Bifurcations and chaos of a two-degree-of-freedom dissipative gyroscope

The dynamic behaviors of a dissipative gyroscope mounted on a vibrating base are investigated qualitatively and numerically. It is shown that the nonlinear system can exhibit regular and chaotic motions. The qualitative behaviors of the system are studied by the center manifold theorem and the normal form theorem. The co-dimension one bifurcation analysis for the Hopf bifurcation is carried out. The pitchfork, Hopf, and saddle connection bifurcations for co-dimension two bifurcation are also found in this study. Regular and chaotic motions are shown to be possible in the parameter space. Numerical methods are used to obtain the time histories, the Poincaré maps, the Liapunov exponents, and the Liapunov dimensions. The effect of the spin speed of the gyroscope on its dynamic behavior is also studied by numerical simulation in conjunction with the Liapunov exponents, and it has been found that the higher spin speed of the gyroscope can quench the chaotic motion.