A new four-dimensional memristive system, synchronization and its application in image encryption

In this paper, a new four-dimensional (4D) memristive system is presented by introducing a memristor equation to a 3D chaotic system. Firstly, the basic characteristics including existence and stability of equilibrium points, and Kaplan–Yoke dimension for the memristive system are analyzed from a theoretical perspective. The description of the 4D memristive system in the integer-order and the fractional-order cases is given. In both cases, the dynamics with the variation of a system parameter or a derivative order is studied by the numerical simulations. The results show that the integer-order and the fractional-order memristive systems have rich dynamics. Secondly, in order to realize the synchronization for the memristive system, two adaptive synchronization schemes are designed, namely, the identical structure and the different structure synchronization. Numerical simulation results show that the designed synchronization controllers are effective. Finally, a novel image encryption algorithm is expanded to the image encryption for color images based on the memristive system. The simulation results demonstrate the image encryption method based on the memristive system has a perfect encryption effect for the color images.