Meso-scale obstructions to stability of 1D center manifolds for networks of coupled differential equations with symmetric Jacobian
Jeremias Epperlein, Anne-Ly Do, Thilo Gross and Stefan Siegmund
Physica D: Nonlinear Phenomena 261, 1-7, 2013
A linear system \(\dot{x} = {Ax}\), \(A \in \mathbb{R}^{n \times n}\), \(x \in \mathbb{R}^n\), with \(\mathrm{rank} (A) = n-1\), 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.