Recently, the nonlinear dynamics of memristor has attracted much attention. In this paper, a novel fourdimensional hyper-chaotic system (4D-HCS) is proposed by introducing a tri-valued memristor to the famous Lü system. Theoretical analysis shows that the 4D-HCS has complex chaotic dynamics such as hidden attritors and coexistent attractors, and it has larger maximum Lyapunov exponent and chaotic parameter space than the original Lü system. We also experimentally analyze the dynamics behaviors of the 4D-HCS in aspects of the phase diagram, Poincaré mapping, bifurcation diagram, Lyapunov exponential spectrum, and the correlation coefficient, and the analysis results show the complex dynamic characteristics of the proposed 4D-HCS. In addition, the comparison with binary-valued memristorbased chaotic system shows that the 4D-HCS has unique characteristics such as hyper-chaos and coexistent attractors. To show the easy implementation of the 4D-HCS, we implement the 4D-HCS in an analogue circuit-based hardware platform, and the implementation results are consistent with the theoretical analysis. Finally, using the 4D-HCS, we design a pseudorandom number generator to explore its potential application in cryptography.