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The Scientific World Journal
Volume 2014 (2014), Article ID 943293, 10 pages
http://dx.doi.org/10.1155/2014/943293
Research Article

Multistage Spectral Relaxation Method for Solving the Hyperchaotic Complex Systems

1Department of Mathematics, Islamic Azad University, Mashhad Branch, Mashhad, Iran
2Departamento de Matemática, Universidade da Beira Interior, 6201-001 Covilhã, Portugal

Received 4 August 2014; Accepted 12 September 2014; Published 16 October 2014

Academic Editor: Fazlollah Soleymani

Copyright © 2014 Hassan Saberi Nik and Paulo Rebelo. 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. G. Chen and T. Ueta, “Yet another chaotic attractor,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 9, no. 7, pp. 1465–1466, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  2. Z. Wei, R. Wang, and A. Liu, “A new finding of the existence of hidden hyperchaotic attractors with no equilibria,” Mathematics and Computers in Simulation, vol. 100, pp. 13–23, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. A. C. Fowler, J. D. Gibbon, and M. J. McGuinness, “The complex Lorenz equations,” Physica D: Nonlinear Phenomena, vol. 4, no. 2, pp. 139–163, 1981/82. View at Publisher · View at Google Scholar · View at MathSciNet
  4. A. G. Vladimirov, V. Y. Toronov, and V. L. Derbov, “Properties of the phase space and bifurcations in the complex Lorenz model,” Technical Physics, vol. 43–48, pp. 877–884, 1998. View at Google Scholar
  5. A. Rauh, L. Hannibal, and N. B. Abraham, “Global stability properties of the complex Lorenz model,” Physica D: Nonlinear Phenomena, vol. 99, no. 1, pp. 45–58, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. R. L. Lang, “A stochastic complex model with random imaginary noise,” Nonlinear Dynamics, vol. 62, no. 3, pp. 561–565, 2010. View at Publisher · View at Google Scholar · View at Scopus
  7. G. M. Mahmoud, M. E. Ahmed, and E. E. Mahmoud, “Analysis of hyperchaotic complex Lorenz systems,” International Journal of Modern Physics C, vol. 19, no. 10, pp. 1477–1494, 2008. View at Publisher · View at Google Scholar · View at Scopus
  8. G. M. Mahmoud, E. E. Mahmoud, and M. E. Ahmed, “On the hyperchaotic complex Lü system,” Nonlinear Dynamics, vol. 58, no. 4, pp. 725–738, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. G. M. Mahmoud, M. A. Al-Kashif, and S. A. Aly, “Basic properties and chaotic synchronization of complex Lorenz system,” International Journal of Modern Physics C, vol. 18, no. 2, pp. 253–265, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. A. K. Alomari, M. S. Noorani, and R. Nazar, “Explicit series solutions of some linear and nonlinear Schrodinger equations via the homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1196–1207, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  11. S. Effati, H. Saberi Nik, and M. Shirazian, “An improvement to the homotopy perturbation method for solving the Hamilton-Jacobi-Bellman equation,” IMA Journal of Mathematical Control and Information, vol. 30, no. 4, pp. 487–506, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. M. S. 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 MathSciNet · View at Scopus
  13. S. M. Goh, M. S. Noorani, and I. Hashim, “On solving the chaotic Chen system: a new time marching design for the variational iteration method using Adomian's polynomial,” Numerical Algorithms, vol. 54, no. 2, pp. 245–260, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. A. Gökdoğan, M. Merdan, and A. Yildirim, “The modified algorithm for the differential transform method to solution of Genesio systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 45–51, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. H. Saberi Nik, S. Effati, S. S. Motsa, and M. Shirazian, “Spectral homotopy analysis method and its convergence for solving a class of nonlinear optimal control problems,” Numerical Algorithms, vol. 65, no. 1, pp. 171–194, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. H. Saberi Nik, S. Effati, S. S. Motsa, and S. Shateyi, “A new piecewise-spectral homotopy analysis method for solving chaotic systems of initial value problems,” Mathematical Problems in Engineering, vol. 2013, Article ID 583193, 13 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  17. S. Effati, H. Saberi Nik, and A. Jajarmi, “Hyperchaos control of the hyperchaotic Chen system by optimal control design,” Nonlinear Dynamics, vol. 73, no. 1-2, pp. 499–508, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. S. S. Motsa, P. G. Dlamini, and M. Khumalo, “Solving hyperchaotic systems using the spectral relaxation method,” Abstract and Applied Analysis, vol. 2012, Article ID 203461, 18 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  19. S. S. Motsa, P. Dlamini, and M. Khumalo, “A new multistage spectral relaxation method for solving chaotic initial value systems,” Nonlinear Dynamics, vol. 72, no. 1-2, pp. 265–283, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. L. N. Trefethen, Spectral Methods in MATLAB, SIAM, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  21. C. Luo and X. Wang, “Hybrid modified function projective synchronization of two different dimensional complex nonlinear systems with parameters identification,” Journal of the Franklin Institute, vol. 350, no. 9, pp. 2646–2663, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. J. Yu, B. Chen, H. Yu, and J. Gao, “Adaptive fuzzy tracking control for the chaotic permanent magnet synchronous motor drive system via backstepping,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 671–681, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. F. Zhang, C. Mu, X. Wang, I. Ahmed, and Y. Shu, “Solution bounds of a new complex PMSM system,” Nonlinear Dynamics, vol. 74, no. 4, pp. 1041–1051, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. X. Wang and H. Zhang, “Backstepping-based lag synchronization of a complex permanent magnet synchronous motor system,” Chinese Physics B, vol. 22, no. 4, Article ID 048902, 2013. View at Publisher · View at Google Scholar · View at Scopus