A memristive hyperchaotic map with extreme multistability: dynamics, complexity, and application in image encryption

The memristor’s capacity to change its resistive state in response to external stimuli allows it to serve as the foundation for generating pseudorandom sequences via appropriate control circuitry. Capitalizing on this property, we present a three-dimensional memristive hyperchaotic map (3D-MHM), formulated by merging a sinusoidal nonlinearity with a discrete memristor model to amplify chaotic complexity. An investigation of its dynamics through Lyapunov exponents reveals that the 3D-MHM undergoes transitions from periodic states to chaos and hyperchaos. The intricacy of its iterative output is further validated using metrics such as permutation entropy and \(C_0\) complexity. A distinctive analysis of this system reveals that it can produce coexisting attractors and multistability, where varying the initial conditions leads to attractors appearing at different spatial positions. Furthermore, the offset boosting behavior of the 3D-MHM displays the coexistence of an infinite number of homogeneous attractors boosted by the memristor initial condition. Leveraging these attributes, a novel color image encryption algorithm based on the 3D-MHM is proposed, presenting a more secure scheme for large-volume data encryption. Statistical and cryptographic analyses confirm the significant potential of the keystream produced by the 3D-MHM for cryptographic applications.