In this paper, the dynamic behaviors and synchronization is investigated for a three-mode system of Couette-Taylor flow, and the corresponding evolution of Couette-Taylor flow is also explained. By constructing a family of generalized radically infinite and positive definite Lyapunov functions, the globally attractive set and positively invariant set of the Couette?Taylor flow system are obtained. It is found that the designed linear feedback controller are effective in globally synchronizing two identical chaotic systems. Eventually, a numerical example is provided to validate the feasibility of the results.