×

zbMATH — the first resource for mathematics

Application of multistage homotopy perturbation method to the chaotic Genesio system. (English) Zbl 1246.65241
Summary: Finding accurate solution of chaotic system by using efficient existing numerical methods is very hard for its complex dynamical behaviors. In this paper, the multistage homotopy-perturbation method (MHPM) is applied to the chaotic Genesio system. The MHPM is a simple reliable modification based on an adaptation of the standard homotopy-perturbation method (HPM). The HPM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions to the chaotic Genesio system. Numerical comparisons between the MHPM and the classical fourth-order Runge-Kutta (RK4) solutions are made. The results reveal that the new technique is a promising tool for the nonlinear chaotic systems of ordinary differential equations.

MSC:
65P20 Numerical chaos
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37N35 Dynamical systems in control
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] R. Genesio and A. Tesi, “Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems,” Automatica, vol. 28, no. 3, pp. 531-548, 1992. · Zbl 0765.93030 · doi:10.1016/0005-1098(92)90177-H
[2] J.-H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37-43, 2000. · Zbl 1068.74618 · doi:10.1016/S0020-7462(98)00085-7
[3] M. J. Ablowitz, B. M. Herbst, and C. Schober, “Homotopy perturbation method and axisymmetric flow over a stretching sheet,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 4, pp. 399-406, 2006.
[4] L. Cveticanin, “The homotopy-perturbation method applied for solving complex-valued differential equations with strong cubic nonlinearity,” Journal of Sound and Vibration, vol. 285, no. 4-5, pp. 1171-1179, 2005. · Zbl 1238.65085 · doi:10.1016/j.jsv.2004.10.026
[5] J. H. He, Non-Perturbative Methods for Strongly Nonlinear Problems, Die Deutsche Bibliothek, Berlin, Germany, 2006.
[6] J.-H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141-1199, 2006. · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[7] M. A. Noor and S. T. Mohyud-Din, “An efficient algorithm for solving fifth-order boundary value problems,” Mathematical and Computer Modelling, vol. 45, no. 7-8, pp. 954-964, 2007. · Zbl 1133.65052 · doi:10.1016/j.mcm.2006.09.004
[8] M. El-Shahed, “Application of He’s homotopy perturbation method to Volterra’s integro-differential equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 163-168, 2005. · Zbl 1401.65150
[9] J. H. He, “Analytical methods for thermal science-an elementary introduction,” Thermal Science, vol. 15, supplement 1, pp. S1-S3, 2011. · doi:10.2298/TSCI11S1001H
[10] H. E. Ji-Huan, “A note on the homotopy perturbation method,” Thermal Science, vol. 14, no. 2, pp. 565-568, 2010.
[11] 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. · Zbl 1198.65135 · doi:10.1016/j.chaos.2007.09.073
[12] 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. · Zbl 1154.65350 · doi:10.1016/j.nonrwa.2007.09.014
[13] I. Hashim and M. S. H. Chowdhury, “Adaptation of homotopy-perturbation method for numeric-analytic solution of system of ODEs,” Physics Letters A, vol. 372, no. 4, pp. 470-481, 2008. · Zbl 1217.81054 · doi:10.1016/j.physleta.2007.07.067
[14] I. Hashim, M. S. H. Chowdhury, and S. Mawa, “On multistage homotopy-perturbation method applied to nonlinear biochemical reaction model,” Chaos, Solitons and Fractals, vol. 36, no. 4, pp. 823-827, 2008. · Zbl 1210.65149 · doi:10.1016/j.chaos.2007.09.009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.