In this work, a new hyper-chaotic system based on memristor is proposed and the anti-synchronization is demonstrated. Through the analysis of the system behaviors, including the stability of equilibrium set, Lyapunov exponents and phase portraits, it demonstrates the proposed system is a new hyper-chaotic system with the features of infinite equilibrium points and complex dynamic characteristics. Besides the anti−synchronization of the memristor hyper-chaotic system is realized. It is proven that the dynamics of the anti-synchronization error is globally and asymptotically stable based on the Lyapunov stability theory. Numerical simulations are given in the end. It provides the theoretical foundation for the aspects of security communication and image encryption.