A nonstandard finite difference scheme for the modeling and nonidentical synchronization of a novel fractional chaotic system

Abstract

The aim of this paper is to introduce and analyze a novel fractional chaotic system including quadratic and cubic nonlinearities. We take into account the Caputo derivative for the fractional model and study the stability of the equilibrium points by the fractional Routh–Hurwitz criteria. We also utilize an efficient nonstandard finite difference (NSFD) scheme to implement the new model and investigate its chaotic behavior in both time-domain and phase-plane. According to the obtained results, we find that the new model portrays both chaotic and nonchaotic behaviors for different values of the fractional order, so that the lowest order in which the system remains chaotic is found via the numerical simulations. Afterward, a nonidentical synchronization is applied between the presented model and the fractional Volta equations using an active control technique. The numerical simulations of the master, the slave, and the error dynamics using the NSFD scheme are plotted showing that the synchronization is achieved properly, an outcome which confirms the effectiveness of the proposed active control strategy.

Fuzzy Differential Subordination of the Atangana–Baleanu Fractional Integral

“…The investigation is performed firstly using a classical Lagrangian approach, which produces the classical equations of motion, and then the generalized form of the fractional Hamilton equations is formulated in the Caputo sense. A novel fractional chaotic system, including quadratic and cubic nonlinearities, is introduced and analyzed in [15], taking into account the Caputo derivative for the fractional model and an efficient nonstandard finite difference scheme in order to investigate its chaotic behavior in both the time domain and the phase plane. For a multi-term time-fractional diffusion equation, the direct and inverse problems are discussed in [16].…”

Synchronization of the Glycolysis Reaction-Diffusion Model via Linear Control Law

“…In this section, we demonstrate some computational examples in one-and twodimensional space to exemplify the practicability of the synchronization scheme proposed in this work. These simulations are carried out using some prepared codes in MAT-LAB based on the finite difference method (FDM), see [56,57] for a full overview of this scheme and how it could be implemented in synchronization problems. First of all, let us take x ∈ Ω = [0, 10] with a step size equal to 0.2, t ∈ [0, 100] with a step size equal to 4, (d 1 , d 2 , a, b) = (0.01, 1, 3.5, 0.25) and the initial conditions associated with the drive system (1) as follows:…”

Solving Fredholm Integral Equations Using Deep Learning

“…The most popular integral equations are the Fredhom integral equations and the Volterra integral equations. The Fredholm integral equation can be considered as a reformulation of the elliptic partial differential equation and the Volterra integral equation is a reformulation of the fractional-order differential equation, which has wide applications in modeling the real problems, for instance, the chaotic system [ 4 ], the dynamics of COVID-19 [ 5 ], the motion of beam on nanowire [ 6 ], the capacitor microphone dynamical system [ 7 ], etc. Since these integral equations usually can not be solved explicitly, numerical methods are necessary to be considered.…”