)
  1. [Abstract / central claim] The assertion that the image of the stability region under the nonlinear Kuramoto vector field is convex is load-bearing for the entire framework (existence condition, ray-intersection definition of K_ℓ, and validity of the polytope bound). The map θ ↦ (∑_j sin(θ_i − θ_j))_i is nonlinear, and the stability region itself may not be convex; a proof that the image is nevertheless convex is therefore required. No such proof is visible in the abstract or the provided outline, and the skeptic correctly flags this as the weakest assumption. If convexity fails, the geometric characterization and the bound K_b collapse.
  2. [Polytope construction] The construction of the polytope from “analytically computable boundary points” is presented as furnishing a closed-form upper bound. It is unclear which specific boundary points are chosen, how they are computed without solving the full nonlinear system, and whether the resulting polytope is guaranteed to contain the convex image (rather than merely intersecting it). A concrete description of the vertex selection and a proof that the polytope outer-approximates the image are needed.
minor comments (2)
  1. [Abstract] The abstract states that the bound “is exact for frequencies aligned with polytope vertices” but does not specify the alignment condition or give an example; a brief illustrative calculation would clarify the claim.
  2. [Introduction / notation] Notation for the reduced co-rotating frame and the rescaled frequency vector ω̂ should be introduced with an explicit equation reference the first time each appears.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments identify points where the manuscript requires greater rigor and explicit detail. We address each below and will revise the paper accordingly.

read point-by-point responses

  1. Referee: [Abstract / central claim] The assertion that the image of the stability region under the nonlinear Kuramoto vector field is convex is load-bearing for the entire framework (existence condition, ray-intersection definition of K_ℓ, and validity of the polytope bound). The map θ ↦ (∑_j sin(θ_i − θ_j))_i is nonlinear, and the stability region itself may not be convex; a proof that the image is nevertheless convex is therefore required. No such proof is visible in the abstract or the provided outline, and the skeptic correctly flags this as the weakest assumption. If convexity fails, the geometric characterization and the bound K_b collapse.

    Authors: We agree that a self-contained proof of convexity is indispensable. The manuscript asserts that the Kuramoto map sends the stability region (defined by negative-definiteness of the reduced Jacobian) to a convex set in frequency space, but the referee is correct that the argument is not presented with sufficient detail or visibility. In the revised version we will insert a dedicated subsection that proves the image is convex. The argument proceeds by showing that any convex combination of two frequency vectors in the image can be realized by a suitable convex combination of the corresponding phase configurations within the stability region, exploiting the oddness and concavity properties of the sine function together with the linear structure of the frequency map. This establishes both the existence criterion and the ray-intersection definition of K_ℓ. revision: yes

  2. Referee: [Polytope construction] The construction of the polytope from “analytically computable boundary points” is presented as furnishing a closed-form upper bound. It is unclear which specific boundary points are chosen, how they are computed without solving the full nonlinear system, and whether the resulting polytope is guaranteed to contain the convex image (rather than merely intersecting it). A concrete description of the vertex selection and a proof that the polytope outer-approximates the image are needed.

    Authors: We will expand the polytope construction section to specify the exact boundary points selected. These points correspond to analytically solvable phase configurations on the boundary of the stability region (for example, configurations in which a subset of oscillators are synchronized at 0 or π relative to the mean, allowing the frequency vector to be expressed in closed form via elementary trigonometric identities without numerical solution of the full system). We will prove that the convex hull of these points contains the entire image by demonstrating that every point in the image lies inside the polytope via supporting hyperplanes derived from the same boundary configurations. The revised manuscript will include both the explicit vertex list and the outer-approximation argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; geometric mapping and polytope are constructed from Jacobian stability analysis

full rationale

The derivation begins by defining the stability region explicitly from the negative definite Jacobian of the reduced system, then states that the Kuramoto vector field maps this region to a convex set whose boundary determines K_ℓ via ray intersection. This is a direct claim about the image of a defined set under the sine-coupling map, not a redefinition of inputs, a fitted parameter renamed as prediction, or a load-bearing self-citation. The polytope bound is built from analytically computable boundary points of the same stability region. No step reduces by construction to prior outputs of the paper itself; the framework is self-contained as a geometric characterization.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the asserted convexity of the image of the stability region and the use of negative definiteness of the Jacobian to define stable locked states; these are domain assumptions from dynamical systems theory.

axioms (1)
  • domain assumption The Kuramoto vector field maps the stability region (Jacobian negative definite) to a convex set in frequency space.
    This convexity is the load-bearing property that enables the ray-intersection definition of K_ℓ and the polytope bound.

pith-pipeline@v0.9.0 · 5592 in / 1468 out tokens · 71573 ms · 2026-05-10T09:21:18.165184+00:00 · methodology

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

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    Prepared and with a preface by Volker Kaibel, Victor Klee and Gunter M

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    Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. 17 [PRK01] Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths.Synchronization: A Uni- versal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge, 2001. [Str00] Steven H. Strogatz. From Kuramoto to Crawford: exploring the onset of synchroniza- tion in popula...

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