Arnold Tongues in Area-Preserving Maps
| Autor: | Mark Levi, Jing Zhou |
|---|---|
| Rok vydání: | 2022 |
| Předmět: |
Mathematics (miscellaneous)
Mathematics - Classical Analysis and ODEs Mechanical Engineering Classical Analysis and ODEs (math.CA) FOS: Mathematics FOS: Physical sciences Dynamical Systems (math.DS) Mathematical Physics (math-ph) 37N05 37C25 34C15 Mathematics - Dynamical Systems Analysis Mathematical Physics |
| DOI: | 10.48550/arxiv.2206.10040 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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