Equilibria, stable states, synchronized states, and spurious minima

PDF p. 3: Section 1 preceding Definition 1.1

Assume countable choice. Let $G=(V,E)$ be a finite undirected graph with nonempty vertex set, allowing self-loops, and consider its homogeneous Kuramoto system with common intrinsic frequency $\omega$. Its phase states are points of $T_V=(S^1{)}^V={\mathbb{R}}^V\mathrm{/}(2\pi \mathbb{Z}{)}^V$, and its energy is ${\mathcal{E}}_G(\theta )=\frac{1}{2}{\sum }_{x,y\in V}{A}_{x,y}(1-\cos ({\theta }_x-{\theta }_y))$. The rotating-frame state is obtained by subtracting $\omega t$ from every phase modulo $2\pi$.

An equilibrium state is a state at which the rotating-frame vector field vanishes, equivalently a critical point of the energy: $\mathrm{\nabla }{\mathcal{E}}_G(\theta )=0$. In the original frame the corresponding trajectory has every phase moving at frequency $\omega$, with all phase differences constant.

A local minimum of ${\mathcal{E}}_G$ is a state $\theta$ having a neighborhood $U$ in $T_V$ such that ${\mathcal{E}}_G(\varphi )\ge {\mathcal{E}}_G(\theta )$ for every $\varphi \in U$. Following the article's terminology, a stable equilibrium state, or stable state, is a local minimum of ${\mathcal{E}}_G$; strict minimality is not required. This terminology is a definition and does not assert a separate Lyapunov-stability theorem.

A fully synchronized state is a state satisfying ${\theta }_x={\theta }_y$ in $S^1$ for every $x,y\in V$. A spurious local minimum is a local minimum that is not fully synchronized. States differing by one common phase rotation are rotation equivalent.

The phase-torus construction makes equality and subtraction of phases modulo $2\pi$ well defined and supplies the topology used in the neighborhood definition. The energy is a well-defined real function on that torus. Thus local minimality, equality of all coordinates, and the spurious-minimum condition are unambiguous (Finite phase tori, local lifts and the flat metric, Homogeneous Kuramoto energy, Point stability, attraction and stability of a set).

Under countable choice, the rotating-frame formulation identifies the vector field with $-\mathrm{\nabla }{\mathcal{E}}_G$. It therefore vanishes exactly at the critical points. A constant rotating-frame trajectory at such a point transforms back by adding $\omega t$ to each phase, giving a trajectory of the homogeneous system whose phase differences remain constant (The Axiom of Countable Choice (AC_ω), Rotating-frame gradient formulation, Graph Kuramoto differential equations, Diagonal phase rotations and relative-phase coordinates).

A common rotation preserves every phase difference. Consequently it preserves the energy, local minima, and equality of all phases; it also preserves the critical-point condition because the gradient components depend only on phase differences. Rotation equivalence is well defined by the diagonal-rotation construction (Diagonal phase rotations and relative-phase coordinates, Homogeneous Kuramoto energy, Rotating-frame gradient formulation).

The terminology is attributed to Expander graphs are globally synchronizing, PDF page 3, preceding Definition 1.1 and footnote 2. The justification uses the stated mathematical suppliers.