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

Abstract

A linear system $\dot{x}=Ax$, $A\in {\mathbb{R}}^{n\times n}$, $x\in {\mathbb{R}}^n$, with $\mathrm{rk}A=n-1$, 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 $x^{\ast }$ then $E^c$ is tangential to $W^c$ with $A=Df(x^{\ast })$ 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.

References in corpus (4)

• Community detection in graphs
• Synchronization in complex networks
• Network synchronization landscape reveals compensatory structures, quantization, and the positive effect of negative interactions
• Graphical notation reveals topological stability criteria for collective dynamics in complex networks

Cited by in corpus (3)

• The Kuramoto model in complex networks
• Dynamical Systems on Networks: A Tutorial
• A note on Graphical Notation Reveals Topological Stability Criteria for Collective Dynamics in Complex Network