English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

A recurrent theme in the description of phase portraits of dynamical systems is that of  elliptic islands in a chaotic sea . Usually this picture is invoked in the context of smooth twist maps of the annulus or the torus, like the standard map. In this setting, ‘elliptic islands’ refer to the topological disks bounded by periodic smooth curves surrounding elliptic periodic points. Establishing the existence of these curves is one of the many achievements of KAM Theory. The aim of this note is to approach the topic from a different angle, namely from the viewpoint of rotation theory in a purely topological setting. We study homeomorphisms of the two‐torus, homotopic to the identity, which have no wandering open sets (as in the area‐preserving case) and whose rotation set has a non‐empty interior. We define local rotation subsets  ρ   U  ( F ) by restricting Misiurewicz and Ziemian's definition of the rotation set to starting points in a small open disk  U . Our main result is the following dichotomy: Either  ρ   U  ( F ) is reduced to a single rational vector and  U  is contained in a periodic topological open disk which contains a periodic point, or  ρ   U  ( F ) is large, in the sense that its convex hull has a non‐empty interior. This allows us to distinguish an ‘elliptic’ and a ‘chaotic’ regime, and as a consequence we obtain that in the chaotic region the dynamics are sensitive with respect to initial conditions.

Elliptic stars in a chaotic night