Abstract
The dynamical behavior of chaotic processes with a noninteger-order operator is considered in this work. A lot of scientific reports have justified that modeling of physical scenarios via non-integer order derivatives is more reliable and accurate than integer-order cases. Motivated by this fact, the standard time derivatives in the model equations are formulated with the novel Caputo fractional-order operator. The choice of using the Caputo derivative among several existing fractional derivatives has to do with the fact that it gives way for both the initial conditions and boundary conditions to be incorporated in the development of the chaotic model. Numerical approximation of fractional derivatives has been the major challenge of many scholars in different areas of engineering and applied sciences. Hence, we developed a numerical approximation technique, which is based on the Chebyshev spectral method for solving the integer-order and non-integer-order chaotic systems which are largely found in physics, finance, biology, engineering, and other areas of applied sciences. The proposed numerical method used here is easy to implement on a digital computer, and capable of solving higher-order problems without reduction to the system of lower-order ordinary differential equations with limited computational costs. Experimental results are presented for different instances of fractional-order parameters.
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References
Abdelhakem, M., Ahmed, A., El-kady, M.: Spectral monic Chebyshev approximation for higher order differential equations. Math. Sci. Lett. 8, 11–17 (2019)
Aguila-Camacho, N., Duarte-Mermoud, M.A., Gallegos, J.A.: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19, 2951–2957 (2014)
Akgul, A., Moroz, I., Pehlivan, I., Vaidyanathan, S.: A new four-scroll chaotic attractor and its engineering applications. Optik 127, 5491–5499 (2016)
Akgul, A., Hussain, S., Pehlivan: A new three-dimensional chaotic system, its dynamical analysis and electronic circuit applications. Optik 127, 7062–7071 (2016)
Atangana, A.: Fractional Operators With Constant and Variable Order with Application to Geo-Hydrology. Academic Press, New York (2017)
Atangana, A., Owolabi, K.M.: New numerical approach for fractional differential equations. Math. Model. Natural Phenom. 13(3), 1–21 (2018)
Bagley, R.L., Calico, R.A.: Fractional order state equations for the control of viscoelastically damped structures. J. Guid. Control Dynam. 14, 304–311 (1991)
Bueno-Orovio, A., Kay, D., Burrage, K.: Fourier spectral methods for fractional-in-space reaction-diffusion equations. BIT Numer. Math. 54, 937–954 (2014)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New York (1988)
Caputo, M.: Linear models of dissipation whose \(Q\) is almost frequency independent II. Geophys. J. Roy. Astron. Soc. 13, 529–539 (1967)
Cavusoglu, U., Akgul, A., Zengin, A., Pehlivan, I.: The design and implementation of hybrid RSA algorithm using a novel chaos based RNG. Chaos Solitons Fractals 104, 655–667 (2017)
Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition using Differential Operators of Caputo Type. Springer, Berlin Heidelberg (2010)
Delavari, H., Baleanu, D., Sadati, J.: Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dynam. 67, 2433–2439 (2012)
Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput. Math. Appl. 62, 2364–2373 (2011)
Elnagar, G.N., Kazemi, M.: Chebyshev spectral solution of nonlinear volterra-hammerstein integral equations. J. Comput. Appl. Math. 76, 147–158 (1996)
Fornberg, B., Driscoll, T.A.: A fast spectral algorithm for nonlinear wave equations with linear dispersion. J. Comput. Phys. 155, 456–467 (1999)
Henry, B.I., Wearne, S.L.: Fractional reaction-diffusion. Phys. A 276, 448–455 (2000)
Hu, W., Ding, D., Zhang, Y., Wang, N., Liang, D.: Hopf bifurcation and chaos in a fractional order delayed memristor-based chaotic circuit system. Optik 130, 189–200 (2017)
Itik, M., Banks, S.P.: Chaos in a three-dimensional cancer model. Int. J. Bifurcat. Chaos 20, 71–79 (2010)
Khader, M.M., Sweilam, N.H., Mahdy, A.M.S., Moniem, N.K.A.: Numerical simulation for the fractional SIRC model and influenza A. Appl. Math. Inform. Sci. 8, 1029–1036 (2014)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, San Diego (2006)
Laskin, N.: Fractional market dynamics. Phys. A 287, 482–492 (2000)
Li, Z., Chen, X., Qui, J., Xia, T.: A novel Chebyshev-collocation spectral method for solving the transport equation. J. Ind. Manag. Optim. 17, 2519–2526 (2021)
Liu, H., Ren, B., Zhao, Q., Li, N.: Characterizing the optical chaos in a special type of small networks of semiconductor lasers using permutation entropy. Opt. Commun. 359, 79–84 (2016)
Lu, J.G.: Chaotic dynamics and synchronization of fractional-order Arneodo’s systems. Chaos Solitons Fractals 26, 1125–1133 (2005)
Ma, Y.C.J.: Study for the bifurcation topological structure and the global complicated character of a kind of non-linear finance system (i). Appl. Math. Mech. 22, 1240–1251 (2001)
Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publishers, Connecticut (2006)
Matignon, D.: Stability results for fractional differential equations with applications to control processing, In: Computational Engineering in Systems Applications, pp. 963-968 (1996)
Miller, K., Ross, B.: An Introduction to the Fractional Calaulus and Fractional Differential Equations. John Wiley & Sons Inc., New York (1993)
Moaddy, K., Radwan, A.G., Salama, K.N., Momani, S., Hashim, I.: The fractional-order modeling and synchronization of electrically coupled neurons system. Comput. Math. Appl. 64, 3329–3339 (2012)
Moutsinga, C.R.B., Pindza, E., Maré, E.: A robust spectral integral method for solving chaotic finance systems. Alex. Eng. J. 59, 601–611 (2020)
Niu, C., Liao, H., Ma, H., Wu, H.: Approximation properties of Chebyshev polynomials in the Legendre norm. Mathematics 2021(9), 3271 (2021)
Owolabi, K.M.: Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order. Commun. Nonlinear Sci. Numer. Simul. 44, 304–317 (2017)
Owolabi, K.M., Atangana, A.: Numerical simulations of chaotic and complex spatiotemporal patterns in fractional reaction-diffusion systems. Comput. Appl. Math. 37, 2166–2189 (2018)
Owolabi, K.M., Atangana, A.: Chaotic behaviour in system of noninteger-order ordinary differential equations. Chaos Solitons Fractals 115, 362–370 (2018)
Owolabi, K.M.: Mathematical analysis and numerical simulation of chaotic noninteger order differential systems with Riemann-Liouville derivative. Numer. Methods Partial Differ. Equ. 34, 274–295 (2018)
Owolabi, K.M.: Numerical solutions and pattern formation process in fractional diffusion-like equations. In: Gómez, J.F., Torres, L., Escobar, R.F. (eds.) Fractional Derivatives with Mittag-Leffler Kernel: Trends and Applications in Science and Engineering, pp. 195–216. Springer, Switzerland (2019)
Owolabi, K.M., Dutta, H.: Numerical Solution of space-time-fractional reaction-diffusion equations via the Caputo and Riesz derivatives. In: Smith, F.T., Dutta, H., Mordeson, J.N. (eds.) Mathematics Applied to Engineering, Modelling, and Social Issues, pp. 161–188. Springer, Switzerland (2019)
Owolabi, K.M., Hammouch, Z.: Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative. Phys. A 523, 1072–1090 (2019)
Owolabi, K.M., Hammouch, Z.: Mathematical modeling and analysis of two-variable system with noninteger-order derivative. Chaos 29, 013145 (2019)
Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)
Petrás, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer, Berlin (2011)
Pindza, E., Owolabi, K.M.: Fourier spectral method for higher order space fractional reaction-diffusion equations. Commun. Nonlinear Sci. Numer. Simul. 40, 112–128 (2016)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego, CA (1999)
Radwan, A.G., Moaddy, K., Salama, K.N., Momani, S., Hashim, I.: Control and switching synchronization of fractional order chaotic systems using active control technique. J. Adv. Res. 5, 125–132 (2014)
Snyder, M.A.: Chebyshev Methods in Numerical Approximations. Prentice-Hall, New Jessey (1966)
Trefethen, L.N.: Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations. Upson Hall Cornell University Ithaca, New York (1996)
Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000)
Trefethen, L.N., Embere, M.: Spectra and Pseudospectra: the behavior of nonnormal matrices and operators. Princeton University Press, New Jersey (2005)
Vaidyanathan, S., Azar, A.T., Rajagopal, K., Sambas, A., Kacar, S., Cavusoglu, U.: A new hyperchaotic temperature fluctuations model, its circuit simulation, FPGA implementation and an application to image encryption. Int. J. Simul. Process Model. 13, 281–296 (2018)
Vargas-De-León, C.: Volterra-type Lyapunov functions for fractional-order epidemic systems. Commun. Nonlinear Sci. Numer. Simul. 24, 75–85 (2015)
Yildirim, A.: Analytical approach to Fokker-Planck equation with space- and time-fractional derivatives by means of the homotopy perturbation method. J. King Saud Univ. Sci. 22, 257–264 (2010)
Zhang, X., Wang, X.: Multiple-image encryption algorithm based on mixed image element and chaos. Comput. Electr. Eng. 62, 401–413 (2017)
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Owolabi, K.M., Pindza, E. Dynamics of Fractional Chaotic Systems with Chebyshev Spectral Approximation Method. Int. J. Appl. Comput. Math 8, 140 (2022). https://doi.org/10.1007/s40819-022-01340-2
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DOI: https://doi.org/10.1007/s40819-022-01340-2
Keywords
- Chaotic dynamics
- Chebyshev spectral method
- Fractional differential equation
- Spatiotemporal oscillations
- Stability analysis