Overdetermined ODEs and rigid periodic states in network dynamics

Abstract

We consider four long-standing Rigidity Conjectures about synchrony and phase patterns for hyperbolic periodic orbits of admissible ODEs for networks. Proofs of stronger local versions of these conjectures, published in 2010–12, are now known to have a gap, but remain valid for a broad class of networks. Using different methods we prove local versions of the conjectures under a stronger condition, ‘strong hyperbolicity’, which is related to a network analogue of the Kupka–Smale Theorem. Under this condition we also deduce global versions of the conjectures and an analogue of the $H/K$ Theorem in equivariant dynamics.We prove the Rigidity Conjectures for all 1- and 2-colourings and all 2- and 3-node networks by showing that strong hyperbolicity is generic in these cases.