Mesoscale obstructions to stability of 1D center manifolds for networks of coupled differential equations with symmetric Jacobian
Abstract
A linear system ẋ=Ax, A∈R, x∈R^{n}, with rkA=n1, has a onedimensional center manifold E^{c}={v∈R^{n}:Av=0}. If a differential equation ẋ=f(x) has a onedimensional 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 C^{+}, i.e. if A is symmetric, it has to be negative semidefinite. We establish a graph theoretical approach to characterize semidefiniteness. Using spanning trees for the graph corresponding to A, we formulate mesoscale conditions with certain principal minors of A which are necessary for semidefiniteness. We illustrate these results by the example of the Kuramoto model of coupled oscillators.
 Publication:

Physica D Nonlinear Phenomena
 Pub Date:
 October 2013
 DOI:
 10.1016/j.physd.2013.05.010
 arXiv:
 arXiv:1207.3736
 Bibcode:
 2013PhyD..261....1E
 Keywords:

 Mathematics  Dynamical Systems
 EPrint:
 doi:10.1016/j.physd.2013.05.010