A NOVEL 4D NO-EQUILIBRIUM HYPERCHAOTIC SYSTEM WITH APPLICATIONS IN SYNCHRONIZATION AND IMAGE ENCRYPTION

Abstract

Four-dimensional chaotic models, such as the no-equilibrium system proposed by Zhang et al. [1], offer significant cryptographic advantages by generating hidden attractors. However, they often fall short of true hyperchaos by producing only a single positive Lyapunov exponent. This research addresses this limitation to significantly expand the available cryptographic key space. We transform a standard 4D chaotic framework into a robust 4D hyperchaotic system. This is achieved by replacing simple polynomial nonlinearities with a custom hybrid activation function. By blending signum and sigmoid traits with layered shifting functions, the proposed system sustains deep hyperchaotic behavior. A comprehensive dynamical analysis confirms the complete absence of fixed points. Crucially, the system exhibits two independent positive Lyapunov exponents. This guarantees the generation of highly complex, unpredictable hidden attractors. The system’s complexity is further validated by a Kaplan–Yorke dimension of approximately 3.174 and severe structural randomness, indicated by a Shannon entropy of 4.6715. The practical viability of the system is demonstrated across two secure communication scenarios. First, an active control synchronization strategy is implemented. This suc cessfully drives synchronization errors to machine precision (10−14) at an exponential rate. Second, the system serves as a keystream generator for a rigorous image encryption algorithm. This algorithm utilizes argsort-based spatial permutation alongside chained XOR diffusion. The resulting ciphertexts achieve a near-ideal information entropy of 7.997 bits, perfectly flat intensity histograms, and essentially zero adjacent pixel correlation. Furthermore, the algorithm achieves an NPCR of 99.4787% against sub-microscopic key perturbations. These results demonstrate that the engineered 4D no-equilibrium hyperchaotic system provides superior resistance to statistical and differential cryptanalysis when compared to standard chaotic maps.