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Application of multistage homotopy-perturbation method for the solutions of the Chen system. (English) Zbl 1154.65350
Summary: In this paper, a new reliable algorithm based on an adaptation of the standard homotopy-perturbation method (HPM) is applied to the Chen system which is a three-dimensional system of ODEs with quadratic nonlinearities. The HPM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions to the Chen system. We shall call this technique as the multistage HPM (for short MHPM). In particular we look at the accuracy of the HPM as the Chen system changes from a nonchaotic system to a chaotic one. Numerical comparisons between the MHPM and the classical fourth-order Runge-Kutta (RK4) numerical solutions reveal that the new technique is a promising tool for the nonlinear chaotic and nonchaotic systems of ODEs.

MSC:
65L99 Numerical methods for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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[1] Chen, G.; Ueta, T., Yet another chaotic attractor, Internat. J. bifurc. chaos, 9, 7, 1465-1466, (1999) · Zbl 0962.37013
[2] Ueta, T.; Chen, G., Bifurcation analysis of chen’s equation, Internat. J. bifurc. chaos, 8, 1917-1931, (2000) · Zbl 1090.37531
[3] Lu, J.; Zhou, T.; Chen, G.; Zhang, S., Local bifurcations of the Chen system, Internat. J. bifurc. chaos, 12, 2257-2270, (2002) · Zbl 1047.34044
[4] Yassen, M.T., The optimal control of Chen chaotic dynamical system, Appl. math. comput., 171, 171-180, (2002) · Zbl 1042.49002
[5] Yassen, M.T., Chaos control of Chen chaotic dynamical system, Chaos solitons fractals, 15, 271-283, (2003) · Zbl 1038.37029
[6] Deng, W.; Li, C., Synchronization of chaotic fractional Chen system, J. phys. soc. jap., 74, 1645-1648, (2005) · Zbl 1080.34537
[7] Plienpanich, T.; Niamsup, P.; Lenbury, Y., Controllability and stability of the perturbed Chen chaotic dynamical system, Appl. math. comput., 171, 927-947, (2005) · Zbl 1121.93309
[8] 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
[9] 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, 32, 1296-1304, (2007) · Zbl 1131.65101
[10] O. Abdulaziz, N.F.M. Noor, I. Hashim, M.S.M. Noorani, Further accuracy tests on Adomian decomposition method for chaotic systems, Chaos Solitons Fractals (2007), doi:10.1016/j.chaos.2006.09.007
[11] Ghosh, S.; Roy, A.; Roy, D., An adaptation of Adomian decomposition for numeric-analytic integration of strongly nonlinear and chaotic oscillations, Comput. methods appl. mech. engrg., 196, 1133-1153, (2007) · Zbl 1120.70303
[12] He, J.H., Approximate analytical solution of Blasius equation, Commun. nonlinear sci. numer. simul., 3, 260-263, (1998) · Zbl 0918.34016
[13] He, J.H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. non-linear mech., 35, 1, 37-43, (2000) · Zbl 1068.74618
[14] He, J.H., A simple perturbation approach to Blasius equation, Appl. math. comput., 140, 217-222, (2003) · Zbl 1028.65085
[15] He, J.H., Application of homotopy perturbation method to nonlinear wave equations, Chaos solitons fractals, 26, 695-700, (2005) · Zbl 1072.35502
[16] He, J.H., Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. nonlinear sci. numer. simul., 6, 207, (2005)
[17] He, J.H., Homotopy perturbation method for solving boundary value problems, Phys. lett. A, 350, 87-88, (2006) · Zbl 1195.65207
[18] J.H. He, Non-Perturbative Methods for Strongly Nonlinear Problems, Die Deutsche bibliothek, Germany, 2006
[19] He, J.H., Some asymptotic methods for strongly nonlinear equations, Int. J. modern phys. B, 20, 1141-1199, (2006) · Zbl 1102.34039
[20] Ghorbani, A.; Saberi-Nadjafi, J., He’s homotopy perturbation method for calculating Adomian polynomials, Int. J. nonlinear sci. numer. simul., 8, 2, 229-232, (2007)
[21] A. Ghorbani, Beyond Adomian polynomials: He polynomials, Chaos Solitons Fractals (2007), doi:1.1016/j/chaos.2007.06.034 · Zbl 1197.65061
[22] El-Shahed, M., Application of he’s homotopy perturbation method to volterra’s integro-differential equation, Int. J. nonlinear sci. numer. simul., 6, 163, (2005)
[23] D.D. Ganji, A. Sadighi, Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations, J. Comput. Appl. Math. (2007), doi:10.1016/j.cam.2006.07.030 · Zbl 1120.65108
[24] Noor, M.A.; Mohyud-Din, S.T., An efficient algorithm for solving fifth-order boundary value problems, Mathl. comput. model., 45, 7-8, 954, (2007) · Zbl 1133.65052
[25] Chowdhury, M.S.H.; Hashim, I., Solutions of a class of singular second-order IVPs by homotopy-perturbation method, Phys. lett. A, 365, 439-447, (2007) · Zbl 1203.65124
[26] Chowdhury, M.S.H.; Hashim, I.; Abdulaziz, O., Application of homotopy-perturbation method to nonlinear population dynamics models, Phys. lett. A, 368, 251-258, (2007) · Zbl 1209.65107
[27] M.S.H. Chowdhury, I. Hashim, Solutions of Emden-Fowler equations by homotopy-perturbation method, Nonlinear Anal. Ser. B: Real World Appl. (2007), in press (doi:10.1016/j.nonrwa.2007.08.07) · Zbl 1209.65106
[28] Chowdhury, M.S.H.; Hashim, I., Solutions of time-dependent emden – fowler type equations by homotopy-perturbation method, Phys. lett. A, 368, 305-313, (2007) · Zbl 1209.65106
[29] M.S.H. Chowdhury, I. Hashim, Application of homotopy-perturbation method to Klein-Gordon and sine-Gordon equations, Chaos Solitons Fractals (2007), doi:10.1016/j.chaos.2007.06.091 · Zbl 1197.65164
[30] M.S.H. Chowdhury, I. Hashim, S. Momani, The multistage homotopy-perturbation method: A powerful scheme for handling the Lorenz system, Chaos Solitons Fractals (in press) · Zbl 1198.65135
[31] I. Hashim, M.S.H. Chowdhury, Adaptation of homotopy-perturbation method for numeric-analytic solution of system of ODEs, Phys. Lett. A (2007), doi:10.1016/j.physleta.2007.07.067 · Zbl 1217.81054
[32] I. Hashim, M.S.H. Chowdhury, S. Mawa, On multistage homotopy-perturbation method applied to nonlinear biochemical reaction model, Chaos Solitons Fractals (2007), in press (doi:10.1016/j.chaos.2007.09.009) · Zbl 1210.65149
[33] Sprott, J.C., Chaos and time-series analysis, (2003), Oxford University Press Oxford
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