An incommensurate ultra-low fractional-order Chen system for sliding mode synchronization and secure communication

Abstract

This study investigates the dynamical characteristics, synchronization properties, and secure communication capability of an incommensurate fractional-order Chen (IFOC) chaotic system. A key contribution of this work is the identification and utilization of extremely low fractional-order values ( \(q_1\) = 0.2, \(q_2\) = 0.3, \(q_3\) = 0.4), close to the critical threshold at which chaotic dynamics vanish. In these regimes, the chaotic bandwidth becomes significantly compressed, and numerical simulation becomes highly challenging due to increased memory effects and reduced Lyapunov divergence. Despite this difficulty, the system is shown to exhibit a persistent chaotic attractor. To provide a transparent synchronization baseline under these chaotic conditions, a conventional full-state active sliding-mode controller with model-based cancellation is adopted, and its nominal closed-loop stability is established through a component-wise vector-Lyapunov framework formulated for incommensurate fractional-order systems. Building on the resulting synchronized dynamics, a projective multiplexed chaotic masking scheme is designed to transmit multiple messages securely. Simulation results demonstrate accurate message recovery, strong masking capability, and high resilience to parameter mismatch, noise, and controller gain perturbation. The ability to achieve secure communication at such low fractional orders distinguishes this work from existing literature and highlights the potential of fractional-order systems in advanced secure communication frameworks.