论文标题
区域保护地图中的阿诺德舌头
Arnold Tongues in Area-Preserving Maps
论文作者
论文摘要
在60年代初期，J。B。Keller和D. Levy发现了一个基本特性：Mathieu型方程中的不稳定性舌头在Mathieu潜力中增加了高频谐波而失去了清晰度。 20年后，阿诺德（V. Arnold）在圆形地图中重新发现了阿诺德舌头清晰度的类似现象（并重新发现了凯勒（Keller）和征费的结果）。在本文中，我们找到了发生类似风味行为的第三类物体：圆柱体的区域保护地图。宽松地说，我们表明，如果地图的电势是三角多项式，那么标准图的周期性轨道相对于增加的漂移（即非出现）是额外脆弱的。也就是说，高频谐波使周期性的轨道在``漂移''方面变得更加健壮。观察是通过对离散的正弦戈登方程的研究进行的研究激励的，进而模拟了各种各样的物理系统。
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.
