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.
Highlights
In the last decade, fractional differential equations started gaining much attention in modeling several real-world problems in different areas including in mathematical epidemiology, physics, engineering, and many others
One of the differences between integer- and fractionalorder derivatives is that the integer-order derivative describes local properties of a certain dynamic system, whereas the fractional-order derivative representation of a dynamic system involves the whole space of the process [5]. at is, applying fractional derivative orders in modeling real-world problems is essential for describing the hereditary specifications and effectiveness of the memory as essential aspects of different mechanisms in the problem [6, 7]
E recent increase in the study of different dynamical systems using fractional-order derivatives is attributed to the fact that most of the dynamic systems associated with complex systems are found to be nonlocal involving long memory in time and intrinsically fractional derivative operators can describe such systems more accurately than the integer derivatives [8]
Summary
Fractional differential equations started gaining much attention in modeling several real-world problems in different areas including in mathematical epidemiology, physics, engineering, and many others. Atangana–Baleanu fractional derivative operators were used for modeling and analysis of different chaotic and hyperchaotic systems, and solutions were approximated using a two-step Adams–Bashforth numerical scheme in [21] It was in 1971 that circuit theorist Chua proposed memristor as a missing two-terminal nonlinear electrical component. Some of the evidence for the originality of this work includes the application of Atangana–Baleanu fractional operator to the memristor-based system considered in this study, application of the newly developed numerical approximation by Toufik and Atangana for the fractionalorder systems, obtaining the phase portraits of the system from the numerical scheme, and performing synchronization of the five-dimensional system using the numerical approximation.
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