Strange non-chaotic attractors in quasiperiodically forced circle maps: Diophantine forcing

TL;DR

This paper demonstrates the existence of strange nonchaotic attractors in quasiperiodically forced circle maps with Diophantine frequencies, providing explicit parameter characterizations and advancing understanding of their hyperbolic behavior.

Contribution

It offers a nonlinear version of Young's results, explicitly characterizes frequency sets, and analyzes the collapse of the Arnold tongue at rotation number 1/2.

Findings

Existence of strange nonchaotic attractors on positive measure parameter sets.

Collapse of the Arnold tongue at rotation number 1/2.

Development of a parameter exclusion method based on multiscale analysis.

Abstract

We study parameter families of quasiperiodically forced (qpf) circle maps with Diophantine frequency. Under certain C1-open conditions concerning their geometry, we prove that these families exhibit nonuniformly hyperbolic behaviour, often referred to as the existence of strange nonchaotic attractors, on parameter sets of positive measure. This provides a nonlinear version of results by Young on quasiperiodic SL (2;R)-cocycles and complements previous results in this direction which hold for sets of frequencies of positive measure, but did not allow for an explicit characterisation of these frequencies. As an application, we study a qpf version of the Arnold circle map and show that the Arnold tongue corresponding to rotation number 1/2 collapses on an open set of parameters. The proof requires to perform a parameter exclusion with respect to some twist parameter and is based on the multiscale analysis of the dynamics on certain dynamically defined critical sets. A crucial ingredient is to obtain good control on the parameter-dependence of the critical sets. Apart from the presented results, we believe that this step will be important for obtaining further information on the behaviour of parameter families like the qpf Arnold circle map.