Bifurcations at ∞ in a model for 1:4 resonance

The equation ż = ez + eiφz|z|2 + bz̄3 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 Z4-equivariant planar vector fields. We study its bifurcations at ∞ 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, φ = 3π/2 and α = 0 in the system parameters and use numerical methods to study global phenomena to complete the description of the behavior near ∞. Our results are evidence in support of the conjecture.