zbMATH — the first resource for mathematics

On solving the chaotic Chen system: A new time marching design for the variational iteration method using Adomian’s polynomial. (English) Zbl 1190.65189
Summary: This paper centres on the effectiveness of the variational iteration method and its modifications for numerically solving the chaotic Chen system [cf. G. Chen and T. Ueta, Int. J. Bifurcation Chaos Appl. Sci. Eng. 9, No. 7, 1465–1466 (1999; Zbl 0962.37013)] which is a three-dimensional system of ordinary differential equations with quadratic nonlinearities. This research implements the multistage variational iteration method with an emphasis on the new multistage hybrid of variational iteration method with Adomian polynomials. Numerical comparisons are made between the multistage variational iteration method, the multistage variational iteration method using the Adomian’s polynomials and the classic fourth-order Runge-Kutta method. Our work shows that the new multistage hybrid provides good accuracy and efficiency with a performance that surpasses that of the multistage variational iteration method.

65P20 Numerical chaos
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
37M05 Simulation of dynamical systems
Zbl 0962.37013
Full Text: DOI
[1] Al-Sawalha, M.M., Noorani, M.S.M., Hashim, I.: On accuracy of Adomian decomposition method for hyperchaotic Rössler system. Chaos Solitons Fractals 40(4), 1801–1807 (2009) · Zbl 1198.65131
[2] Hashim, I., Noorani, M.S.M., Ahmad, R., Bakar, S.A., Ismail, E.S., Zakaria, A.M.: Accuracy of the Adomian decomposition method applied to the Lorenz system. Chaos Solitons Fractals 28, 1149–1158 (2006) · Zbl 1096.65066
[3] Noorani, M.S.M., Hashim, I., Ahmad, R., Bakar, S.A., Ismail, E.S., Zakaria, A.M.: Comparing numerical methods for the solutions of the Chen system. Chaos Solitons Fractals 321, 296–1304 (2007) · Zbl 1131.65101
[4] Li, C.P., Wang, Y.H.: Numerical algorithm based on Adomian decomposition for fractional differential equations. Comput. Math. Appl. 57, 1672–1681 (2009) · Zbl 1186.65110
[5] Al-Sawalha, M.M., Noorani, M.S.M.: Application of the differential transformation method for the solution of the hyperchaotic Rössler system. Commun. Non. Sci. Numer. Simul. 14, 1509–1514 (2009)
[6] Goh, S.M., Noorani, M.S.M., Hashim, I.: Efficacy of variational iteration method for chaotic Genesio system–Classical and multistage approach. Chaos Solitons Fractals 40(5), 2152–2159 (2009) · Zbl 1198.65140
[7] Goh, S.M., Noorani, M.S.M., Hashim, I., Al-Sawalha, M.M.: Variational iteration method as a reliable treatment for the hyperchaotic Rössler system. Int. J. Non. Sci. Numer. Simul. 10(3), 363–371 (2009) · Zbl 06942409
[8] Allan, F.M.: Construction of analytic solution to chaotic dynamical systems using the Homotopy analysis method. Chaos Solitons Fractals 39(4), 1744–1752 (2009) · Zbl 1197.65102
[9] Chowdhury, M.S.H., Hashim, I.: Application of multistage homotopy-perturbation method for the solutions of the Chen system. Non. Anal. Real World Appl. 10(1), 381–391 (2009) · Zbl 1154.65350
[10] Yamaguti, M., Ushiki, S.: Chaos in numerical analysis of ordinary differential equations. Physica D 3, 618–626 (1981) · Zbl 1194.37064
[11] Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurcat. Chaos 9(7), 1465–1466 (1999) · Zbl 0962.37013
[12] Lu, J., Zhou, T., Chen, G., Zhang, S.: Local bifurcations of the Chen system. Int. J. Bifurcat. Chaos 12, 2257–2270 (2002) · Zbl 1047.34044
[13] Ueta, T., Chen, G.: Bifurcation analysis of Chen’s equation. Int. J. Bifurcat. Chaos 8, 1917–1931 (2000) · Zbl 1090.37531
[14] Li, C.P., Chen, G.R.: A note one Hopf bifurcation in Chen’s system. Int. J. Bifurcat. Chaos 13, 1609–1615 (2003) · Zbl 1074.34045
[15] Inokuti, M., Sekine, H., Mura, T.: General use of the Lagrange multiplier in nonlinear mathematical physics. In: Nemat-Nassed, S. (ed.). Variational Method in the Mechanics of Solids. Pergamon, Oxford (1978)
[16] He, J.H.: Variational iteration method–some recent results and new interpretations. J. Comput. Appl. Math. 207(1), 3–17 (2007) · Zbl 1119.65049
[17] He, J.H., Wu, X.H.: Variational iteration method: new development and applications. Int. J. Comput. Math. Appl. 54(7–8), 881–894 (2007) · Zbl 1141.65372
[18] He, J.H.: An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering. Int. J. Mod. Phys. B 22(21), 3487–3578 (2008) · Zbl 1149.76607
[19] He, J.H.: Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B 20(10), 1141–1199 (2007) · Zbl 1102.34039
[20] Abulwafa, E.M., Abdou, M.A., Mahmoud, A.A.: The solution of nonlinear coagulation problem with mass loss. Chaos Soliton Fractals 29(2), 313–330 (2006) · Zbl 1101.82018
[21] He, J.H.: Variational iteration method for autonomous ordinary differential systems. Appl. Math. Comput. 114(2–3), 115–123 (2000) · Zbl 1027.34009
[22] He, J.H.: A simple perturbation approach to Blasius equation. Appl. Math. Comput. 140(2–3), 217–222 (2003) · Zbl 1028.65085
[23] He, J.H., Wan, Y.Q., Guo, Q.: An iteration formulation for normalized diode characteristics. Int. J. Circ. Theory Appl. 32(6), 629–632 (2004) · Zbl 1169.94352
[24] Momani, S., Abuasad, S.: Application of He’s variational iteration method to Helmholtz equation. Chaos Solitons Fractals 27(5), 1119–1123 (2006) · Zbl 1086.65113
[25] Shou, D.H., He, J.H.: Beyond Adomian method: The variational iteration method for solving heat-like and wave-like equations with variable coefficients. Phys. Lett. A 372, 233–237 (2008) · Zbl 1217.35091
[26] Goh, S.M., Ismail, A.I.M., Noorani, M.S.M., Hashim, I.: Dynamics of the Hantavirus infection through variational iteration method. Non. Anal. Real World Appl. 10(4), 2171–2176 (2009) · Zbl 1163.92326
[27] Goh, S.M., Noorani, M.S.M., Hashim, I.: Prescribing a multistage analytical method to a prey-predator dynamical system. Phys. Lett. A 373, 107–110 (2008) · Zbl 1227.34017
[28] Ghorbani, A.: Beyond Adomian polynomials: He polynomials. Chaos Solitons Fractals 39(3) 1486–1492 (2009) · Zbl 1197.65061
[29] Mohyud-Din, S.T., Noor, M.A., Noor, K.I.: Traveling wave solutions of seventh-order generalized KdV equations using He’s polynomials. Int. J. Non. Sci. Numer. Simul. 10, 227–233 (2009) · Zbl 1168.35427
[30] Noor, M.A., Mohyud-Din, S.T.: Variational iteration method for solving higher-order nonlinear boundary value problems using He’s polynomials. Int. J. Non. Sci. Numer. Simul. 9, 141–156 (2008) · Zbl 1151.65334
[31] Ghorbani, A., Saberi-Nadjafi, J.: Exact solutions for nonlinear integral equations by a modified homotopy perturbation method. Comput. Math. Appl. 56, 1032–1039 (2008) · Zbl 1155.45300
[32] Batiha, B., Noorani, M.S.M., Hashim, I., Ismail, E.S.: The multistage variational iteration method for a class of nonlinear system of ODEs. Phys. Scr. 76, 1–5 (2007) · Zbl 1138.81393
[33] Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer, Boston (1994) · Zbl 0802.65122
[34] Wazwaz, A.M.: A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl. Math. Comput. 111, 53–69 (2000) · Zbl 1028.65138
[35] Ghorbani, A., Saberi-Nadjafi, J.: He’s homotopy perturbation method for calculating adomian polynomials. Int. J. Non. Sci. Numer. Simul. 8(2), 229–232 (2007) · Zbl 1401.65056
[36] Abbasbandy, S.: A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials. J. Comput. Appl. Math. 207, 59–63 (2007) · Zbl 1120.65083
[37] El-Wakil, S.A., Abdou, M.A.: New applications of variational iteration method using Adomian polynomials. Nonlinear Dynamics 52, 41–49 (2008) · Zbl 1170.76356
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.