Nonconformal perturbations of z → z2+c:: the 1: 3 resonance

We study a family of non-conformal maps of the plane, as a perturbation of the quadratic map z --> z(2) + c. In particular, a neigbbourhood in phase-parameter space of the 1 : 3 resonance of the unperturbed map is analysed, by theoretical and numerical means, mostly in a local setting, but some more global aspects a re discussed as well. Certain topological constructions, like the Mandelbrot and Julia sets, and external rays, can be carried through to the nonanalytic setting. Other familiar properties of the quadratic map, like the number and possible types of periodic points, are lost under the perturbation. A bifurcation analysis shows complicated dynamics, where the 1 : 3 resonance point as well as cusp and Bogdanov-Takens points act as organizing centres. Arnol'd tongues and invariant circles-originating from Neimarck-Sacker bifurcations-also play an important role in structuring the dynamics. Finally, we discuss a planar vector field approximation of the family of maps that can explain part of these phenomena.