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Synchronization of fractional order chaotic systems

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Page 1: Synchronization of fractional order chaotic systems

Physics Letters A 363 (2007) 426–432

www.elsevier.com/locate/pla

Synchronization of fractional order chaotic systems

Guojun Peng

Department of Mathematics, Hengyang Normal University, Hengyang 421008, China

Received 23 June 2006; accepted 20 November 2006

Available online 14 December 2006

Communicated by A.R. Bishop

Abstract

The chaotic dynamics of fractional order systems have attracted much attentions recently. In this Letter, we study the synchronization of thefractional order chaotic systems with a unidirectional linear error feedback coupling. The numerical results show that the fractional order chaoticsystems can also be synchronized.© 2006 Elsevier B.V. All rights reserved.

PACS: 05.45.-a

Keywords: Fractional order; Chaos; Synchronization

1. Introduction

Although the fractional calculus has existed for 300 years,the applications of fractional calculus to physics and engineer-ing are just a recent focus of interest [1–3]. The fractional cal-culus has proved to be valuable tool in the modeling of manyphysical phenomena [4–6]. Most recently, many authors beginto investigate the chaotic dynamics of fractional dynamical sys-tems [7–11].

On the other hand, since Pecora and Carroll showed that itis possible to synchronize two identical chaotic systems [12],chaos synchronization has been intensively and extensivelystudied. Naturally, the synchronization of fractional orderchaotic systems becomes a topic [13]. In this Letter, we studythe synchronization of fractional order chaotic systems also.Because the analysis of fractional order systems is not suffi-cient at present, we will numerically investigate this topic here.

There are many definitions of fractional derivative [1–3].Many authors formally use the Riemann–Liouville fractionalderivative, defined by

(1)Dα∗ f (t) = dm

dtmJm−αf (t), α > 0,

E-mail address: [email protected] (G. Peng).

0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2006.11.053

where m = �α�, i.e., m is the first integer which is not less thanα. Jβ is the β-order Riemann–Liouville integral operator whichis described as following

(2)Jβg(t) = 1

�(β)

t∫0

(t − τ)β−1g(τ) dτ, β > 0.

In this Letter, the following definition is used

(3)Dαf (t) = Jm−αf (m)(t), α > 0,

where m = �α�. It is common practice to call operator Dα theCaputo differential operator of order α because it has appar-ently first been used for the solution of practical problems byCaputo [14]. As a matter of fact, it had already been introducedin a 19th century paper of Liouville [1]. A typical feature ofdifferential equations (classical or fractional) is that one needsto specify additional conditions to make sure that the solutionis unique. For an initial value problem, it is easily seen thatthe number of initial conditions that one needs to specify inorder to obtain a unique solution is m = �α�. For the Riemann–Liouville case, one would have to specify the values of certainfractional derivatives (and integrals) of the unknown solution atthe initial point t = 0 [3]. However, in practical applications,these values are frequently not available, and it may not evenbe clear what their physical meaning is. When we deal with

Page 2: Synchronization of fractional order chaotic systems

G. Peng / Physics Letters A 363 (2007) 426–432 427

Caputo derivative, the situation is different. We may specify theinitial values f (0), f ′(0), . . . , f (m−1)(0). These data typicallyhave a well understood physical meaning and can be measured.It is known that under homogeneous conditions the equationswith Riemann–Liouville operators are equivalent to those withCaputo operators [2]. We choose the Caputo version becauseit allows us to specify inhomogeneous initial conditions also ifthis is desired. As we have described above, for the Riemann–Liouville approach this generalization is connected with majorpractical difficulties [15].

2. Numerical algorithm for fractional order differentialequations

Most authors study fractional order chaotic systems usingthe Riemann–Liouville derivative [7–11]. The numerical algo-rithm is proposed in [16]. Following is the basic idea. One ap-plies the Laplace transform to the Riemann–Liouville fractionalderivative and has the form because of the homogeneous condi-tions: L{ dαf (t)

dtα} = sαL{f (t)}. Then, approximate the fractional

operators by using the standard integer order operators. To real-ize it, one effects the approximations in the Laplace s-variableand makes it approximate the system behavior in the frequencydomain. Finally, one can establish a table of the integer orderapproximate expressions of 1

sq , 0 < q < 1. When q , 0 < q < 1,is not in the table, one has to execute the program from beginto end. However, in this Letter, we choose the Caputo ver-sion and use a predictor–corrector algorithm for fractional orderdifferential equations [17,18], which is the generalization ofAdams–Bashforth–Moulton one. When α > 0, the algorithm isuniversal. The following is a brief introduction of the algorithm.

The differential equation

(4)

{Dαy(t) = f (t, y(t)), 0 � t � T ,

yk(0) = yk0 , k = 0,1, . . . ,m − 1,

is equivalent to the Volterra integral equation

(5)y(t) =�α�−1∑k=0

tk

k!y(k)0 + 1

�(α)

t∫0

(t − τ)α−1f(τ, y(τ )

)dτ.

Set h = T/N , N ∈ Z, tn = nh, n = 0,1, . . . ,N . Then (5) canbe discreted as following

yh(tn+1) =�α�−1∑k=0

tkn+1

k! y(k)0 + hα

�(α + 2)f

(tn+1, y

ph (tn+1)

)

(6)+ hα

�(α + 2)

n∑j=0

aj,n+1f(tj , yh(tj )

),

where

aj,n+1 =

⎧⎪⎨⎪⎩

nα+1 − (n − α)(n + 1)α, if j = 0,

(n − j + 2)α+1 + (n − j)α+1 − 2(n − j + 1)α+1,

if 1 � j � n,

yph (tn+1) =

�α�−1∑ tkn+1

k! y(k)0 + 1

�(α)

n∑bj,n+1f

(tj , yh(tj )

),

k=0 j=0

bj,n+1 = hα

α

((n + 1 − j)α − (n − j)α

).

The error estimate is

maxj=0,1,...,N

∣∣y(tj ) − yh(tj )∣∣ = O

(hp

), where

p = min(2,1 + α).

3. Synchronization of fractional order chaos

Consider the master–slave (or drive–response) synchroniza-tion scheme of two autonomous n-dimensional fractional orderchaotic systems

(7)

{M: dαx

dtα= f (x),

S: dαxdtα

= f (x) + D(x − x),

where α = (α1, α2, . . . , αn) ∈ Rn, αi > 0, is the fractional or-der and the systems are chaotic. D is a coupling matrix. Forsimplicity, we let D have the form: D = diag(d1, d2, . . . , dn),where di � 0. The error is e = x − x and the aim of the synchro-nization is to design the coupling matrix such that ‖e(t)‖ → 0as t → +∞.

First of all, similar to [19], we derive a simple synchro-nization criterion for integer order chaotic systems, i.e., αi = 1(1 � i � n), with unidirectional linear error feedback coupling.At this case, we can write (7) in the form

(8)

{M: x = Ax + F(x),

S: ˙x = Ax + F(x) + D(x − x),

where A is the Jacobian matrix of f (x) at 0, and F(x) =f (x) − Ax. The following error system can be obtainedfrom (8)

(9)e = Ae + Mx,xe − De,

where Mx,xe = F(x) − F(x) and the elements of Mx,x are de-pendent on x and x. When choosing the Lyapunov function

V = eT e,

we have

V = eT e + eT e

= eT[(A + Mx,x − D)T + (A + Mx,x − D)

]e

= eT He.

Since H = (A+Mx,x −D)T + (A+Mx,x −D) is a Hermitematrix, there exist a unitary matrix U and a real matrix Λ =diag(λ1, λ2, . . . , λn) such that

H = U∗ΛU.

Subsequently,

V = eT U∗ΛUe = eT Λe.

Noticing that H = Q−2D, where Q = (A+Mx,x)T +(A+

Mx,x), so, when the elements of D satisfy

(10)di >1

2

(Qi,i +

n∑|Qi,j |

),

j=1,j =i

Page 3: Synchronization of fractional order chaotic systems

428 G. Peng / Physics Letters A 363 (2007) 426–432

(a) (b)

Fig. 1. The double scroll attractors of Chua’s systems with m = 10, n = 14.87, a = −1.27, b = −0.68 and (a): α = (0.90,0.95,0.96), (b): α = (1,1,1).

where Qi,j denotes the ind row, jnd column element of Q,from the Gersgorin disk theorem, we know the elements of Λ

have the following properties

λi < 0, 1 � i � n.

Therefore,

V < 0,

which implies the systems in (8) are synchronized.Here, we study the synchronization of fractional order

chaotic systems by numerical simulations. Simultaneously, wealso consider the corresponding integer order systems. TheChua, Rössler, and Chen systems are examples.

First, consider the Chua system [20]

(11)

⎧⎪⎪⎨⎪⎪⎩

dα1xdtα1 = m(y − x − f (x)),

dα2ydtα2 = x − y + z,

dα3zdtα3 = −ny,

where m > 0, n > 0, f (x) is a piecewise linear function

f (x) = bx + 1

2(a − b)

(|x + 1| − |x − 1|), in which

a < b < 0.

When m = 10, n = 14.87, a = −1.27, b = −0.68, and (α1, α2,

α3) = (0.90,0.95,0.96), system (11) exhibits chaotic behavior(see Fig. 1(a)). The slave system corresponding to (11) is

(12)

⎧⎪⎪⎨⎪⎪⎩

dα1 xdtα1 = m(y − x − f (x)) + d1(x − x),

dα2 ydtα2 = x − y + z + d2(y − y),

dα3 zdtα3 = −ny + d3(z − z).

For the purpose of comparison, we also consider the integerorder forms of (11) and (12), i.e., (α1, α2, α3) = (1,1,1), otherparameters are the same as fractional order’s. The phase of in-teger order form corresponding to (11) is provided in Fig. 1(b).

From (10), we know that when

(13)

⎧⎪⎨⎪⎩

d1 > 12 (1 − m − 2am),

d2 > 12 (m − 1 + |1 − n|),

d > 1 (|1 − n|),

3 2

the integer order Chua systems are asymptotically synchro-nized. By choosing the coupling parameters as d1 = 9, d2 =12, d3 = 8, which satisfy (13), the integer order forms of(11) and (12) are asymptotically synchronized, as shown inFig. 2(b). We find that the corresponding fractional order Chuasystems are also asymptotically synchronized at these couplingparameters (see Fig. 2(a)). Simultaneously, comparing Fig. 2(a)with Fig. 2(b), we can know that the synchronization rate of thefractional order Chua systems is slower than its integer orderforms.

We also plot the curves of synchronization error of the frac-tional and integer order Chua systems when d1 = 20, d2 = 25,d3 = 15 in Fig. 3(a) and (b), respectively. From the graphs, wehave the same conclusion as above.

Next, consider the Rössler system [21]

(14)

⎧⎪⎪⎨⎪⎪⎩

dα1xdtα1 = −(y + z),

dα2ydtα2 = x + ay,

dα3zdtα3 = b + z(x − c),

where a > 0, b > 0, c > 0. When a = 0.2, b = 0.2, c = 5, and(α1, α2, α3) = (1,1,0.8), system (14) can produce chaotic so-lutions (see Fig. 4). The phases of corresponding integer orderRössler system are plotted in Fig. 5. The slave system corre-sponding to (14) is

(15)

⎧⎪⎪⎨⎪⎪⎩

dα1 xdtα1 = −(y + z) + d1(x − x),

dα2 ydtα2 = x + ay + z + d2(y − y),

dα3 zdtα3 = b + z(x − c) + d3(z − z).

From (10), we know that when

(16)

⎧⎪⎨⎪⎩

d1 > 12 (|z − 1|),

d2 > a,

d3 > 12 (|z − 1| + 2x − 2c),

the integer order Rössler systems are asymptotically synchro-nized. From Figs. 4 and 5, we can know the bounds of Rösslerattractors satisfy |x| < 12, and |z| < 20. By selecting d1 =15, d2 = 0.5, d3 = 20, which satisfy (16), the integer orderforms of (14) and (15) are asymptotically synchronized, as

Page 4: Synchronization of fractional order chaotic systems

G. Peng / Physics Letters A 363 (2007) 426–432 429

(a) (b)

Fig. 2. Synchronization errors of Chua’s systems with d1 = 9, d2 = 12, d3 = 8 and (a): α = (0.90,0.95,0.96), (b): α = (1,1,1).

(a) (b)

Fig. 3. Synchronization errors of Chua’s systems with d1 = 20, d2 = 25, d3 = 15 and (a): α = (0.90,0.95,0.96), (b): α = (1,1,1).

(a) (b)

Fig. 4. The attractor of Rössler’s system with a = 0.2, b = 0.2, c = 5 and α = (1,1,0.8).

shown in Fig. 6(b). The corresponding fractional order Rösslersystems are also asymptotically synchronized at these cou-pling parameters (see Fig. 6(a)). Comparing Fig. 6(a) withFig. 6(b), we can know that the synchronization rate of thefractional order Rössler systems is slightly slower than its in-

teger order forms. When d1 = 40, d2 = 5, d3 = 40, the curvesof synchronization error of the fractional and integer orderRössler systems are provided in Fig. 7(a) and (b), respec-tively. From the graphs, we also have the same conclusion asabove.

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430 G. Peng / Physics Letters A 363 (2007) 426–432

(a) (b)

Fig. 5. The attractor of Rössler’s system with a = 0.2, b = 0.2, c = 5 and α = (1,1,1).

(a) (b)

Fig. 6. Synchronization errors of Rössler’s systems with d1 = 15, d2 = 0.5, d3 = 20 and (a): α = (1,1,0.8), (b): α = (1,1,1).

(a) (b)

Fig. 7. Synchronization errors of Rössler’s systems with d1 = 40, d2 = 5, d3 = 40 and (a): α = (1,1,0.8), (b): α = (1,1,1).

Finally, consider the Chen system [22]

(17)

⎧⎪⎨⎪⎩

dα1xdtα1 = a(y − x),

dα2ydtα2 = (c − a)x − xz + cy,

dα3zdtα3 = xy − bz.

When a = 35, b = 3, c = 27 and (α1, α2, α3) = (0.76,0.96,

0.94), this system is chaotic (see Fig. 8). The phases of corre-

sponding integer order Chen system are plotted in Fig. 9. Itsslave system is

(18)

⎧⎪⎪⎨⎪⎪⎩

dα1 xdtα1 = a(y − x) + d1(x − x),

dα2 ydtα2 = (c − a)x − xz + cy + d2(y − y),

dα3 zdtα3 = xy − bz + d3(z − z).

Page 6: Synchronization of fractional order chaotic systems

G. Peng / Physics Letters A 363 (2007) 426–432 431

(a) (b)

Fig. 8. The attractor of Chen’s system with a = 35, b = 3, c = 27 and α = (0.76,0.96,0.94).

(a) (b)

Fig. 9. The attractor of Chen’s system with a = 35, b = 3, c = 27 and α = (1,1,1).

(a) (b)

Fig. 10. Synchronization errors of Chen’s systems with d1 = 60, d2 = 40, d3 = 20 and (a): α = (0.76,0.96,0.94), (b): α = (1,1,1).

From the analysis of (10), we know that when

(19)

⎧⎪⎨⎪⎩

d1 > 12 (2a + |c − z| + |y|),

d2 > 12 (|c − z| + 2c),

d3 > 12 (|y| + 2b),

the integer order Chen systems are synchronized. From Figs. 8and 9, we can know the bounds of Chen attractors satisfy|y| < 30, and |z| < 45. By selecting d1 = 60, d2 = 40, d3 = 20,which satisfy (19), the integer order forms of (17) and (18) aresynchronized, as shown in Fig. 10(b). The corresponding frac-tional order Chen systems are also synchronized at these cou-

Page 7: Synchronization of fractional order chaotic systems

432 G. Peng / Physics Letters A 363 (2007) 426–432

(a) (b)

Fig. 11. Synchronization errors of Chen’s systems with d1 = 100, d2 = 50, d3 = 30 and (a): α = (0.76,0.96,0.94), (b): α = (1,1,1).

pling parameters (see Fig. 10(a)). Comparing Fig. 10(a) withFig. 10(b), we can know that the synchronization rate of thefractional order Chen systems is also slower than its integer or-der counterpart.

When d1 = 100, d2 = 50, d3 = 30, the curves of synchro-nization error of the fractional and integer order Chen sys-tems are provided in Fig. 11(a) and (b), respectively. From thegraphs, we also have the same conclusion as above.

4. Conclusions

In this Letter, we investigate the synchronization of the frac-tional order chaotic systems with a unidirectional linear errorfeedback coupling. We use the Caputo’s definition of fractionalderivative and a predictor–corrector algorithm of fractional or-der differential equations. Simulations have shown that sometypical chaotic systems can also be synchronized. We can alsoapply the frequency domain techniques [16] to simulate the syn-chronization phenomena, however, we would prefer to use thefractional calculus techniques since these can be theoreticallyanalyzed, such as convergence, stability and error estimate,which are omitted in this Letter.

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