Dynamics analysis and predefined-time sliding mode synchronization of multi-scroll systems based on a single memristor model

To overcome the limitations of conventional designs for memristive multi-scroll chaotic systems, this paper introduces a novel memristor that relies solely on a single memristor function and a single state variable function to generate both odd and even numbers of double-scroll attractors. This design not only simplifies the memristor structure but also offers a new approach to constructing multi-scroll chaotic systems. The proposed memristor is integrate into a modified Sprott-C system to develop the one-dimensional memristive multi-scroll Sprott-C systems (1D-MMSCS), two-dimensional memristive multi-scroll Sprott-C systems (2D-MMSCS), and the three-dimensional memristive multi-scroll Sprott-C systems (3D-MMSCS). The complex dynamics of these memristive multi-scroll systems are analyzed using equilibrium points, Poincaré maps, bifurcation diagrams, and Lyapunov exponents. Interestingly, the constructed MMSCS exhibits extreme multi-stability, indicating its high sensitivity to initial conditions and enhanced unpredictability. To verify the practical feasibility of the system, it is developed a digital hardware platform based on a Field-Programmable Gate Array (FPGA) and successfully implemented both the 1D-MMSCS and 2D-MMSCS. Finally, leveraging Lyapunov stability theory and predefined-time stability theory, a novel predefined-time sliding mode control scheme (PTSMS) is proposed. This scheme is applied to achieve synchronization in the more complex 3D-MMSCS. Simulation results confirm that the proposed method ensures rapid synchronization and exhibits strong robustness against internal uncertainties and external disturbances.