Dynamics of nonholomorphic singular continuations: A case in radial symmetry

Brett Bozyk, Bruce B Peckham

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

This paper is primarily a study of special families of rational maps of the real plane of the form: z → zn + β/zn, where the dynamic variable z ∈ C, and C is identified with R2. The parameter β is complex; n is a positive integer. For β small, this family can be considered a nonholomorphic singular perturbation of the holomorphic family z → zn, although we consider large values of β as well. Compared to the more general family z → zn + c + β/2d, the special case where n = d and c = 0 is easier to analyze because the radial component in polar coordinates decouples from the angular component. This reduces a significant part of the analysis to the study of a family of one-real-dimensional unimodal maps. For each fixed n, the β parameter plane separates into three major regions, corresponding to maps which have one of the following behaviors: (i) all orbits go off to infinity, (ii) only an annulus of points stays bounded, and (iii) only a Cantor set of circles stays bounded. In cases (ii) and (iii), there is a transitive invariant set; this set is an attractor in case (ii). Some comparisons are made between z n + β/zn and the holomorphic singularly perturbed maps: z → zn + λ/zn, studied by Devaney and coauthors over the last decade. Additional results and observations are made about the more general family where c ≠ 0 and n ≠ d.

Original languageEnglish (US)
Article number1330036
JournalInternational Journal of Bifurcation and Chaos
Volume23
Issue number11
DOIs
StatePublished - Nov 2013

Keywords

  • Chaotic dynamics
  • Rational maps of the plane
  • Singular maps

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