Global rotation equivalence of phase configurations

PDF p. 3: Footnote 2; PDF p. 11: Section 2.1 phase convention

Let $G=(V,E)$ be a finite undirected graph with nonempty vertex set and symmetric adjacency matrix $A$. A phase is a real number modulo $2\pi$, and a configuration is an element $\theta \in T_V={\mathbb{R}}^V\mathrm{/}(2\pi \mathbb{Z}{)}^V$. For a phase $c$, the global rotation $R_c$ sends $\theta$ to the configuration with coordinates $(R_c\theta {)}_x={\theta }_x+c$ modulo $2\pi$. Configurations $\theta$ and ${\theta }^{\mathrm{\prime }}$ are equivalent up to global rotation if there is a common real $c$ such that ${\theta }_x-{\theta }_x^{\mathrm{\prime }}=c$ modulo $2\pi$ for every $x\in V$. Each phase may be represented by its unique real representative in $(-\pi ,\pi ]$.

The homogeneous Kuramoto energy ${\mathcal{E}}_G(\theta )=\frac{1}{2}{\sum }_{x,y\in V}{A}_{x,y}(1-\cos ({\theta }_x-{\theta }_y))$ is invariant under $R_c$. Assuming countable choice, the homogeneous Kuramoto dynamics ${\dot{\theta }}_x=\omega -{\sum }_{y\in V}{A}_{x,y}\sin ({\theta }_x-{\theta }_y)$, with common real frequency $\omega$, are equivariant: the trajectory starting at $R_c\theta$ is the rotation by $R_c$ of the trajectory starting at $\theta$.

Finite phase tori, local lifts and the flat metric

Diagonal phase rotations and relative-phase coordinates

Homogeneous Kuramoto energy

Graph Kuramoto differential equations

Given: The graph, configurations and rotation above; for the trajectory assertion assume countable choice and fix a common real frequency $\omega$.

Goal: Verify the equivalence relation and representative convention, energy invariance, and equivariance of the dynamics.

1.1 The relation ${\theta }_x-{\theta }_x^{\mathrm{\prime }}=c$ for all $x$ says precisely that $\theta =R_c{\theta }^{\mathrm{\prime }}$; the orbit relation is therefore an equivalence relation by [F1]. For any real lift, [F2] gives a congruent $r\in [0,2\pi )$; use $r$ if $r\le \pi$ and $r-2\pi$ otherwise. This yields a representative in $(-\pi ,\pi ]$. Two representatives in that interval differ by less than $2\pi$ in absolute value, so if their difference is an integer multiple of $2\pi$ it is zero. Thus the representative is unique. [F1, F2, algebra]

2.1 A common rotation leaves every phase difference unchanged modulo $2\pi$. By [F3], all sine and cosine terms in the displayed formulas are consequently unchanged. The finite energy sum is therefore unchanged by [F4]. If $\theta (t)$ is a trajectory, adding the fixed phase $c$ to each local lift leaves its derivative unchanged and gives the same right-hand side at the rotated configuration. Hence $R_c\theta (t)$ is a trajectory with initial configuration $R_c\theta (0)$. Uniqueness in [F5] identifies it with the trajectory starting there, establishing equivariance. [F3, F4, F5, step 1.1, algebra] ∎

The rotation equivalence and the phase convention are attributed to Expander graphs are globally synchronizing, PDF page 3, footnote 2, and page 11, Section 2.1. The justification uses the mathematical inputs above.