Application of Multistage Homotopy-perturbation Method for the

ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.13(2012) No.1,pp.3-14

Application of Multistage Homotopy-perturbation Method for the Solutions of the Chaotic Fractional Order Systems Sha Wang ∗ , Yongguang Yu Department of Mathematics, Beijing Jiaotong University, Beijing 100044, P.R.China (Received 30 March 2011, accepted 28 November 2011)

Abstract:In this paper, the multistage homotopy-perturbation method is extended to solve the chaotic fractional order systems. The multistage homotopy-perturbation method is only a simple modification of the standard homotopy-perturbation method, in which it is treated as an algorithm in a sequence of intervals for finding accurate approximate analytical solutions. The fractional derivatives are described in the Caputo sense. The solutions are obtained in the form of rapidly convergent infinite series with easily computable terms. Numerical results reveal that the multistage method is a promising tool for the chaotic and hyperchaotic fractional order systems. Keywords: chaos; Hyperchaos; fractional order systems; homotopy-perturbation method; multistage method

1

Introduction

Over the last decades, fractional order differential equations (FODEs) have been used to describe a variety of systems in interdisciplinary fields, such as viscoelasticity, biology, physiology, medicine, hydraulics geology and engineering [1–3]. Based on the extension of applications of FODEs, the chaotic fractional order systems become a new topic due to its potential applications especially in secure communication, encryption and control processing [3–5]. Now, it is demonstrated that many fractional order differential systems behave chaotically or hyperchaotically, like the fractional order Chua circuit [6], the fractional order Duffing system [7], the fractional order Arneodo system [8], the fractional order Rössler system [9] and the fractional order unified system [10]. For better understanding the dynamic behavior of a chaotic fractional order system, the solution of the chaotic fractional order system is much involved. In general, it is difficult to obtain the exact solution for a nonlinear FODEs. Finding accurate and efficient methods for solving FODEs has been an active research undertaking. Some analytical and numerical methods have been proposed for the solutions of FODEs. The classical approaches include Laplace transform method, Mellin transform method, fractional Green’s function method and power series method [2]. Recently, variation iteration method (VIM) [11–14], Adomian decomposition method (ADM) [15–17] and homotopy perturbation method (HPM) [18–25] have been applied for the numeric-analytic solutions of FODEs. However, one notable difficulty inherent in ADM is the cumbersome calculation of Adomian polynomials. The VIM and HPM are first provided by Prof. He in 1998 and 1999, respectively [11, 18]. Both of the methods overcome the disadvantages of the ADM. And they are no need to discretization of the variables and no requirement of large computer memory. The HPM is widely used for the solutions of the nonlinear integer-order equations, such as Blasius equation [26], quadratic Riccati differential equation [27], Thomas-Fermi equation [28], Fisher’s equation [29] and so on. In [30], Hashim points that the HPM for the ordinary differential equations is only valid in a short time and the advised multistage homotopy-perturbation method (MHPM) can overcome this lack of global convergence. Moreover, the MHPM is successfully used to solve the integer order chaotic and hyperchaotic systems [31–34]. Very recently, the applicability of the HPM is extended to fractional quadratic Riccati differential equation [20], fractional KdV-Burgers equation [21], fractional diffusion and wave equations [22], the spacetime fractional advection-dispersion equation [23] and a class of FODEs [24]. Nevertheless, the HPM and the MHPM have not been applied to solve the chaotic fractional order systems. ∗ Corresponding

author.

E-mail address: [email protected] c Copyright⃝World Academic Press, World Academic Union IJNS.2012.02.15/569

4

International Journal of Nonlinear Science, Vol.13(2012), No.1, pp. 3-14

In the literatures of fractional chaos field, two approximation methods have been advised for the numerical solutions of the fractional order systems. One method is based on the approximation of the fractional order system behavior in the frequency domain and time domain [35, 36]. The other method is the well-known predictor-correctors scheme [36–38]. According to the theory of factional calculus, our concern in this work is to extend the MHPM to consider the approximate numeric-analytic solutions of the chaotic fractional order systems. Compared with the HPM, the MHPM is a very effective and simple method for the accurate approximate solutions of the chaotic and hyperchaotic fractional order systems for a long time. The paper is organized as follows. In Section 2, some definitions and properties of fractional calculus are introduced. Section 3 is devoted to describe the standard HPM and the MHPM. Simulations are achieved in Section 4. Numerical comparisons with the HPM show that the MHPM is a simple, yet powerful method to give the approximate solutions for the chaotic and hyperchaotic fractional order system. Finally, conclusion is presented in Section 5.

2

Basic definition and preliminary

We give some basic definitions and properties of fractional calculus which are used further in this letter [2]. Definition 1 A real function h(t), t > 0, is said to be in the space Cµ , µ ∈ R, if there exists a real number p(> µ), such that h(t) = tp h1 (t), where h1 (t) ∈ C[0, ∞), and it is said to be in the space Cµn if and only if h(n) ∈ Cµ , n ∈ N . Definition 2 The Riemann-Liouville fractional integral operator (a Jtα ) of order α ≥ 0 of a function h ∈ Cµ , µ ≥ −1, is defined as α a Jt h(t) 0 a Jt h(t)

1 = Γ(α)

∫

t

(t − τ )α−1 h(τ )dτ

(α > 0),

a

= h(t),

where t ≥ a ≥ 0, Γ(·) is the well-known Gamma function. Some of the properties of the operator a Jtα , which will be needed here, are given as follows. For h ∈ Cµ , µ ≥ −1, a, α, β ≥ 0, γ ≥ −1: (1) a Jtα a Jtβ h(t) = a Jtα+β h(t), (2) a Jtα a Jtβ h(t) = a Jtβ a Jtα h(t), (3) a Jtα (t − a)γ =

Γ(γ + 1) (t − a)α+γ . Γ(α + γ + 1)

Definition 3 The Caputo fractional derivative (a Dtα ) of h(t) is defined as α a Dt h(t)

=

1 Γ(n − α)

∫

t

(t − τ )n−α−1 h(n) (τ )dτ, a

n for n − 1 < α ≤ n, n ∈ N , t ≥ a ≥ 0 and h ∈ C−1 .

Hence, we have the following properties: (1) If n − 1 < α ≤ n, n ∈ N , and h ∈ Cµn , µ ≥ −1, then a Dtα a Jtα h(t) = h(t), and α α a Jt a Dt h(t)

= h(t) −

n−1 ∑ k=0

h(k) (a)

(t − a)k . k!

n (2) Let h(t) ∈ C−1 , n ∈ N . Then a Dtα h, 0 ≤ α ≤ n is well defined and a Dtα h ∈ C−1 .

IJNS email for contribution: [email protected]

S. Wang,Y. Yu: Application of Multistage Homotopy-perturbation Method for the Solutions of the Chaotic Fractional · · ·

3

5

Analysis of the method

In this section, we extend the application of the MHPM to the fractional order differential equations in the following form: α1 a Dt y1 (t) α2 a Dt y2 (t)

= f1 (t, y1 , y2 , . . . , yn ),

(1)

= f2 (t, y1 , y2 , . . . , yn ),

(2)

= fn (t, y1 , y2 , . . . , yn ),

(3)

.. . αn a Dt yn (t)

subject to the following initial conditions: yi (a) = ci ,

i = 1, 2, . . . , n,

where 0 < αi ≤ 1, t ≥ a ≥ 0, fi is arbitrary linear or nonlinear function.

3.1

Solution by HPM

The HPM is a universal one which can be applied to various kinds of linear and nonlinear equations. It usually needs only a few iterations to lead to the active approximate analytical solutions for a given system. In view of the HPM [18, 19], we construct a homotopy for the Eqs. (1)-(3) which satisfies the following relations: αi a Dt yi (t)

= pfi (t, y1 , y2 , . . . , yn ),

(4)

where i = 1, 2, . . . , n, p ∈ [0, 1] is an embedding parameter. When p = 0, Eq. (4) becomes linear equation a Dtαi yi = 0, and when p = 1, Eq. (4) turns out to be the original equation in Eqs. (1)-(3). The basic assumption is that the solution of Eq. (4) can be expanded as a power series in p yi (t) = yi0 + pyi1 + p2 yi2 + p3 yi3 + · · · ,

(5)

yi0 (a) = yi (a) = ci , yik (a) = 0, k = 1, 2, 3, . . . ,

(6)

and the initial conditions are taken as

where yij (t), j = 0, 1, 2, . . . are the functions to be determined later. Substituting Eq. (5) into Eq. (4), collecting the terms of the same powers of p, we obtain p0 p1 p2 p3

: : : :

αi a Dt yi0 αi a Dt yi1 αi a Dt yi2 αi a Dt yi3

= 0, = fi1 (t, y10 , y20 , . . . , yn0 ), = fi2 (t, y10 , y20 , . . . , yn0 , y11 , y21 , . . . , yn1 ), = fi3 (t, y10 , y20 , . . . , yn0 , y11 , y21 , . . . , yn1 , y12 , y22 , . . . , yn2 ),

(7)

.. .

where the functions fi1 , fi2 , . . . satisfy the following equation: fi (t, y10 + py11 + p2 y12 + · · · , . . . , yn0 + pyn1 + p2 yn2 + · · · ) = fi1 (t, y10 , y20 , . . . , yn0 ) + pfi2 (t, y10 , y20 , . . . , yn0 , y11 , y21 , . . . , yn1 ) +p2 fi3 (t, y10 , y20 , . . . , yn0 , y11 , y21 , . . . , yn1 , y12 , y22 , . . . , yn2 ) + · · · .

(8)

Applying the integral operator a Jtαi (·) on both sides of the above fractional order equations in (7), with considering the initial conditions (6), by using the properties of the Caputo fractional derivative, we can determine the ∑∞unknown functions yij (t). By setting p = 1 in (5), the HPM series solutions to Eqs. (1)-(3) are given as yi (t) = j=0 yij (t), i = 1, 2, . . . , n. The asymptotic behavior of the series was discussed in Refs. [39, 40]. The convergence of the series was illustrated in [41]. The N -term approximation of the HPM series solution can be expressed as yi (t) ≈ ΦiN (t) =

N −1 ∑

yij (t),

i = 1, 2, . . . , n.

j=0

IJNS homepage: http://www.nonlinearscience.org.uk/

(9)

6

3.2


Solution by MHPM

The approximate solutions (9) are generally, as shall be shown in the numerical experiments of this paper, not valid for large t. The HPM is treated as an algorithm in a sequence of intervals for finding accurate approximate solution to the Eqs. (1)-(3). The modified HPM, i.e. the MHPM, can give the valid solutions for a long time. The time interval [a, t) can be divided into a sequence of subintervals [t0 , t1 ), [t1 , t2 ), . . ., [tj−1 , tj ), in which t0 = a, tj = t. Without loss of generality, the subintervals can be chosen as the same length △t, i.e. △t = tl − tl−1 , (l = 1, 2, · · · , j). Furthermore, the equations in (7) can be solved by the HPM in every sequential interval [tl−1 , tl ), l = 1, 2, · · · , j, choosing the initial approximations as yi0 (t∗ ) = yi (t∗ ) = c∗i , yik (t∗ ) = 0, i = 1, 2, . . . , n, k = 1, 2, 3, . . . , where t∗ is the left-end point of each subinterval. But, in general, we only have the initial values at the point t∗ = t0 = a. A simple way to obtain the other necessary values could be by means of the previous N -term approximate solutions ΦiN (t), i = 1, 2, · · · , n of the preceding subinterval [tl−2 , tl−1 ), (l = 2, 3, · · · , j), i.e. ∗ ∗ yi0 (t ) = ΦiN (t∗ ).

Finally, the unknown functions yij (t), i = 1, 2, . . . , n, j = 0, 1, 2, . . . can be easily obtained by the fractional integral operator ∫ t 1 αi (t − τ )αi −1 yij (τ )dτ. t∗ Jt yij (t) = Γ(αi ) t∗

4

Numerical experiments

In order to manifest the advantages and the accuracy of the MHPM, we have applied the MHPM and the HPM to the chaotic factional order Chen system and the hyperchaotic fractional order Lorenz system respectively. The Maple software is used and its environment variable Digits controlling the number of significant digits is set to 35 in all the calculations done in this paper.

4.1

MHPM usage in the chaotic fractional order Chen system

The fractional order Chen system is described as α1 0 Dt x(t) α2 0 Dt y(t) α3 0 Dt z(t)

= a(y − x), = (c − a)x − xz + cy, = xy − bz,

(10)

subject to the initial conditions x(0) = c1 , y(0) = c2 , z(0) = c3 , where 0 < αi < 1, i = 1, 2, 3, x, y, z are state vectors, a, b, c are positive parameters. According to the HPM, we construct a homotopy for system (10) which satisfies the following relations: α1 0 Dt x(t) α2 0 Dt y(t) α3 0 Dt z(t)

= p(ay − ax), = p((c − a)x − xz + cy), = p(xy − bz),

(11)

where p ∈ [0, 1] is an embedding parameter. Using the parameter p, we expand the solutions of system (11) in the following form: x(t) = x0 (t) + px1 (t) + p2 x2 (t) + p3 x3 (t) + · · · , y(t) = y0 (t) + py1 (t) + p2 y2 (t) + p3 y3 (t) + · · · , z(t) = z0 (t) + pz1 (t) + p2 z2 (t) + p3 z3 (t) + · · · ,

(12)

x0 (0) = x(0) = c1 , y0 (0) = y(0) = c2 , z0 (0) = z(0) = c3 , xi (0) = 0, yi (0) = 0, zi (0) = 0, i = 1, 2, 3, . . . ,

(13)

with the initial conditions



7

where xi , yi , zi , i = 0, 1, 2, · · · are the functions to be determined later. Substituting (12) into (11) and collecting the terms of the same powers of p, we obtain the following series of fractional order equations α1 0 Dt x 0 α1 0 Dt x 1 α1 0 Dt x 2 α1 0 Dt x 3

p0 : 1

p : 2

p : 3

p : .. .

α2 0 Dt y 0 α2 0 Dt y 1 α2 0 Dt y 2 α2 0 Dt y 3

p0 : 1

p : 2

p : 3

p : .. .

α3 0 Dt z 0 α3 0 Dt z 1 α3 0 Dt z 2 α3 0 Dt z 3

p0 : 1

p : 2

p : 3

p : .. .

= 0, = ay0 − ax0 , = ay1 − ax1 , = ay2 − ax2 ,

= 0, = (c − a)x0 − x0 z0 + cy0 , = (c − a)x1 − x0 z1 − x1 z0 + cy1 , = (c − a)x2 − x0 z2 − x1 z1 − x2 z0 + cy2 ,

= 0, = x0 y0 − bz0 , = x0 y1 + x1 y0 − bz1 , = x0 y2 + x1 y1 + x2 y0 − bz2 ,

Finally, we solve the above equations for the unknown functions xi , yi , zi , i = 0, 1, 2, . . . by applying the integral operator 0 Jtαi , i = 1, 2, 3, therefore, x0 = x0 (0) = c1 , x1 y1 z1 x2

y0 = y0 (0) = c2 , z0 = z0 (0) = c3 , t α1 = (ac2 − ac1 ) · , Γ(α1 + 1) t α2 , = (cc1 − ac1 + cc2 − c1 c3 ) · Γ(α2 + 1) tα3 = (c1 c2 − bc3 ) · , Γ(α3 + 1) tα1 +α2 = (acc1 − a2 c1 + acc2 − ac1 c3 ) · Γ(α1 + α2 + 1) 2α1 t , − (a2 c2 − a2 c1 ) · Γ(2α1 + 1)

y2 = (acc2 − acc1 − a2 c2 + a2 c1 − ac2 c3 + ac1 c3 ) · + (c2 c1 − acc1 + c2 c2 − cc1 c3 ) ·

t2α2 Γ(2α2 + 1)

tα2 +α3 , z2 Γ(α2 + α3 + 1) tα1 +α3 + (ac22 − ac1 c2 ) · Γ(α1 + α3 + 1) t2α3 − (bc1 c2 − b2 c3 ) · . Γ(2α3 + 1) − (c21 c2 − bc1 c3 ) ·

tα1 +α2 Γ(α1 + α2 + 1)

= (cc21 − ac21 + cc1 c2 − c21 c3 ) ·


tα2 +α3 Γ(α2 + α3 + 1)

8


Hence, the N-term HPM approximate solutions to the chaotic fractional order Chen system (10) are obtained as x(t) ≈ xN (t) =

N −1 ∑

xk (t),

y(t) ≈ yN (t) =

k=0

N −1 ∑

yk (t),

z(t) ≈ zN (t) =

k=0

N −1 ∑

zk (t).

(14)

k=0

Based on the MHPM, to carry out the iterations on every subinterval [ti−1 , ti ), i = 1, 2, . . . j of equal length △t, we need to know the values of the following initial conditions xi0 (t∗ ) = c∗1 ,

yi0 (t∗ ) = c∗2 ,

zi0 (t∗ ) = c∗3 .

Usually, we have the values at the initial point t∗ = t0 = 0. And we can obtain the other necessary initial values by means of the previous N -term approximations of the preceding subinterval [ti−2 , ti−1 ), i = 2, 3, . . . j given by Eq. (14), i.e. x∗i0 (t∗ ) = xi−1,N (t∗ ) =

N −1 ∑

xi−1,k (t∗ ),

k=0 ∗ ∗ yi0 (t ) = yi−1,N (t∗ ) =

N −1 ∑

yi−1,k (t∗ ),

k=0 ∗ ∗ zi0 (t ) = zi−1,N (t∗ ) =

N −1 ∑

zi−1,k (t∗ ).

k=0

When (a, b, c) = (35, 3, 28), αi = 0.98, i = 1, 2, 3, x(0) = −10, y(0) = 0, z(0) = 37, the 15-term HPM approximate series solutions to system (10) are given as follows: x(t) ≈ −10 + 352.926t0.98 + 1633.703t1.96 − 46144.635t2.94 + 59962.710t3.92 + 4346980.875t4.90 − 37482871.327t5.88 − 308090416.301t6.86 + 5908609653.887t7.84 + 463892582.682t8.82 − 627436322216.374t9.80 + 3837725770026.378t10.78 + 44399716721631.604t11.76 − 697006346473392.856t12.74 − 524270599230792.939t13.72 , y(t) ≈ 443.679t0.98 − 2173.084t1.96 − 39592.798t2.94 + 650719.273t3.92 − 1741346.262t4.90 − 95672099.150t5.88 + 963657094.177t6.86 + 6020656559.892t7.84 − 167549742693.793t8.82 + 500729494154.365t9.80 + 18050476981400.532t10.78 − 196908986847879.348t11.76 − 892174210352015.224t12.74 + 31009861603412525.142t13.72 , z(t) ≈ 37 − 111.928t0.98 − 2109.292t1.96 + 63948.645t2.94 + 42356.398t3.92 − 9383166.519t4.90 + 59351681.161t5.88 + 734067944.743t6.86 − 14193941951.415t7.84 + 6884409879.795t8.82 + 1842475897692.222t9.80 − 13843346818646.202t10.78 − 135707257607649.604t11.76 + 2711959166611633.821t12.74 − 2043932915875380.863t13.72 . Based on the preliminary calculations, we fix the number of terms N=15 and select the time step size △t = 0.001 for the MHPM. In Table 1, we present the numerical solutions which are applied by the 15-term HPM and the 15-term MHPM on time step △t = 0.001. Evidently, the classical HPM solutions are not valid for a long time. The MHPM overcomes this lack of the HPM. The well-known chaotic attractor of the fractional order Chen system in the time range of [0, 20] is shown in Fig. 1. Obviously, the MHPM is an effective method for a large t.

4.2

MHPM usage in the hyperchaotic fractional order Lorenz system

The fractional order Lorenz system is defined as α1 0 Dt x(t) = a(y − x) + w, α2 0 Dt y(t) = cx − y − xz, α3 0 Dt z(t) = xy − bz, α4 0 Dt w(t) = −yz + rw,


(15)


Table 1 Comparisons between 15-term HPM solutions and 15-term MHPM solutions for system (10)

t 0 0.05 0.1 1 5 10 15 20

|x| 10 7.842 47.428 1.174E+15 2.591E+24 3.132E+28 7.838E+30 3.974E+32

HPM |y| 0 14.943 339.081 2.994E+16 1.199E+26 1.622E+30 4.232E+32 2.193E+34

|z| 37 32.588 228.353 5.203E+14 5.794E+24 9.244E+28 2.535E+31 1.346E+33

|x| 10 9.023 11.841 11.352 12.620 8.052 3.835 1.222

MHPM |y| 0 15.555 7.983 11.511 9.804 6.630 6.248 3.259

|z| 37 32.678 36.232 29.042 29.376 24.010 21.810 21.017

Figure 1: The chaotic attractor of the fractional order Chen system (10) with (a, b, c) = (35, 3, 28), αi = 0.98, i = 1, 2, 3.


9

10


subject to the initial conditions x(0) = c1 , y(0) = c2 , z(0) = c3 , w(0) = c4 , where 0 < αi < 1, i = 1, 2, 3, 4, x, y, z, w are state vectors. According to (4), we construct the following homotopy: α1 0 Dt x(t) = p(ay − ax + w), α2 0 Dt y(t) = p(cx − y − xz), α3 0 Dt z(t) = p(xy − bz), α4 0 Dt w(t) = p(−yz + rw),

(16)

where p ∈ [0, 1] is an embedding parameter. Using the parameter p, we expand the solutions of system (16) in the following form: x(t) =

∞ ∑

pk xk (t),

y(t) =

k=0

∞ ∑

pk yk (t),

k=0

z(t) =

∞ ∑

pk zk (t),

w(t) =

k=0

∞ ∑

pk wk (t),

(17)

k=0

with the initial conditions x0 (0) = x(0) = c1 , y0 (0) = y(0) = c2 , z0 (0) = z(0) = c3 , w0 (0) = w(0) = c4 , xi (0) = 0, yi (0) = 0, zi (0) = 0, wi (0) = 0,

(18) i = 1, 2, 3, . . .

where xi , yi , zi , wi , i = 0, 1, 2, · · · are the functions to be determined later. Substituting (17) into (16) and collecting the terms of the same powers of p, we obtain the following series of fractional order equations p0 p1 p2 p3 .. .

: : : :

α1 0 Dt x0 α1 0 Dt x1 α1 0 Dt x2 α1 0 Dt x3

= 0, = ay0 − ax0 + w0 , = ay1 − ax1 + w1 , = ay2 − ax2 + w2 ,

p0 p1 p2 p3 .. .

: : : :

α2 0 Dt y0 α2 0 Dt y1 α2 0 Dt y2 α2 0 Dt y3

= 0, = cx0 − x0 z0 − y0 , = cx1 − x0 z1 − x1 z0 − y1 , = cx2 − x0 z2 − x1 z1 − x2 z0 − y2 ,

p0 p1 p2 p3 .. .

: : : :

α3 0 Dt z0 α3 0 Dt z1 α3 0 Dt z2 α3 0 Dt z3

= 0, = x0 y0 − bz0 , = x0 y1 + x1 y0 − bz1 , = x0 y2 + x1 y1 + x2 y0 − bz2 ,

p0 p1 p2 p3 .. .

: : : :

α4 0 Dt w0 α4 0 Dt w1 α4 0 Dt w2 α4 0 Dt w3

= 0, = −y0 z0 + rw0 , = −y0 z1 − y1 z0 + rw1 , = −y0 z2 − y1 z1 − y2 z0 + rw2 ,

Finally, we solve the above equations for the unknown functions xi , yi , zi , wi , i = 0, 1, 2, . . . by applying the integral operator 0 Jtαi , i = 1, 2, 3, 4. The first few terms of the HPM series solutions for system (15) are obtained as follows: x0 = x(0) = c1 , y0 = y(0) = c2 , z0 = z(0) = c3 , w0 = w(0) = c4 , tα1 t α2 t α3 t α4 x1 = x ˜1 · , y1 = y˜1 · , z1 = z˜1 · , w1 = w ˜1 · , Γ(α1 + 1) Γ(α2 + 1) Γ(α3 + 1) Γ(α4 + 1)



11

Figure 2: The hyperchaotic attractor of the fractional order Lorenz system (15) with αi = 0.98, i = 1, 2, 3, 4, (a, b, c, r) = (10, 8/3, 28, −1).

tα1 +α2 t2α1 tα1 +α4 − a˜ x1 · +w ˜1 · , Γ(α1 + α2 + 1) Γ(2α1 + 1) Γ(α1 + α4 + 1) Γ(α1 + α3 + 1)tα1 +α2 +α3 y2 = −˜ x1 z˜1 · Γ(α1 + 1)Γ(α3 + 1)Γ(α1 + α2 + α3 + 1) tα1 +α2 t2α2 + c˜ x1 · − y˜1 · , Γ(α1 + α2 + 1) Γ(2α2 + 1) Γ(α1 + α2 + 1)tα1 +α2 +α3 t2α3 z2 = −b˜ z1 · +x ˜1 y˜1 · , Γ(2α3 + 1) Γ(α1 + 1)Γ(α2 + 1)Γ(α1 + α2 + α3 + 1) t2α4 Γ(α2 + α3 + 1)tα2 +α3 +α4 w2 = rw ˜1 · − y˜1 z˜1 · , Γ(2α4 + 1) Γ(α2 + 1)Γ(α3 + 1)Γ(α2 + α3 + α4 + 1) x2 = a˜ y1 ·

where x ˜1 = ac2 − ac1 + c4 , y˜1 = cc1 − c2 − c1 c3 , z˜1 = c1 c2 − bc3 , w ˜1 = −c2 c3 + rc4 . Hence, the N-term approximate analytical solutions to the hyperchaotic fractional order Lorenz system (15) are x(t) ≈

N −1 ∑

xk (t),

y(t) ≈

k=0

N −1 ∑

yk (t),

k=0

z(t) ≈

N −1 ∑

zk (t),

w(t) ≈

N −1 ∑

k=0

wk (t).

(19)

k=0

Based on the MHPM, to carry out the iterations on every subinterval [ti−1 , ti ), i = 1, 2, . . . j of equal length △t, we need to know the values of the following initial conditions xi0 (t∗ ) = c∗1 ,

yi0 (t∗ ) = c∗2 ,

zi0 (t∗ ) = c∗3 ,

wi0 (t∗ ) = c∗4 .

Usually, we have the values at the initial point t∗ = t0 = 0. And we can obtain the other necessary initial values by means of the previous N -term approximations of the preceding subinterval [ti−2 , ti−1 ), i = 2, 3, . . . j given by Eq. (19), i.e. x∗i0 (t∗ ) = xi−1,N (t∗ ) =

N −1 ∑

xi−1,k (t∗ ),

∗ ∗ yi0 (t ) = yi−1,N (t∗ ) =

k=0 ∗ ∗ zi0 (t ) = zi−1,N (t∗ ) =

N −1 ∑

N −1 ∑

yi−1,k (t∗ ),

k=0

zi−1,k (t∗ ),

∗ ∗ wi0 (t ) = wi−1,N (t∗ ) =

k=0


N −1 ∑ k=0

wi−1,k (t∗ ).

12

International Journal of Nonlinear Science, Vol.13(2012), No.1, pp. 3-14 Table 2 Comparisons between 15-term HPM solutions and 15-term MHPM solutions for system (15)

t 0 0.05 0.1 1 5 10 15 20

|x| 1.232 3.167 6.685 4.904E+11 1.944E+21 2.625E+25 6.842E+27 3.543E+29

HPM |y| 4.536 5.478 10.033 7.608E+12 3.206E+22 4.372E+26 1.143E+29 5.931E+30

|z| 10.18 9.133 9.857 3.980E+12 1.866E+22 2.582E+26 6.788E+28 3.530E+30

|w| 50 45.086 39.101 1.263E+13 5.463E+22 7.476E+26 1.957E+29 1.016E+31

|x| 1.232 3.471 7.414 0.344 0.404 13.241 6.185 4.099

MHPM |y| 4.536 5.693 11.025 6.239 1.043 16.832 0.072 2.901

|z| 10.18 9.095 10.269 24.624 28.504 30.059 17.704 23.790

|w| 50 44.678 37.854 88.116 52.575 26.337 96.654 36.961

When (a, b, c, r) = (10, 8/3, 28, −1), αi = 0.98, i = 1, 2, 3, 4, x(0) = 1.232, y(0) = −4.536, z(0) = 10.18, w(0) = −50, the 15-term HPM approximate series solutions to system (15) are obtained as bellow: x(t) ≈ 1.232 − 108.580t0.98 + 745.735t1.96 − 6096.930t2.94 + 26418.828t3.92 − 88124.062t4.90 + 68470.376t5.88 + 2007097.820t6.86 − 21442510.665t7.84 + 161169021.505t8.82 − 951518798.253t9.80 + 4534710218.249t10.78 − 14970918002.876t11.76 + 1431019797.809t12.74 + 500277064898.955t13.72 , y(t) ≈ −4.536 + 26.712t0.98 − 988.010t1.96 + 3568.690t2.94 − 13017.268t3.92 − 67321.958t4.90 + 1510297.905t5.88 − 15043351.088t6.86 + 120024410.607t7.84 − 751010146.800t8.82 + 3665597576.211t9.80 − 10339853267.387t10.78 − 36199256629.485t11.76 + 856114127806.253t12.74 − 8420692320073.729t13.72 , z(t) ≈ 10.18 − 33.009t0.98 + 315.520t1.96 − 2888.990t2.94 + 43656.473t3.92 − 323647.440t4.90 + 2108006.641t5.88 − 8574718.102t6.86 + 5784711.506t7.84 + 322629284.048t8.82 − 4152519262.027t9.80 + 35015563420.827t10.78 − 235541420110.977t11.76 + 1276439800025.071t12.74 − 5052588860388.937t13.72 , w(t) ≈ −50 + 96.981t0.98 − 266.752t1.96 + 4376.832t2.94 − 24802.465t3.92 + 181235.465t4.90 − 1150411.982t5.88 + 8972226.773t6.86 − 52183196.098t7.84 + 203939092.487t8.82 + 485811679.874t9.80 − 18985787869.104t10.78 + 229623420094.840t11.76 − 2028574632656.812t12.74 + 14455911360954.396t13.72 . Based on the preliminary calculations, we fix the number of terms N=15 and select the time step size △t = 0.001 for the MHPM. In Table 2, we present the numerical solutions which are applied by the 15-term HPM and the 15-term MHPM on time step △t = 0.001. Evidently, the HPM solutions are not valid for a long time. In the time range of [0, 20], the well-known hyperchaotic attractor of the fractional order Lorenz system (15) is shown in Fig. 2. Obviously, the MHPM can be an effective method for a large t.

5

Conclusion

In this work, we apply the MHPM to give the approximate numeric-analytic solutions of the chaotic fractional order systems. It is shown that the solutions obtained by the standard HPM are not valid for a long time, while the ones gain by the proposed MHPM are highly accurate for a long time. The MHPH, which treats the HPM as an algorithm in a



13

sequence of intervals, yields a very rapid convergence of the solution series, usually only a few iterations leading to very accurate solutions. Moreover, the numerical experiments show that the MHPM is an efficient and powerful mathematic tool which is not only valid for the chaotic fractional order systems, but also for the hyperchaotic fractional order systems. The application of the MHPM in the control and synchronization of the chaotic fractional order systems will be some interesting problems for future research.

Acknowledgement This work is supported by the National Nature Science Foundation of China (Grant No. 10901014) and the Science Foundation of Beijing Jiaotong University (No. 2011YJS076, No. 2011JBM130).

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