Nonlinear stability and bifurcations of symmetric heavy gyroscope

The complete solutions of the upright and oblique permanent rotations of a symmetric heavy gyroscope with perfect dissipation are given. The asymptotic stability criteria and unstability criteria for these rotations in the sense of Liapunov and the sense of Movchan are also given on the basis of exact nonlinear motion equations respectively. The related oblique rotations are non-isolated. The main subdomains of the regions of asymptotic stability are obtained. The related bifurcation phenomena are discussed in detail. The project is supported by the National Natural Science Foundation of China.