Graph Kuramoto differential equations

PDF p. 2: Section 1, equation (1.1)

Assume countable choice. Let $G=(V,E)$ be a finite undirected graph with nonempty vertex set, possibly with self-loops, and symmetric adjacency matrix $A=(A_{x,y}{)}_{x,y\in V}$, where $A_{x,y}=1$ when $\{x,y\}\in E$ and is zero otherwise. Put $n=\mathrm{\mid }V\mathrm{\mid }$. The state space is the $n$-torus $T_V={\mathbb{R}}^V\mathrm{/}(2\pi \mathbb{Z}{)}^V$, identified with $(S^1{)}^V$, where $S^1=\{(a,b)\in {\mathbb{R}}^2:a^2+b^2=1\}$. A phase is a coordinate ${\theta }_x$ of a state in this quotient, and a lift is a real representative of it.

For prescribed real numbers ${\omega }_x$, called intrinsic frequencies, the graph Kuramoto system is the autonomous differential equation

${\dot{\theta }}_x={\omega }_x-{\sum }_{y\in V}{A}_{x,y}\sin ({\theta }_x-{\theta }_y),x\in V\mathrm{.}$

Here ${\dot{\theta }}_x$ means the derivative of a local real lift of the circle-valued curve ${\theta }_x$, and the sine is evaluated using any real lifts of the phases. A trajectory is a differentiable curve $\theta :\mathbb{R}\to T_V$ satisfying these equations in local angular lifts. Thus each coordinate is a map ${\theta }_x:\mathbb{R}\to S^1$. The homogeneous case is the specialization ${\omega }_x=\omega$ for every $x\in V$, for one real number $\omega$.

Adjacency, degrees, graph Laplacian, and spectral notation

Finite phase tori, local lifts and the flat metric

Phasors, trigonometric identities and periodic derivatives

The Axiom of Countable Choice (AC_ω)

Given: Countable choice, the graph and frequencies in the definition, and an initial phase state in $T_V$.

Goal: Show that the displayed equation is independent of lifts and defines a unique trajectory through the initial state for all real time.

1.1 Replacing any lift by another changes it by an integer multiple of $2\pi$. By [F3], every sine in the finite sum is unchanged. The chart transitions in [F2] are locally translations, so their derivatives are the identity and a curve's local lift derivatives agree on overlaps. Regard the displayed right-hand side as the components of a tangent vector in each angular chart. These vectors agree on overlaps and are based at the phase state, so they define a section $X:T_V\to T{T}_V$ of the tangent-bundle projection. Repeated use of the derivative identities in [F3] shows that sine is smooth; finite sums and composition with the linear coordinate differences give smooth component functions. Thus $X$ is a smooth vector field in the sense of [F6]. A loop term vanishes because its phase difference is zero and $\sin 0=0$. [F1, F2, F3, F6, given]

2.1 In each angular chart, [F5] supplies local existence and uniqueness for the smooth coordinate field. A differentiable coordinate solution is smooth by repeated differentiation of its equation with smooth right-hand side. Its coordinate derivative is the component vector of $X$, so it is an integral curve as defined in [F7]; conversely, that integral-curve identity gives the displayed coordinate equations. The coordinate solutions agree on overlaps by uniqueness and the compatibility established in step 1.1. By [F2] the torus is compact, so [F4] makes $X$ complete. By [F8], its maximal integral curve through the initial state is therefore defined on all of $\mathbb{R}$ and is a trajectory in the Definition. Local uniqueness along any two curves with the same initial state makes them identical. The homogeneous case is the stated choice of the frequency parameters. [F2, F4, F5, F7, F8, step 1.1] ∎

The equation, phase interpretation, and homogeneous specialization are attributed to Expander graphs are globally synchronizing, PDF page 2, Section 1. The justification uses the mathematical suppliers above, not the article as proof.

The equation, phase interpretation, and homogeneous specialization are attributed to Expander graphs are globally synchronizing, PDF page 2, Section 1. The justification uses the mathematical suppliers above, not the article as proof.