Arnold Tongues in Area-Preserving Maps

Mark Levi, Jing Zhou

Research output: Contribution to journalArticlepeer-review

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Abstract

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. Twenty years later, V. Arnold discovered a similar phenomenon on the sharpness of Arnold tongues in circle maps (and rediscovered the result of Keller and Levy). In this paper we find a third class of object where a similar type of behavior takes place: area-preserving maps of the cylinder. loosely speaking, 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". This 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.

Original languageEnglish (US)
Article number32
JournalArchive for Rational Mechanics and Analysis
Volume247
Issue number3
DOIs
StatePublished - Jun 2023

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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