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Solving Hyperchaotic Systems Using the Spectral Relaxation Method

Author

Listed:
  • S. S. Motsa
  • P. G. Dlamini
  • M. Khumalo

Abstract

A new multistage numerical method based on blending a Gauss‐Siedel relaxation method and Chebyshev pseudospectral method, for solving complex dynamical systems exhibiting hyperchaotic behavior, is presented. The proposed method, called the multistage spectral relaxation method (MSRM), is applied for the numerical solution of three hyperchaotic systems, namely, the Chua, Chen, and Rabinovich‐Fabrikant systems. To demonstrate the performance of the method, results are presented in tables and diagrams and compared to results obtained using a Runge‐Kutta‐(4,5)‐based MATLAB solver, ode45, and other previously published results.

Suggested Citation

  • S. S. Motsa & P. G. Dlamini & M. Khumalo, 2012. "Solving Hyperchaotic Systems Using the Spectral Relaxation Method," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:203461
    DOI: 10.1155/2012/203461
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    References listed on IDEAS

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    1. Younghae Do & Bongsoo Jang, 2012. "Enhanced Multistage Differential Transform Method: Application to the Population Models," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    2. Goh, S.M. & Noorani, M.S.M. & Hashim, I., 2009. "Efficacy of variational iteration method for chaotic Genesio system – Classical and multistage approach," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2152-2159.
    3. Chowdhury, M.S.H. & Hashim, I. & Momani, S., 2009. "The multistage homotopy-perturbation method: A powerful scheme for handling the Lorenz system," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1929-1937.
    4. Asad Freihat & Shaher Momani, 2012. "Adaptation of Differential Transform Method for the Numeric‐Analytic Solution of Fractional‐Order Rössler Chaotic and Hyperchaotic Systems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    5. Abdulaziz, O. & Noor, N.F.M. & Hashim, I. & Noorani, M.S.M., 2008. "Further accuracy tests on Adomian decomposition method for chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 36(5), pages 1405-1411.
    6. Al-Sawalha, M. Mossa & Noorani, M.S.M. & Hashim, I., 2009. "On accuracy of Adomian decomposition method for hyperchaotic Rössler system," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1801-1807.
    7. Asad Freihat & Shaher Momani, 2012. "Adaptation of Differential Transform Method for the Numeric-Analytic Solution of Fractional-Order Rössler Chaotic and Hyperchaotic Systems," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-13, April.
    8. M. S. H. Chowdhury & I. Hashim & S. Momani & M. M. Rahman, 2012. "Application of Multistage Homotopy Perturbation Method to the Chaotic Genesio System," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-10, May.
    9. S. S. Motsa & Y. Khan & S. Shateyi, 2012. "Application of Piecewise Successive Linearization Method for the Solutions of the Chen Chaotic System," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    10. S. S. Motsa & Y. Khan & S. Shateyi, 2012. "Application of Piecewise Successive Linearization Method for the Solutions of the Chen Chaotic System," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-12, July.
    11. Younghae Do & Bongsoo Jang, 2012. "Enhanced Multistage Differential Transform Method: Application to the Population Models," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-14, June.
    12. M. S. H. Chowdhury & I. Hashim & S. Momani & M. M. Rahman, 2012. "Application of Multistage Homotopy Perturbation Method to the Chaotic Genesio System," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
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    Cited by:

    1. S. S. Motsa & P. G. Dlamini & M. Khumalo, 2014. "Spectral Relaxation Method and Spectral Quasilinearization Method for Solving Unsteady Boundary Layer Flow Problems," Advances in Mathematical Physics, John Wiley & Sons, vol. 2014(1).
    2. repec:plo:pone00:0107622 is not listed on IDEAS
    3. P. G. Dlamini & S. S. Motsa & M. Khumalo, 2013. "Higher Order Compact Finite Difference Schemes for Unsteady Boundary Layer Flow Problems," Advances in Mathematical Physics, John Wiley & Sons, vol. 2013(1).
    4. Vusi Mpendulo Magagula, 2019. "On the Multidomain Bivariate Spectral Local Linearisation Method for Solving Systems of Nonsimilar Boundary Layer Partial Differential Equations," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2019, pages 1-18, June.

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