Feedback control of the Kuramoto model defined on uniform graphs I: Deterministic natural frequencies

Kazuyuki Yagasaki

arxiv: 2505.02196 · v4 · pith:TPPWEFTAnew · submitted 2025-05-04 · 🧮 math.DS

Feedback control of the Kuramoto model defined on uniform graphs I: Deterministic natural frequencies

Kazuyuki Yagasaki This is my paper

Pith reviewed 2026-05-22 17:12 UTC · model grok-4.3

classification 🧮 math.DS

keywords Kuramoto modelfeedback controlsynchronized solutionsbifurcationsstability analysisuniform graphsoscillator networks

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The pith

In the controlled Kuramoto model on uniform graphs with uniform natural frequencies, exactly 2^n synchronized solutions exist for n nodes, and only one is stable while converging to the target rotation as feedback gain grows.

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

The paper applies feedback control to the Kuramoto model so that all oscillators settle into one shared constant rotation rather than their individual natural speeds. For three or more nodes it proves the controlled system always possesses precisely 2^n synchronized solutions and classifies the saddle-node and pitchfork bifurcations that create or destroy them. Stability calculations then show that a single solution remains attractive while the rest are unstable, and this stable solution approaches the exact desired rotation when the feedback strength is made arbitrarily large. The same pattern carries over to the continuous limit of infinitely many nodes and continues to describe the behavior on large random dense or sparse graphs.

Core claim

For the controlled Kuramoto model with uniformly spaced natural frequencies defined on uniform graphs, when the number of nodes n is at least 3, there exist exactly 2^n synchronized solutions whose saddle-node and pitchfork bifurcations are established and whose stability is determined, with only the solution converging to the desired motion in the limit of infinite feedback gain being stable and the others unstable.

What carries the argument

The controlled Kuramoto model (CKM) obtained by adding feedback to enforce a common constant rotational motion, together with the explicit count and stability classification of its synchronized solutions.

If this is right

As feedback gain tends to infinity the unique stable synchronized solution converges to the exact desired common rotation.
The asymptotically stable solution identified in the continuous-limit model remains asymptotically stable in the finite-node model once the node count is large enough.
The long-term behavior of the model on large random dense or sparse uniform graphs is captured by the continuous-limit solution.
All but one of the 2^n synchronized solutions are unstable for any n at least 3.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same feedback law may therefore be used to enforce uniform rotation on large random networks without needing to know the precise edge set in advance.
Adding more nodes increases the number of unstable equilibria but leaves the existence and stability of the target solution unchanged.
The bifurcation diagram could be used to predict the minimal gain needed to reach the stable regime for any given n.

Load-bearing premise

Natural frequencies are uniformly spaced and the underlying graphs are uniform, either complete or random with constant edge probability.

What would settle it

For n=3 a numerical search that locates any number of synchronized solutions other than exactly eight, or a simulation that finds more than one stable synchronized solution at finite gain, would contradict the stated count and stability result.

Figures

Figures reproduced from arXiv: 2505.02196 by Kazuyuki Yagasaki.

**Figure 1.** Figure 1: Function ¯χ σ (ξ) for n = 4: (a) and (b) β = 1; (c) and (d) 0.1. In plates (a) and (c) (resp. plates (b) and (d)) the sign ‘+’ (resp. ‘−’) is taken in (3.8). See the text for more details. Corollary 3.3. Fix the value of pK > 0. If ξ ∈ (0, 1] satisfies (3.8) for σ ∈ Σn and b1 > 0, then v σ given by (3.5) with φi = ± arcsin 2i − n − 1 n − 1 ξ , instead of (3.6), is an equilibrium in (3.2), where the uppe… view at source ↗

**Figure 2.** Figure 2: Bifurcation diagrams of equilibria in (3.18): (a) c1/c2 < 0; (b) c1/c2 > 0. for i ∈ [n] when b1 = pK/χ¯ σ (ξ), where fi(v; b1) is the ith element of f(v; b1) and the upper or lower sign is taken simultaneously, depending on whether pKχσ (ξ)+b1 is positive or not. Suppose that (d ¯χ σ/dξ)(ξ) has a zero at ξ = ξ∗ and let b1∗ = pK/χ¯ σ (ξ∗) > 0 be bounded. Then the Jacobian matrix Dvf(v σ (ξ∗); b1∗) has a sim… view at source ↗

**Figure 3.** Figure 3: Pixel pictures of sampled weighted matrices for the random undirected graphs given by w n ij = 1, i, j ∈ [n] with probability (6.1) and (6.2) for n = 1000: (b) Dense graph with p = 0.5; (c) Sparse graph with p = 0.5 and γ = 0.3. The color of the corresponding pixel is blue if wij = 1 and it is light blue otherwise. We specifically take p = 0.5 in case (ii) and p = 0.5 and γ = 0.3 in case (iii). Figures … view at source ↗

**Figure 4.** Figure 4: Numerical simulation results of the CKM (1.7) with n = 1000, K = 0.5, V1, b0 = 1, V0 = 1 and b1 = 0.2: (a) (a, p) = (1, 1) in case (i); (b) (0.5, 0.5) in case (ii); (c) (a, p, γ) = (0.5, 0.5, 0.3) in case (iii). The time-history of every 100th node (from 50th to 950th) is plotted with different colors. Figures 6(a), (b) and (c) display the maximal and minimal deviations of the steady-state responses from t… view at source ↗

**Figure 5.** Figure 5: Deviations of steady-state responses from the desired motion in the CKM (1.7) with n = 1000, K = 0.5, V1, b0 = 1, V0 = 1 and b1 = 0.2: (a) (a, p) = (1, 1) in case (i); (b) (0.5, 0.5) in case (ii); (c) (a, p, γ) = (0.5, 0.5, 0.3) in case (iii). Here u n i (t) − V (t), i ∈ [n], with t = 100 are plotted as red dots. The blue line represents the corresponding theoretical predictions computed from the synchroni… view at source ↗

**Figure 6.** Figure 6: Maximal and minimal deviations of the steady-state responses from the desired motion in the CKM (1.7) with n = 1000, K = 0.5, V1, b0 = 1 and V0 = 1 when the feedback gain b1 is changed: (a) (a, p) = (1, 1) in case (i); (b) (0.5, 0.5) in case (ii); (c) (a, p, γ) = (0.5, 0.5, 0.3) in case (iii). Here maxi∈[n](u n i (t) − V (t)) and mini∈[n](u n i (t) − V (t)), i ∈ [n], for t > 0 sufficiently large are plott… view at source ↗

read the original abstract

We study feedback control of the Kuramoto model with uniformly spaced natural frequencies defined on uniform graphs which may be complete, random dense or random sparse. The control objective is to drive all nodes to the same constant rotational motion. For the case of node number $n\ge 3$, we establish the existence of exactly $2^n$ synchronized solutions in the controlled Kuramoto model (CKM) and their saddle-node and pitchfork bifurcations, and determine their stability. In particular, we show that only a solution converging to the desired motion in the limit of infinite feedback gain is stable and the others are unstable. Based on the previous results, it is shown that (i) the solution to which the stable synchronized solution in the CKM converge as $n\to\infty$ is always asymptotically stable in the continuous limit (CL) if it exists, and (ii) the asymptotically stable solution of the CL captures the asymptotic behavior of the CKM when the node number is sufficiently large, even if the graphs are random dense or sparse. We demonstrate the theoretical results by numerical simulations for the CKM on complete simple, and uniform random dense and sparse graphs.

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.

Desk Editor's Note private letter to a colleague

Exactly 2^n synchronized solutions with only the target one stable under uniform frequencies and graphs, but the random cases lean on how far the symmetry carries.

read the letter

The one or two things to know are that under uniform natural frequencies and uniform graphs the controlled Kuramoto model has exactly 2^n synchronized solutions for n greater than or equal to 3, with only the one converging to the target at high gain being stable, and that this stable behavior transfers to the continuous limit as n grows large. The paper does well by delivering the explicit count and the bifurcation details for this setting. It applies standard techniques to the controlled equations and backs them with simulations on complete, dense random, and sparse random graphs. This gives a clearer picture than previous control studies that did not enumerate all solutions this way. The soft spots are around the uniformity requirement. The 2^n count and clean stability classification use the regular spacing to simplify the locked-state equations. For the random graphs the paper includes, the same counting may not be exact on every sample, though the stability of the target solution and the large-n limit might still hold approximately. The stress test note highlights this, and it seems accurate based on the abstract. This work is for people studying control of coupled oscillators in networks. A reader wanting precise results on symmetric cases or large-scale limits will get something from it. It deserves a serious referee because the claims are specific and the approach is reproducible enough to review.

Referee Report

2 major / 2 minor

Summary. The manuscript studies feedback control of the Kuramoto model with uniformly spaced natural frequencies on uniform graphs (complete, random dense, or random sparse). For n ≥ 3 it claims to establish the existence of exactly 2^n synchronized solutions in the controlled Kuramoto model, together with their saddle-node and pitchfork bifurcations and a complete stability classification; only the solution that converges to the target rotational motion as feedback gain tends to infinity is stable. The paper then shows that the stable solution of the continuous limit (n → ∞) is asymptotically stable when it exists and that this limit captures the large-n behavior of the finite-n system even for random graphs, with numerical simulations on complete, dense, and sparse graphs.

Significance. If the derivations hold, the work supplies an explicit combinatorial enumeration of equilibria and a sharp stability result under the stated symmetry assumptions, which is a concrete strength for the deterministic-frequency case. The passage to the continuous limit and the claim that the same stability picture persists for random graphs (dense or sparse) would extend the utility to large-scale networks. The numerical illustrations provide direct verification of the bifurcation and stability predictions.

major comments (2)

[Sections treating random graphs (likely §3–4)] The extension of the exact 2^n count and the stability classification to random dense and sparse graphs is load-bearing for the headline statements. The counting argument relies on the uniform spacing of frequencies and the regularity of the graph; it is not immediate that the same algebraic reduction and root count survive pathwise (or even in expectation) once the adjacency matrix is random. A precise statement of whether the result holds almost surely, in probability, or only for the expectation of the system is required.
[Continuous-limit analysis (likely §5)] The claim that the asymptotically stable solution of the continuous limit captures the asymptotic behavior of the controlled Kuramoto model for sufficiently large but finite n, even on random graphs, needs quantitative support. Convergence rates, uniform error bounds between the finite-n equilibria and the limit, or a precise mode of convergence (almost sure, in probability) should be supplied; without them the passage from the finite-n stability result to the continuous-limit statement remains formal.

minor comments (2)

[Notation and model equations] Notation for the feedback gain and the common rotation frequency should be introduced once and used consistently; occasional redefinition of symbols across sections makes the stability arguments harder to follow.
[Numerical simulations] The numerical section would benefit from explicit reporting of the random-graph parameters (edge probability for dense/sparse cases) and from overlaying the theoretically predicted bifurcation values on the simulation plots.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the careful reading and the recommendation for minor revision. The comments help us clarify the scope of our results on random graphs and the continuous limit. We respond to each major comment below.

read point-by-point responses

Referee: [Sections treating random graphs (likely §3–4)] The extension of the exact 2^n count and the stability classification to random dense and sparse graphs is load-bearing for the headline statements. The counting argument relies on the uniform spacing of frequencies and the regularity of the graph; it is not immediate that the same algebraic reduction and root count survive pathwise (or even in expectation) once the adjacency matrix is random. A precise statement of whether the result holds almost surely, in probability, or only for the expectation of the system is required.

Authors: We thank the referee for this important remark. The exact count of 2^n synchronized solutions and the complete stability classification are proven for the case of uniform regular graphs, where the algebraic structure allows a direct reduction to a polynomial equation with exactly 2^n roots. For random dense and sparse graphs, the manuscript does not claim a pathwise exact count for every realization; instead, the results on stability and the capture of asymptotic behavior are established through the convergence to the continuous limit as n → ∞. In the revised version, we will explicitly state that the 2^n enumeration holds deterministically for regular uniform graphs, while for random graphs the stability properties hold in probability, leveraging the fact that random regular graphs concentrate around the mean-field limit. revision: yes
Referee: [Continuous-limit analysis (likely §5)] The claim that the asymptotically stable solution of the continuous limit captures the asymptotic behavior of the controlled Kuramoto model for sufficiently large but finite n, even on random graphs, needs quantitative support. Convergence rates, uniform error bounds between the finite-n equilibria and the limit, or a precise mode of convergence (almost sure, in probability) should be supplied; without them the passage from the finite-n stability result to the continuous-limit statement remains formal.

Authors: We agree that a more quantitative justification would be desirable. The paper establishes the asymptotic stability of the limiting solution in the continuous limit and supports the approximation for large finite n through both theoretical arguments based on the law of large numbers for the graph structure and extensive numerical simulations on complete, dense, and sparse random graphs. In the revision, we will add a clarification specifying that the convergence is in probability for random graphs and that the finite-n equilibria approach the continuous-limit equilibrium as n increases. While we do not provide explicit convergence rates in the current work, the numerical evidence and the stability analysis in the limit provide strong support for the claims. We will also note that deriving sharp error bounds is a natural extension for future work. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation uses symmetry assumptions for explicit count but remains self-contained

full rationale

The paper derives the exact count of 2^n synchronized solutions, their bifurcations, and stability directly from the controlled Kuramoto equations under the stated uniform-frequency and uniform-graph assumptions, applying standard algebraic reduction and bifurcation criteria without any self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claims to prior unverified results. The extension to random dense/sparse graphs is justified by the persistence of the symmetry in the large-n limit or pathwise, which is an independent modeling step rather than a circular reduction. The analysis is therefore self-contained against the model equations and external mathematical tools.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The analysis relies on standard dynamical-systems assumptions (existence of equilibria, bifurcation theory) and the modeling choice of uniform spacing and uniform graphs; no free parameters or invented entities are mentioned.

axioms (2)

standard math Standard results from bifurcation theory and stability analysis for finite-dimensional dynamical systems apply to the controlled Kuramoto equations.
Invoked implicitly to classify saddle-node and pitchfork bifurcations and to determine stability of the 2^n solutions.
domain assumption The graphs are uniform (complete, random dense, or random sparse) and natural frequencies are deterministically uniformly spaced.
This regularity is required to obtain the exact count of synchronized solutions and to pass to the continuous limit.

pith-pipeline@v0.9.0 · 5733 in / 1482 out tokens · 38260 ms · 2026-05-22T17:12:58.198208+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

For each σ ∈ Σ_n ... v^σ_i = φ_i or π-φ_i ... exactly 2^n synchronized solutions ... saddle-node and pitchfork bifurcations
IndisputableMonolith/Foundation/DimensionForcing alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

uniformly spaced natural frequencies ω^n_i = a(2i-n-1)/(2n) and uniform graphs W(x,y)=p

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