On some properties of invariant sets of two-dimensional noninvertible maps

Christos E. Frouzakis, Laura Gardini, Ioannis G. Kevrekidis, Gilles Millerioux, Christian Mira

Research output: Contribution to journalArticle

53 Scopus citations

Abstract

We study the nature and dependence on parameters of certain invariant sets of noninvertible maps of the plane. The invariant sets we consider are unstable manifolds of saddle-type fixed and periodic points, as well as attracting invariant circles. Since for such maps a point may have more than one first-rank preimages, the geometry, transitions, and general properties of these sets are more complicated than the corresponding sets for diffeomorphisms. The critical curve(s) (locus of points having at least two coincident preimages) as well as its antecedent(s), the curve(s) where the map is singular (or "curve of merging preimages") play a fundamental role in such studies. We focus on phenomena arising from the interaction of one-dimensional invariant sets with these critical curves, and present some illustrative examples.

Original languageEnglish (US)
Pages (from-to)1167-1194
Number of pages28
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Volume7
Issue number6
DOIs
StatePublished - Jun 1997

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Engineering (miscellaneous)
  • General
  • Applied Mathematics

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