Density of mode-locking property for quasi-periodically forced Arnold circle maps

TL;DR

This paper proves that the mode-locking region is dense in quasi-periodically forced Arnold circle maps with generic forcing functions, confirming numerical observations and extending previous results to broader conditions.

Contribution

It establishes the density of mode-locking in a broad class of forced circle maps, generalizing earlier findings and providing rigorous verification of numerical results.

Findings

Density of mode-locking region proven for generic forcing functions.

Generalization of previous results to broader classes of base maps.

Rigorous confirmation of numerical observations in prior studies.

Abstract

We show that the mode-locking region of the family of quasi-periodically forced Arnold circle maps with a topologically generic forcing function is dense. This gives a rigorous verification of certain numerical observations in \cite{DGO} for such forcing functions. More generally, under some general conditions on the base map, we show the density of the mode-locking property among dynamically forced maps (defined in \cite{Zha}) equipped with a topology that is much stronger than the $C^0$ topology, compatible with smooth fiber maps. For quasi-periodic base maps, our result generalizes the main results in \cite{ABD, WZJ, Zha}.