TY - JOUR
T1 - Meso-scale obstructions to stability of 1D center manifolds for networks of coupled differential equations with symmetric Jacobian
AU - Epperlein, J.
AU - Do, A. L.
AU - Gross, T.
AU - Siegmund, S.
N1 - Funding Information:
All authors are members of the Center for Dynamics, Dresden, Germany. This work is partly supported by the German Research Foundation (DFG) through the Cluster of Excellence ‘Center for Advancing Electronics Dresden’ (cfaed).
PY - 2013/1/1
Y1 - 2013/1/1
N2 - A linear system ẋ = Ax, A ∈ ℝn×n, x ∈ ℝn, with rkA = n - 1, has a one-dimensional center manifold Ec = {v ∈ ℝn : Av = 0}. If a differential equation ẋ = f (x) has a one-dimensional center manifold Wc at an equilibrium x* then Ec 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 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.
AB - A linear system ẋ = Ax, A ∈ ℝn×n, x ∈ ℝn, with rkA = n - 1, has a one-dimensional center manifold Ec = {v ∈ ℝn : Av = 0}. If a differential equation ẋ = f (x) has a one-dimensional center manifold Wc at an equilibrium x* then Ec 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 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.
KW - Definiteness
KW - Minors
KW - Positive spanning tree
KW - Stability
UR - http://www.scopus.com/inward/record.url?scp=84885374058&partnerID=8YFLogxK
U2 - 10.1016/j.physd.2013.05.010
DO - 10.1016/j.physd.2013.05.010
M3 - Article
AN - SCOPUS:84885374058
SN - 0167-2789
VL - 261
SP - 1
EP - 7
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
ER -