On a Memristor-Based Hyperchaotic Circuit in the Context of Nonlocal and Nonsingular Kernel Fractional Operator

Abstract

Memristor is a nonlinear and memory element that has a future of replacing resistors for nonlinear circuit computation. It exhibits complex properties such as chaos and hyperchaos. A five-dimensional memristor-based circuit in the context of a nonlocal and nonsingular fractional derivative is considered for analysis. The Banach fixed point theorem and contraction principle are utilized to verify the existence and uniqueness of the solution of the five-dimensional system. A numerical method developed by Toufik and Atangana is used to get approximate solutions of the system. Local stability analysis is examined using the Matignon fractional-order stability criteria, and it is shown that the trivial equilibrium point is unstable. The Lyapunov exponents for different fractional orders exposed that the nature of the five-dimensional fractional-order system is hyperchaotic. Bifurcation diagrams are obtained by varying the fractional order and two of the parameters in the model. It is shown using phase-space portraits and time-series orbit figures that the system is sensitive to derivative order change, parameter change, and small initial condition change. Master-slave synchronization of the hyperchaotic system was established, the error analysis was made, and the simulation results of the synchronized systems revealed a strong correlation among themselves.