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Dynamics of Fractional Chaotic Systems with Chebyshev Spectral Approximation Method

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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

  1. Abdelhakem, M., Ahmed, A., El-kady, M.: Spectral monic Chebyshev approximation for higher order differential equations. Math. Sci. Lett. 8, 11–17 (2019)

    Article  Google Scholar 

  2. 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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Akgul, A., Moroz, I., Pehlivan, I., Vaidyanathan, S.: A new four-scroll chaotic attractor and its engineering applications. Optik 127, 5491–5499 (2016)

    Article  Google Scholar 

  4. Akgul, A., Hussain, S., Pehlivan: A new three-dimensional chaotic system, its dynamical analysis and electronic circuit applications. Optik 127, 7062–7071 (2016)

    Article  Google Scholar 

  5. Atangana, A.: Fractional Operators With Constant and Variable Order with Application to Geo-Hydrology. Academic Press, New York (2017)

    MATH  Google Scholar 

  6. Atangana, A., Owolabi, K.M.: New numerical approach for fractional differential equations. Math. Model. Natural Phenom. 13(3), 1–21 (2018)

    MathSciNet  MATH  Google Scholar 

  7. 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)

    Article  Google Scholar 

  8. Bueno-Orovio, A., Kay, D., Burrage, K.: Fourier spectral methods for fractional-in-space reaction-diffusion equations. BIT Numer. Math. 54, 937–954 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New York (1988)

    Book  MATH  Google Scholar 

  10. Caputo, M.: Linear models of dissipation whose \(Q\) is almost frequency independent II. Geophys. J. Roy. Astron. Soc. 13, 529–539 (1967)

    Article  Google Scholar 

  11. 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)

    Article  Google Scholar 

  12. Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition using Differential Operators of Caputo Type. Springer, Berlin Heidelberg (2010)

    Book  MATH  Google Scholar 

  13. Delavari, H., Baleanu, D., Sadati, J.: Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dynam. 67, 2433–2439 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Elnagar, G.N., Kazemi, M.: Chebyshev spectral solution of nonlinear volterra-hammerstein integral equations. J. Comput. Appl. Math. 76, 147–158 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fornberg, B., Driscoll, T.A.: A fast spectral algorithm for nonlinear wave equations with linear dispersion. J. Comput. Phys. 155, 456–467 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Henry, B.I., Wearne, S.L.: Fractional reaction-diffusion. Phys. A 276, 448–455 (2000)

    Article  MathSciNet  Google Scholar 

  18. 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)

    Article  Google Scholar 

  19. Itik, M., Banks, S.P.: Chaos in a three-dimensional cancer model. Int. J. Bifurcat. Chaos 20, 71–79 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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)

    Article  Google Scholar 

  21. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, San Diego (2006)

    MATH  Google Scholar 

  22. Laskin, N.: Fractional market dynamics. Phys. A 287, 482–492 (2000)

    Article  MathSciNet  Google Scholar 

  23. 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)

    Article  MathSciNet  MATH  Google Scholar 

  24. 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)

    Article  Google Scholar 

  25. Lu, J.G.: Chaotic dynamics and synchronization of fractional-order Arneodo’s systems. Chaos Solitons Fractals 26, 1125–1133 (2005)

    Article  MATH  Google Scholar 

  26. 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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publishers, Connecticut (2006)

    Google Scholar 

  28. Matignon, D.: Stability results for fractional differential equations with applications to control processing, In: Computational Engineering in Systems Applications, pp. 963-968 (1996)

  29. Miller, K., Ross, B.: An Introduction to the Fractional Calaulus and Fractional Differential Equations. John Wiley & Sons Inc., New York (1993)

    Google Scholar 

  30. 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)

    Article  MathSciNet  MATH  Google Scholar 

  31. 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)

    Article  Google Scholar 

  32. Niu, C., Liao, H., Ma, H., Wu, H.: Approximation properties of Chebyshev polynomials in the Legendre norm. Mathematics 2021(9), 3271 (2021)

    Article  Google Scholar 

  33. 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)

    Article  MathSciNet  MATH  Google Scholar 

  34. 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)

    Article  MathSciNet  MATH  Google Scholar 

  35. Owolabi, K.M., Atangana, A.: Chaotic behaviour in system of noninteger-order ordinary differential equations. Chaos Solitons Fractals 115, 362–370 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. 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)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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)

    Chapter  Google Scholar 

  38. 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)

    Chapter  MATH  Google Scholar 

  39. 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)

    Article  MathSciNet  Google Scholar 

  40. Owolabi, K.M., Hammouch, Z.: Mathematical modeling and analysis of two-variable system with noninteger-order derivative. Chaos 29, 013145 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  41. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  42. Petrás, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  43. 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)

    Article  MathSciNet  MATH  Google Scholar 

  44. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego, CA (1999)

    MATH  Google Scholar 

  45. 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)

    Article  Google Scholar 

  46. Snyder, M.A.: Chebyshev Methods in Numerical Approximations. Prentice-Hall, New Jessey (1966)

    MATH  Google Scholar 

  47. Trefethen, L.N.: Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations. Upson Hall Cornell University Ithaca, New York (1996)

    Google Scholar 

  48. Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  49. Trefethen, L.N., Embere, M.: Spectra and Pseudospectra: the behavior of nonnormal matrices and operators. Princeton University Press, New Jersey (2005)

    Book  Google Scholar 

  50. 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)

    Article  Google Scholar 

  51. Vargas-De-León, C.: Volterra-type Lyapunov functions for fractional-order epidemic systems. Commun. Nonlinear Sci. Numer. Simul. 24, 75–85 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  52. 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)

    Article  Google Scholar 

  53. Zhang, X., Wang, X.: Multiple-image encryption algorithm based on mixed image element and chaos. Comput. Electr. Eng. 62, 401–413 (2017)

    Article  Google Scholar 

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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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