Meso-scale obstructions to stability of 1D center manifolds for networks of coupled differential equations with symmetric Jacobian

TL;DR

This paper introduces a graph-theoretic method to analyze the stability of one-dimensional center manifolds in coupled differential equations, focusing on the semi-definiteness of symmetric Jacobians using principal minors and spanning trees.

Contribution

It develops meso-scale conditions based on graph theory to determine semi-definiteness of symmetric matrices, aiding stability analysis of coupled systems.

Findings

Meso-scale conditions for semi-definiteness are formulated using principal minors.

Graph spanning trees are used to characterize stability conditions.

The approach is demonstrated on the Kuramoto model of coupled oscillators.

Abstract

A linear system $\dot x = Ax$, $A \in \mathbb{R}^{n \times n}$, $x \in \mathbb{R}^n$, with $\mathrm{rk} A = n-1$, has a one-dimensional center manifold $E^c = \{v \in \mathbb{R}^n : Av=0\}$. If a differential equation $\dot x = f(x)$ has a one-dimensional center manifold $W^c$ at an equilibrium $x^*$ then $E^c$ is tangential to $W^c$ with $A = Df(x^*)$ and for stability of $W^c$ it is necessary that $A$ has no spectrum in $\mathbb{C}^+$, i.e.\ if $A$ is symmetric, it has to be negative semi-definite. We establish a graph theoretical approach to characterize semi-definiteness. Using spanning trees for the graph corresponding to $A$, we formulate meso-scale conditions with certain principal minors of $A$ which are necessary for semi-definiteness. We illustrate these results by the example of the Kuramoto model of coupled oscillators.