Abstract and Applied Analysis

Solving Hyperchaotic Systems Using the Spectral Relaxation Method

S. S. Motsa, P. G. Dlamini, and M. Khumalo

Full-text: Open access

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.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 203461, 18 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364476003

Digital Object Identifier
doi:10.1155/2012/203461

Mathematical Reviews number (MathSciNet)
MR3004907

Zentralblatt MATH identifier
1256.65102

Citation

Motsa, S. S.; Dlamini, P. G.; Khumalo, M. Solving Hyperchaotic Systems Using the Spectral Relaxation Method. Abstr. Appl. Anal. 2012 (2012), Article ID 203461, 18 pages. doi:10.1155/2012/203461. https://projecteuclid.org/euclid.aaa/1364476003


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References

  • E. Lorenz, “Deterministic nonperiodic flow,” Journal of Atmospheric Sciences, vol. 20, pp. 130–141, 1963.
  • O. E. Rössler, “An equation for hyperchaos,” Physics Letters A, vol. 71, no. 2-3, pp. 155–157, 1979.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, Germany, 1988.
  • B. Fornberg, A Practical Guide to Pseudospectral Methods, vol. 1, Cambridge University Press, Cambridge, UK, 1996.
  • L. N. Trefethen, Spectral Methods in MATLAB, vol. 10, SIAM, Philadelphia, Pa, USA, 2000.
  • L. O. Chua, “Genesis of Chua's circuit,” Archiv für Elektronik und Ubertragungstechnik, vol. 46, no. 4, pp. 250–257, 1992.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.