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Positive-semidefinite-equilibrium criterion for global synchrony

PDF p. 4: Section 1, first paragraph

Statement

Assume countable choice. Let $G = (V, E)$ be a finite undirected graph with nonempty vertex set, at most one edge between any two vertices, and possible self-loops. Let $A$ be its adjacency matrix and let $T_{V} = R^{V} / (2 π Z)^{V}$ be the phase torus. The homogeneous Kuramoto energy is $E_{G} (θ) = \frac{1}{2} \sum_{x, y \in V} A_{x, y} (1 - \cos (θ_{x} - θ_{y}))$ . An equilibrium is a critical point of this energy, and it is fully synchronized when all its phase coordinates are equal modulo $2 π$ . For its real symmetric angular Hessian $H = \nabla^{2} E_{G} (θ)$ , write $H ⪰ 0$ when $z^{*} H z \geq 0$ for every $z \in C^{V}$ .

If every equilibrium $θ$ satisfying $\nabla^{2} E_{G} (θ) ⪰ 0$ is fully synchronized, then $G$ is globally synchronizing: for normalized phase-volume almost every initial phase, the rotating-frame homogeneous Kuramoto trajectory converges to a fully synchronized state.

Statement references

The Axiom of Countable Choice (AC_ω)

Homogeneous Kuramoto energy

Finite phase tori, local lifts and the flat metric

Equilibria, stable states, synchronized states, and spurious minima

Loewner order, Hermitian inner product, and operator norm

Globally synchronizing graph — Definition 1.1

Normalized phase volume and almost-everywhere assertions

Facts & Assumptions

[F1]

The energy is a real analytic function on the flat phase torus: it is a finite real linear combination of cosine differences and the constant $1$ , itself $\cos (θ_{x} - θ_{x})$ for any fixed vertex $x$ (Analytic potentials on a finite phase torus). It is $C^{2}$ in angular coordinates because the first two derivatives of every cosine term exist and are continuous, and finite sums may be differentiated termwise (Phasors, trigonometric identities and periodic derivatives). Under countable choice, the homogeneous system has a unique all-time trajectory for each initial state, and its rotating-frame trajectory is the downward gradient trajectory of this energy. Equilibria are exactly its critical points (Homogeneous Kuramoto energy, Finite phase tori, local lifts and the flat metric, Rotating-frame gradient formulation, Equilibria, stable states, synchronized states, and spurious minima, The Axiom of Countable Choice (AC_ω)).

[F2]

Every homogeneous Kuramoto trajectory converges in the rotating frame to a single equilibrium. The set of initial phases whose rotating-frame trajectories converge to any strict saddle is phase-null, even if the strict saddles are not isolated. A strict saddle is a critical point whose angular Hessian has a negative eigenvalue (Single-point convergence for analytic gradient flows, Measure-zero convergence to strict saddles, Strict saddle critical point).

[F3]

A real symmetric matrix is Hermitian and hence normal. A normal endomorphism of a finite-dimensional complex inner-product space has an orthonormal eigenbasis. The Hessian of a $C^{2}$ real function is symmetric by equality of mixed partials (Loewner order, Hermitian inner product, and operator norm, Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely, Clairaut--Schwarz theorem for continuous second partial derivatives).

[F4]

Normalized phase volume is a complete probability measure; phase-null means measurable with volume zero. Global synchrony means convergence in the rotating frame to a fully synchronized point outside such a null set (Normalized phase volume and almost-everywhere assertions, Globally synchronizing graph — Definition 1.1).

Proof

Given: The graph, countable choice, and the hypothesis that every equilibrium with positive-semidefinite Hessian is fully synchronized.

Goal: Prove that $G$ is globally synchronizing.

1.1givenF1F2F3choosealgebra

Let $p$ be an equilibrium that is not fully synchronized. The given hypothesis implies that its Hessian $H$ is not positive semidefinite. The energy is $C^{2}$ by [F1], so $H$ is real symmetric by [F3]. Choose an orthonormal complex eigenbasis of $H$ using [F3]. Its eigenvalues are real: for an eigenvector $v$ , self-adjointness makes $v^{*} H v$ real and it equals the eigenvalue times $v^{*} v$ . If every eigenvalue were nonnegative, expanding any vector in this orthonormal basis would give $z^{*} H z \geq 0$ , contrary to the choice of $p$ . Hence $H$ has a negative eigenvalue, and $p$ is a strict saddle by [F2].

2.1F2F4step 1.1∎

Let $N$ be the set of initial phases whose rotating-frame trajectory converges to a strict saddle. It is phase-null by [F2]. For any initial phase outside $N$ , [F2] gives convergence to an equilibrium $p$ ; step 1.1 shows that $p$ must be fully synchronized, since otherwise the initial phase would belong to $N$ . Thus convergence to a fully synchronized point holds outside a phase-null set, and [F4] gives global synchrony.

Source notes

The criterion is attributed to the cited article, PDF p. 4, Section 1. The argument above uses the stated mathematical suppliers; the article is not used to prove its own criterion.

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Sources

Expander graphs are globally synchronizing · PDF p. 4, Section 1, first paragraph