epperlein2013mesoscale
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.