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, Stefan Siegmund

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)
233 Downloads (Pure)

Abstract

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.
Original languageEnglish
Pages (from-to)1-7
Number of pages7
JournalPhysica D: Nonlinear Phenomena
Volume261
Early online date11 Jun 2013
DOIs
Publication statusPublished - 15 Oct 2013

Keywords

  • Definiteness
  • Positive spanning tree
  • Stability
  • Minors

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