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Analytic local minima and Lyapunov stability

PDF p. 3: Footnote 1

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. On $T_{V} = (S^{1})^{V} = R^{V} / (2 π Z)^{V}$ , let $E_{G} (θ) = \frac{1}{2} \sum_{x, y \in V} A_{x, y} (1 - \cos (θ_{x} - θ_{y}))$ . Consider the rotating-frame flow $\dot{θ} = - \nabla E_{G} (θ)$ with the flat metric. A state $θ$ is a local minimum of $E_{G}$ exactly when it is a Lyapunov-stable equilibrium of this flow.

Here a local minimum has a neighborhood on which the energy is at least its value at $θ$ ; an equilibrium remains fixed under the flow. Lyapunov stability means that for every $ε > 0$ there is $δ > 0$ such that $d (θ (0), θ) \leq δ$ implies $d (θ (t), θ) \leq ε$ for all $t \geq 0$ , where $d$ is the flat torus distance. No strict minimum or attraction to the individual equilibrium is asserted.

Statement references

The Axiom of Countable Choice (AC_ω)

Homogeneous Kuramoto energy

Equilibria, stable states, synchronized states, and spurious minima

Finite phase tori, local lifts and the flat metric

Point stability, attraction and stability of a set

Rotating-frame gradient formulation

Facts & Assumptions

[F1]

The phase torus has angular charts with locally translational transitions, Euclidean metric coefficients and a distance equal locally to Euclidean distance in the chart (Finite phase tori, local lifts and the flat metric).

[F2]

The energy is a finite real linear combination of a constant and cosine functions of phase differences, hence is real analytic in every angular chart. In this context real analytic means locally represented by a convergent real power series (Homogeneous Kuramoto energy, Analytic potentials on a finite phase torus).

[F3]

Under countable choice the smooth downward-gradient flow exists uniquely for all real time, and the homogeneous system in the rotating frame has this gradient equation. Equilibria are precisely the critical points of the energy (The Axiom of Countable Choice (AC_ω), Gradient trajectories and complete compact phase dynamics, Rotating-frame gradient formulation, Equilibria, stable states, synchronized states, and spurious minima).

[F4]

For a real analytic function on a Euclidean neighborhood, an equilibrium of its downward-gradient system is Lyapunov stable exactly when it is a local minimum. A differentiable function at an interior Euclidean local minimum has zero gradient (Łojasiewicz inequality and analytic point stability, Fermat's theorem: an interior differentiable local extremum has zero gradient).

[F5]

Point stability uses the same quantified condition with strict distance inequalities; local minimality and equilibrium have the meanings stated above (Point stability, attraction and stability of a set, Equilibria, stable states, synchronized states, and spurious minima).

Proof

Given: The graph, energy, flat torus and rotating-frame flow in the Statement, and a state $θ$ .

Goal: Establish the equivalence and the displayed non-strict epsilon-delta formulation.

1.1F1F2F3F4

Choose an angular chart centered at $θ$ and a smaller Euclidean ball whose image lies in the chart and on which distance from $θ$ is the Euclidean coordinate distance. By [F1] and [F2], the chart expression of the energy is real analytic and the flat downward-gradient equation is its Euclidean downward-gradient equation. Local minimality is preserved by the chart homeomorphism. If $θ$ is a local minimum, Fermat's theorem in [F4] makes its coordinate gradient zero; by [F3] it is an equilibrium.

2.1F1F3F4F5step 1.1

For an equilibrium, apply the analytic stability equivalence in [F4] to the chart expression. The local and torus stability conditions agree: for a desired radius choose a smaller radius contained in the chart; trajectories starting sufficiently near $θ$ remain in that smaller ball by the relevant stability condition, so their coordinate and torus descriptions agree for all forward times. Conversely, torus stability with such a radius keeps the trajectory in the chart and yields the Euclidean local condition. Uniqueness and complete forward existence are supplied by [F3]. Together with step 1.1 this proves both implications.

3.1F5step 2.1∎

The strict-distance definition in [F5] is equivalent to the non-strict formulation in the Statement: apply strict stability with $ε / 2$ and then choose a positive non-strict initial radius smaller than its strict radius. Conversely, apply the non-strict condition with $ε / 2$ and use a strictly smaller initial radius. Neither implication requires strict minimality or convergence to $θ$ .

Source notes

The equivalence and the non-strict stability formulation are attributed to Expander graphs are globally synchronizing, PDF p. 3, Footnote 1, which cites Absil and Kurdyka. The proof above uses the independent analytic-gradient supplier, not the article assertion.

Depends on

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Sources

Expander graphs are globally synchronizing · PDF p. 3, Footnote 1