The bifurcation set for the {$1:4$} resonance problem

Abstract

We study the bifurcation set in $(b,\phi,\alpha)$-space of the equation $\dot{z} = e^{i\alpha} z +e^{i\phi} z \,|z|^2 + b \bar{z}^3$. This $\mZ_4$-equivariant planar vector field is equivalent to the model equation that has been considered in the study of the 1:4 resonance problem.

We present a three-dimensional model of the bifurcation set that describes the known properties of the system in a condensed way, and, under certain assumptions for which there is strong numerical evidence, is topologically correct and complete. In this model, the bifurcation set consists of surfaces of codimension-one bifurcations that divide $(b,\phi,\alpha)$-space into fifteen regions of generic phase portraits. The model also offers further insight into the question of versality of the system. All bifurcation phenomena seem to unfold generically for $\phi \neq \pi/2, \, 3\pi/2$.