A simple four-dimensional memristive chaotic system with multistability and amplitude modulation characteristics is constructed based on three-dimensional Sprott-R chaotic system. Firstly, the system's stability characteristics are analyzed, and it is found that the system has infinite unstable equilibrium points. And then, the complex dynamical behaviors of the memristive chaotic system are studied using Lyapunov exponential spectrum, bifurcation diagram and the projection of chaotic attractors on the phase plane. The results show that when the system parameters change, the system will change from chaotic state to periodic state through inverse periodic bifurcation. And under different initial conditions, the system can generate three kinds of coexistence attractors, namely chaotic attractors coexistence, periodic limit cycle and chaotic attractor coexistence, and periodic limit cycles coexistence. In addtion, when the initial conditions change, the amplitude of the four-dimensional chaotic signals will change. Finally, the circuit design and realization of the system are carried out to further verify the existence of the new system.