Arnold Tongues in Area-Preserving Maps

TL;DR

This paper investigates the fragility of Arnold tongues in area-preserving maps, revealing that higher harmonics in the potential make periodic orbits more robust against drift, extending classical results to a new class of dynamical systems.

Contribution

It demonstrates a novel phenomenon where higher harmonics enhance the stability of periodic orbits in area-preserving maps, a behavior analogous to known results in circle maps and Mathieu equations.

Findings

Higher harmonics increase robustness of periodic orbits against drift.

Fragility of Arnold tongues depends on the harmonic content of the potential.

Results are motivated by applications to discretized sine-Gordon equations.

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. 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.