This paper is a small review devoted to the dynamics of a point on a paraboloid. Specifically, it is concerned with the motion both under the action of a gravitational field and without it. It is assumed that the paraboloid can rotate about a vertical axis with constant angular velocity. The paper includes both well-known results and a number of new results.We consider the two most widespread friction (resistance) models: dry (Coulomb) friction and viscous friction. It is shown that the addition of external damping (air drag) can lead to stability of equilibrium at the saddle point and hence to preservation of the region of bounded motion in a neighborhood of the saddle point. Analysis of three-dimensional Poincaré sections shows that limit cycles can arise in this case in the neighborhood of the saddle point.

中文翻译：

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抛物型Chaplygin钟摆和Paul Trap：不可积性，稳定性和有界性

本文是一篇有关抛物面的动力学的小型综述。具体地说，它既涉及在重力场作用下的运动，也涉及没有重力场的运动。假定抛物面可以恒定角速度绕垂直轴旋转。本文同时包含了著名的结果和许多新的结果。我们考虑了两种最广泛的摩擦（阻力）模型：干（库仑）摩擦和粘性摩擦。结果表明，外部阻尼（空气阻力）的增加可导致在鞍点处保持平衡的稳定性，并因此保留在鞍点附近的有界运动区域。三维庞加莱截面的分析表明，在这种情况下，在鞍点附近会出现极限环。