TY - JOUR
T1 - Meso-scale obstructions to stability of 1D center manifolds for networks of coupled differential equations with symmetric Jacobian
AU - Epperlein, Jeremias
AU - Do, Anne-Ly
AU - Gross, Thilo
AU - Siegmund, Stefan
PY - 2013/10/15
Y1 - 2013/10/15
N2 - A linear system x' = Ax, with A in n×n, x in R, has a one-dimensional center manifold Ec = {v in Rn : Av = 0}. If a differential equation x' = f(x) has a one-dimensional center manifold Wc at an equilibrium x* then Ec is tangential to Wc with A = Df(x?) and for stability of Wc 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 mesoscale 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 x' = Ax, with A in n×n, x in R, has a one-dimensional center manifold Ec = {v in Rn : Av = 0}. If a differential equation x' = f(x) has a one-dimensional center manifold Wc at an equilibrium x* then Ec is tangential to Wc with A = Df(x?) and for stability of Wc 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 mesoscale 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 - Positive spanning tree
KW - Stability
KW - Minors
U2 - 10.1016/j.physd.2013.05.010
DO - 10.1016/j.physd.2013.05.010
M3 - Article (Academic Journal)
VL - 261
SP - 1
EP - 7
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
SN - 0167-2789
ER -