Multistability and synchronization of discrete maps via memristive coupling

Due to its internal state, the memristive nonlinearity has different dynamic characteristics from the traditional resistive nonlinearity. It has been proved that the memristive nonlinearity used as a nonlinear coupler to connect two dynamical systems has rich multi-stability phenomena and complex synchronous behaviors. However, few reports have extended memristive couplers and dynamical systems from continuous-time domain to discrete-time domain. To this end, this article proposes a simple memristor-coupled Logistic map (MCLM) model by coupling two identical Logistic maps through a memristive coupler. The MCLM model owns two line fixed point sets related to memristor initial condition and their stability distributions are discussed based on three eigenvalues. The chaotic/hyperchaotic attractors with outstanding performance indicators are revealed using numerical methods, and the initial-related heterogeneous multistability and memristor initial-boosting homogeneous multistability are demonstrated by basins of attraction that have complex and fractal evolutions. Afterwards, by inspecting the synchronous behaviors of the two Logistic maps in the MCLM model, the lag and complete synchronization behaviors dependent on the coupling strength and memristor initial condition, especially the homogeneous synchronization behavior boosted by the memristor initial condition, are disclosed in succession. In addition, an MCU-based hardware platform is fabricated to experimentally validate the numerical results. Of particular interest, to the best knowledge of the authors, the initial-boosting synchronization has not been reported in the literature.