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

Abstract

A linear system ẋ=Ax, A∈Rn×n, x∈Rn, with rkA=n−1, has a one-dimensional center manifold Ec={v∈Rn:Av=0}. If a differential equation ẋ=f(x) has a one-dimensional center manifold Wc at an equilibrium x∗ then Ec is tangential to Wc with A=Df(x∗) and for stability of Wc it is necessary that A has no spectrum in 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.