Overdetermined constraints and rigid synchrony patterns for network equilibria (Q2025696)

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.