  Hopf bifurcation in networks of coupled ODEs creates periodic states in which
the relative phases of nodes are well defined near bifurcation. When the
network is a fully inhomogeneous nearest-neighbour coupled unidirectional ring,
and node spaces are 1-dimensional, we derive constraints on these phase shifts
that apply to any ODE that respects the ring topology. We begin with a 3-node
ring and generalise the results to any number of nodes. The main point is that
such constraints exist even when the only structure present is the network
topology. We also prove that the usual nondegeneracy conditions in the
classical Hopf Bifurcation Theorem are valid generically for ring networks, by
perturbing only coupling terms.
