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

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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  • 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.

    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 $

    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 $

    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

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

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