Manifold decomposition conjecture for Kuramoto graph equilibria

Let $G$ be a graph, and consider the set of its Kuramoto equilibria, namely the phase configurations satisfying the equilibrium equation

Manifold decomposition conjecture. For every graph, the set of equilibria of the Kuramoto network is a finite union of manifolds. The question arises because the equilibrium set is a finite union of algebraic varieties, which can have singular points and therefore need not themselves be manifolds; the conjecture asserts that these singularities can nevertheless be decomposed into finitely many manifold pieces.

Progress summary

Solved

An unverified posted proof claims the conjecture follows from triangulating the algebraic equilibrium set, while the published literature records no confirmed solution.

Sclosa’s 2022 paper records the conjecture that every graph’s Kuramoto equilibrium set is a finite union of manifolds. It explains that algebraic-variety decompositions alone do not establish this conclusion.

Known results

Sclosa, 2022: for every $d\geq 1$ , some connected graph has a $d$ -dimensional manifold of stable equilibria; this does not settle the decomposition conjecture.

Posted attempt

A posted complete proof models the phase torus algebraically and applies semialgebraic triangulation, whose simplex interiors are smooth manifold pieces. The argument appears to establish the conjecture, but it has not been independently verified and is not supported by a published source.

Current status (as of August 2026): The conjecture has a complete-proof claim, but no published or independently verified proof or counterexample is recorded; accordingly, its mathematical resolution remains unconfirmed.

Sources

Sources & referencesView supporting material

Primary source

Davide Sclosa, “Kuramoto Networks with Infinitely Many Stable Equilibria”, arXiv:2207.08182 (2022).

dynamical-systems kuramoto-networks

1 week ago

Complete proof. Let $G=(V,E)$ be any finite graph, with $V=\{1,\ldots,n\}$ . Its equilibrium equations are

\sum_{j\in N(i)}\sin(\theta_j-\theta_i)=0, \qquad 1\le i\le n.

Set $c_i=\cos\theta_i$ and $s_i=\sin\theta_i$ . The torus is real-analytically identified with

T=\{(c_1,s_1,\ldots,c_n,s_n)\in\mathbb R^{2n}: c_i^2+s_i^2=1\text{ for every }i\}.

Since

\sin(\theta_j-\theta_i)=s_jc_i-c_js_i,

the equilibrium set corresponds exactly to

Z_G= \left\{(c,s)\in\mathbb R^{2n}: \begin{array}{ll} c_i^2+s_i^2=1,&1\le i\le n,\\ \displaystyle\sum_{j\in N(i)}(s_jc_i-c_js_i)=0,&1\le i\le n. \end{array} \right\}.

Thus $Z_G$ is a real algebraic, hence semialgebraic, set. It is compact because it is a closed subset of $T$ .

The semialgebraic Whitney-triangulation theorem provides a finite simplicial complex $K$ and a semialgebraic triangulation $h:|K|\to Z_G$ such that

M_\sigma=h(\operatorname{relint}\sigma),\qquad \sigma\in K,

are real-analytic embedded manifolds forming a Whitney stratification. Consequently,

\boxed{\displaystyle Z_G=\coprod_{\sigma\in K}M_\sigma,}

a finite disjoint union of connected smooth manifolds. Pulling back under the torus embedding proves the conjecture for every finite graph. Singular equilibria simply occur in lower-dimensional strata.

The argument also applies to arbitrary real edge weights and intrinsic frequencies, since the equilibrium equations remain polynomial in the coordinates $c_i,s_i$ .

Reference: M. Shiota, Whitney triangulations of semialgebraic sets, Annales Polonici Mathematici 87 (2005), 237–246.

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