Complexity of Motions of Gyroscope

The motions of a gravity symmetric gyroscope, whose outer ring's axis is placed horizontally, are dealt with. Firstly, the motions of a free gyroscope are studied and explained in the sense of mechanics under a certain condition. The Melnikov's methods and KAM theory can be used to study the motions of a non-free gyroscope if its rotor's gravity center can be treated as a perturbation. It is shown that the motions of the non free gyroscope are chaotic in the sense of Smale horseshoe, and the KAM invariant tori and closed curves exist in the Hamiltonian flows of the perturbed system if the gravity center of the gyroscope's rotor doesn't coincide with the center of its supports, and the distance between them is sufficiently small, or its energy is sufficiently large in comparison with its potential energy.