Dynamical analysis of a fractional discrete-time Chua’s circuit system

In this paper, a fractional discrete Chua’s system exhibiting both chaotic and hyperchaotic behaviors is proposed. Stability analysis is conducted for both commensurate and incommensurate fractional orders at equilibrium points, supported by numerical calculations and simulations. The critical point of Hopf bifurcation is determined for commensurate orders. To understand how order asymmetry influences the dynamic complexity of a fractional system, a principle named Synchronous Order Maximum Entropy Theorem is proposed. The study focuses on diverse attractors, including quasi-periodic, chaotic, periodic, and hyperchaotic types, as well as the coexistence of multiple attractors under specific parameter settings. Notably, the system demonstrates the coexistence of offset-boosted and initial-switched boosting behaviors, which are rarely observed in discrete systems. Furthermore, synchronization of the fractional discrete Chua’s system is achieved. The results demonstrate that the proposed fractional discrete Chua’s system exhibits remarkably rich and complex dynamical characteristics. Finally, an image encryption application based on this system is presented to illustrate its practical potential.