A memristive Ikeda map and its application in image encryption

Existing research indicates that discrete-time chaotic systems are more likely to achieve hyperchaotic states in lower dimensions compared to continuous-time chaotic systems. Recently, introducing discrete memristors into chaotic map to enhance system dynamics performance has become a hot topic in the field of chaos research. In this paper, a memristive Ikeda map (MIKM) based on discrete memristors is proposed and the system dynamics behavior is analyzed in depth by chaotic attractor phase diagrams, Lyapunov exponent spectrum, bifurcation diagrams, spectral entropy (SE), distributional properties and fractal dimensions. Numerical simulation results indicate that the introduction of discrete memristor enriches the dynamic characteristics of the Ikeda map, such as expanding the range of chaos, enhancing the ergodicity, and prompting the transition from chaotic to hyperchaotic states. We further studied the influence of coupling strength $K$ on the dynamic behavior of the system. We explored the use of the discrete memristor as internal perturbations to achieve parameter-controlled symmetric attractors and the introduction of constant controllers to achieve signal polarity adjustment. At the same time, we implemented the improved Ikead map on the STM32 hardware platform and developed a pseudo-random number generator (PRNG). Finally, an image encryption algorithm was designed based on the proposed improved Ikeda map. The experimental results show that the proposed algorithm is robust.