Design and implement a nested coexisting multi-vortex hyperchaotic system with multiple stability memristors

Abstract This paper proposes a nested coexisting multi-vortex hyperchaotic system based on memristors. Analysis of the system’s equilibrium points, Lyapunov exponents and bifurcations, coexisting attractors, complexity, etc. The system has infinitely many equilibrium points, and the number of system vortices can be expanded with parameters. The most significant feature of this system is the existence of nested coexisting attractors and mega-stability phenomena. Additionally, it exhibits phenomena of homogeneous multi-stability, heterogeneous multi-stability, and extreme multi-stability. Further analysis was conducted on the relationship between these phenomena and the basins of attraction. Under certain parameters, the system (6) exhibits burst oscillations, transient chaos, and intermittent chaos, demonstrating diverse dynamic characteristics. To verify the feasibility of the system theory, design and build a simulation circuit using Multisim, and implement digital circuit simulation on FPGA. By realizing analog and digital circuits, further broaden its application value. Finally, the system is applied to adaptive synchronization and anti-synchronization. Simulation results indicate that the time required for achieving complete synchronization or complete anti-synchronization is shorter. This significant reduction in synchronization and anti-synchronization time not only enhances the system’s operational efficiency and performance but also strengthens its adaptability, thus carrying important theoretical and practical value for system applications.