Overdetermined constraints and rigid synchrony patterns for network equilibria
type: Item
instance of: scholarly article
title: Overdetermined constraints and rigid synchrony patterns for network equilibria
zbMATH Open document ID: 1468.34049
DOI: 10.4171/PM/2048
published in: Portugaliae Mathematica. Nova Série
publication date: 2021-05-14T00:00:00Z
review text: Summary: 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 [\textit{M. Golubitsky} et al., SIAM J. Appl. Dyn. Syst. 4, No. 1, 78--100 (2005; Zbl 1090.34030)] 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.
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34D06
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34D09
Mathematics Subject Classification ID: 34D10
Mathematics Subject Classification ID: 05C99
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 20L05
zbMATH DE Number: 7348212
zbMATH Keywords: network
zbMATH Keywords: equilibrium
zbMATH Keywords: rigid
zbMATH Keywords: synchrony
zbMATH Keywords: balanced colouring
zbMATH Keywords: hyperbolic
zbMATH Keywords: overdetermined
author: Ian Stewart
MaRDI profile type: Publication
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