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Research Papers

An Efficient Nonstandard Finite Difference Scheme for a Class of Fractional Chaotic Systems

[+] Author and Article Information
Mojtaba Hajipour

Department of Mathematics,
Sahand University of Technology,
Tabriz 5376138461, Iran
e-mail: hajipour@sut.ac.ir

Amin Jajarmi

Department of Electrical Engineering,
University of Bojnord,
P.O. Box 94531-1339,
Bojnord 9453155111, Iran
e-mail: a.jajarmi@ub.ac.ir

Dumitru Baleanu

Department of Mathematics,
Faculty of Arts and Sciences,
Cankaya University,
Ankara 06530, Turkey;
Institute of Space Sciences,
P.O. Box, MG-23, R 76900,
Magurele-Bucharest 76900, Romania
e-mail: dumitru@cankaya.edu.tr

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 2, 2016; final manuscript received November 1, 2017; published online December 7, 2017. Assoc. Editor: Firdaus Udwadia.

J. Comput. Nonlinear Dynam 13(2), 021013 (Dec 07, 2017) (9 pages) Paper No: CND-16-1533; doi: 10.1115/1.4038444 History: Received November 02, 2016; Revised November 01, 2017

In this paper, we formulate a new nonstandard finite difference (NSFD) scheme to study the dynamic treatments of a class of fractional chaotic systems. To design the new proposed scheme, an appropriate nonlocal framework is applied for the discretization of nonlinear terms. This method is easy to implement and preserves some important physical properties of the considered model, e.g., fixed points and their stability. Additionally, this scheme is explicit and inexpensive to solve fractional differential equations (FDEs). From a practical point of view, the stability analysis and chaotic behavior of three novel fractional systems are provided by the proposed approach. Numerical simulations and comparative results confirm that this scheme is also successful for the fractional chaotic systems with delay arguments.

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Figures

Grahic Jump Location
Fig. 1

Phase portraits of the fractional-order finance system (13) with αi = α by using the NSFD scheme (23) with h = 0.1: (a) fractional-order α = 0.84, (b) fractional-order α = 0.87, (c) fractional-order α = 0.88, and (d) fractional-order α = 1

Grahic Jump Location
Fig. 2

Phase portraits of the fractional-order Finance system (13) with (a, b) α1 = α4 = 0.89, α2 = α3 = 0.91 and (c, d) α1 = α4 = 0.89, α2 = α3 = 1 by using the NSFD scheme (23) with h = 0.1

Grahic Jump Location
Fig. 3

Phase portraits of the commensurate Bloch system (24) without delay on [0, 2000] using the NSFD scheme (26) with h = 0.05: (a) a limit cycle for α = 0.81, (b) two limit cycles for α = 0.91, (c) four limit cycles for α = 0.94, and (d) chaotic behavior for α = 0.949

Grahic Jump Location
Fig. 4

(a) Time series (x, y, z) and (b) xy-phase portrait of the incommensurate Bloch system (24) without delay when α1 = 0.78 and α2 = α3 = 0.19 by using the NSFD scheme (26) where h = 0.05

Grahic Jump Location
Fig. 5

(a) Chaotic behavior, (b) two limit cycles, (c) chaotic behavior, and (d) one limit cycle for the commensurate Bloch system (24) with α = 0.88 and τ = 0.1, 0.45, 1, 1.4 using the NSFD scheme (26) with h = 0.05

Grahic Jump Location
Fig. 6

The comparison of numerical solutions of (a) the NSFD scheme given by Eq. (28) and (b) the PECE method [18,34] for the Memristive system with α1 = α2 = 0.67 and τ = 0, where the time-step size h is 0.1, 0.05 and 0.01

Grahic Jump Location
Fig. 7

The comparison of numerical solutions of (a) the NSFD scheme given by Eq. (28) and (b) the RK4 method [35] for the Memristive system with α1 = α2 = 0.99 and τ = 0, where the time-step size h is 0.25, 0.2 and 0.1

Grahic Jump Location
Fig. 8

(a) One limit cycle, (b) two limit cycles, (c) six limit cycles, and (d) chaotic behavior for the Memristive system (27) with α1 = α2 = 0.99 and τ = 0.66, 0.99, 1.06, 1.3 on [0, 1000] using the NSFD scheme (28) with h = 0.05

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