×

zbMATH — the first resource for mathematics

Comparing numerical methods for the solutions of the Chen system. (English) Zbl 1131.65101
Summary: The Adomian decomposition method (ADM) is applied to the Chen system [cf. G. Chen and T. Ueta, Int. 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. The ADM yields an analytical solution in terms of a rapidly convergent infinite power series with easily computable terms. Comparisons between the decomposition solutions and the classical fourth-order Runge-Kutta numerical solutions are made. In particular we look at the accuracy of the ADM as the Chen system changes from a non-chaotic system to a chaotic one. To highlight some computational difficulties due to a high Lyapunov exponent, a comparison with the Lorenz system is given.

MSC:
65P20 Numerical chaos
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
37M05 Simulation of dynamical systems
Citations:
Zbl 0962.37013
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adomian, G., Nonlinear stochastics systems theory and application to physics, (1989), Kluwer Dordrecht
[2] Adomian, G., A review of the decomposition method in applied mathematics, J math anal appl, 135, 501-544, (1988) · Zbl 0671.34053
[3] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Boston · Zbl 0802.65122
[4] Adomian, G., Analytic solution of nonlinear integral equations of Hammerstein type, Appl math lett, 11, 127-130, (1998) · Zbl 0933.65148
[5] Biazar, J.; Babolian, E.; Islam, R., Solution of the system of ordinary differential equations by Adomian decomposition method, Appl math comput, 147, 713-719, (2004) · Zbl 1034.65053
[6] Biazar, J.; Montazeri, R., A computational method for solution of the prey and predator problem, Appl math comput, 163, 841-847, (2005) · Zbl 1060.65612
[7] Chen, G.; Ueta, T., Yet another chaotic attractor, Int J bifurcat chaos, 9, 7, 1465-1466, (1999) · Zbl 0962.37013
[8] Deng, W.; Li, C., Synchronization of chaotic fractional Chen system, J phys soc jpn, 74, 1645-1648, (2005) · Zbl 1080.34537
[9] Evans, D.J.; Raslan, K.R., The Adomian decomposition method for solving delay differential equations, Int J comput math, 82, 49-54, (2005) · Zbl 1069.65074
[10] Guellal, S.; Grimalt, P.; Cherruault, Y., Numerical study of lorenz’s equation by the Adomian method, Comput math appl, 33, 25-29, (1997) · Zbl 0869.65044
[11] Hashim I. Adomian decomposition method for solving BVPs for fourth-order integro-differential equations. J Comp Appl Math, in press, doi:10.1016/j.cam.2005.05.034. · Zbl 1093.65122
[12] Hashim I. Comments on: A new algorithm for solving classical Blasius equation by L. Wang. Appl Math Comput, in press, doi:10.1016/j.amc.2005.10.016. · Zbl 1331.65103
[13] Hashim, I.; Noorani, M.S.M.; Ahmad, R.; Bakar, S.A.; Ismail, E.S.I.; Zakaria, A.M., Accuracy of the Adomian decomposition method applied to the Lorenz system, Chaos, solitons & fractals, 28, 1149-1158, (2006) · Zbl 1096.65066
[14] Kaya, D.; El-Sayed, S.M., Numerical soliton-like solutions of the potential Kadomtsev-Petviashvili equation by the decomposition method, Phys lett A, 320, 192-199, (2003) · Zbl 1065.35219
[15] 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
[16] Olek, S., An accurate solution to the multispecies Lotka-Volterra equations, SIAM rev., 36, 3, 480-488, (1994) · Zbl 0802.92018
[17] Park, J.H., Chaos synchronization between two different chaotic dynamical systems, Chaos, solitons & fractals, 27, 549-554, (2006) · Zbl 1102.37304
[18] Park, J.H., Chaos, synchronization of nonlinear Bloch equations, Chaos, solitons & fractals, 27, 357-361, (2006) · Zbl 1091.93029
[19] 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
[20] Répaci, A., Nonlinear dynamical systems: on the accuracy of adomian’s decomposition method, Appl math lett, 3, 4, 35-39, (1990) · Zbl 0719.93041
[21] Shawagfeh, N.; Adomian, G., Non-perturbative analytical solution of the general Lotka-Volterra three-species system, Appl math comput, 76, 251-266, (1996) · Zbl 0846.65034
[22] Shawagfeh, N.; Kaya, D., Comparing numerical methods for the solutions of systems of ordinary differential equations, Appl math lett, 17, 323-328, (2004) · Zbl 1061.65062
[23] Sprott, J.C., Chaos and time-series analysis, (2003), Oxford University Press Oxford · Zbl 1012.37001
[24] Ueta, T.; Chen, G., Bifurcation analysis of chen’s equation, Int J bifurcat chaos, 8, 1917-1931, (2000) · Zbl 1090.37531
[25] Vadasz, P.; Olek, S., Convergence and accuracy of adomian’s decomposition method for the solution of Lorenz equation, Int J heat mass transfer, 43, 1715-1734, (2000) · Zbl 1015.76075
[26] Yassen, M.T., The optimal control of Chen chaotic dynamical system, Appl math comput, 131, 171-180, (2002) · Zbl 1042.49002
[27] Yassen, M.T., Chaos control of Chen chaotic dynamical system, Chaos, solitons & fractals, 15, 271-283, (2003) · Zbl 1038.37029
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.