On a new four-dimensional model of memristor-based chaotic circuit in the context of nonsingular Atangana–Baleanu–Caputo operators

• Main Authors: Chernet Tuge Deressa, Sina Etemad, Shahram Rezapour
• Format: Article
• Language: English
• Published: SpringerOpen 2021-10-01
• Series: Advances in Difference Equations
• Subjects: Memristor-based chaotic system; ABC operator; Bifurcation diagram; Lyapunov exponent; Sensitivity to initial values; Fixed point
• Online Access: https://doi.org/10.1186/s13662-021-03600-9
• ISSN: 1687-1847

Abstract A memristor is naturally a nonlinear and at the same time memory element that may substitute resistors for next-generation nonlinear computational circuits that can show complex behaviors including chaos. A four-dimensional memristor system with the Atangana–Baleanu fractional nonsingular operator in the sense of Caputo is investigated. The Banach fixed point theorem for contraction principle is used to verify the existence–uniqueness of the fractional representation of the given system. A newly developed numerical scheme for fractional-order systems introduced by Toufik and Atangana is utilized to obtain the phase portraits of the suggested system for different fractional derivative orders and different parameter values of the system. Analysis on the local stability of the fractional model via the Matignon criteria showed that the trivial equilibrium point is unstable. The dynamics of the system are investigated using Lyapunov exponents for the characterization of the nature of the chaos and to verify the dissipativity of the system. It is shown that the supposed system is chaotic and it is significantly sensitive to parameter variation and small initial condition changes.