Abundance of mode-locking for quasiperiodically forced circle maps

TL;DR

This paper demonstrates that in quasiperiodically forced circle maps, mode-locking plateaus are abundant and have positive measure under certain conditions, extending the understanding of Arnold tongues in these systems.

Contribution

It establishes the positive measure of the union of mode-locking plateaus in quasiperiodically forced circle maps using multiscale analysis and parameter exclusion methods.

Findings

Positive measure of mode-locking plateaus under certain conditions

Existence of infinitely many Arnold tongues in the system

Methods applicable to classical and forced Arnold circle maps

Abstract

We study the phenomenon of mode-locking in the context of quasiperiodically forced non-linear circle maps. As a main result, we show that under certain C1-open condition on the geometry of twist parameter families of such systems, the closure of the union of modelocking plateaus has positive measure. In particular, this implies the existence of infinitely many mode-locking plateaus (open Arnold tongues). The proof builds on multiscale analysis and parameter exclusion methods in the spirit of Benedicks and Carleson, which were previously developed for quasiperiodic SL(2,R)-cocycles by Young and Bjerkl\"ov. The methods apply to a variety of examples, including a forced version of the classical Arnold circle map.