Phase-locked long-term behavior

PDF p. 2: Section 1, homogeneous phase locking and rotating frame; PDF p. 3: Section 1, equilibrium interpretation

Assume countable choice. Let $G=(V,E)$ be a finite undirected graph with nonempty vertex set $V$ and at most one edge between each pair of vertices, allowing self-loops. Thus $E$ is a set of unordered pairs $\{x,y\}$ with $x,y\in V$, where $x=y$ represents a self-loop. Write $A=(A_{x,y}{)}_{x,y\in V}$ for its symmetric binary adjacency matrix, with $A_{x,y}=1$ when $\{x,y\}\in E$ and zero otherwise. Let $\theta :\mathbb{R}\to T_V={\mathbb{R}}^V\mathrm{/}(2\pi \mathbb{Z}{)}^V$ be a trajectory of the graph Kuramoto system. Each ${\theta }_x(t)$ is a phase, meaning a real angle modulo $2\pi$. The trajectory has phase-locked long-term behavior if, for every $x,y\in V$, the circle-valued difference ${\theta }_x(t)-{\theta }_y(t)$ converges to a constant phase as $t\to \mathrm{\infty }$. Convergence is in the circle topology: equivalently, the distance modulo $2\pi$ to that phase tends to zero.

In the homogeneous case all intrinsic frequencies equal one real number $\omega$. Its rotating-frame state is $\varphi (t)=R_{-\omega t}\theta (t)$, i.e. ${\varphi }_x(t)={\theta }_x(t)-\omega t$ modulo $2\pi$. An equilibrium of the rotating-frame equation is a state at which its vector field vanishes. The trajectory starting at such a state is constant in that frame; in the original frame each oscillator has angular velocity $\omega$, and its phase differences are constant.

Given: The data and trajectory in the definition, and, for the equilibrium assertion, homogeneous frequency $\omega$ and an equilibrium ${\varphi }^{\ast }$ of the rotating-frame equation.

Goal: Verify that phase locking is well defined and establish the equilibrium interpretation.

1.1 By [F2], changing a representative of either phase changes their real difference by an integer multiple of $2\pi$. Its circle class and its distance to a prescribed limiting phase are therefore unchanged. Thus the convergence condition is independent of representatives. By [F3], diagonal rotation preserves each difference, so the condition is also unchanged by passage to the rotating frame. [F2, F3, algebra]

1.2 With $A$ as in [F5], in local real lifts write ${\varphi }_x={\theta }_x-\omega t$. Differentiating and using [F1] gives ${\dot{\varphi }}_x=-{\sum }_{y\in V}{A}_{x,y}\sin ({\theta }_x-{\theta }_y)=-{\sum }_{y\in V}{A}_{x,y}\sin ({\varphi }_x-{\varphi }_y)$. The last equality uses [F3] and [F4]. Conversely, adding $\omega t$ to a solution of this equation gives a solution of the homogeneous original equation. [F1, F3, F4, F5, algebra]

2.1 Since the rotating-frame vector field vanishes at ${\varphi }^{\ast }$, the constant curve $\varphi (t)={\varphi }^{\ast }$ solves its equation. Its inverse-frame curve is ${\theta }_x(t)={\varphi }_x^{\ast }+\omega t$ modulo $2\pi$ and solves the original equation by step 1.2. By uniqueness in [F1], it is the trajectory through ${\varphi }^{\ast }$ at time zero. Its local angular velocity is $\omega$ at every vertex and its differences equal ${\varphi }_x^{\ast }-{\varphi }_y^{\ast }$ for all time. This proves the equilibrium interpretation. [F1, step 1.2, algebra] ∎

The phase-locking terminology and equilibrium interpretation are attributed to Expander graphs are globally synchronizing, PDF pp. 2–3, Section 1. The justification above uses the stated mathematical suppliers. The assertion that every homogeneous trajectory converges is a separate claim.