July  2021, 14(7): 2455-2469. doi: 10.3934/dcdss.2021060

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)

Received  June 2019 Revised  July 2019 Published  July 2021 Early access  May 2021

Fund Project: 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

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.

Citation: Kolade M. Owolabi, Abdon Atangana, Jose Francisco Gómez-Aguilar. Fractional Adams-Bashforth scheme with the Liouville-Caputo derivative and application to chaotic systems. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2455-2469. doi: 10.3934/dcdss.2021060
References:
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[2] A. Atangana, Fractional Operators With Constant and Variable Order with Application to Geo-Hydrology, Academic Press, London, 2018. 
[3]

A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.

[4]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763–769. doi: 10.2298/TSCI160111018A.

[5]

A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), Paper No. 3, 21 pp. doi: 10.1051/mmnp/2018010.

[6]

D. Baleanu, R. Caponetto and J. A. T. Machado, Challenges in fractional dynamics and control theory, J. Vib. Control, 22 (2016), 2151–2152. doi: 10.1177/1077546315609262.

[7]

H. M. BaskonusT. MekkaouiZ. Hammouch and H. Bulut, Active control of a chaotic fractional order economic system, Entropy, 17 (2015), 5771-5783. 

[8]

J. CaoC. Li and Y. Chen, Compact difference method for solving the fractional reaction-subdiffusion equation with Neumann boundary value condition, Int. J. Comput. Math., 92 (2015), 167-180.  doi: 10.1080/00207160.2014.887702.

[9]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progress in Fractional Differentiation and Applications, 2 (2016), 1-11.  doi: 10.18576/pfda/020101.

[10]

A. Coronel-EscamillaJ. F. Gómez-AguilarM. G. López-LópezV. M. Alvarado-Martínez and G. V. Guerrero-Ramírez, Triple pendulum model involving fractional derivatives with different kernels, Chaos Solitons Fractals, 91 (2016), 248-261.  doi: 10.1016/j.chaos.2016.06.007.

[11]

E. Demirci and N. Ozalp, A method for solving differential equations of fractional order, J. Comput. Appl. Math., 236 (2012), 2754-2762.  doi: 10.1016/j.cam.2012.01.005.

[12]

J. Deng and L. Ma, Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations, Appl. Math. Lett., 23 (2010), 676-680.  doi: 10.1016/j.aml.2010.02.007.

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K. DiethelmN. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31-52.  doi: 10.1023/B:NUMA.0000027736.85078.be.

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R. DuW. R. Cao and and Z. Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model., 34 (2010), 2998-3007.  doi: 10.1016/j.apm.2010.01.008.

[15]

R. Garrappa, On some explicit Adams multistep methods for fractional differential equations, J. Comput. Appl. Math., 229 (2009), 392-399.  doi: 10.1016/j.cam.2008.04.004.

[16]

J. F. Gómez-Aguilar, L. Torres, H. Yépez-Martínez, D. Baleanu, J. M. Reyes and I. O. Sosa, Fractional Liénard type model of a pipeline within the fractional derivative without singular kernel, Adv. Difference Equ., 2016 (2016), Paper No. 173, 13 pp. doi: 10.1186/s13662-016-0908-1.

[17]

J. F. Gómez-Aguilar, M. G. López-López, V. M. Alvarado-Martínez, J. Reyes-Reyes and M. Adam-Medina, Modeling diffusive transport with a fractional derivative without singular kernel, Phys. A, 447 (2016), 467–481. doi: 10.1016/j.physa.2015.12.066.

[18]

J. F. Gómez-Aguilar and Abdon Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, The European Physical Journal Plus, 132 (2017). doi: 10.1140/epjp/i2017-11293-3.

[19]

R. Gorenflo and E. A. Abdel-Rehim, Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion, J. Comput. Appl. Math., 205 (2007), 871-881.  doi: 10.1016/j.cam.2005.12.043.

[20]

Z. Hammouch and T. Mekkaoui, Control of a new chaotic fractional-order system using Mittag-Leffler stability, Nonlinear Stud., 22 (2015), 565-577. 

[21]

Z. Hammouch and T. Mekkaoui, Chaos synchronization of a fractional nonautonomous system, Nonauton. Dyn. Syst., 1 (2014), 61-71.  doi: 10.2478/msds-2014-0001.

[22]

X. Hu and L. Zhang, Implicit compact difference schemes for the fractional cable equation, Appl. Math. Model., 36 (2012), 4027-4043.  doi: 10.1016/j.apm.2011.11.027.

[23]

A. Q. M. KhaliqX. Liang and K. M. Furati, A fourth-order implicit-explicit scheme for the space fractional nonlinear Schrödinger equations, Numer. Algorithms, 75 (2017), 147-172.  doi: 10.1007/s11075-016-0200-1.

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V. Lakshmikantham and A. S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal., 11 (2007), 395-402. 

[26]

C. Li and F. Zeng, The finite difference methods for fractional ordinary differential equations, Numer. Funct. Anal. Optim., 34 (2013), 149-179.  doi: 10.1080/01630563.2012.706673.

[27] C. Li and F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Taylor and Francis Group, London, 2015. 
[28]

X. LiangA. Q. M. KhaliqH. Bhatt and K. M. Furati, The locally extrapolated exponential splitting scheme for multi-dimensional nonlinear space-fractional Schrödinger equations, Numer. Algorithms, 76 (2017), 939-958.  doi: 10.1007/s11075-017-0291-3.

[29]

K. M. Owolabi, Numerical solution of diffusive HBV model in a fractional medium, Springer Plus, 5 (2016), 1643. doi: 10.1186/s40064-016-3295-x.

[30]

K. M. Owolabi, Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems, Chaos Solitons Fractals, 93 (2016), 89-98.  doi: 10.1016/j.chaos.2016.10.005.

[31]

K. M. Owolabi and A. Atangana, Numerical approximation of nonlinear fractional parabolic differential equations with Caputo-Fabrizio derivative in Riemann-Liouville sense, Chaos Solitons Fractals, 99 (2017), 171-179.  doi: 10.1016/j.chaos.2017.04.008.

[32]

K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 304-317.  doi: 10.1016/j.cnsns.2016.08.021.

[33]

K. M. Owolabi, Mathematical modelling and analysis of two-component system with Caputo fractional derivative order, Chaos Solitons Fractals, 103 (2017), 544-554.  doi: 10.1016/j.chaos.2017.07.013.

[34]

K. M. Owolabi and A. Atangana, Analysis and application of new fractional Adams-Bashforth scheme with Caputo-Fabrizio derivative, Chaos Solitons Fractals, 105 (2017), 111-119.  doi: 10.1016/j.chaos.2017.10.020.

[35]

K. M. Owolabi, Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense, Math. Model. Nat. Phenom., 13 (2018), Paper No. 7, 17 pp. doi: 10.1051/mmnp/2018006.

[36]

K. M. Owolabi and A. Atangana, Modelling and formation of spatiotemporal patterns of fractional predation system in subdiffusion and superdiffusion scenarios, The European physical Journal Plus, 133 (2018), Article number: 43. doi: 10.1140/epjp/i2018-11886-2.

[37]

K. M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, The European physical Journal Plus, 133 (2018), Article number: 15. doi: 10.1140/epjp/i2018-11863-9.

[38]

K. M. Owolabi, Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator, The European Physical Journal Plus, 133 (2018), Article number: 98. doi: 10.1140/epjp/i2018-11951-x.

[39]

K. M. Owolabi and A. Atangana, Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations, Chaos Solitons Fractals, 111 (2018), 119-127.  doi: 10.1016/j.chaos.2018.04.019.

[40]

K. M. Owolabi and Z. Hammouch, Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative, Phys. A, 523 (2019), 1072-1090.  doi: 10.1016/j.physa.2019.04.017.

[41] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. 
[42]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

[43]

J. SinghD. KumarZ. Hammouch and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316 (2018), 504-515.  doi: 10.1016/j.amc.2017.08.048.

[44]

T. A. Sulaimana, M. Yavuz, H. Bulut and H. M. Baskonus, Investigation of the fractional coupled viscous Burgers-equation involving Mittag-Leffler kernel, Phys. A, 527 (2019), 121126, 20 pp. doi: 10.1016/j.physa.2019.121126.

[45]

M. Yavuz, N. Ozdemir and H. M. Baskonus, Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, The European Physical Journal Plus, 133, (2018), Article number: 215. doi: 10.1140/epjp/i2018-12051-9.

[46]

M. Yavuz and E. Bonyah, New approaches to the fractional dynamics of schistosomiasis disease model, Phys. A, 525 (2019), 373-393.  doi: 10.1016/j.physa.2019.03.069.

[47]

X. Zhao and Z.-Z. Sun, A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys., 230 (2011), 6061-6074.  doi: 10.1016/j.jcp.2011.04.013.

[48]

A. T. Azar and S. Vaidyanathan, Advances in Chaos Theory and Intelligent Control, Springer, Switzerland, 2016.

show all references

References:
[1] A. Atangana, Derivative with a New Parameter: Theory, Methods and Applications, Academic Press, New York, 2016.  doi: 10.1016/B978-0-08-100644-3.00001-5.
[2] A. Atangana, Fractional Operators With Constant and Variable Order with Application to Geo-Hydrology, Academic Press, London, 2018. 
[3]

A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.

[4]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763–769. doi: 10.2298/TSCI160111018A.

[5]

A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), Paper No. 3, 21 pp. doi: 10.1051/mmnp/2018010.

[6]

D. Baleanu, R. Caponetto and J. A. T. Machado, Challenges in fractional dynamics and control theory, J. Vib. Control, 22 (2016), 2151–2152. doi: 10.1177/1077546315609262.

[7]

H. M. BaskonusT. MekkaouiZ. Hammouch and H. Bulut, Active control of a chaotic fractional order economic system, Entropy, 17 (2015), 5771-5783. 

[8]

J. CaoC. Li and Y. Chen, Compact difference method for solving the fractional reaction-subdiffusion equation with Neumann boundary value condition, Int. J. Comput. Math., 92 (2015), 167-180.  doi: 10.1080/00207160.2014.887702.

[9]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progress in Fractional Differentiation and Applications, 2 (2016), 1-11.  doi: 10.18576/pfda/020101.

[10]

A. Coronel-EscamillaJ. F. Gómez-AguilarM. G. López-LópezV. M. Alvarado-Martínez and G. V. Guerrero-Ramírez, Triple pendulum model involving fractional derivatives with different kernels, Chaos Solitons Fractals, 91 (2016), 248-261.  doi: 10.1016/j.chaos.2016.06.007.

[11]

E. Demirci and N. Ozalp, A method for solving differential equations of fractional order, J. Comput. Appl. Math., 236 (2012), 2754-2762.  doi: 10.1016/j.cam.2012.01.005.

[12]

J. Deng and L. Ma, Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations, Appl. Math. Lett., 23 (2010), 676-680.  doi: 10.1016/j.aml.2010.02.007.

[13]

K. DiethelmN. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31-52.  doi: 10.1023/B:NUMA.0000027736.85078.be.

[14]

R. DuW. R. Cao and and Z. Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model., 34 (2010), 2998-3007.  doi: 10.1016/j.apm.2010.01.008.

[15]

R. Garrappa, On some explicit Adams multistep methods for fractional differential equations, J. Comput. Appl. Math., 229 (2009), 392-399.  doi: 10.1016/j.cam.2008.04.004.

[16]

J. F. Gómez-Aguilar, L. Torres, H. Yépez-Martínez, D. Baleanu, J. M. Reyes and I. O. Sosa, Fractional Liénard type model of a pipeline within the fractional derivative without singular kernel, Adv. Difference Equ., 2016 (2016), Paper No. 173, 13 pp. doi: 10.1186/s13662-016-0908-1.

[17]

J. F. Gómez-Aguilar, M. G. López-López, V. M. Alvarado-Martínez, J. Reyes-Reyes and M. Adam-Medina, Modeling diffusive transport with a fractional derivative without singular kernel, Phys. A, 447 (2016), 467–481. doi: 10.1016/j.physa.2015.12.066.

[18]

J. F. Gómez-Aguilar and Abdon Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, The European Physical Journal Plus, 132 (2017). doi: 10.1140/epjp/i2017-11293-3.

[19]

R. Gorenflo and E. A. Abdel-Rehim, Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion, J. Comput. Appl. Math., 205 (2007), 871-881.  doi: 10.1016/j.cam.2005.12.043.

[20]

Z. Hammouch and T. Mekkaoui, Control of a new chaotic fractional-order system using Mittag-Leffler stability, Nonlinear Stud., 22 (2015), 565-577. 

[21]

Z. Hammouch and T. Mekkaoui, Chaos synchronization of a fractional nonautonomous system, Nonauton. Dyn. Syst., 1 (2014), 61-71.  doi: 10.2478/msds-2014-0001.

[22]

X. Hu and L. Zhang, Implicit compact difference schemes for the fractional cable equation, Appl. Math. Model., 36 (2012), 4027-4043.  doi: 10.1016/j.apm.2011.11.027.

[23]

A. Q. M. KhaliqX. Liang and K. M. Furati, A fourth-order implicit-explicit scheme for the space fractional nonlinear Schrödinger equations, Numer. Algorithms, 75 (2017), 147-172.  doi: 10.1007/s11075-016-0200-1.

[24]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amersterdam, 2006.

[25]

V. Lakshmikantham and A. S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal., 11 (2007), 395-402. 

[26]

C. Li and F. Zeng, The finite difference methods for fractional ordinary differential equations, Numer. Funct. Anal. Optim., 34 (2013), 149-179.  doi: 10.1080/01630563.2012.706673.

[27] C. Li and F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Taylor and Francis Group, London, 2015. 
[28]

X. LiangA. Q. M. KhaliqH. Bhatt and K. M. Furati, The locally extrapolated exponential splitting scheme for multi-dimensional nonlinear space-fractional Schrödinger equations, Numer. Algorithms, 76 (2017), 939-958.  doi: 10.1007/s11075-017-0291-3.

[29]

K. M. Owolabi, Numerical solution of diffusive HBV model in a fractional medium, Springer Plus, 5 (2016), 1643. doi: 10.1186/s40064-016-3295-x.

[30]

K. M. Owolabi, Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems, Chaos Solitons Fractals, 93 (2016), 89-98.  doi: 10.1016/j.chaos.2016.10.005.

[31]

K. M. Owolabi and A. Atangana, Numerical approximation of nonlinear fractional parabolic differential equations with Caputo-Fabrizio derivative in Riemann-Liouville sense, Chaos Solitons Fractals, 99 (2017), 171-179.  doi: 10.1016/j.chaos.2017.04.008.

[32]

K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 304-317.  doi: 10.1016/j.cnsns.2016.08.021.

[33]

K. M. Owolabi, Mathematical modelling and analysis of two-component system with Caputo fractional derivative order, Chaos Solitons Fractals, 103 (2017), 544-554.  doi: 10.1016/j.chaos.2017.07.013.

[34]

K. M. Owolabi and A. Atangana, Analysis and application of new fractional Adams-Bashforth scheme with Caputo-Fabrizio derivative, Chaos Solitons Fractals, 105 (2017), 111-119.  doi: 10.1016/j.chaos.2017.10.020.

[35]

K. M. Owolabi, Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense, Math. Model. Nat. Phenom., 13 (2018), Paper No. 7, 17 pp. doi: 10.1051/mmnp/2018006.

[36]

K. M. Owolabi and A. Atangana, Modelling and formation of spatiotemporal patterns of fractional predation system in subdiffusion and superdiffusion scenarios, The European physical Journal Plus, 133 (2018), Article number: 43. doi: 10.1140/epjp/i2018-11886-2.

[37]

K. M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, The European physical Journal Plus, 133 (2018), Article number: 15. doi: 10.1140/epjp/i2018-11863-9.

[38]

K. M. Owolabi, Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator, The European Physical Journal Plus, 133 (2018), Article number: 98. doi: 10.1140/epjp/i2018-11951-x.

[39]

K. M. Owolabi and A. Atangana, Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations, Chaos Solitons Fractals, 111 (2018), 119-127.  doi: 10.1016/j.chaos.2018.04.019.

[40]

K. M. Owolabi and Z. Hammouch, Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative, Phys. A, 523 (2019), 1072-1090.  doi: 10.1016/j.physa.2019.04.017.

[41] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. 
[42]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

[43]

J. SinghD. KumarZ. Hammouch and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316 (2018), 504-515.  doi: 10.1016/j.amc.2017.08.048.

[44]

T. A. Sulaimana, M. Yavuz, H. Bulut and H. M. Baskonus, Investigation of the fractional coupled viscous Burgers-equation involving Mittag-Leffler kernel, Phys. A, 527 (2019), 121126, 20 pp. doi: 10.1016/j.physa.2019.121126.

[45]

M. Yavuz, N. Ozdemir and H. M. Baskonus, Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, The European Physical Journal Plus, 133, (2018), Article number: 215. doi: 10.1140/epjp/i2018-12051-9.

[46]

M. Yavuz and E. Bonyah, New approaches to the fractional dynamics of schistosomiasis disease model, Phys. A, 525 (2019), 373-393.  doi: 10.1016/j.physa.2019.03.069.

[47]

X. Zhao and Z.-Z. Sun, A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys., 230 (2011), 6061-6074.  doi: 10.1016/j.jcp.2011.04.013.

[48]

A. T. Azar and S. Vaidyanathan, Advances in Chaos Theory and Intelligent Control, Springer, Switzerland, 2016.

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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