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 L¨u 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 L¨u system. We also experimentally analyze the dynamics behaviors of the 4D-HCS in aspects of the phase diagram, Poincar´e 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.