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Nonconformal perturbations of \(z\mapsto z^ 2+c\): The \(1:3\) resonance. (English) Zbl 1055.37046

Summary: We study a family of nonconformal maps of the plane, as a perturbation of the quadratic map \(z\mapsto z^ 2+c\). In particular, a neighbourhood 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 are 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 Neimark-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.

MSC:

37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
37G05 Normal forms for dynamical systems
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