Control of chaos in the burke-shaw system of fractal-fractional order in the sense of Caputo-Fabrizio

• Autorzy: Saber Sayed

Identyfikatory

DOI

10.17512/jamcm.2024.1.07

• Języki publikacji: EN

Abstrakty

EN

For the Burke-Shaw system, we propose a fractal-fractional order in the sense of the Caputo-Fabrizio derivative. The proposed system is solved by utilizing the fractal-fractional derivative operator with an exponential decay kernel. Time-fractional Caputo-Fabrizio fractal fractional derivatives are applied to the Burke-Shaw-type nonlinear chaotic systems.Based on fixed point theory, it has been demonstrated that a fractal-fractional-order model under the Caputo-Fabrizio operator exists and is unique. Using a numerical power series method, we solve the fractional Burke-Shaw model. Using Newton’s interpolation polynomial, we solve the equation numerically by implementing a novel numerical scheme based on an efficient polynomial.

Słowa kluczowe

EN

fractional derivatives; nonlinear equations; simulation; numerical results; iterative method

PL

pochodne ułamkowe; równania nieliniowe; symulacja; metoda iteracyjna

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2024
• Tom: Vol. 23, nr 1
• Strony: 83--96
• Opis fizyczny: Bibliogr. 43 poz., rys.

Twórcy

autor

Saber Sayed

• Department of Mathematics, Al-Baha University Baljurashi, Saudi Arabia

Bibliografia

• [1] Grigorenko, I., & Grigorenko, E. (2003). Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett., 91, 034101.
• [2] Wang, X.Y., & Song, J.M. (2009). Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control. Commun. Nonlinear Sci. Numer. Simulat., 14,3351-3357.
• [3] Hartley, T.T., Lorenzo, C.F., & Qammer, H.K. (1995). Chaos in a fractional order Chua’s system. IEEE Trans. Circ. Syst. I, 42, 485-490.
• [4] Li, C.P., Deng, W.H., & Xu, D. (2006). Chaos synchronization of the Chua system with a fractional order. Physica A, 360, 171-185.
• [5] Zhu, H., Zhou, S., & Zhang, J. (2009). Chaos and synchronization of the fractional-order Chua’s system. Chaos Solitons Fract., 39, 1595-1603.
• [6] Agarwal, R.P., Ahmed, El-Sayed, M.A., & Sanaa, M.S. (2013). Fractional-order Chua’s system: discretization, bifurcation and chaos. Advances in Difractal-Fractionalerence Equations, 320.
• [7] Li, C.G., & Chen, G. (2004). Chaos and hyperchaos in the fractional-order Rössler equations. Physica A, 341, 55-61.
• [8] Yu, Y., & Li, H.X. (2008). The synchronization of fractional-order R¨ossler hyperchaotic systems. Physica A, 387, 1393-1403.
• [9] Shao, S. (2009). Controlling general projective synchronization of fractional order Rössler systems. Chaos Solitons Fract., 39, 1572-1577.
• [10] Wang, X., & Tian, L. (2006). Bifurcation analysis and linear control of the Newton-Leipnik system. Chaos Solitons Fract., 27, 31-38.
• [11] Almutairi, N., & Saber, S. (2023). Chaos control and numerical solution of time-varying fractional Newton-Leipnik system using fractional Atangana-Baleanu derivatives. AIMS Mathematics, 8(11), 25863-25887.
• [12] Wang, X.Y., & Wang, M.J. (2007). Dynamic analysis of the fractional-order Liu system and its synchronization. Chaos, 17, 033106.
• [13] Matouk, A.E. (2008). Dynamical analysis feedback control and synchronization of Liu dynamical system. Nonlinear Anal. Theor. Meth. Appl., 69, 3213-3224.
• [14] Liu, C., Liu, L., & Liu, T. (2009). A novel three-dimensional autonomous chaos system. Chaos Solitons Fract., 39(4), 1950-1958.
• [15] Deng, W.H., & Li, C.P. (2005). Chaos synchronization of the fractional Lü system. Physica A,353, 61-72.
• [16] Lin, W. (2007). Global existence theory and chaos control of fractional difractal-fractionalerential equations. J. Math. Anal. Appl., 332, 709-726.
• [17] Zhu, H., Zhou, S., & He, Z. (2009). Chaos synchronization of the fractional-order Chen’s system. Chaos Solitons Fract., 41, 2733-2740.
• [18] Li, C.P., & Peng, G.J. (2004). Chaos in Chen’s system with a fractional order. Chaos Solitons Fract., 22, 443-450.
• [19] Li, C.G., & Chen, G. (2004). Chaos in the fractional order Chen system and its control. Chaos Solitons Fract., 22, 549-554.
• [20] Shaw, R. (1981). Strange Attractors, Chaotic Behavior, and Information Flow. Z. Naturforschung A, 36, 80-112.
• [21] Chen, G., & Ueta, T. (1999). Yet another chaotic attractor. Int. J. Bifurcat. Chaos, 9(07), 1465-1466.
• [22] Atangana, A., Akgül, A., & Owolabi, K.M. (2020). Analysis of fractal fractional difractal fractional erential equations. Alex. Eng. J., 59(3), 1117-1134.
• [23] Atangana, A. & Qureshi, S. (2019). Modeling attractors of chaotic dynamical systems with fractal-fractional operators. Chaos Solitons Fract., 123, 320-337.
• [24] Owolabi, K.M., Atangana, A., & Akgul, A. (2020). Modelling and analysis of fractal-fractional partial difractal-fractionalerential equations: application to reaction-difractal-fractionalusion model. Alex. Eng. J., 59(4), 2477-2490.
• [25] Khalid, I.A., Haroon, D.S., Youssif, M.Y., & Saber, S. (2023). Different strategies for diabetes by mathematical modeling: Modified minimal model. Alex. Eng. J., 80, 74-87.
• [26] Khalid, I.A., Haroon, D.S., Youssif, M.Y., & Saber, S. (2023). Different strategies for diabetes by mathematical modeling: Applications of fractal-fractional derivatives in the sense of Atangana-Baleanu. Res. Phys., 52, 106892.
• [27] Almutairi, N., Saber, S., & Hijaz, A. (2023). The fractal-fractional Atangana-Baleanu operator for pneumonia disease: stability, statistical and numerical analyses[J]. AIMS Mathematics, 8(12),29382-29410.
• [28] Almutairi, N., & Saber, S. (2023). On chaos control of nonlinear fractional Newton-Leipnik system via fractional Caputo-Fabrizio derivatives. Sci. Rep., 13, 22726.
• [29] Almutairi, N., & Saber, S. (2023). Application of a time-fractal fractional derivative with a power-law kernel to the Burke-Shaw system based on Newton’s interpolation polynomials. MethodsX.
• [30] Qureshi, S., & Zaib-un-Nisa, M. (2020). Monotonically decreasing behavior of measles epidemic well captured by Atangana-Baleanu-Caputo fractional operator under real measles data of Pakistan. Chaos Solitons Fract., 131, 109478.
• [31] Qureshi, S., & Atangana, A. (2020). Fractal-fractional difractal-fractionalerentiation for the modeling and mathematical analysis of nonlinear diarrhea transmission dynamics under the use of real data. Chaos Solitons Fract., 136, 109812.
• [32] Qureshi, S., Rangaig, N.A., & Baleanu, D. (2019). New numerical aspects of Caputo-Fabrizio fractional derivative operator. Mathematics, 7(4), 374.
• [33] Qureshi, S., Abdullahi, Y., Shaikh, A.A., Inc, M., & Baleanu, D. (2020). Mathematical modeling for adsorption process of dye removal nonlinear equation using power law and exponentially decaying kernels. Chaos, 30(4), 043106.
• [34] Sambas, A., Mohammadzadeh, A., Vaidyanathan, S. et al. (2023). Investigation of chaotic behavior and adaptive type-2 fuzzy controller approach for Permanent Magnet Synchronous Generator (PMSG) wind turbine system [J]. AIMS Mathematics, 8(3), 5670-5686.
• [35] Johansyah, M.D., Sambas, A., Vaidyanathan, S. et al. (2023). Multistability analysis and adaptive feedback control on a new financial risk system. Int. J. Appl. Comput. Math., 9, 88.
• [36] Benkouider, K. et al. (2022). A new 5-D multistable hyperchaotic system with three positive Lyapunov exponents: Bifurcation analysis, circuit design, FPGA realization and image encryption. IEEE Access, 10, 90111-90132.
• [37] Alquran, M. (2023). The amazing fractional Maclaurin series for solving different types of fractional mathematical problems that arise in physics and engineering. Partial Differ. Equ. Appl. Math., 7, 100506.
• [38] Alquran, M. (2023). Investigating the revisited generalized stochastic potential-KdV equation: fractional time-derivative against proportional time-delay. Roman. J. Phys., 68, 1-12, 106.
• [39] Almutairi, N., & Saber, S. (2023). Chaos control and numerical solution of time-varying fractional Newton-Leipnik system using fractional Atangana-Baleanu derivatives. AIMS Mathematics, 8(11), 25863-25887.
• [40] Alquran, M. et al. (2017). Mathematical and numerical study of existence of bifurcations of the generalized fractional Burgers-Huxley equation. Nonlinear Stud., 24(1), 235-244.
• [41] Caputo, M., & Fabrizio, M. (2015). A new definition of fractional derivative without singular kernel. Prog. Fract. Difractal-Fractionaler. Appl., 1, 73-85.
• [42] Caputo, M., & Fabrizio, M. (2017). On the notion of fractional derivative and applicatios to the hysteresis phenomena. Mecc., 52, 13, 3043-3052.
• [43] Schlage-Puchta, J.-Ch. (2021). Optimal version of the Picard-Lindelöf theorem. Electronic Journal of Qualitative Theory of Differential Equations, 39, 1-8.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).