Construction of a nonlinear cyclically symmetric conservative chaotic system and its application in image encryption

Abstract Conservative chaotic systems lack attractors compared to dissipative chaotic systems and are less susceptible to reconstruction. Each state equation in a cyclic symmetric system shares a similar structure. By combining cyclic symmetric systems with conservative chaotic systems, the positions of x 1 , x 2 ,…, x n follow the sequence x 1 → x 2 →···→ x n → x 1 . This constructs a conservative chaotic system incorporating nonlinear cyclic symmetry. Based on the model of a cyclically symmetric conservative system, this paper constructs four-dimensional and five-dimensional conservative chaotic systems incorporating a cos x nonlinear term, each exhibiting distinct dynamical characteristics. The two constructed chaotic systems are compared in terms of the range of Lyapunov exponents, bifurcation diagram range, and distinctive features. The four-dimensional cyclic symmetric conservative chaotic system exhibits broader Lyapunov exponent ranges and bifurcation map ranges, along with higher efficiency. When applied to image encryption algorithms, the method employs interactive scrambling and XOR diffusion to generate ciphertext image. Finally, the safety analysis revealed an average information entropy of 7.9993, with correlations approaching 0. The NPCR and UACI rates were approximately 99.61% and 33.46%, respectively. These results demonstrate the algorithm’s robust performance against statistical and differential attacks.