2025/06/03 by Philip Boyland, Boyland, Philip
Mathematics · #Mathematical Dynamics and Fractals #Geometric and Algebraic Topology #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2506.02988
The family of circle maps fb, ω (x) = x + ω+ b ϕ(x) is used as a simple model for a periodically forced oscillator. The parameter ω represents the unforced frequency, b the coupling, and ϕ the forcing. When ϕ= (1)/(2 π) sin(2 πx) this is the classical Arnol'd standard family. Such families are often studied in the (ω,b)-plane via the so-called tongues Tβ consisting of all (ω,b) such that fb, ω has rotation number β. The interior of the rational tongues Tp/q represent the system mode-locked into a p/q-periodic response. Campbell, Galeeva, Tresser, and Uherka proved that when the forcing is a PL map with k=2 breakpoints, all Tp/q pinch down to a width of a single point at multple values when q large enough. In contrast, we prove that it generic amongst PL forcings with a given k≥ 3 breakpoints that there is no such pinching of any of the rational tongues. We also prove that the absence of pinching is generic for Lipschitz and Cr (r>0) forcing.