A numeric–analytic method for approximating the chaotic Chen system M. Mossa Al-sawalha * , M.S.M. Noorani Center for Modelling & Data Analysis, School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia article info Article history: Accepted 20 March 2009 abstract The epitome of this paper centers on the application of the differential transformation method (DTM) the renowned Chen system which is described as a three-dimensional sys- tem of ODEs with quadratic nonlinearities. Numerical comparisons are made between the DTM and the classical fourth-order Runge–Kutta method (RK4). Our work showcases the precision of the DTM as the Chen system transforms from a non-chaotic system to a chaotic one. Since the Lyapunov exponent for this system is much higher compared to other cha- otic systems, we shall highlight the difficulties of the simulations with respect to its accu- racy. We wrap up our investigations to reveal that this direct symbolic–numeric scheme is effective and accurate. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Not one scientist who deals with nonlinear dynamical systems can elude the experience of Chaos, a phenomenon where its theory has been rigorously studied for more than two decades, yet still, as captivating as ever. Many scientists who are interested have struggled to find analytical solutions for these chaotic systems, but due to its complexities, such tasks always meet with a stumbling block. Numerical methods are available to provide approximate solutions but accuracies on its long- term solutions are sometimes questionable. Lately, much attention has been given to analytical techniques such as the Ado- mian decomposition method [1–5], Variational iteration method (VIM) [6–12] and more recently the Homotopy analysis method (HAM) and the Homotopy–perturbation method (HPM) [13–16]. These semi-numerical–analytic methods provides some promising results in solving a wide array of problems with rapid and efficient convergence. However, each of these methods has its own strengths and weaknesses. We note that the ADM and the HAM has some cumbersome calculations of the Adomian polynomials and Homotopy polynomials, respectively. VIM is relatively simple and straightforward to use but one may face longer computational time due to possible exponential coefficients in its iterations. To overcome some of the problems mentioned above, there is a new method called the Differential transformation method (DTM) which was introduced by Zhou [17], who solved linear and nonlinear initial value problems in electric circuit analysis. The DTM [18–21] presents its solution in the form of polynomials. This is different from the traditionally high-ordered Taylor series method, which requires symbolic computation of the necessary derivatives of the data functions. The Taylor series method is known to take long computational time for large orders. The present method reduces the size of computational domain and is applicable to many problems easily. When dealing with nonlinear systems of ordinary differential equations, such as the Chen system, it is often difficult to obtain a closed form of the analytic solution. In the absence of such a solution, the accuracy of the DTM method is then tested against classical numerical methods, such as the Runge–Kutta method (RK4). RK4 has been widely and commonly used for simulating solutions for chaotic systems [22–25]. The Chen system can exhibit both chaotic and non-chaotic behavior for distinct parameter values. As such, the objective of this paper is essentially two fold. First, we shall give a comparison in the case of a fixed time step between the DTM and RK4 for the solution of the chaotic Chen system. And secondly, we look 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.03.096 * Corresponding author. E-mail address: [email protected](M. Mossa Al-sawalha). Chaos, Solitons and Fractals 42 (2009) 1784–1791 Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Transcript
Chaos, Solitons and Fractals 42 (2009) 1784–1791
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals
journal homepage: www.elsevier .com/locate /chaos
A numeric–analytic method for approximating the chaotic Chen system
M. Mossa Al-sawalha *, M.S.M. NooraniCenter for Modelling & Data Analysis, School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia
a r t i c l e i n f o
Article history:Accepted 20 March 2009
0960-0779/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.chaos.2009.03.096
The epitome of this paper centers on the application of the differential transformationmethod (DTM) the renowned Chen system which is described as a three-dimensional sys-tem of ODEs with quadratic nonlinearities. Numerical comparisons are made between theDTM and the classical fourth-order Runge–Kutta method (RK4). Our work showcases theprecision of the DTM as the Chen system transforms from a non-chaotic system to a chaoticone. Since the Lyapunov exponent for this system is much higher compared to other cha-otic systems, we shall highlight the difficulties of the simulations with respect to its accu-racy. We wrap up our investigations to reveal that this direct symbolic–numeric scheme iseffective and accurate.
� 2009 Elsevier Ltd. All rights reserved.
1. Introduction
Not one scientist who deals with nonlinear dynamical systems can elude the experience of Chaos, a phenomenon whereits theory has been rigorously studied for more than two decades, yet still, as captivating as ever. Many scientists who areinterested have struggled to find analytical solutions for these chaotic systems, but due to its complexities, such tasks alwaysmeet with a stumbling block. Numerical methods are available to provide approximate solutions but accuracies on its long-term solutions are sometimes questionable. Lately, much attention has been given to analytical techniques such as the Ado-mian decomposition method [1–5], Variational iteration method (VIM) [6–12] and more recently the Homotopy analysismethod (HAM) and the Homotopy–perturbation method (HPM) [13–16]. These semi-numerical–analytic methods providessome promising results in solving a wide array of problems with rapid and efficient convergence.
However, each of these methods has its own strengths and weaknesses. We note that the ADM and the HAM has somecumbersome calculations of the Adomian polynomials and Homotopy polynomials, respectively. VIM is relatively simple andstraightforward to use but one may face longer computational time due to possible exponential coefficients in its iterations.To overcome some of the problems mentioned above, there is a new method called the Differential transformation method(DTM) which was introduced by Zhou [17], who solved linear and nonlinear initial value problems in electric circuit analysis.The DTM [18–21] presents its solution in the form of polynomials. This is different from the traditionally high-ordered Taylorseries method, which requires symbolic computation of the necessary derivatives of the data functions. The Taylor seriesmethod is known to take long computational time for large orders. The present method reduces the size of computationaldomain and is applicable to many problems easily.
When dealing with nonlinear systems of ordinary differential equations, such as the Chen system, it is often difficult toobtain a closed form of the analytic solution. In the absence of such a solution, the accuracy of the DTM method is then testedagainst classical numerical methods, such as the Runge–Kutta method (RK4). RK4 has been widely and commonly used forsimulating solutions for chaotic systems [22–25]. The Chen system can exhibit both chaotic and non-chaotic behavior fordistinct parameter values. As such, the objective of this paper is essentially two fold. First, we shall give a comparison inthe case of a fixed time step between the DTM and RK4 for the solution of the chaotic Chen system. And secondly, we look
M. Mossa Al-sawalha, M.S.M. Noorani / Chaos, Solitons and Fractals 42 (2009) 1784–1791 1785
into the effect of time steps on the accuracy of the DTM as the Chen system transform from a non-chaotic system to a chaoticone.
2. Differential transformation method
The differential transform method is a semi-numerical–analytic-technique that formalizes the Taylor series in a totallydifferent manner. With this technique, the given differential equation and related initial conditions are transformed intoa recurrence equation that finally leads to the solution of a system of algebraic equations as coefficients of a power seriessolution. This method is useful for obtaining exact and approximate solutions of linear and nonlinear differential equations.There is no need for linearization or perturbations, large computational work and round-off errors are avoided. It has beenused to solve effectively, easily and accurately a large class of linear and nonlinear problems with approximations. The meth-od is well addressed in [26–36].
The basic definitions of differential transformation are introduced as follows:Let xðtÞ be analytic in a domain D and let t ¼ ti represent any point in D. The function xðtÞ is then represented by one
power series whose center is located at ti. The Taylor series expansion function of xðtÞ is of the form:
x tð Þ ¼X1k¼0
t � tið Þk
k!
dkxðtÞdtk
" #t¼ti
; 8t 2 D ð1Þ
The particular case of Eq. (1) when ti ¼ 0 is referred to as the Maclaurin series of xðtÞ and is expressed as:
x tð Þ ¼X1k¼0
tk
k!
dkxðtÞdtk
" #t¼0
; 8t 2 D ð2Þ
From Zhou [17], the differential transform of function xðtÞ is defined as:
XðkÞ � Hk
k!
dkxðtÞdtk
" #t¼0
; k ¼ 0; 1; 2; . . . ;1 ð3Þ
where XðkÞ represents the transformed function (commonly referred to as the T-function) and xðtÞ is the original function.The differential spectrum of XðkÞ is confined within the interval t 2 ½0; H�, where H is a constant.
The differential inverse transform of XðkÞ is defined as follows:
xðtÞ ¼X1k¼0
tH
� �k
XðkÞ ð4Þ
It is clear that the concept of differential transformation is based upon the Taylor series expansion. The original functions aredenoted by lowercase letters, while their transformed functions (i.e., their T-functions) are indicated by the correspondinguppercase letter. The values of function XðkÞ at values of argument k are referred to as discretes, i.e., Xð0Þ is known as the zerodiscrete, Xð1Þ as the first discrete etc. The more discretes available, the more precise it is possible to restore the unknownfunction. The function xðtÞ consists of the T-function XðkÞ, and its value is given by the sum of the T-function with ðt=HÞk
as its coefficient. In real applications, at the right choice of constant H, the larger values of argument k the discretes of spec-trum reduce rapidly. The function xðtÞ is expressed by a finite series and Eq. (3) can be written as:
xðtÞ ¼Xn
k¼0
tH
� �k
XðkÞ ð5Þ
Eq. (5) implies that the value ofP1
k¼nþ1tH
� �kXðkÞ is negligible.
3. The Chen system
The Chen dynamical system, first found by Chen and Ueta [37], is defined as:
dxdt¼aðy� xÞ; ð6Þ
dydt¼ðc � aÞx� xzþ cy; ð7Þ
dzdt¼xy� bz; ð8Þ
where x, y and z are state variables and a, b and c are positive parameters. Bifurcation studies shows that with the parametersa ¼ 35 and c ¼ 28, system (6)–(8) exhibits non-chaotic behavior and chaotic behavior when b ¼ 12 and b ¼ 3, respectively,(see [38]). For other aspects of this dynamical system, see for example, Yassen [23], Deng and Li [39] and Plienpanich et al.[40].
1786 M. Mossa Al-sawalha, M.S.M. Noorani / Chaos, Solitons and Fractals 42 (2009) 1784–1791
By taking the differential transform of Eqs. (6)–(8), with respect to time t, gives:
kþ 1H
Xðkþ 1Þ ¼ � aXðkÞ þ aYðkÞ; ð9Þ
kþ 1H
Yðkþ 1Þ ¼ðc � aÞXðkÞ �Xk
‘¼0
Xð‘ÞZðk� ‘Þ þ cYðkÞ; ð10Þ
kþ 1H
Zðkþ 1Þ ¼Xk
‘¼0
Xð‘ÞYðk� ‘Þ � bZðkÞ: ð11Þ
where XðkÞ, YðkÞ and ZðkÞ are the differential transformations of the corresponding functions xðtÞ; yðtÞ and zðtÞ, respectively.Eqs. (9)–(11) can be rewritten in the following forms:
Xðkþ 1Þ ¼ Hkþ 1
�aXðkÞ þ aYðkÞ½ �; ð12Þ
Yðkþ 1Þ ¼ Hkþ 1
ðc � aÞXðkÞ �Xk
l¼0
XðlÞZðk� lÞ þ cYðkÞ" #
; ð13Þ
Zðkþ 1Þ ¼ Hkþ 1
Xk
‘¼0
Xð‘ÞYðk� ‘Þ � bZðkÞ" #
: ð14Þ
where the initial conditions are given by Xð0Þ ¼ �10, Yð0Þ ¼ 0 and Zð0Þ ¼ 37. The difference equations presented in Eqs.(12)–(14) describe the Chen system, from a process of inverse differential transformation, it can be shown that the solutionsof each sub-domain take nþ 1 terms for the power series like Eq. (5), i.e.,
xiðtÞ ¼Xn
k¼0
tHi
� �k
XiðkÞ; 0 6 t 6 Hi; ð15Þ
yiðtÞ ¼Xn
k¼0
tHi
� �k
YiðkÞ; 0 6 t 6 Hi; ð16Þ
ziðtÞ ¼Xn
k¼0
tHi
� �k
ZiðkÞ; 0 6 t 6 Hi: ð17Þ
where k ¼ 0; 1; 2; . . . ; n represents the number of terms of the power series, i ¼ 0; 1; 2; . . . expresses the ith sub-domain andHi is the sub-domain interval.
4. Results and discussion
The accuracy of the DTM is demonstrated against the Maple’s built-in fourth-order.Runge–Kutta procedure RK4 for the solutions of both non-chaotic and chaotic systems. The DTM algorithm is coded in the
computer algebra package Maple where its environment variable Digits, which controls the number of significant digits, isset to 35 in all the calculations done in this paper. We also fixed the values of the parameters such that a ¼ 35, c ¼ 28 andb ¼ 12 (for non-chaotic) and with b ¼ 3 (for chaotic). The initial conditions are set to be xð0Þ ¼ �10, yð0Þ ¼ 0 and zð0Þ ¼ 37.The simulations done in this paper are for the time span of t 2 ½0; 7� (refer [3,13]). Based on our preliminary calculations, wehave decided to use 15 terms in the DTM series of solutions.
4.1. Non-chaotic solutions
First, we shall consider the non-chaotic case where a ¼ 35, b ¼ 12 and c ¼ 28. The following approximate solutions areobtained from Eqs. (15)–(17):
xðtÞ ¼ �10þ 350t þ 1575t2 � 1867253
t3 þ 756647512
t4 � 73548692414023124997
t5 � . . . :
yðtÞ ¼ 440t � 3760t2 þ 9820t3 þ 8829503
t4 � 128000173
t5 þ . . . :
zðtÞ ¼ 37� 444t þ 4864t2 � 2499683
t3 þ 430268t4 þ 276512845
t5 � . . . :
The accuracy of RK4 has to be determined first for the solution of (6)–(8) at different time steps before comparing with theDTM. The results of this analysis can be found in Table 1 in [3]. The maximum difference between the RK4 solutions on timesteps Dt ¼ 0:001 and Dt ¼ 0:0001 is of the order of magnitude of 10�5. This level of accuracy is matched by the 15-term DTMsolutions on the smaller time step Dt ¼ 0:01, as shown in Table 1. We move a step further to investigate its accuracy at asmaller time step, Dt ¼ 0:001, where comparisons are made in Table 2 and visually shown in Fig. 1. It can be observed from
Fig. 1. A non-chaotic case with parameters a ¼ 35, b ¼ 12, c ¼ 28: (left) 15-term DTM (Dt ¼ 0:001) vs RK4 (Dt ¼ 0:001) and (right) 15-term DTM(Dt ¼ 0:001) vs RK4 (Dt ¼ 0:0001) for t 2 ½0; 7�.
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1788 M. Mossa Al-sawalha, M.S.M. Noorani / Chaos, Solitons and Fractals 42 (2009) 1784–1791
these figures that we could increase the accuracy of the RK4 solutions by decreasing the time step, and this brought the DTMsolutions and the RK4 solutions closer to each other up to a maximum difference of order j 10�8 j.
4.2. Chaotic solutions
System (6)–(8) with a ¼ 35, b ¼ 3 and c ¼ 28 will exhibit chaotic behaviour, [37,38]. In this case the following approxi-mate solutions are obtain from Eqs. (15)–(17):
Since the system is chaotic, we will expect solutions which are highly sensitive to time step. Referring to Table 3 presented in[3], we could see that the maximum difference between the RK4 solutions on time steps Dt ¼ 0:001 and t ¼ 0:0001 is of theorder j 10�2 j, which is much larger than in the non-chaotic case. In the previous section we have found that for the non-cha-otic case, the 15-term DTM on time step Dt ¼ 0:001 out performs the RK4 on a much smaller step size Dt ¼ 0:0001. From theleft figures of Fig. 2 we demonstrate that both the DTM (on Dt ¼ 0:001) and RK4 (on Dt ¼ 0:01) solutions begin to deviategreatly from each other when t is roughly greater than 4 (see also Table 2). We also show in the right figures of Fig. 2 thatthe two solutions on the same time step Dt ¼ 0:001 seem to coincide and overlapping each other on the curves.
g. 2. A chaotic case with parameters a ¼ 35, b ¼ 3, c ¼ 28: 15-term DTM (Dt ¼ 0:001) vs (left) RK4 (Dt ¼ 0:01) and (right) RK4 (Dt ¼ 0:001).
Fig. 3. A chaotic case with parameters a ¼ 35, b ¼ 3, c ¼ 28: (left) 15-term DTM (Dt ¼ 0:001) vs RK4 (Dt ¼ 0:001) and (right) 15-term DTM (Dt ¼ 0:001) vsRK4 (Dt ¼ 0:0001) for t 2 ½0; 7�.
M. Mossa Al-sawalha, M.S.M. Noorani / Chaos, Solitons and Fractals 42 (2009) 1784–1791 1789
Further comparisons are shown in Fig. 3. It can be observed from Fig. 3 that increasing the accuracy of the RK4 solutionsby decreasing the time step brings the DTM solutions and the RK4 solutions closer to each other, but up to a maximum dif-ference between the two of order j 10�6 j, which is evidently bigger than in the non-chaotic case. In Fig. 4 we reproduce thewell-known x�y, x�z, y�z and x�y�z phase portraits of the chaotic Chen system using the 15-term DTM solutions onDt ¼ 0:001.
4.3. Comparing with the Lorenz system
When it comes to chaotic systems, its level of sensitivity lies in the Lyapunov exponent number. The Chen systemexhibits a higher Lyapunov exponent compared to the legendary Lorenz system. This is one of the main reasons forthe care taken in choosing the computational parameters. We immediately note that the time span used in both studiesare significantly different. For the Lorenz system [2,18], the time span used was t 2 ½0; 20�, whereas the current Chensystem only uses t 2 ½0; 7�. The rapid build-up of errors due to sensitive dependence on initial conditions (i.e., due tohigh Lyapunov exponent) casts doubts on the accuracy of the solutions, thus our rationale to use a shorter time spanfor the highly chaotic Chen system [4]. We are not concerned about the time span used for non-chaotic cases as it isnot an issue at all. However, another matter of importance is the choice of time steps. Comparing the results from Table3 [3] and the results from Table 3 of [4], we observed that the RK4 solutions on time step Dt ¼ 0:001 for the Chen sys-tem are much less accurate than RK4 for the Lorenz system. We deduced that this trend will apply for the DTM too.
5. Conclusions
This investigation was carried out using the DTM to approximate the solutions of the well-known Chen system. Compar-isons between the DTM solutions and RK4’s numerical solutions were made. For both non-chaotic and chaotic cases studied,
Fig. 4. Phase portraits using 15-term DTM on Dt ¼ 0:001 for a ¼ 35, b ¼ 3 and c ¼ 28.
1790 M. Mossa Al-sawalha, M.S.M. Noorani / Chaos, Solitons and Fractals 42 (2009) 1784–1791
we found that the 15-term DTM solutions on a larger time step achieved comparable accuracy compared with the RK4 solu-tions on a much smaller time step. However, we note that the orders of magnitude of the errors in the non-chaotic and cha-otic cases differ (cf. Figs. 1 and 3). The DTM’s significant advantages are noted in this study. This method provides an iterativeprocedure to calculate the spectrum of exact solution with straightforward applications. Therefore, it is not necessary to car-ry out complicated symbolic computation. It has been shown that this effective technique could yield reasonably accurateapproximate solutions with great ease.
Acknowledgement
This work is financially supported by the Malaysian Ministry of Higher Education Grant: UKM-ST-06-FRGS0008-2008.
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