Rich dynamics and multiple transitions of a discrete memristive coupled map

This paper proposes a five-dimensional coupled Logistic-memristor chaotic map (CL-MCM) by connecting two pairs of Logistic maps via dual discrete memristive couplers with distinct nonlinearities. Line fixed-point sets related to the memristor initial condition are identified, and their stability is analyzed. Numerical investigations, including bifurcation analysis, Lyapunov-exponent spectra, and complexity measures, demonstrate that CL-MCM can generate abundant periodic, chaotic, and hyperchaotic dynamics with high randomness. Basin analysis reveals rich multistability phenomena, including initial-related heterogeneous coexistence and homogeneous extreme multistability induced by a periodic initial-boosting effect of the memristor. Moreover, synchronization behaviors between subsystems are studied, and complete and lag synchronization windows depending on coupling strength and memristor initial condition are disclosed, highlighting an initial-boosting synchronization mechanism. Furthermore, the map is extended to a two-dimensional coupled network to generate ordered spatiotemporal patterns. Finally, FPGA implementation and applications to pseudorandom number generation and image encryption validate the feasibility and effectiveness of the proposed model.