Pinched Arnol'd tongues for Families of circle maps

TL;DR

This paper investigates the structure of Arnol'd tongues in families of circle maps, showing that for certain piecewise linear and smooth forcings, the tongues do not pinch into points, contrasting with previous results for simpler cases.

Contribution

It demonstrates that for piecewise linear forcings with three or more breakpoints, the Arnol'd tongues do not pinch, and this non-pinching property is also generic for Lipschitz and smooth forcings.

Findings

Forcing with at least 3 breakpoints prevents pinching of rational tongues.

Pinching occurs only in specific cases with fewer breakpoints.

Non-pinching is a generic property among Lipschitz and smooth forcings.

Abstract

The family of circle maps \begin{equation*} f_{b, \omega} (x) = x + \omega + b\, \phi(x) \end{equation*} is used as a simple model for a periodically forced oscillator. The parameter $\omega$ represents the unforced frequency, $b$ the coupling, and $\phi$ the forcing. When $\phi = \frac{1}{2 \pi} \sin(2 \pi x)$ this is the classical Arnol'd standard family. Such families are often studied in the $(\omega,b)$-plane via the so-called tongues $T_\beta$ consisting of all $(\omega,b)$ such that $f_{b, \omega}$ has rotation number $\beta$. The interior of the rational tongues $T_{p/q}$ 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 $T_{p/q}$ 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 $k\geq 3$ 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 $C^r$ ($r>0$) forcing.