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

• Semi-definiteness of symmetric matrices is related to the structure of the Coates graph.
• Topological criteria allow testing high-dimensional dynamical systems for stability.
• We derive a necessary stability criterion that can be checked in linear time.
• The criterion allows to determine meso-scale obstructions to stability.

A linear system ẋ=Ax, A∈Rn×nA∈Rn×n, x∈Rnx∈Rn, with rkA=n−1, has a one-dimensional center manifold Ec={v∈Rn:Av=0}Ec={v∈Rn:Av=0}. If a differential equation ẋ=f(x) has a one-dimensional center manifold WcWc at an equilibrium x∗x∗ then EcEc is tangential to WcWc with A=Df(x∗)A=Df(x∗) and for stability of WcWc it is necessary that AA has no spectrum in C+C+, i.e. if AA 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 AA, we formulate meso-scale conditions with certain principal minors of AA which are necessary for semi-definiteness. We illustrate these results by the example of the Kuramoto model of coupled oscillators.