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Rotating-frame gradient formulation

PDF p. 2: Section 1, rotating frame and gradient flow; PDF p. 3: Section 1, rotating frame and gradient flow

Statement

Assume countable choice. Let $G = (V, E)$ be a finite undirected graph with nonempty vertex set and symmetric adjacency matrix $A$ , allowing self-loops. Let $θ : R \to T_{V} = R^{V} / (2 π Z)^{V}$ be a trajectory of the graph Kuramoto system with common intrinsic frequency $ω \in R$ . Thus, in local real angular lifts, ${\dot{θ}}_{x} = ω - \sum_{y \in V} A_{x, y} \sin (θ_{x} - θ_{y})$ .

Define the rotating-frame state $ϕ (t) = R_{- ω t} θ (t)$ , meaning $ϕ_{x} (t) = θ_{x} (t) - ω t$ modulo $2 π$ for every $x$ . Equip $T_{V}$ with its flat metric, and define the homogeneous Kuramoto energy by $E_{G} (ϕ) = \frac{1}{2} \sum_{x, y \in V} A_{x, y} (1 - \cos (ϕ_{x} - ϕ_{y}))$ . Then $ϕ$ is a downward gradient trajectory: $\dot{ϕ} = - \nabla E_{G} (ϕ)$ . Conversely, adding $ω t$ to every coordinate of such a trajectory gives a trajectory of the homogeneous Kuramoto system.

Statement references

The Axiom of Countable Choice (AC_ω)

Graph Kuramoto differential equations

Finite phase tori, local lifts and the flat metric

Diagonal phase rotations and relative-phase coordinates

Homogeneous Kuramoto energy

Gradient trajectories and complete compact phase dynamics

Facts & Assumptions

[F1]

In local angular lifts the homogeneous system has components ${\dot{θ}}_{x} = ω - \sum_{y} A_{x, y} \sin (θ_{x} - θ_{y})$ ; under countable choice its trajectories are defined for all real time (Graph Kuramoto differential equations, The Axiom of Countable Choice (AC_ω)).

[F2]

The phase torus is the quotient by coordinatewise integer multiples of $2 π$ ; its angular chart transitions are locally translations and its flat metric has Euclidean coefficients (Finite phase tori, local lifts and the flat metric). Diagonal rotation adds one common phase to every coordinate (Diagonal phase rotations and relative-phase coordinates).

[F3]

Sine is unchanged by shifting an argument by an integer multiple of $2 π$ , and differentiable real functions satisfy the sum and scalar multiplication rules (Phasors, trigonometric identities and periodic derivatives).

[F4]

For this graph and energy, the flat gradient has components $(\nabla E_{G} (ϕ))_{x} = \sum_{y \in V} A_{x, y} \sin (ϕ_{x} - ϕ_{y})$ (Homogeneous Kuramoto energy, Gradient and Hessian formulas).

[F5]

A downward gradient trajectory is an integral curve of the negative gradient; for a smooth potential on this compact phase torus, countable choice gives a unique such maximal trajectory defined on $R$ (Gradient trajectories and complete compact phase dynamics, The Axiom of Countable Choice (AC_ω)).

Proof

Given: The graph, common frequency, homogeneous trajectory $θ$ and rotating-frame state $ϕ$ in the Statement.

Goal: Prove the negative-gradient equation for $ϕ$ and the converse transformation.

1.1givenF1F2F3

Fix a time and choose local real angular lifts $u_{x}$ of $θ_{x}$ . By [F2], $v_{x} (t) = u_{x} (t) - ω t$ locally lifts $ϕ_{x}$ , and $v_{x} - v_{y} = u_{x} - u_{y}$ . The derivative rules and homogeneous equation give ${\dot{v}}_{x} = {\dot{u}}_{x} - ω = - \sum_{y} A_{x, y} \sin (v_{x} - v_{y})$ . This calculation is independent of lift choices by the translation transitions and sine periodicity in [F2] and [F3].

2.1F1F2F3F4F5step 1.1∎

By [F4], the right side in step 1.1 is $- (\nabla E_{G} (ϕ))_{x}$ in every coordinate. Thus $ϕ$ is an integral curve of the negative gradient. It is defined for all real time by [F1], so it is maximal; [F5] identifies it as the downward gradient trajectory through its initial state. Conversely, for a downward gradient trajectory with local lifts $v_{x}$ , put $u_{x} = v_{x} + ω t$ . The same derivative and difference identities yield ${\dot{u}}_{x} = ω - \sum_{y} A_{x, y} \sin (u_{x} - u_{y})$ , which is the homogeneous system. The quotient construction makes these local equations independent of lifts.

Source notes

The rotating-frame formulation is attributed to Expander graphs are globally synchronizing, PDF pages 2–3. The proof above uses the stated mathematical suppliers.

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Used by · 10

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Sources

Expander graphs are globally synchronizing · PDF pp. 2–3, Section 1, rotating frame and equation (1.2)