The Bifurcation Set for the 1:4 Resonance Problem

Bernd Krauskopf

Abstract

We study the bifurcation set in (b, ϕ, α)-space of the equation ż = e^{iα}z + e^{iϕ}z |z|^{2} + bz̄^{3}. This ℤ_{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, ϕ, α)-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 ϕ ≠ π/2, 3π/2.