A quasiperiodically forced skew-product on the cylinder without fixed-curves

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Abstract

© 2016 Elsevier Ltd In Fabbri et al. (2005) the Sharkovskiĭ Theorem was extended to periodic orbits of strips of quasiperiodic skew products in the cylinder. In this paper we deal with the following natural question that arises in this setting: Does Sharkovskiĭ Theorem hold when restricted to curves instead of general strips? We answer this question in the negative by constructing a counterexample: We construct a map having a periodic orbit of period 2 of curves (which is, in fact, the upper and lower circles of the cylinder) and without any invariant curve. In particular this shows that there exist quasiperiodic skew products in the cylinder without invariant curves.
Original languageEnglish
Pages (from-to)199-263
JournalNonlinear Analysis, Theory, Methods and Applications
Volume145
DOIs
Publication statusPublished - 1 Nov 2016

Keywords

  • Invariant strips
  • Quasiperiodically forced systems on the cylinder

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