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 θ _(cd) ∈ [0, 1) such that x _d(t) = x _c(t + θ _(cd)T). 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.
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机译：我们证明了斯图尔特和帕克的刚性相位猜想。然后，从先前的结果（斯图尔特和帕克以及我们自己的结果）可以得出，传递网络上周期解中的刚性相移是由商网络上的循环对称性产生的。更准确地说，令X（t）=（x _1（t），？，x _n（t））是n节点网络上可容许系统的双曲T周期解。如果存在相移θ_（cd）∈[0，1），使得x _d（t）= x _c（t +θ_（cd）T），则两个节点c和d与相位相关。该猜想指出，如果在所有小的容许扰动下相位关系仍然存在（即，相位关系是刚性的），那么对于每对与相位相关的单元，它们的输入信号也与相同的相移相关。对于传递网络，刚性相位关系也可以抽象地描述为商网络的Z _m置换对称性。我们讨论了相移同步的模式如何导致刚性同步，刚性相位同步和刚性多重节奏，并且我们显示出，对于每个相位模式，都存在一个具有该相位模式的周期解的可容许系统。最后，我们将结果推广到非传递网络，在非传递网络中，表明产生刚性相移的对称性发生在商网络的扩展上。
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