The memristor is a nonlinear element and intrinsically possesses memory function. When it works as nonlinear part of a chaotic system, the complexity and the randomness of signal will be enhanced. In this paper memristor is introduced into a three-dimensional chaotic system based on the augmented L system. The interesting and promising behaviors of complex single, double and four-scroll chaotic attractors generated only by varying a parameter have not been reported in memristive chaotic system and thus they deserve to be further investigated. It is also obvious that such a simple change of one parameter could be used to generate a variety of quite complex attractors. Therefore, as a nonlinear device the memristor plays an important role in this system. Firstly, some basic dynamical properties of the memristive chaotic system, including symmetry and in-variance, the existence of attractor, equilibrium, and stability are investigated in detail. By numerically simulating the power spectrum, Lyapunov exponent, Poincare map and bifurcation diagram, in this paper we verify that the proposed system has abundant dynamical behaviors. The sensitivities of system parameters to the chaotic behaviors are further explored by calculating, in detail, its Lyapunov exponent spectrum and bifurcation diagrams. The results of simulation and experiment are in good agreement, thereby proving the veracity of analysis. The memristive chaotic circuit is designed using the memristor, operational amplifier, analog multiplier and other conventional components. The circuit implementation of the memristive system is simulated using SPICE (simulation program with integrated circuit emphasis). The SPICE simulation results and the theoretical analysis are found to be in good agreement, and thus verifying that the system can produce chaos. Pulse synchronization has the following characteristics: low energy consumption, fast synchronization and easy-to-implement single-channel transmission. Therefore, it is more practical in chaotic secure communication. Subsequently the pulse chaos synchronization is realized from the perspective of the maximum Lyapunov exponent, and numerical simulations show the existence of new memristive chaotic system and the feasibility of pulse synchronization control, and also provide an experimental basis for further studying the applications of the memristive chaotic system in voice secure communication and information processing.