Pinched Arnol'd tongues for Families of circle maps

Abstract

We prove that generically for a family of circle maps \begin{equation*} f_{b, ω} (x) = x + ω+ b\, ϕ(x) \end{equation*} with $ϕ$ a piecewise linear forcing with $k>2$ breakpoints there is no pinching in any of its Arnol'd tongues. This is in contrast to a theorem of Campbell, Galeeva, Tresser, and Uherka who showed that with two break points there are always multiple pinchpoints in its rational tongues. We also prove that the absence of pinching is generic for Lipschitz and $C^r$ ($r>0$) forcing. The family $f_{b, ω}$ is used as a simple model for a periodically forced oscillator. The rational tongue $T_{p/q}$ represents parameter values where the system is mode-locked into a $p/q$-periodic response. The pinching of the tongues to a point represents parameter values where the system's periodic response is unstable to all perturbations in the frequency parameter $ω$. The theorems in this paper show that typically this type of instability does not occur in the families under consideration.