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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 203461, 18 pages
http://dx.doi.org/10.1155/2012/203461
Research Article

Solving Hyperchaotic Systems Using the Spectral Relaxation Method

1School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X01, Pietermaritzburg, Scottsville 3209, South Africa
2Department of Mathematics, University of Johannesburg, P.O. Box 17011, Doornfontein 2028, South Africa

Received 10 October 2012; Accepted 28 November 2012

Academic Editor: Narcisa C. Apreutesei

Copyright © 2012 S. S. Motsa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. E. Lorenz, “Deterministic nonperiodic flow,” Journal of Atmospheric Sciences, vol. 20, pp. 130–141, 1963.
  2. O. E. Rössler, “An equation for hyperchaos,” Physics Letters A, vol. 71, no. 2-3, pp. 155–157, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. O. Abdulaziz, N. F. M. Noor, I. Hashim, and M. S. M. Noorani, “Further accuracy tests on Adomian decomposition method for chaotic systems,” Chaos, Solitons and Fractals, vol. 36, no. 5, pp. 1405–1411, 2008. View at Publisher · View at Google Scholar · View at Scopus
  4. S. Ghosh, A. Roy, and D. Roy, “An adaptation of Adomian decomposition for numeric-analytic integration of strongly nonlinear and chaotic oscillators,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 4–6, pp. 1133–1153, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. G. González-Parra, A. J. Arenas, and L. Jódar, “Piecewise finite series solutions of seasonal diseases models using multistage Adomian method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 11, pp. 3967–3977, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. M. M. Al-Sawalha, M. S. M. Noorani, and I. Hashim, “On accuracy of Adomian decomposition method for hyperchaotic Rössler system,” Chaos, Solitons and Fractals, vol. 40, no. 4, pp. 1801–1807, 2009. View at Publisher · View at Google Scholar · View at Scopus
  7. A. K. Alomari, M. S. M. Noorani, and R. Nazar, “Adaptation of homotopy analysis method for the numeric analytic solution of Chen system,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 2336–2346, 2009.
  8. A. K. Alomari, M. S. M. Noorani, and R. Nazar, “Homotopy approach for the hyperchaotic Chen system,” Physica Scripta, vol. 81, no. 4, Article ID 045005, 2010. View at Publisher · View at Google Scholar · View at Scopus
  9. A. Freihat and S. Momani, “Adaptation of differential transform method for the numeric-analytic solution of fractional-order Rössler chaotic and hyperchaotic systems,” Abstract and Applied Analysis, vol. 2012, Article ID 934219, 13 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. Y. Do and B. Jang, “Enhanced multistage differential transform method: application to the population models,” Abstract and Applied Analysis, vol. 2012, Article ID 253890, 14 pages, 2012. View at Zentralblatt MATH
  11. Z. M. Odibat, C. Bertelle, M. A. Aziz-Alaoui, and G. H. E. Duchamp, “A multi-step differential transform method and application to non-chaotic or chaotic systems,” Computers & Mathematics with Applications, vol. 59, no. 4, pp. 1462–1472, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. B. Batiha, M. S. M. Noorani, I. Hashim, and E. S. Ismail, “The multistage variational iteration method for a class of nonlinear system of ODEs,” Physica Scripta, vol. 76, no. 4, pp. 388–392, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. S. M. Goh, M. S. M. Noorani, and I. Hashim, “Efficacy of variational iteration method for chaotic Genesio system—classical and multistage approach,” Chaos, Solitons and Fractals, vol. 40, no. 5, pp. 2152–2159, 2009. View at Publisher · View at Google Scholar · View at Scopus
  14. S. M. Goh, M. S. N. Noorani, I. Hashim, and M. M. Al-Sawalha, “Variational iteration method as a reliable treatment for the hyperchaotic rössler system,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 3, pp. 363–371, 2009. View at Scopus
  15. M. S. H. Chowdhury, I. Hashim, and S. Momani, “The multistage homotopy-perturbation method: a powerful scheme for handling the Lorenz system,” Chaos, Solitons and Fractals, vol. 40, no. 4, pp. 1929–1937, 2009. View at Publisher · View at Google Scholar · View at Scopus
  16. M. S. H. Chowdhury, I. Hashim, S. Momani, and M. M. Rahman, “Application of multistage homotopy perturbation method to the chaotic Genesio system,” Abstract and Applied Analysis, vol. 2012, Article ID 974293, 10 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. M. S. H. Chowdhury and I. Hashim, “Application of multistage homotopy-perturbation method for the solutions of the Chen system,” Nonlinear Analysis. Real World Applications, vol. 10, no. 1, pp. 381–391, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. S. Wang and Y. Yu, “Application of multistage homotopy-perturbation method for the solutions of the chaotic fractional order systems,” International Journal of Nonlinear Science, vol. 13, no. 1, pp. 3–14, 2012.
  19. A. Ghorbani and J. Saberi-Nadjafi, “A piecewise-spectral parametric iteration method for solving the nonlinear chaotic Genesio system,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 131–139, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. S. Motsa, “A new piecewise-quasilinearization method for solving chaotic systems of initial value problems,” Central European Journal of Physics, vol. 10, pp. 936–946, 2012.
  21. S. S. Motsa, Y. Khan, and S. Shateyi, “Application of piecewise successive linearization method for the solutions of the Chen chaotic system,” Journal of Applied Mathematics, vol. 2012, Article ID 258948, 12 pages, 2012. View at Publisher · View at Google Scholar
  22. S.S. Motsa and P. Sibanda, “A multistage linearisation approach to a fourdimensional hyper-chaotic system with cubic nonlinearity,” Nonlinear Dynamics, vol. 70, no. 1, pp. 651–657, 2012.
  23. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, Germany, 1988.
  24. B. Fornberg, A Practical Guide to Pseudospectral Methods, vol. 1, Cambridge University Press, Cambridge, UK, 1996. View at Publisher · View at Google Scholar
  25. L. N. Trefethen, Spectral Methods in MATLAB, vol. 10, SIAM, Philadelphia, Pa, USA, 2000. View at Publisher · View at Google Scholar
  26. L. O. Chua, “Genesis of Chua's circuit,” Archiv für Elektronik und Ubertragungstechnik, vol. 46, no. 4, pp. 250–257, 1992.
  27. P. C. Rech and H. A. Albuquerque, “A Hyperchaotic Chua system,” International Journal of Bifurcation and Chaos, vol. 19, no. 11, pp. 3823–3828, 2009. View at Publisher · View at Google Scholar · View at Scopus
  28. H. Cheng, J. Zhou, and Q. Wu, “Adaptive synchronization of coupled hyperchaotic Chua systems,” in Proceedings of the Control and Decision Conference (CCDC '11), pp. 143–148, Mianyang, China, 2011.
  29. G. Qi, S. Du, G. Chen, Z. Chen, and Z. Yuan, “On a four-dimensional chaotic system,” Chaos, Solitons and Fractals, vol. 23, no. 5, pp. 1671–1682, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. M. Danca and G. Chen, “Bifurcation and chaos in a complex model of dissipative medium,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 14, no. 10, pp. 3409–3447, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. X. Luo, M. Small, Marius-F. Danca, and G. Chen, “On a dynamical system with multiple chaotic attractors,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 17, no. 9, pp. 3235–3251, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. M. I. Rabinovich and A. L. Fabrikant, “Stochastic self-modulation of waves in nonequlibrium media,” Journal of Experimental and Theoretical Physics, vol. 77, pp. 617–629, 1979.