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

Bruin, H. and Van Noort, M. (2004) Nonconformal perturbations of z to z2+ c: the 1 : 3 resonance Nonlinearity, 17. pp. 765-789.

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Abstract

We study a family of non-conformal maps of the plane, as a perturbation of the quadratic map z to z2 + 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 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.

Item Type: Article
Additional Information: This is a pre-press version of the article. The citation details for the published version are: Bruin, H., & Van Noort, M. (2004). Nonconformal perturbations of z to z2 + c: the 1 : 3 resonance. Nonlinearity, 17, 765-789. © 2004 IOP Publishing Ltd.
Divisions: Faculty of Engineering and Physical Sciences > Mathematics
Depositing User: Mr Adam Field
Date Deposited: 27 May 2010 14:41
Last Modified: 23 Sep 2013 18:33
URI: http://epubs.surrey.ac.uk/id/eprint/1505

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