Adaptive Asymptotic Shape Synchronization of a Chaotic System with Applications for Image Encryption

In contrast to previous research that has primarily focused on distance synchronization of states in chaotic systems, shape synchronization emphasizes the geometric shape of the attractors of two chaotic systems. Diverging from the existing work on shape synchronization, this paper introduces the application of adaptive control methods to achieve asymptotic shape synchronization for the first time. By designing an adaptive controller using the proposed adaptive rule, the response system under control is able to attain asymptotic synchronization with the drive system. This method is capable of achieving synchronization for models with parameters requiring estimation in both the drive and response systems. The control approach remains effective even in the presence of uncertainties in model parameters. The paper presents relevant theorems and proofs, and simulation results demonstrate the effectiveness of adaptive asymptotic shape synchronization. Due to the pseudo-random nature of chaotic systems and their extreme sensitivity to initial conditions, which make them suitable for information encryption, a novel channel-integrated image encryption scheme is proposed. This scheme leverages the shape synchronization method to generate pseudo-random sequences, which are then used for shuffling, scrambling, and diffusion processes. Simulation experiments demonstrate that the proposed encryption algorithm achieves exceptional performance in terms of correlation metrics and entropy, with a competitive value of 7.9971. Robustness is further validated through key space analysis, yielding a value of 10210×2512, as well as visual tests, including center and edge cropping. The results confirm the effectiveness of adaptive asymptotic shape synchronization in the context of image encryption.