Nonlinear fractional differential equations and chaotic systems can be modeled with variable-order differential operators. We propose a generalized numerical scheme to simulate variable-order fractional differential operators. Fractional calculus' fundamental theorem and Lagrange polynomial interpolation are used. Two methods, Atangana-Baleanu-Caputo and Atangana-Seda derivatives, were used to solve a chaotic Newton-Leipnik system problem with fractional operators. Our scheme examined the existence and uniqueness of the solution. We analyze the model qualitatively using its equivalent integral through an iterative convergence sequence. This novel method is illustrated with numerical examples. Simulated and analytical results agree. We contribute to real-world mathematical applications. Finally, we applied a numerical successive approximation method to solve the fractional model.
Citation: Najat Almutairi, Sayed Saber. Chaos control and numerical solution of time-varying fractional Newton-Leipnik system using fractional Atangana-Baleanu derivatives[J]. AIMS Mathematics, 2023, 8(11): 25863-25887. doi: 10.3934/math.20231319
Nonlinear fractional differential equations and chaotic systems can be modeled with variable-order differential operators. We propose a generalized numerical scheme to simulate variable-order fractional differential operators. Fractional calculus' fundamental theorem and Lagrange polynomial interpolation are used. Two methods, Atangana-Baleanu-Caputo and Atangana-Seda derivatives, were used to solve a chaotic Newton-Leipnik system problem with fractional operators. Our scheme examined the existence and uniqueness of the solution. We analyze the model qualitatively using its equivalent integral through an iterative convergence sequence. This novel method is illustrated with numerical examples. Simulated and analytical results agree. We contribute to real-world mathematical applications. Finally, we applied a numerical successive approximation method to solve the fractional model.
| [1] |
B. Eleonora, P. Pietro, The language of chaos, Int. J. Bifurcat. Chaos, 16 (2006), 523–557. https://doi.org/10.1142/S0218127406014988 doi: 10.1142/S0218127406014988
|
| [2] |
R. B. Leipnik, T. A. Newton, Double strange attractors in rigid body motion, Phys. Lett. A, 86 (1981), 63–67. https://doi.org/10.1016/0375-9601(81)90165-1 doi: 10.1016/0375-9601(81)90165-1
|
| [3] |
X. Wang, L. Tian, Bifurcation analysis and linear control of the Newton-Leipnik system, Chaos Soliton. Fract., 27 (2006), 31–38. https://doi.org/10.1016/j.chaos.2005.04.009 doi: 10.1016/j.chaos.2005.04.009
|
| [4] |
H. K. Chen, C. I. Lee, Anti-control of chaos in rigid body motion, Chaos Soliton. Fract., 21 (2004), 957–965. https://doi.org/10.1016/j.chaos.2003.12.034 doi: 10.1016/j.chaos.2003.12.034
|
| [5] |
H. Richter, Controlling chaotic system with multiple strange attractors, Phys. Lett. A, 300 (2002), 182–188. https://doi.org/10.1016/S0375-9601(02)00183-4 doi: 10.1016/S0375-9601(02)00183-4
|
| [6] |
L. J. Sheu, H. K. Chen, J. H. Chen, L. M. Tam, W. C. Chen, K. T. Lin, et al., Chaos in the Newton-Leipnik system with fractional order, Chaos Soliton. Fract., 36 (2008), 98–103. https://doi.org/10.1016/j.chaos.2006.06.013 doi: 10.1016/j.chaos.2006.06.013
|
| [7] |
K. M. S. Tavazoei, M. Haeri, A necessary condition for double scroll attractor existence in fractional order systems, Phys. Lett. A, 367 (2007), 102–113. https://doi.org/10.1016/j.physleta.2007.05.081 doi: 10.1016/j.physleta.2007.05.081
|
| [8] |
H. K. Chen, Chaos and chaos synchronization of a symmetric gyro with linear-plus-cubic damping, J. Sound. Vib., 255 (2002), 719–740. https://doi.org/10.1006/jsvi.2001.4186 doi: 10.1006/jsvi.2001.4186
|
| [9] |
A. Khalid, Splines solutions of boundary value problems that arises in sculpturing electrical process of motors with two rotating mechanism circuit, Phys. Scripta, 96 (2021), 104001. https://doi.org/10.1088/1402-4896/ac0bd0 doi: 10.1088/1402-4896/ac0bd0
|
| [10] | M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Dier. Appl., 2 (2015), 1–13. Available from: https://digitalcommons.aaru.edu.jo/pfda/vol1/iss2/1 |
| [11] |
W. Deng, C. Li, J. Lu, Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dynam., 48 (2007), 409–416. https://doi.org/10.1007/s11071-006-9094-0 doi: 10.1007/s11071-006-9094-0
|
| [12] |
S. Rashid, K. T. Kubra, S. Sultana, P. Agarwal, M. S. Osman, An approximate analytical view of physical and biological models in the setting of Caputo operator via Elzaki transform decomposition method, J. Comput. Appl. Math., 413 (2022), 114378. https://doi.org/10.1016/j.cam.2022.114378 doi: 10.1016/j.cam.2022.114378
|
| [13] |
V. D. Gejji, Y. Sukale, S. Bhalekar, A new predictor-corrector method for fractional differential equations, Appl. Math. Comput., 244 (2014), 158–182. https://doi.org/10.1016/j.amc.2014.06.097 doi: 10.1016/j.amc.2014.06.097
|
| [14] |
K. M. Owolabi, Mathematical modelling and analysis of two-component system with Caputo fractional derivative order, Chaos Soliton. Fract., 103 (2017), 544–554. https://doi.org/10.1016/j.chaos.2017.07.013 doi: 10.1016/j.chaos.2017.07.013
|
| [15] |
C. Li, C. Tao, On the fractional adams method, Comput. Math. Appl., 58 (2009), 1573–1588. https://doi.org/10.1016/j.camwa.2009.07.050 doi: 10.1016/j.camwa.2009.07.050
|
| [16] |
V. D. Gejji, H. Jafari, Analysis of a system of non autonomous fractional differential equations involving Caputo derivatives, J. Math. Anal. Appl., 328 (2007), 1026–1033. https://doi.org/10.1016/j.jmaa.2006.06.007 doi: 10.1016/j.jmaa.2006.06.007
|
| [17] |
A. Atangana, J. F. G. Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus, 133 (2018), 1–23. https://doi.org/10.1140/epjp/i2018-12021-3 doi: 10.1140/epjp/i2018-12021-3
|
| [18] |
A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A, 505 (2018), 688–706. https://doi.org/10.1016/j.physa.2018.03.056 doi: 10.1016/j.physa.2018.03.056
|
| [19] |
M. S. Tavazoei, M. Haeri, Chaotic attractors in incommensurate fractional order systems, Physica D, 237 (2008), 2628–2637. https://doi.org/10.1016/j.physd.2008.03.037 doi: 10.1016/j.physd.2008.03.037
|
| [20] |
H. M. Baskonus, T. Mekkaoui, Z. Hammouch, H. Bulut, Active control of a chaotic fractional order economic system, Entropy, 17 (2015), 5771–5783. https://doi.org/10.3390/e17085771 doi: 10.3390/e17085771
|
| [21] |
M. Toufik, A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, Eur. Phys. J. Plus, 132 (2017), 1–16. https://doi.org/10.1140/epjp/i2017-11717-0 doi: 10.1140/epjp/i2017-11717-0
|
| [22] |
L. Galeone, R. Garrappa, Fractional adams-moulton methods, Math. Comput. Simulat., 79 (2008), 1358–1367. https://doi.org/10.1016/j.matcom.2008.03.008 doi: 10.1016/j.matcom.2008.03.008
|
| [23] |
K. Hattaf, Stability of fractional differential equations with new generalized hattaf fractional derivative, Math. Probl. Eng., 2021 (2021), 8608447. https://doi.org/10.1155/2021/8608447 doi: 10.1155/2021/8608447
|
| [24] |
K. Hattaf, Z. Hajhouji, M. A. Ichou, N. Yousfi, A Numerical method for fractional differential equations with new generalized hattaf fractional derivative, Math. Probl. Eng., 2022 (2022). https://doi.org/10.1155/2022/3358071 doi: 10.1155/2022/3358071
|
| [25] |
K. Hattaf, On the stability and numerical scheme of fractional differential equations with application to biology, Computation, 10 (2022), 97. https://doi.org/10.3390/computation10060097 doi: 10.3390/computation10060097
|
| [26] |
K. Hattaf, A new generalized definition of fractional derivative with non-singular kernel, Computation, 8 (2020), 49. https://doi.org/10.3390/computation8020049 doi: 10.3390/computation8020049
|
| [27] |
K. Hattaf, A new class of generalized fractal and fractal-fractional derivatives with non-singular kernels, Fractal Fract., 7 (2023), 395. https://doi.org/10.3390/fractalfract7050395 doi: 10.3390/fractalfract7050395
|
| [28] |
M. H. Alshehri, S. Saber, F. Z. Duraihem, Dynamical analysis of fractional-order of IVGTT glucose-insulin interaction, Int. J. Nonlin. Sci. Num., 24 (2023), 1123–1140. https://doi.org/10.1515/ijnsns-2020-0201 doi: 10.1515/ijnsns-2020-0201
|
| [29] |
M. H. Alshehri, F. Z. Duraihem, A. Alalyani, S. Saber, A Caputo (discretization) fractional-order model of glucose-insulin interaction: Numerical solution and comparisons with experimental data, J. Taibah Univ. Sci., 15 (2021), 26–36. https://doi.org/10.1080/16583655.2021.1872197 doi: 10.1080/16583655.2021.1872197
|
| [30] |
S. Saber, A. M. Alghamdi, G. A. Ahmed, K. M. Alshehri, Mathematical modelling and optimal control of pneumonia disease in sheep and goats in Al-Baha region with cost-effective strategies, AIMS Math., 7 (2022), 12011–12049. https://doi.org/10.3934/math.2022669 doi: 10.3934/math.2022669
|
| [31] |
S. Saber, A. Alalyani, Stability analysis and numerical simulations of IVGTT glucose-insulin interaction models with two time delays, Math. Model. Anal., 27 (2022), 383–407. https://doi.org/10.3846/mma.2022.14007 doi: 10.3846/mma.2022.14007
|
| [32] |
A. Alalyani, S. Saber, Stability analysis and numerical simulations of the fractional COVID-19 pandemic model, Int. J. Nonlin. Sci. Num., 24 (2023), 989–1002. https://doi.org/10.1515/ijnsns-2021-0042 doi: 10.1515/ijnsns-2021-0042
|
| [33] |
T. W. Zhang, L. L. Xiong, Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative, Appl. Math. Lett., 101 (2020), 106072. https://doi.org/10.1016/j.aml.2019.106072 doi: 10.1016/j.aml.2019.106072
|
| [34] |
K. I. A. Ahmed, H. D. S. Adam, M. Y. Youssif, S. Saber, Different strategies for diabetes by mathematical modeling: Modified Minimal Model, Alex. Eng. J., 80 (2023), 74–87. https://doi.org/10.1016/j.aej.2023.07.050 doi: 10.1016/j.aej.2023.07.050
|
| [35] |
K. I. A. Ahmed, H. D. S. Adam, M. Y. Youssif, S. Saber, Different strategies for diabetes by mathematical modeling: Applications of fractal-fractional derivatives in the sense of Atangana-Baleanu, Results Phys., 2023, 106892. https://doi.org/10.1016/j.rinp.2023.106892 doi: 10.1016/j.rinp.2023.106892
|
| [36] |
S. G. Samko, Fractional integration and differentiation of variable order, Anal, Math., 21 (1995), 213–236. https://doi.org/10.1007/s11071-012-0485-0 doi: 10.1007/s11071-012-0485-0
|
| [37] |
J. E. S. Pérez, J. F. G. Aguilar, A. Atangana, Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws, Chaos Soliton. Fract., 114 (2018), 175–185. https://doi.org/10.1016/j.chaos.2018.06.032 doi: 10.1016/j.chaos.2018.06.032
|
| [38] |
B. S. T. Alkahtani, I. Koca, A. Atangana, A novel approach of variable order derivative: Theory and methods, J. Nonlinear Sci. Appl., 9 (2016), 4867–4876. http://dx.doi.org/10.22436/jnsa.009.06.122 doi: 10.22436/jnsa.009.06.122
|
| [39] |
A. Atangana, On the stability and convergence of the time-fractional variable-order telegraph equation, J. Comput. Phys., 293 (2015), 104–114. https://doi.org/10.1016/j.jcp.2014.12.043 doi: 10.1016/j.jcp.2014.12.043
|
| [40] |
A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.48550/arXiv.1602.03408 doi: 10.48550/arXiv.1602.03408
|
| [41] |
S. Kumar, A. Kumar, D. Baleanu, Two analytical methods for time-fractional nonlinear coupled Boussinesq-Burger's equations arise in propagation of shallow water waves, Nonlinear Dyn., 1 (2016), 1–17. https://doi.org/10.1007/s11071-016-2716-2 doi: 10.1007/s11071-016-2716-2
|
| [42] |
P. Zhuang, F. Liu, V. Anh, I. Turner, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal., 47 (2009), 1760–1781. https://doi.org/10.1137/080730597 doi: 10.1137/080730597
|
| [43] |
A. H. Bhrawy, M. A. Zaky, Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation, Nonlinear Dyn., 80 (2015), 101–116. https://doi.org/10.1007/s11071-014-1854-7 doi: 10.1007/s11071-014-1854-7
|
| [44] |
S. Djennadi, N. Shawagfeh, M. Inc, M. S. Osman, The Tikhonov regularization method for the inverse source problem of time fractional heat equation in the view of ABC-fractional technique, Phys. Scr., 96 (2021), 094006. https://doi.org/10.1088/1402-4896/ac0867 doi: 10.1088/1402-4896/ac0867
|
| [45] |
B. P. Moghaddam, S. Yaghoobi, J. T. Machado, An extended predictor-corrector algorithm for variable-order fractional delay differential equations, J. Comput. Nonlinear Dyn., 1 (2016), 1–11. https://doi.org/10.1115/1.4032574 doi: 10.1115/1.4032574
|
| [46] |
M. F. Danca, Lyapunov exponents of a discontinuous 4D hyperchaotic system of integer or fractional order, Entropy, 20 (2018), 337. https://doi.org/10.3390/e20050337 doi: 10.3390/e20050337
|
| [47] |
M. F. Danca, N. Kuznetsov, Matlab code for Lyapunov exponents of fractional-order systems, Int. J. Bif. Chaos, 28 (2018), 1850067. https://doi.org/10.1142/S0218127418500670 doi: 10.1142/S0218127418500670
|
| [48] |
L. Shi, S. Tayebi, O. A. Arqub, M. S. Osman, P. Agarwal, W. Mahamoud, et al., The novel cubic B-spline method for fractional Painleve and Bagley-Trovik equations in the Caputo, Caputo-Fabrizio, and conformable fractional sense, Alex. Eng. J., 65 (2023), 413–426. https://doi.org/10.1016/j.aej.2022.09.039 doi: 10.1016/j.aej.2022.09.039
|
| [49] |
A. Atangana, K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), 1–17. https://doi.org/10.48550/arXiv.1707.08177 doi: 10.48550/arXiv.1707.08177
|
| [50] |
A. Atangana, I. S. Araz, New numerical method for ordinary differential equations: Newton polynomial, J. Comput. Appl. Math., 372 (2019). https://doi.org/10.1016/j.cam.2019.112622 doi: 10.1016/j.cam.2019.112622
|
| [51] | A. Atangana, I. S. Araz, New numerical scheme with newton polynomial, theory, methods, and applications, 1 Eds., Academic Press, 2021. |
| [52] |
B. S. T. Alkahtani, A new numerical scheme based on Newton polynomial with application to fractional nonlinear differential equations, Alex. Eng. J., 59 (2019), 1893–1907. https://doi.org/10.1016/j.aej.2019.11.008 doi: 10.1016/j.aej.2019.11.008
|
| [53] |
T. W. Zhang, Y. K. Li, Exponential Euler scheme of multi-delay Caputo-Fabrizio fractional-order differential equations, Appl. Math. Lett., 124 (2022), 107709. https://doi.org/10.1016/j.aml.2021.107709 doi: 10.1016/j.aml.2021.107709
|
| [54] |
K. K. Ali, M. A. A. Salam, E. M. H. Mohamed, B. Samet, S. Kumar, Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series, Adv. Differ. Equ., 494 (2020). https://doi.org/10.1186/s13662-020-02951-z doi: 10.1186/s13662-020-02951-z
|
| [55] | S. Rashid, K. T. Kubra, S. Sultana, P. Agarwal, M. S. Osman, An approximate analytical view of physical and biological models in the setting of Caputo operator via Elzaki transform decomposition method, J. Comput. Appl. Math., 413 (2022), 114378 |
| [56] |
S. Qureshi, A. Soomro, E. Hincal, J. R. Lee, C. Park, M. S. Osman, An efficient variable stepsize rational method for stiff, singular and singularly perturbed problems, Alex. Eng. J., 61 (2022), 10953–10963. https://doi.org/10.1016/j.aej.2022.03.014 doi: 10.1016/j.aej.2022.03.014
|
| [57] |
O. A. Arqub, M. S. Osman, C. Park, J. R. Lee, H. Alsulam, M. Alhodaly, Development of the reproducing kernel Hilbert space algorithm for numerical pointwise solution of the time-fractional nonlocal reaction-diffusion equation, Alex. Eng. J., 61 (2022), 10539–10550. https://doi.org/10.1016/j.aej.2022.04.008 doi: 10.1016/j.aej.2022.04.008
|
| [58] |
O. A. Arqub, S. Tayebi, D. Baleanu, M. S. Osman, W. Mahmoud, H. Alsulami, A numerical combined algorithm in cubic B-spline method and finite difference technique for the time-fractional nonlinear diffusion wave equation with reaction and damping terms, Results Phys., 41 (2022), 105912. https://doi.org/10.1016/j.rinp.2022.105912 doi: 10.1016/j.rinp.2022.105912
|
| [59] |
N. Djeddi, S. Hasan, M. A. Smadi, S. Momani, Modified analytical approach for generalized quadratic and cubic logistic models with Caputo-Fabrizio fractional derivative, Alex. Eng. J., 59 (2020), 5111–5122. https://doi.org/10.1016/j.aej.2020.09.041 doi: 10.1016/j.aej.2020.09.041
|
| [60] |
A. Khalid, A. S. A. Alsubaie, M. Inc, A. Rehan, W. Mahmoud, M. S. Osman, Cubic spline solutions of the higher-order boundary value problems arise in sandwich panel theory, Results Phys., 39 (2022), 105726. https://doi.org/10.1016/j.rinp.2022.105726 doi: 10.1016/j.rinp.2022.105726
|
Figures(16) / Tables(1)
Najat Almutairi, Sayed Saber. Chaos control and numerical solution of time-varying fractional Newton-Leipnik system using fractional Atangana-Baleanu derivatives[J]. AIMS Mathematics, 2023, 8(11): 25863-25887. doi: 10.3934/math.20231319
DownLoad: