Hopf Bifurcation in Asymmetric Ring Networks: Constraints on Phase Shifts

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.