Generalized Hénon maps and smale horseshoes of new types

Sergey Gonchenko*, Ming-Chia Li, Mikhail Malkin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

We study hyperbolic dynamics and bifurcations for generalized Hénon maps in the form x̄ = y, ȳ = γ y(1 - y) - bx + αxy (with b, α small and γ > 4). Hyperbolic horseshoes with alternating orientation, called half-orientable horseshoes, are proved to represent the nonwandering set of the maps in certain parameter regions. We show that there are infinitely many classes of such horseshoes with respect to the local topological conjugacy. We also study transitions from the usual orientable and nonorientable horseshoes to half-orientable ones (and vice versa) as parameters vary.

Original languageEnglish
Pages (from-to)3029-3052
Number of pages24
JournalInternational Journal of Bifurcation and Chaos
Volume18
Issue number10
DOIs
StatePublished - 1 Jan 2008

Keywords

  • Half-orientable horseshoe
  • Hyperbolic dynamics
  • Hénon map
  • Nonwandering set
  • Singular bifurcation
  • Smale horseshoe

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