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Abstract: 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. [Copyright &y& Elsevier]

Abstract: Nonlinear control is an effective method for making two identical chaotic systems or two different chaotic systems be synchronized. However, this method assumes that the Lyapunov function of error dynamic (e) of synchronization is always formed as V(e)=1/2eTe. In this paper, modification based on Lyapunov stability theory to design a controller is proposed in order to overcome this limitation. The method has been applied successfully to make two identical new systems and two different chaotic systems (new system and Lorenz system) globally asymptotically synchronized. Since the Lyapunov exponents are not required for the calculation, this method is effective and convenient to synchronize two identical systems and two different chaotic systems. Numerical simulations are also given to validate the proposed synchronization approach. [Copyright &y& Elsevier]