Pinched Arnol'd tongues for Families of circle maps
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910983500333056 |
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| author | Boyland, Philip |
| author_facet | Boyland, Philip |
| contents | The family of circle maps \begin{equation*} f_{b, ω} (x) = x + ω+ b\, ϕ(x) \end{equation*} 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 $ϕ= \frac{1}{2 π} \sin(2 πx)$ this is the classical Arnol'd standard family. Such families are often studied in the $(ω,b)$-plane via the so-called tongues $T_β$ consisting of all $(ω,b)$ such that $f_{b, ω}$ has rotation number $β$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_02988 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pinched Arnol'd tongues for Families of circle maps Boyland, Philip Dynamical Systems 37E10 (Primary) 37E45 (Secondary) The family of circle maps \begin{equation*} f_{b, ω} (x) = x + ω+ b\, ϕ(x) \end{equation*} 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 $ϕ= \frac{1}{2 π} \sin(2 πx)$ this is the classical Arnol'd standard family. Such families are often studied in the $(ω,b)$-plane via the so-called tongues $T_β$ consisting of all $(ω,b)$ such that $f_{b, ω}$ has rotation number $β$. 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. |
| title | Pinched Arnol'd tongues for Families of circle maps |
| topic | Dynamical Systems 37E10 (Primary) 37E45 (Secondary) |
| url | https://arxiv.org/abs/2506.02988 |