Theoretical analysis demonstrates dense mode-locking in quasi-periodically forced Arnold circle maps, highlighting rigorous proof for numerical observations in dynamically forced maps.
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 {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 {Zha}) equipped with a topology that is much stronger than the C⁰ topology, compatible with smooth fiber maps. For quasi-periodic base maps, our result generalizes the main results in {ABD, WZJ, Zha}.
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Wang et al. (2024) studied this question.
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