Fractional Adams-Bashforth scheme with the Liouville-Caputo derivative and application to chaotic systems

Kolade M. Owolabi; Abdon Atangana; Jose Francisco Gómez-Aguilar

doi:10.3934/dcdss.2021060

Article Contents

2021, Volume 14, Issue 7: 2455-2469. Doi: 10.3934/dcdss.2021060

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Fractional Adams-Bashforth scheme with the Liouville-Caputo derivative and application to chaotic systems

1.
Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa
2.
Department of Mathematical Sciences, Federal University of Technology, PMB 704, Akure, Ondo State, Nigeria
3.
CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, México

^* Corresponding author: kmowolabi@futa.edu.ng (Kolade M. Owolabi)
^* Corresponding author: kmowolabi@futa.edu.ng (Kolade M. Owolabi)

Received: June 2019

Revised: July 2019

Early access: May 2021

Published: July 2021

A. Atangana would like to thank NRF for their support, J. F. Gómez Aguilar acknowledges the support provided by CONACyT: Cátedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT

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Abstract

A recently proposed numerical scheme for solving nonlinear ordinary differential equations with integer and non-integer Liouville-Caputo derivative is applied to three systems with chaotic solutions. The Adams-Bashforth scheme involving Lagrange interpolation and the fundamental theorem of fractional calculus. We provide the existence and uniqueness of solutions, also the convergence result is stated. The proposed method is applied to several examples that are shown to have unique solutions. The scheme converges to the classical Adams-Bashforth method when the fractional orders of the derivatives converge to integers.

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Mathematics Subject Classification: Primary: 34A34, 35A05, 35K57; Secondary: 65L05, 65M06, 93C10.

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Figure 1. Numerical simulation for Eq. (31). In (a) depicts the classical chaotic attractor and (b-d) shows the classical phase portrait of system (31). In (e) depicts the strange chaotic attractor and (f-h) shows the phase portrait of system (31) for $ p = 0.9 $

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Figure 2. Numerical simulation for Eq. (32). In (a) depicts the classical macroeconomic model with foreign capital investments and (b-d) shows the classical phase portrait of system (32). In (e) depicts the strange chaotic attractor and (f-h) shows the phase portrait of system (32) for $ p = 0.95 $

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Figure 3. Numerical simulation for Eq. (33). In (a)-(c) depicts the time series and the phase portrait of system (33), respectively. In (d)-(f) depicts the time series and the phase portrait of system (33) for $ p = 0.95 $, respectively

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Figure 4. Two and three dimensional projections for the fractional jerk system for $ p = 0.93 $

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