Design and characteristic analysis of incommensurate-order fractional discrete memristor-based hyperchaotic system

The discrete memristive chaotic system is characterized by discontinuous phase trajectories. To address the limitations of the ideal integer-order discrete memristor model, which fails to accurately reflect the characteristics of practical devices, this study introduces a Grunwald-Letnikov type quadratic trivariate fractional discrete memristor model to enhance the nonlinearity and memory properties of memristors. Simultaneously, it is demonstrated that our model satisfies the essential characteristics of the generalized memristor. Based on this newly proposed fractional discrete memristor, a new four-dimensional fractional discrete memristive hyperchaotic system is constructed by coupling non-uniform, incommensurate-order memristors. This system advances the structure of existing discrete chaotic systems and provides a more flexible strategy for optimizing memory effects. The dynamical behaviors are analyzed using attractor phase diagrams, bifurcation diagrams, Lyapunov exponent spectra, and permutation entropy complexity. Numerical simulation results show that the system can exhibit a larger hyperchaotic region, higher complexity, and rich multistable behaviors, such as the coexistence of infinitely symmetric attractors and enhanced offset. Additionally, the impact of the incommensurate-order parameter on the system's chaotic behavior is revealed, with order serving as a tunable control variable that dynamically reconfigures bifurcation paths as needed, thereby enabling transitions between hyperchaotic, chaotic, and non-chaotic states. Furthermore, a simulation circuit was designed to validate the numerical simulation results.