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A new multistage spectral relaxation method for solving chaotic initial value systems

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Abstract

In this paper, we present a new pseudospectral method application for solving nonlinear initial value problems (IVPs) with chaotic properties. The proposed method, called the multistage spectral relaxation method (MSRM) is based on a novel technique of extending Gauss–Siedel type relaxation ideas to systems of nonlinear differential equations and using the Chebyshev pseudo-spectral methods to solve the resulting system on a sequence of multiple intervals. In this new application, the MSRM is used to solve famous chaotic systems such as the such as Lorenz, Chen, Liu, Rikitake, Rössler, Genesio–Tesi, and Arneodo–Coullet chaotic systems. The accuracy and validity of the proposed method is tested against Runge–Kutta and Adams–Bashforth–Moulton based methods. The numerical results indicate that the MSRM is an accurate, efficient, and reliable method for solving very complex IVPs with chaotic behavior.

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References

  1. Abdulaziz, O., Noor, N.F.M., Hashim, I., Noorani, M.S.M.: Further accuracy tests on Adomian decomposition method for chaotic systems. Chaos Solitons Fractals 36, 1405–1411 (2008)

    Article  Google Scholar 

  2. Alomari, A.K., Noorani, M.S.M., Nazar, R.: Adaptation of homotopy analysis method for the numeric analytic solution of Chen system. Commun. Nonlinear Sci. Numer. Simul. 14, 2336–2346 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alomari, A.K., Noorani, M.S.M., Nazar, R.: Homotopy approach for the hyperchaotic Chen system. Phys. Scr. 81, 045005 (2010). doi:10.1088/0031-8949/81/04/045005

    Article  Google Scholar 

  4. Arneodo, A., Coullet, P., Tresser, C.: Possible new strange attractors with spiral structure. Commun. Math. Phys. 79, 573–579 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Batiha, B., Noorani, M.S.M., Hashim, I., Ismail, E.S.: The multistage variational iteration method for a class of nonlinear system of ODEs. Phys. Scr. 76, 388–392 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bogacki, P., Shampine, L.F.: A 3(2) pair of Runge–Kutta formulas. Appl. Math. Lett. 2, 1–9 (1989)

    Article  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  8. Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9(7), 1465–1466 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chowdhury, M.S.H., Hashim, I., Momani, S.: The multistage homotopy-perturbation method: a powerful scheme for handling the Lorenz system. Chaos Solitons Fractals 40, 1929–1937 (2009)

    Article  MATH  Google Scholar 

  10. Chowdhury, M.S.H., Hashim, I., Momani, S., Rahman, M.M.: Application of multistage homotopy perturbation method to the chaotic genesio system. Abstr. Appl. Anal. 2012, 974293 (2012). doi:10.1155/2012/974293

    Article  MathSciNet  Google Scholar 

  11. Chowdhury, M.S.H., Hashim, I.: Application of multistage homotopy-perturbation method for the solutions of the Chen system. Nonlinear Anal., Real World Appl. 10, 381–391 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Do, Y., Jang, B.: Enhanced multistage differential transform method: application to the population models. Abstr. Appl. Anal. 2012, 253890 (2012). doi:10.1155/2012/253890

    MathSciNet  Google Scholar 

  13. Dormand, J.R., Prince, P.J.: A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6(1), 19–26 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  15. Freihat, A., Momani, S.: Adaptation of differential transform method for the numeric-analytic solution of fractional-order Rössler chaotic and hyperchaotic systems. Abstr. Appl. Anal. 2012, 253890 (2012). doi:10.1155/2012/253890

    Article  MathSciNet  Google Scholar 

  16. Genesio, R., Tesi, A.: A harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 28, 531–548 (1992)

    Article  MATH  Google Scholar 

  17. Ghorbani, A., Saberi-Nadja, J.: A piecewise-spectral parametric iteration method for solving the non-linear chaotic Genesio system. Math. Comput. Model. 54, 131–139 (2011)

    Article  MATH  Google Scholar 

  18. Goh, S.M., Noorani, M.S.M., Hashim, I.: Efficacy of variational iteration method for chaotic Genesio system—classical and multistage approach. Chaos Solitons Fractals 40, 2152–2159 (2009)

    Article  MATH  Google Scholar 

  19. Goh, S.M., Noorani, M.S.M., Hashim, I., Al-Sawalha, M.M.: Variational iteration method as a reliable treatment for the hyperchaotic Rössler system. Int. J. Nonlinear Sci. Numer. Simul. 10, 363–371 (2009)

    Article  Google Scholar 

  20. Gonzalez, G., Arenas, A.J., Jodar, L.: Piecewise finite series solutions of seasonal diseases models using multistage Adomian method. Commun. Nonlinear Sci. Numer. Simul. 14, 3967–3977 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, C., Liu, T., Liu, L., Liu, K.: A new chaotic attractor. Chaos Solitons Fractals 22, 1031–1038 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lorenz, E.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  23. Mossa Al-Sawalha, M., Noorani, M.S.M., Hashim, I.: On accuracy of Adomian decomposition method for hyperchaotic Rössler system. Chaos Solitons Fractals 40, 1801–1807 (2009)

    Article  MATH  Google Scholar 

  24. Motsa, S.: A new piecewise-quasilinearization method for solving chaotic systems of initial value problems. Cent. Eur. J. Phys. 10, 936–946 (2012)

    Article  Google Scholar 

  25. Motsa, S.S., Khan, Y., Shateyi, S.: Application of piecewise successive linearization method for the solutions of the Chen chaotic system. J. Appl. Math. 2012, 258948 (2012). doi:10.1155/2012/258948

    MathSciNet  Google Scholar 

  26. Motsa, S.S., Sibanda, P.: A multistage linearisation approach to a four-dimensional hyper-chaotic system with cubic nonlinearity. Nonlinear Dyn. (2012). doi:10.1007/s11071-012-0484-1

    MathSciNet  Google Scholar 

  27. Nour, H.M., Elsaid, A., Elsonbaty, A.: Comparing the multistage and PADE techniques for iterative methods in solving nonlinear food chain model. Math. Comput. Sci. 2, 810–823 (2012)

    MathSciNet  Google Scholar 

  28. Odibat, Z.M., Bertelle, C., Aziz-Alaoui, M.A., Duchamp, G.H.E.: A multi-step differential transform method and application to non-chaotic or chaotic systems. Comput. Math. Appl. 59, 1462–1472 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rikitake, T.: Oscillations of a system of disk dynamos. Proc. Camb. Philos. Soc. 54, 89–105 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)

    Article  Google Scholar 

  31. Shampine, L.F., Gordon, M.K.: Computer Solution of Ordinary Differential Equations: The Initial Value Problem. Freeman, San Francisco (1975)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  33. Wang, S., Yu, Y.: Application of Multistage Homotopy-perturbation Method for the Solutions of the Chaotic Fractional Order Systems. Int. J. Nonlinear Sci. 13, 3–14 (2012)

    MathSciNet  MATH  Google Scholar 

  34. Xiang-Jun, W., Jing-Sen, L., Guan-Rong, C.: Chaos synchronization of Rikitake chaotic attractor using the passive control technique. Nonlinear Dyn. 53, 45–53 (2008)

    Article  MathSciNet  Google Scholar 

  35. Yao, L.S.: Computed chaos or numerical errors. Nonlinear Anal. Model. Control 15, 109–126 (2010)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg, 3209, South Africa

    S. S. Motsa

  2. Department of Mathematics, University of Johannesburg, P.O. Box 17011, Doornfontein, 2028, South Africa

    P. Dlamini & M. Khumalo

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  1. S. S. Motsa
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  2. P. Dlamini
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  3. M. Khumalo
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Correspondence to S. S. Motsa.

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Motsa, S.S., Dlamini, P. & Khumalo, M. A new multistage spectral relaxation method for solving chaotic initial value systems. Nonlinear Dyn 72, 265–283 (2013). https://doi.org/10.1007/s11071-012-0712-8

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  • DOI: https://doi.org/10.1007/s11071-012-0712-8

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