Reconstruction of Second-Order Discrete Chaotic Maps

Discrete chaotic maps are widely utilized in engineering applications ranging from signal processing to computational intelligence. However, existing research predominantly focuses on first-order maps. Higher-order maps offer a superior capacity to characterize the system’s historical evolution processes and inherent memory effects, rendering their investigation of profound theoretical significance. Starting from the construction of second-order discrete maps, this paper systematically elucidates the dynamical reconstruction methodology for such systems. Furthermore, the feasibility of implementing offset boosting within the second-order regulation framework is explored to achieve flexible attractor control. The systems are also validated on the Gowin GW5A-LV25UG FPGA platform. Finally, the computational potential of the proposed systems is evaluated through a reservoir computing task. Comparative experiments on the Mackey–Glass chaotic time-series prediction demonstrate that the intrinsic memory effect endows the second-order maps with expanded memory capacity and can significantly enhance the prediction accuracy compared to their memoryless first-order prototypes.