M. Golubitsky, D. Romano and Y. Wang
Network periodic solutions: patterns of phase-shift synchrony
Nonlinearity.
25 (2012) 1045-1074.
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)=(x_1(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 theta_{cd} in [0,1) such that x_d(t) = x_c(t+theta_{cd} T).
The conjecture states that if phase-relations persists 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 with 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.