Theoretical and numerical investigation of a memristor system with a piecewise memductance under fractal–fractional derivatives

Abstract This research deals with the theoretical and numerical investigations of a memristor system with memductance function. Stability, dissipativity, and Lyapunov exponents are extensively investigated and the chaotic tendencies of the system are studied in depth. The memristor model, where a piecewise memductance function is incorporated, is modified with fractal–fractional derivatives with exponential decay, power law, and Mittag–Leffler kernels, which provide powerful tools for modeling complex systems with memory effects, long-range interactions, and fractal-like behavior. Employing the Krasnoselskii–Krein uniqueness theorem and the fixed point theorem, the existence and uniqueness of the solutions of the model including fractal–fractional derivatives with the Mittag–Leffler kernel are proven. The fractal–fractional derivative model is solved numerically using the Lagrange polynomial approach, and the chaotic tendencies of the system are exhibited through numerical simulations. The findings indicated that the memristor model with fractal–fractional derivatives was observed to exhibit chaotic behavior.