On accuracy of Adomian decomposition method for hyperchaotic Rössler system

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Chaos, Solitons and Fractals 40 (2009) 1801–1807

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On accuracy of Adomian decomposition method forhyperchaotic Rossler system

M. Mossa Al-Sawalha, M.S.M. Noorani *, I. Hashim

School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

Accepted 17 September 2007

Abstract

The aim of this paper is to investigate the accuracy of the Adomian decomposition method for solving the hyper-chaotic Rossler system, which is a four-dimensional system of ODEs with quadratic nonlinearities. Comparisonsbetween the decomposition solutions and the fourth-order Runge–Kutta (RK4) solutions are made. It is found thatthe comparable accuracy of the ADM with the RK4 for the chaotic Lorenz system does not carry to the hyperchaoticRossler system.� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

The Adomain decomposition method (ADM) [1] is a simple and powerful method for solving systems of nonlineardifferential equations. It has recently been identified as being efficient for certain nonlinear problems and comparableto the fourth-order Runge–Kutta (RK4) numerical scheme. No transformation, linearization, perturbation or discretiza-tion is required for the ADM and it yields an analytical solution in the form of a rapidly convergent infinite power serieswith easily computable terms.

The accuracy of the ADM method is usually examined with the conventional numerical method of Runge–Kutta orits variants. Moreover, since the global solution (via ADM or any numerical method) for nonlinear systems is a bit farfetched to hope for, it is crucial for researchers to understand the effect of time step on such methods. Vadasz and Olek[2] made numerical comparisons between the ADM and the variable timestep Runge–Kutta–Verner method for solvingthe extended Lorenz system. Guellal et al. [3] compared the ADM with RK4 for solving the classical chaotic Lorenzsystem. Hashim et al. [4] presented a more comprehensive comparative study of ADM and RK4 for the classical chaoticand non-chaotic Lorenz systems. In [5], Abdulaziz et al. gave further confirmation on the high accuracy of the ADMcompared to the variable timestep seventh- and eighth-order Runge–Kutta method. The accuracy assessment of ADMfor the Chen system was undertaken by Noorani et al. [6]. Ghosh et al. [7] demonstrated the high accuracy of ADM forstrongly nonlinear and chaotic oscillators. Cafagna and Grassi [8] applied ADM to achieve hyperchaos synchronizationof two coupled Chua’s circuits.

0960-0779/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2007.09.062

* Corresponding author. Tel.: +60 3 8921 3706; fax: +60 3 8925 4519.E-mail address: [email protected] (M.S.M. Noorani).

1802 M.M. Al-Sawalha et al. / Chaos, Solitons and Fractals 40 (2009) 1801–1807

Again, we are interested in the accuracy of the ADM for nonlinear systems of ODEs capable of exhibiting chaoticbehavior. The system which is of interest to us in this paper is the hyperchaotic Rossler system [9],

TableA dete

t

2468101214161820

dxdt¼ �y � z; ð1Þ

dydt¼ xþ ay þ w; ð2Þ

dzdt¼ bþ xz; ð3Þ

dwdt¼ �czþ dw; ð4Þ

where x; y; z;w are the state variables and a; b; c; d are positive constants. This system exhibits a hyperchaotic behaviorwhen a ¼ 0:25; b ¼ 3; c ¼ 0:5 and d ¼ 0:05. This system is similar to the Lorenz system in the sense that both have non-linear terms in the form of products of two of the dependent variables (i.e. quadratic nonlinearities). Nonetheless, it isdifferent from the Lorenz system as it has two positive Lyapunov exponents [10], k1 ¼ 0:109 and k2 ¼ 0:024. TheLyapunov exponent for the Lorenz system is 0.9056.

In this paper, the ADM is again treated as an algorithm for approximating the solutions of the hyperchaotic Rosslersystem in a sequence of time intervals. Comparison between the decomposition solutions and the RK4 solutions shall bemade. It is demonstrated that the comparable accuracy of ADM (as compared to RK4) for the chaotic Lorenz systemdoes not carry to the hyperchaotic Rossler system.

2. Adomian decomposition solution

Following [4,6], the explicit analytical solution via ADM for the four-dimensional hyperchaotic Rossler system(1)–(4) can be written as:

x ¼X1

m¼0

amðt � t�Þm

m!; ð5Þ

y ¼X1

m¼0

bmðt � t�Þm

m!; ð6Þ

z ¼X1

m¼0

cmðt � t�Þm

m!; ð7Þ

w ¼X1

m¼0

dmðt � t�Þm

m!; ð8Þ

1rmination of the accuracy of RK4

D ¼ jRK40:01 �RK40:001j D ¼ jRK40:001 �RK40:0001jDx Dy Dz Dw Dx Dy Dz Dw

1.090E�09 3.174E�10 1.094E�09 1.131E�10 6.800E�13 5.680E�13 9.520E�14 4.800E�131.025E�09 2.609E�09 2.214E�09 1.848E�10 1.070E�12 3.080E�13 1.871E�13 7.800E�133.443E�09 3.196E�09 4.358E�10 1.106E�10 7.70E�13 2.620E�13 3.330E�14 1.180E�128.014E�09 3.929E�09 4.033E�09 2.412E�10 1.000E�12 8.400E�13 3.736E�13 1.170E�121.320E�09 1.324E�08 3.971E�09 5.173E�10 2.380E�12 6.020E�13 3.137E�13 8.300E�131.749E�08 7.631E�09 3.235E�09 3.805E�10 1.410E�12 4.400E�13 2.632E�13 7.400E�132.423E�08 2.133E�08 1.162E�08 5.833E�10 1.680E�12 2.100E�12 1.034E�12 6.000E�141.260E�08 3.402E�08 9.461E�09 1.320E�08 5.280E�13 2.499E�12 1.743E�12 1.780E�122.087E�08 3.787E�08 1.564E�08 1.267E�08 2.200E�13 3.690E�12 1.187E�12 1.390E�128.441E�08 3.390E�08 2.298E�08 1.409E�08 7.130E�12 2.250E�12 1.848E�12 1.580E�12

Fig. 1. (Left) RK4(h ¼ 0:01) vs RK4 (h ¼ 0:001) and (right) RK4 (h ¼ 0:001) vs RK4 (h ¼ 0:0001) for t 2 ½0; 20�.

M.M. Al-Sawalha et al. / Chaos, Solitons and Fractals 40 (2009) 1801–1807 1803

1804 M.M. Al-Sawalha et al. / Chaos, Solitons and Fractals 40 (2009) 1801–1807

where the coefficients are given by the recurrence relations,

TableDiffere

t

2468101214161820

a0 ¼ xðt�Þ; b0 ¼ yðt�Þ; c0 ¼ zðt�Þ; d0 ¼ wðt�Þ; ð9Þam ¼ �bm�1 � cm�1; m P 1; ð10Þbm ¼ am�1 þ abm�1 þ dm�1; m P 1; ð11Þ

cm ¼ bþ ðm� 1Þ!Xm�1

k¼0

akcm�k�1

k!ðm� k � 1Þ! ; m P 1; ð12Þ

dm ¼ �ccm�1 þ ddm�1; m P 1: ð13Þ

As first hinted in [1] and applied in [2–6], we treat the ADM as an algorithm for approximating the dynamical responsein a sequence of time intervals (i.e. time step), ½0; t1Þ; ½t1; t2Þ; . . . ; ½tm�1; T Þ, such that the initial condition in ½t�; tmþ1Þ istaken to be the condition at t�. For practical computations, a finite number of terms in the series (5)–(8) are used ina time step procedure just outlined.

3. Results and discussions

We shall demonstrate the accuracy of the ADM against RK4. The Adomian algorithm is coded in the computeralgebra package Maple with the Maple environment variable Digits controlling the number of significant digitsset to 16 in all the calculations done in this paper. We also fix the values of the parameters a ¼ 0:25;b ¼ 3; c ¼ 0:5 and d ¼ 0:05, and take the initial conditions xð0Þ ¼ �20; yð0Þ ¼ 0; zð0Þ ¼ 0 and wð0Þ ¼ 15 as consid-ered in [9]. The simulations done in this paper are for the time span t 2 ½0; 20�. Based on our preliminary calcula-tions, we decided to use 10 terms in the Adomian decomposition series solutions. We note that increasing thenumber of terms improves the accuracy of the ADM solutions, but at the expense of increased computationalefforts.

First we determine the accuracy of RK4 for the solution of (1)–(4) for different sizes of timestep h. From the resultspresented in Table 1 and Fig. 1 we see that the maximum differences between the RK4 (h ¼ 0:01) and RK4 (h ¼ 0:001)solutions is of the order of magnitude of 10�8 and that for the RK4 (h ¼ 0:001) and RK4 (h ¼ 0:0001) solutions is of theorder of magnitude of 10�12. Hence we can conclude that the RK4 (h ¼ 0:001) is sufficiently accurate for our compar-ison purposes.

We note that the RK4 (h ¼ 0:001) for the hyperchaotic Rossler system is much more accurate that the RK4(h ¼ 0:001) for the chaotic Lorenz system (cf. Tables 1 and 3 of [4]). This could be due to the lower Lyapunov exponentsof the hyperchaotic Rossler system than that of the Lorenz system. This observation leads us to the expectation that theADM for the hyperchaotic Rossler system is more accurate than the ADM for the chaotic Lorenz system. However,this expectation is premature. Comparing the results presented in Table 2 and Table 4 of [4] we observe overall thatthe 10-term ADM (h ¼ 0:001) for the hyperchaotic Rossler system is less accurate than the same method for the chaoticLorenz system (see also Figs. 2–4).

2nces between the 10-term decomposition and RK4 solutions

D ¼ jADM0:01 �RK40:001j D ¼ jADM0:001 �RK40:001jDx Dy Dz Dw Dx Dy Dz Dw

3.360E�04 1.993E�03 1.170E�03 8.921E�04 3.334E�05 1.989E�04 1.144E�04 8.900E�053.861E�03 4.301E�03 1.206E�03 2.440E�03 3.852E�04 4.286E�04 1.179E�04 2.433E�046.942E�03 9.320E�04 5.572E�04 3.503E�03 6.918E�04 9.321E�05 5.328E�05 3.493E�041.485E�04 2.037E�03 8.445E�04 4.490E�03 1.482E�05 2.026E�04 8.195E�05 4.477E�044.509E�03 9.661E�03 1.556E�03 6.496E�03 4.502E�04 9.631E�04 1.529E�04 6.477E�042.009E�02 1.562E�03 3.960E�04 7.966E�03 2.003E�03 1.550E�04 3.722E�05 7.943E�042.175E�03 1.337E�02 5.315E�04 9.192E�03 2.163E�04 1.332E�03 5.073E�05 9.165E�044.056E�03 1.786E�02 4.769E�03 1.252E�02 4.043E�04 1.781E�03 4.762E�04 1.249E�034.736E�02 1.900E�02 3.091E�04 1.539E�02 4.725E�03 1.896E�03 2.857E�05 1.536E�031.763E�02 4.295E�02 3.181E�04 1.726E�02 1.760E�03 4.285E�03 2.946E�05 1.722E�03

Fig. 2. (Left) 10-term ADM (h ¼ 0:01) vs RK4 (h ¼ 0:001) and (right) 10-term ADM (h ¼ 0:001) vs RK4 (h ¼ 0:001) for t 2 ½0; 20�.

M.M. Al-Sawalha et al. / Chaos, Solitons and Fractals 40 (2009) 1801–1807 1805

Fig. 3. Differences between the 10-term ADM (h ¼ 0:001) and RK4 (h ¼ 0:001) of the hyperchaotic Rossler system.

Fig. 4. Differences between the 10-term ADM (h ¼ 0:001) and RK4 (h ¼ 0:001) for the chaotic Lorenz system.

1806 M.M. Al-Sawalha et al. / Chaos, Solitons and Fractals 40 (2009) 1801–1807

4. Concluding remarks

The ADM was treated as an algorithm for approximating the solutions of the hyperchaotic Rossler system in asequence of time intervals. The ADM was demonstrated to be of comparable accuracy with the RK4 on the same time-step h ¼ 0:001 for the chaotic Lorenz system by Hashim et al. [4]. However, in this paper it is demonstrated that theADM is of less accuracy compared to the RK4 on the same timestep h ¼ 0:001 even if the hyperchaotic Rossler systemhas small Lyapunov exponents.

Acknowledgements

The authors would like to acknowledge the financial supports received from the Academy of Sciences Malaysia un-der the SAGA Grant No. P24c (STGL-011-2006) and Ministry of Science, Technology and Innovation, Malaysia underthe Science Fund 04-01-02-SF0177.

M.M. Al-Sawalha et al. / Chaos, Solitons and Fractals 40 (2009) 1801–1807 1807

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