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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2206.10040 |
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| _version_ | 1866912396192251904 |
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| author | Zhou, Jing Levi, Mark |
| author_facet | Zhou, Jing Levi, Mark |
| contents | In the early 60's J. B. Keller and D. Levy discovered a fundamental property: the instability tongues in Mathieu-type equations lose sharpness with the addition of higher-frequency harmonics in the Mathieu potentials. 20 years later V. Arnold rediscovered a similar phenomenon on sharpness of Arnold tongues in circle maps (and rediscovered the result of Keller and Levy). In this paper we find a third class of objects where a similarly flavored behavior takes place: area-preserving maps of the cylinder. Speaking loosely, we show that periodic orbits of standard maps are extra fragile with respect to added drift (i.e. non-exactness) if the potential of the map is a trigonometric polynomial. That is, higher-frequency harmonics make periodic orbits more robust with respect to ``drift". The observation was motivated by the study of traveling waves in the discretized sine-Gordon equation which in turn models a wide variety of physical systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_10040 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Arnold Tongues in Area-Preserving Maps Zhou, Jing Levi, Mark Dynamical Systems Mathematical Physics Classical Analysis and ODEs 37N05, 37C25, 34C15 In the early 60's J. B. Keller and D. Levy discovered a fundamental property: the instability tongues in Mathieu-type equations lose sharpness with the addition of higher-frequency harmonics in the Mathieu potentials. 20 years later V. Arnold rediscovered a similar phenomenon on sharpness of Arnold tongues in circle maps (and rediscovered the result of Keller and Levy). In this paper we find a third class of objects where a similarly flavored behavior takes place: area-preserving maps of the cylinder. Speaking loosely, we show that periodic orbits of standard maps are extra fragile with respect to added drift (i.e. non-exactness) if the potential of the map is a trigonometric polynomial. That is, higher-frequency harmonics make periodic orbits more robust with respect to ``drift". The observation was motivated by the study of traveling waves in the discretized sine-Gordon equation which in turn models a wide variety of physical systems. |
| title | Arnold Tongues in Area-Preserving Maps |
| topic | Dynamical Systems Mathematical Physics Classical Analysis and ODEs 37N05, 37C25, 34C15 |
| url | https://arxiv.org/abs/2206.10040 |