SYSTEM
A.K. ALOMARI
Abstract. A novel solution to the fraction chaotic Chen system is presented in this paper by using the step homotopy analysis method. This method yields a continuous solution in terms of a rapidly convergent infinite power series with easily computable terms. Moreover, the residual error of the SHAM solution is defined and computed for each time interval. Via the computing of the residual error we observe that the accuracy of the present method tends to 10−11which is very high.
1. Introduction
Nature is intrinsically nonlinear. So, it is not surprising that most of the systems we encounter in the real world are nonlinear. And what is interesting is that some of these nonlinear systems can be described by fractional-order differential equation which can display a variety of behaviors including chaos and hyperchaos. The purpose of this paper is to obtain a continuous solution for fractional chaotic Chen system [1, 2, 3].
Dαt1x = a(y−x), (1.1)
Dαt2y = (c−a)x−xz+cy, (1.2)
Dαt3z = xy−bz, (1.3)
x(0) = c1 y(0) =c2, z(0) =c3, (1.4)
whereDαi, i= 1,2,3 are Caputo fractional derivatives,a, b, cfromℜand 0< αi≤ 1.
Finding accurate and efficient methods for solving FDEs has been an active re- search undertaking. Exact solutions of most of the FDEs cannot be found easily, thus analytical and numerical methods must be used. For example, generalized Adams-Bashforth-Moulton method (GABMM) is one of the most used method to solve fractional differential equations [4, 5, 6, 7]. Some of the recent analytic meth- ods for solving nonlinear problems include the Adomian decomposition method (ADM) [8, 9], homotopy-perturbation method (HPM) [10, 11] and variational iter- ation method (VIM) [12, 13].
Recently, homotopy analysis method (HAM) becomes one of the most famous technique to solve such nonlinear problems. the method was proposed by Liao [14, 15]. Many researchers have applied this method for different class of differential equations [17, 18, 19, 20, 21, 22, 23, 24]. Alomari et al. [25] used the idea of time step
1991Mathematics Subject Classification. 65P20, 26A33, 34A08.
Key words and phrases. Chaotic system, Fractional Chen system, Homotopy analysis method, Step homotopy analysis method, Residual error.
1
in the algorithm of HAM to get multistage homotopy analysis method (MSHAM) and apply to Chen system. Recently, Alomari et al. [28] introduce new algorithm to obtain the solution of fractional chaotic system using HAM.
This paper investigates for the first time the applicability and effectiveness of HAM when we hybrid the numerical with analytical in a sequence of intervals (i.e. time step) for finding accurate approximate solutions to the fractional Chen system. To the best of our knowledge, this is also the first time that the residual error can be calculated for the analytical solution at each subinterval of fractional Chen system. Numerical results are presented graphically and are found to be in excellent agreement with the GABMM solution.
2. Preliminaries and notations
In this section, we give some definitions and properties of the fractional calculus and homotopy-derivative
2.1. Fractional calculus. The following properties can found in [26].
Definition 1 A real function f(t),t >0, is said to be in the space Cµ, µ∈ ℜ, if there exists a real numberp > µ, such thatf(t) =tpf1(t), where f1(t)∈C(0,∞), and it is said to be in the space Cµn if and only ifh(n)∈Cµ,n∈N.
Definition 2The Riemann-Liouville fractional integral operator(Jα)of order α≥ 0, of a functionf ∈Cµ,µ≥ −1, is defined as
Jαf(t) = 1 Γ(α)
t
Z
0
(t−s)α−1f(t)s. (α >0), (2.1)
J0f(t) = f(t), (2.2)
whereΓ(α)is the well-known gamma function.
Some of the properties of the operatorJα, which we will need here, are as follows:
Forf ∈Cµ,µ≥ −1,α, β≥0 andγ≥ −1:
(1) JαJβf(t) =Jα+βf(t), (2) JαJβf(t) =JβJαf(t), (3) Jαtγ= Γ(α+γ+1)Γ(γ+1) tα+γ.
Definition 3The fractional derivative(Dα)off(t), in the Caputo sense is defined as
Dαf(t) = 1 Γ(n−α)
t
Z
0
(t−s)n−α−1f(n)(t)s., (2.3)
for n−1< α < n, n∈N, t >0, f ∈C−n1.
The following are two basic properties of the Caputo fractional derivative:
(1) Let f ∈C−n1, n∈N, thenDαf,0≤α≤nis well defined andDαf ∈C−1. (2) Let n−1≤α≤n, n∈N and f∈Cµn, µ≥ −1. Then
(JαDα)f(t) =f(t)−
n−1
X
k=0
f(k)(0+)tk k!. (2.4)
2.2. homotopy-derivative. The following properties can found in [19]
Definition 4Let φbe a function of the homotopy-parameter q, then Dm(φ) = 1
m!
∂mφ
∂qm q=0
(2.5) .
is called themth-order homotopy-derivative of φ, where m≥0 is an integer.
propertiesFor homotopy-series φ1=
+∞
X
i=0
uiqi, φ2=
+∞
X
i=0
viqi it holds
(1) Dm(φ1) =um. (2) Dm(qφ1) =Dm−1(φ1)
(3) If L be a linear operator independent of the homotopy-parameterq. For homotopy-series, thenDm(Lφ1) =LDm(φ1).
(4) If f and g be functions independent of the homotopy-parameter q, then Dm(f φ1±gφ2) =f Dm(φ1)±gDm(φ2).
(5) Dm(φ1φ2) =Pm
i=0φ1,iφ2,m−i
3. Solution approaches To solve (1.1)–(1.3), we choose the base function as
{tnα1+mα2+kα3|n≥0, m≥0, k≥0}, (3.1)
so the solutions are in the form x(t) =a0,0,0+
+∞
X
n=0 +∞
X
m=0 +∞
X
k=0
an,m,ktnα1+mα2+kα3, (3.2)
y(t) =b0,0,0+
+∞
X
n=0 +∞
X
m=0 +∞
X
k=0
bn,m,ktnα1+mα2+kα3, (3.3)
z(t) =c0,0,0+
+∞
X
n=0 +∞
X
m=0 +∞
X
k=0
cn,m,ktnα1+mα2+kα3, (3.4)
wherean,m,k, bn,m,kandcn,m,kare the coefficients. It is straightforward to choose x0(t) =c1, y0(t) =c2, z0(t) =c3,
(3.5)
as our initial approximations ofx(t), y(t) and z(t), and the linear operator should be
Lα1[ˆx] = Dα1x,ˆ (3.6)
Lα2[ˆy] = Dα2y,ˆ (3.7)
Lα3[ˆz] = Dα3z,ˆ (3.8)
since we used Caputo fractional derivative then we have the property Lα1[A1] =Lα2[A2] =Lα3[A3] = 0,
(3.9)
whereAi i= 1,2,3 are the integration constants that will be determined by the initial conditions.
If q ∈ [0,1] and ~ indicate the embedding and non-zero auxiliary parameters, respectively, then thezeroth-order deformation problems are of the following form:
(1−q)Lα1[ˆx(t;q)−x0(t)] = q~Nx[ˆx(t;q),y(t;ˆ q)], (3.10)
(1−q)Lα2[ˆy(t;q)−y0(t)] = q~Ny[ˆx(t;q),y(t;ˆ q),z(t;ˆ q)], (3.11)
(1−q)Lα3[ˆz(t;q)−z0(t)] = q~Nz[ˆx(t;q),y(t;ˆ q),z(t;ˆ q)], (3.12)
subject to the initial conditions ˆ
x(0;q) =c1, ˆy(0;q) =c2, z(0;ˆ q) =c3, (3.13)
in which we define the nonlinear operatorsNx, Ny andNz as Nx[ˆx(t;q),y(t;ˆ q)] = ∂α1ˆx(t;q)
∂tα1 −a(ˆy(t;q)−x(t;ˆ q)), Ny[ˆx(t;q),y(t;ˆ q),z(t;ˆ q)] = ∂α2y(t;ˆ q)
∂tα2 −(c−a)ˆx(t;q) + ˆx(t;q)ˆz(t;q)−cy(t;ˆ q), Nz[ˆx(t;q),y(t;ˆ q),z(t;ˆ q)] = ∂α3ˆz(t;q)
∂tα3 −x(t;ˆ q)ˆy(t;q) +bz(t;ˆ q).
For q = 0 and q = 1, the above zeroth-order equations (3.10)-(3.12) have the solutions
(3.14) x(t; 0) =ˆ x0(t), y(t; 0) =ˆ y0(t), z(t; 0) =ˆ z0(t), and
(3.15) x(t; 1) =ˆ x(t), y(t; 1) =ˆ y(t), z(t; 1) =ˆ z(t).
Whenqincreases from 0 to 1, then ˆx(t;q),y(t;ˆ q) and ˆz(t;q) vary fromx0(t), y0(t) andz0(t) tox(t), y(t) andz(t), respectively. Expanding ˆx,yˆand ˆz in Taylor series with respect to q, we have
ˆ
x(t;q) = x0(t) +
∞
X
m=1
xm(t)qm, (3.16)
ˆ
y(t;q) = y0(t) +
∞
X
m=1
ym(t)qm, (3.17)
ˆ
z(t;q) = z0(t) +
∞
X
m=1
zm(t)qm, (3.18)
in which
(3.19) xm(t) =Dm(ˆx(t;q)), ym(t) =Dm(ˆy(t;q)), zm(t) =Dm(ˆz(t;q)),
where~is chosen in such a way that these series are convergent atq= 1. Therefore, through Eqs. (3.14)–(3.19), we have
x(t) = x0(t) +
∞
X
m=1
xm(t), (3.20)
y(t) = y0(t) +
∞
X
m=1
ym(t), (3.21)
z(t) = z0(t) +
∞
X
m=1
zm(t).
(3.22)
Take themth-order homotopy-derivative ofzeroth-orderequations (3.10)-(3.12) and used the properties (1)–(4), then we have themth-order deformation equations
Lα1[xm(t)−χmxm−1(t)] = ~Rmx(t), (3.23)
Lα2[ym(t)−χmym−1(t)] = ~Rmy(t), (3.24)
Lα3[zm(t)−χmzm−1(t)] = ~Rmz(t), (3.25)
with the following initial conditions:
(3.26) xm(0) = 0, ym(0) = 0, zm(0) = 0,
whereRxm(t), Rmy(t) andRzm(t) can be found by used the properties (1),(4) and (5) as
Rmx(t) = Dαt1xm−1−a(ym−1−xm−1), (3.27)
Rmy(t) = Dαt2ym−1−(c−a)xm−1+
m−1
X
i=0
xi(t)zm−1−i(t)−cym−1(t), (3.28)
Rmz(t) = Dαt3zm−1−
m−1
X
i=0
xi(t)ym−1−i(t) +bzm−1(t).
(3.29) and
χm=
0, m≤1, 1, m >1.
In this way, it is easy to solve the linear non-homogeneous Eqs. (3.23)–(3.25) at initial conditions (3.26) for allm≥1, and now we successfully obtain
x1(t) = ~tα1a(−c2+c1) Γ (α1+ 1) ,
y1(t) = ~tα2(−c1c+c1a−cc2+c3c1) Γ (α2+ 1) , z1(t) = ~tα3c2(b−c1)
Γ (α3+ 1) ,
¯ h x(10−7)
-0.6 -0.8 -1 -1.2 -1.4
1641.7 1641.65 1641.6 1641.55 1641.5 1641.45 1641.4 1641.35 1641.3
¯ h y(10−7)
-0.6 -0.8 -1 -1.2 -1.4
2064 2063.8
2063.6
2063.4 2063.2
2063
¯ h z(10−7)
-0.6 -0.8 -1 -1.2 -1.4
-655.1 -655.2 -655.3 -655.4 -655.5 -655.6 -655.7
Figure 1. ~-curves of 13th-order approximation for (0.9,0.9,0.9)
etc. Then the 13-term of the approximate solutions of Eqs. (1.1)–(1.4) are
x(t) = x0(t) +
12
X
m=1
xm(t), (3.30)
y(t) = y0(t) +
12
X
m=1
ym(t), (3.31)
z(t) = z0(t) +
12
X
m=1
zm(t).
(3.32)
To determine the value of~we plot the ~-curves for Eqs. (3.30)–(3.32) in Fig. 1.
From this figure, it is noted that the valid regions of ~ correspond to the line segments nearly parallel to the horizontal axis.
If ~ = −1 we get the homotopy perturbation method (HPM) solution when α1 =α2 =α3 = 1 which is not effective for large values of t (for more detail see [27]).
3.1. SHAM. The HAM solution for Eqs. (3.30)–(3.32) is not effective for largert.
In case if we need the solution for [0,7], then the simple idea is to divide the interval [0,7] to subintervals with time step ∆tand we get the solution at each subinterval.
So in this case we have to satisfy the initial condition at each of the subinterval.
Accordingly, the initial values x0, y0, z0 will be changed for each subinterval, i.e.
x(t∗) = c∗1 = x0, y(t∗) = c∗2 =y0 and z(t∗) = c∗3 =z0 and we should satisfy the
initial conditionsxm(t∗) = 0,ym(t∗) = 0 andzm(t∗) = 0 for allm≥1 so x1(t) = ~(t−t∗)α1a(−c2+c1)
Γ (α1+ 1) ,
y1(t) = ~(t−t∗)α2(−c1c+c1a−cc2+c3c1)
Γ (α2+ 1) ,
z1(t) = ~(t−t∗)α3c2(b−c1) Γ (α3+ 1) , ...
So, the solution will be as follows:
x(t) = c∗1+
12
X
m=1
xm(t−t∗), (3.33)
y(t) = c∗2+
12
X
m=1
ym(t−t∗), (3.34)
z(t) = c∗3+
12
X
m=1
zm(t−t∗), (3.35)
where t∗ starting from t0 = 0 untiltn =T = 7. To carry out the solution on every subinterval of equal length ∆t, we need to know the values of the following initial conditions:
c1=x(t∗), c2=y(t∗), c3=z(t∗).
In general, we do not have these information at our clearance except at the initial pointt∗=t0= 0, but we can obtain these values by assuming that the new initial condition is the solution in the previous interval. (i.e. If we need the solution in interval [ti, ti+1], then the initial conditions of this interval will be as
c1 = x(ti) =
12
X
m=0
xm(ti−ti−1), (3.36)
c2 = y(ti) =
12
X
m=0
ym(ti−ti−1), (3.37)
c3 = z(ti) =
12
X
m=0
zm(ti−ti−1), (3.38)
wherec1, c2and c3 are the initial conditions in the interval [ti, ti+1]). By this way we get modified homotopy perturbation method (MHPM) solution as a special case when~=−1 andα1=α2=α3= 1 [27].
3.2. Error analysis for SHAM. The different between the exact solution and the given solution which we will so-call residual error can be define as
Ex = Dαt1X−a(Y −X), (3.39)
Ey = Dαt2Y −(c−a)X+XZ−cY, (3.40)
Ez = Dαt3Z−XY +bZ.
(3.41)
where X, Y and Z are the HAM solution for the equations (1.1–1.3) respectively.
Since the SHAM solution is analytic at each time step then it is easy to obtain the residual error at each time step. According Eqs. (3.39-3.41), we can find the residual error on each time step by applying the that equations and usingc1, c2and c3which is defined as in SHAM solution. We noted that the orders of magnitude of the errors in SHAM solution depend on the order of approximation and the length of the subintervals.
4. Results and discussion
In this part, we set a = 35, c = 28, b = 3 and we take the initial conditions x(0) = −10, y(0) = 0 and z(0) = 37 as in [25] at the standard case (1,1,1). To observe the convergent of the solution, we plot the 10-term and 12-term of SHAM solution with ∆t= 0.005 in Fig. 2. It is clear that the solution of 10-term like the
12-term 10-term
t
x(t)
7 6 5 4 3 2 1 0 30 25 20 15 10 5 0 -5 -10 -15 -20
12-term 10-term
t
y(t)
7 6 5 4 3 2 1 0 30 20 10 0 -10 -20 -30
12-term 10-term
t
x(t)
7 6 5 4 3 2 1 0 30 25 20 15 10 5 0 -5 -10 -15 -20
Figure 2. Time series of the SHAM solution Using 10 and 12 order of approximation for (0.95,0.95,0.95)
solution of 12-term then we can consider 12-term as good approximate solution.
The phase portraits of the SHAM solution and GABMM solution are given in Fig.
3 and Fig. 4 at different fractional derivative.The figure gives that SHAM solution have good agreement with GABMM solution.
The residual error of the SHAM solution is presented in Fig.5 and 6 for (0.95,0.95,0.95) and (0.99,0.99,0.99) respectively. Table 1 and 2 give the residual error for the given solution at several points. We observe that a higher accuracy of the given solution is cited which is not extended than 10−10. On the other hand the GABMM usually used accuracy with 10−6.
5. Conclusions
In this present work continuous solution for fractional Chen system is obtained by SHAM. The modified method has the advantage of giving an analytical form of
(a)
z
x y
z
4540 3530 2520 1510 5
2025 1015 05 -10-5 -20-15 30 -25 10 20 -10 0 -30 -20
(b)
z
x y
(c)
z
4540 3530 2520 1510 5
2025 1015 05 -10-5 -20-15 30 -25 10 20 -10 0 -30 -20
Figure 3. Comparison the phase portraits ofx−y−zusing 12- term SHAM in (b) with GABMM in (a) when (α1, α2, α3) as (0.95,0.95,0.95)
(a)
z
x y
z
4035 3025 2015 105
1520 510 -50 -15-10 25 -20 1520 510 -5 0 -15-10 -25-20
(b)
z
x y
z 40 35 3025 20 1510
2025 1015 05 -10-5 -20-15 25 -25 1520 510 -5 0 -15-10 -25-20
Figure 4. Comparison the phase portraits ofx−y−zusing 12- term SHAM in (b) with GABMM in (a) when (α1, α2, α3) as (0.9,0.9,0.9)
(a)
0 1 2 3 4 5 6 7
-4.´10-11 -2.´10-11 0 2.´10-11 4.´10-11
t Ex
(b)
0 1 2 3 4 5 6 7
-1.´10-10 -5.´10-11 0 5.´10-11 1.´10-10 1.5´10-10
t Ey
(c)
0 1 2 3 4 5 6 7
-1.´10-10 -5.´10-11 0 5.´10-11 1.´10-10 1.5´10-10
t Ez
Figure 5. Residual error for SHAM solution using 8-terms with
△t= 0.001 when (α1, α2, α3) as (0.95,0.95,0.95)
the solution within each time interval which is not possible in purely numerical tech- niques like fourth–order Runge–Kutta method RK4 or ABMM. The residual error for subintervals solution is defined and calculated. We also note that the SHAM solutions were computed via a simple algorithm without any need for perturbation
(a)
0 1 2 3 4 5 6 7
-5.´10-12 0 5.´10-12
t Ez
(b)
0 1 2 3 4 5 6 7
-2.´10-12 -1.´10-12 0 1.´10-12 2.´10-12
t Ez
(c)
0 1 2 3 4 5 6 7
-5.´10-12 0 5.´10-12
t Ez
Figure 6. Residual error for SHAM solution using 8-terms with
△t= 0.001 when (α1, α2, α3) as (0.99,0.99,0.99)
Table 1. The Residual error forαi= 0.95 with 8-terms and ∆t= 0.001
t Ex Ey Ez
1 5.68434E-14 -7.81597E-14 -2.84217E-14 2 1.13687E-13 -1.27898E-13 -1.56319E-13
3 -7.10543E-15 0 -1.42109E-14
4 0 5.68434E-14 -3.55271E-15
5 -8.52651E-14 0 -3.01981E-14
6 -8.52651E-14 -2.84217E-14 -6.12843E-14 7 -7.10543E-15 -5.68434E-14 -7.10543E-15
Table 2. The Residual error forαi= 0.99 with 8-terms and ∆t= 0.001
t Ex Ey Ez
1 -2.27374E-13 -5.68434E-14 3.97904E-13 2 -8.9706E-14 -1.06581E-13 -7.10543E-14 3 2.84217E-14 1.42109E-13 1.42109E-14 4 -2.8777E-13 -4.05009E-13 -5.40012E-13
5 0 -5.68434E-14 0
6 -1.42109E-14 -2.84217E-14 1.42109E-14 7 8.52651E-14 3.97904E-13 6.53699E-13
techniques, special transformations, linearization or discretization. The SHAM so- lutions are in excellent agreement with the GABMM solution. Moreover The HPM and MHPM solution is a special case when~=−1 andα1=α2=α3= 1.
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Department of Mathematics, Faculty of Science, Yarmouk University, 211-63 Irbid, Jordan
E-mail address: [email protected]
