Arnold Tongues in Area-Preserving Maps

In the early 60’s J. B. Keller and D. Levy discovered a fundamental property: instability tongues in Mathieu-type equations lose sharpness with addition of higher-frequency harmonics Mathieu potentials. Twenty years later, V. Arnold similar phenomenon on circle maps (and rediscovered result Levy). this paper we find third class object where type behavior takes place: area-preserving cylinder. loosely speaking, show that periodic orbits standard are extra fragile respect to added drift (i.e. non-exactness) if potential map is trigonometric polynomial. That is, make more robust “drift". This observation was motivated by study traveling waves discretized sine-Gordon equation which turn models wide variety physical systems.