Bifurcations at infinity in a model for 1: 4 resonance

The equation (z) over dot = e(i alpha)z + e(i phi)z\z\(2) + bz(-3) models a map near a Hopf bifurcation with 1:4 resonance. It is a conjecture by V. I. Arnol'd that this equation contains all versal unfoldings of Z(4)-equivariant planar vector fields. We study its bifurcations at infinity and show that the singularities of codimension two unfold versally in a neighborhood. We give an unfolding of the codimension-three singularity for b = 1, phi = 3 pi/2 and alpha = 0 in the system parameters and use numerical methods to study global phenomena to complete the description of the behavior near co. Our results are evidence in support of the conjecture.