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

A linear system x ̇ = A x , A ∈ R n × n , x ∈ R n , with rk A = n − 1 , has a one-dimensional center manifold E c = { v ∈ R n : A v = 0 } . If a differential equation 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 = D f ( x ∗ ) and for stability of W c it is necessary that A has no spectrum in C + , i.e. if A is symmetric, it has to be negative semi-definite. ablish 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.