JournalspmVol. 77 , No. 2pp. 163–196

Overdetermined constraints and rigid synchrony patterns for network equilibria

  • Ian Stewart

    University of Warwick, Coventry, UK
Overdetermined constraints and rigid synchrony patterns for network equilibria cover
Download PDF

A subscription is required to access this article.

Abstract

In network dynamics, synchrony between nodes defines an equivalence relation, usually represented as a colouring. If the colouring is balanced, meaning that nodes of the same colour have colour-isomorphic inputs, it determines a subspace that is flow-invariant for any ODE compatible with the network structure. Therefore any state lying in such a subspace has the synchrony pattern determined by that balanced colouring. In 2005 Golubitsky and coworkers proved a strong converse for synchronous equilibria: every rigid synchrony colouring for a hyperbolic equilibrium is balanced, where rigidity means that the pattern persists under small admissible perturbations. We give a different proof of this theorem, based on overdetermined constraint equations, Sard’s Theorem, bump functions, and groupoid symmetrisation.

Cite this article

Ian Stewart, Overdetermined constraints and rigid synchrony patterns for network equilibria. Port. Math. 77 (2020), no. 2 pp. 163–196

DOI 10.4171/PM/2048