A quasiperiodically forced skew-product on the cylinder without fixed-curves. (November 2016)

• Record Type: Journal Article
• Authors: Alsedà, Lluís; Mañosas, Francesc; Morales, Leopoldo
• Is Part Of: Nonlinear analysis. Volume 145(2016)
• Page Start: 199
• Page End: 263
• Publication Date: 2016-11
• Subjects: primary 37C55 37C70; Quasiperiodically forced systems on the cylinder -- Invariant strips; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.08.011
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: 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.