This paper introduces a novel memristor-based hyperchaotic system in which the integer-order derivatives are replaced by a variable-order fractal-fractional operator. The dynamical properties of the system, including equilibrium points, Lyapunov exponents, bifurcation diagrams with respect to the variable orders, and the Kaplan–Yorke dimension, are analyzed. A synchronization scheme based on active control is designed for the master–slave configuration, and global Mittag–Leffler stability of the error dynamics is established using a suitable variable-order Lyapunov function. The synchronized states are then applied to an image encryption algorithm. Numerical simulations, security analyses, and NIST randomness tests demonstrate the effectiveness and enhanced performance of the proposed framework compared to existing fixed-order and classical fractional-order methods.
Ren et al. (2026) studied this question.