Overdetermined ODEs and rigid periodic states in network dynamics

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 proving that strong hyperbolicity is generic in these cases.