陀螺仪运动的混沌与KAM理论

本文建立了外环轴水平放置的重力对称陀螺仪的运动方程．将非自由陀螺仪转子的质心位置作为扰动，利用Ｍｅｌｎｉｋｏｖ方法和ＫＡＭ理论研究了陀螺仪的运动，研究结果表明：在 螺仪转子的质心与支架中心不重合且充分接近时，或 螺仪能量充分大时，陀螺仪运动的Ｐｏｉｎｃａｒｅ映射既出现Ｓｍａｌｅ马蹄意义下的混沌，又存在ＫＡＭ不变环面和不变闭曲线。

THE CHAOS AND KAM THEORY OF THE GYROSCOPE

This paper deals with the motion of a gravity-symmetric gyroscope which outer ring is placed on horizontal plane. The Melnikov's mathods and KAM theory can be used to describe the motion of a non-free gyroscope if the gravity center of its rotor is treated as a perturbation. It is shown that its Poincare's map have the chaos in the sense of Smale horseshoe, and the KAM invariant ton and closed curves either if the gravity center of the gyroscope's rotor doesn't coincide with the center of its support, and the distance between them is sufficiently small, or its energy is sufficiently large in comparision with its potential energy.