Network periodic solutions: patterns of phase-shift synchrony

We prove the rigid phase conjecture of Stewart and Parker. It then follows from previous results (of Stewart and Parker and our own) that rigid phase-shifts in periodic solutions on a transitive network are produced by a cyclic symmetry on a quotient network. More precisely, let X(t) = (x₁(t), ..., x_n(t)) be a hyperbolic T-periodic solution of an admissible system on an n-node network. Two nodes c and d are phase-related if there exists a phase-shift θ_cd ∊ [0, 1) such that x_d(t) = x_c(t + θ_cdT). The conjecture states that if phase relations persist under all small admissible perturbations (that is, the phase relations are rigid), then for each pair of phase-related cells, their input signals are also phase-related to the same phase-shift. For a transitive network, rigid phase relations can also be described abstractly as a Z_m permutation symmetry of a quotient network. We discuss how patterns of phase-shift synchrony lead to rigid synchrony, rigid phase synchrony, and rigid multirhythms, and we show that for each phase pattern there exists an admissible system with a periodic solution with that phase pattern. Finally, we generalize the results to nontransitive networks where we show that the symmetry that generates rigid phase-shifts occurs on an extension of a quotient network.