Physica A 387 (2008) 3738–3746www.elsevier.com/locate/physa

Generalized projective synchronization of fractional orderchaotic systems

Guojun Penga,∗, Yaolin Jianga, Fang Chenb

a Department of Mathematics, Xi’an Jiaotong University, Xi’an 710049, Chinab Department of Mathematics and Physics, Xi’an University of Post and Telecommunication, Xi’an 710121, China

Received 14 July 2007; received in revised form 14 September 2007Available online 25 February 2008

Abstract

In this paper, based on the idea of a nonlinear observer, a new method is proposed and applied to “generalized projectivesynchronization” for a class of fractional order chaotic systems via a transmitted signal. This synchronization approach istheoretically and numerically studied. By using the stability theory of linear fractional order systems, suitable conditions forachieving synchronization are given. Numerical simulations coincide with the theoretical analysis.c© 2008 Elsevier B.V. All rights reserved.

Keywords: Synchronization; Fractional order system; Chaos

1. Introduction

Although the fractional calculus was introduced in the early 17th century [1,2], the applications of the fractionalcalculus to physics and engineering are just a recent focus of interest [3–5]. Most recently, many authors begin toinvestigate the chaotic dynamics and synchronization of fractional dynamical systems [6–15].

In this paper, first, a new synchronization called “generalized projective synchronization” is proposed. Next,a drive-response synchronization method is developed, which is used to “generalize projective synchronization”for a class of fractional order chaotic systems via a transmitted signal. The proposed technique is simple andtheoretically rigorous. It enables the synchronization of fractional order chaotic systems to be achieved in a systematicway. Simultaneously, a necessary and sufficient condition for the existence of a feedback gain matrix such that asynchronization between the fractional order drive system and response system can be achieved is shown. Finally, thefractional order Arneodo and Chen systems are used to illustrate the effectiveness of the proposed synchronizationmethod.

In the rest of this section, the definitions of fractional derivative and generalized projective synchronization areintroduced.

∗ Corresponding author.E-mail address: pgjat163@163.com (G. Peng).

0378-4371/$ - see front matter c© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2008.02.057

G. Peng et al. / Physica A 387 (2008) 3738–3746 3739

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

Dαx(t) =dm

dtm J m−αx(t), α > 0, (1)

where m = dαe, i.e., m is the first integer which is not less than α. Jβ is the β-order Riemann–Liouville integraloperator, which is described as the following

Jβ y(t) =1

Γ (β)

∫ t

0(t − τ)β−1 y(τ )dτ, β > 0. (2)

In this paper, the following definition is used:

Dα∗ x(t) = J m−αx (m)(t), α > 0, (3)

where m = dαe. It is common practice to call the operator Dα∗ the Caputo differential operator of order α because

it has apparently first been used for the solution of practical problems by Caputo [16]. For a initial problem ofRiemann–Liouville type, one would have to specify the values of certain fractional derivatives (and integrals) ofthe unknown solution at the initial point t = 0 [3]. However, it is not clear what the physical meanings of fractionalderivatives of x are when we are dealing with a concrete physical application, and hence it is also not clear howsuch quantities can be measured. When we deal with a initial problem of Caputo type, the situation is different.We may specify the initial values x (n)(0), n = 0, 1, . . . , m − 1, i.e., the function value itself and integer-orderderivatives [16]. These data typically have well understood physical meanings and can be measured. Furthermore,Riemann–Liouville initial problems require homogeneous initial conditions, whereas Caputo initial problems allowus to specify inhomogeneous initial conditions too if this is desired [17]. It is known [1] that under those homogeneousconditions the problems with Riemann–Liouville operators are equivalent to those with Caputo operators. These arethe reasons for us to choose the Caputo derivative and not the Riemann–Liouville derivative.

We’ll introduce the projective synchronization and the generalized synchronization at first. For simplicity, we takethe notation Dα

∗ (·), which is described as

Dα∗ x(t) =

Dα

∗ x1(t)Dα

∗ x2(t)· · ·

Dα∗ xn(t)

, (4)

where x(t) = (x1(t), x2(t), . . . , xn(t))T.For two identical chaotic systems which are coupled through the variable z in the formDα

∗ xm = A(z)xm,

Dα∗ xs = A(z)xs,

Dα∗ z = f(xm, z),

(5)

the subscripts m and s stand for the drive (or master) and the response (or slave) systems, respectively. The variable zis nonlinearly related to the variable xm . If there exists a constant σ ∈ R − {0}, such that

limt→∞

‖σxm − xs‖ = 0, (6)

then we regard that xm synchronize to xs up to a scaling factor σ , and call such synchronization “projectivesynchronization”, which has some topological invariants, such as Lyapunov exponents and fractal dimensions [18–21].

For the following two chaotic systems in unidirectional coupling form{Dα

∗ x = f(x),

Dβ∗ y = g(y, uµ(x)),

(7)

where x ∈ Rn , y ∈ Rk , f : Rn→ Rn , u : Rn

→ Rk , and g : Rk+k→ Rk , if µ = 0, y has no relation to x. When

µ 6= 0, the two systems are said to be in “generalized synchronization” if there exists a transformation 8 : x → y

3740 G. Peng et al. / Physica A 387 (2008) 3738–3746

which maps asymptotically the trajectories of the drive attractor into those of the response attractor, regardless on theinitial conditions in the basin of the synchronization manifold M = {(x, y) : y = 8(x)} [22–25]. In general, thetransformation 8 is difficult to find.

From above two synchronization definitions, a new one is derived [26].Consider the following chaotic systems{

Dα∗ x = f(x),

Dα∗ y = g(y, h(x, y)),

(8)

where x ∈ Rn , y ∈ Rn , f : Rn→ Rn , h : Rn+n

→ Rn , g : Rn+n→ Rn , and g(x, 0) ≡ f(x). If there exists a constant

σ ∈ R − {0} such that

limt→∞

‖σx − y‖ = 0, (9)

then we regard the two systems are synchronized. Such synchronization is called a “generalized projectivesynchronization”. Comparing (8) with (7), we find that this definition is very similar to that of generalizedsynchronization. At the same time, if the two systems in (8) are synchronized up to a scaling factor σ , the Lyapunovexponents and fractal dimensions remain invariant. The function h in the second system of (8) can be regardedas a feedback controller. If such a feedback controller h is applied to the slave system, generalized projectivesynchronization may occur.

2. Generalized projective synchronization of fractional order chaotic systems

Assume the fractional order chaotic drive systems under study can be written as

Dα∗ x = f(x) = Ax + BF(x), (10)

where A is the Jacobian matrix of f(x) about state vector x ∈ Rn at x = 0, B ∈ Rn×m , F : Rn→ Rm and all

elements of F(x) are nonlinear functions. Many fractional order chaotic systems belong to the class characterized by(10). Examples include the fractional order Chua circuit with a cubic term [6], fractional order Arneodo system [27],the unified chaotic system of fractional order version (including the Lorenz, Lu, and Chen systems) [28].

If system (10) has the output F(x), based on the design idea of the nonlinear observer [29,30], we configure thetransmitted synchronizing signal as

G(x) = F(x) + K x, (11)

where K ∈ Rm×n is a feedback gain matrix to be decided. When the nonlinear vector function F(x) defined by (10)belongs to Rn

→ R1, that is to say, m=1, the transmitted synchronizing signal G(x) defined by (11) is a scalar.Construct the following fractional order response system

Dα∗ y = f(y) + B(σG(x) − G(y)), (12)

where σ is a scaling factor of the generalized projected synchronization. Then, we have

Proposition 1. For fractional order drive system (10) and response system (12), defining the error e = σx − yaccording to (9), we get the error system which can be expressed as

Dα∗ e = (A − BK )e. (13)

Proof. From (10) and (12), we can obtain

Dα∗ e = Dα

∗ (σx − y)

= σ f(x) − f(y) − BσG(x) + BG(y)

= σ Ax + σ BF(x) − Ay − BF(y) − BσF(x) − Bσ K x + BF(y) + BK y

= A(σx − y) − BK (σx − y)

= (A − BK )e.

The proof is completed. �

G. Peng et al. / Physica A 387 (2008) 3738–3746 3741

For a given autonomous linear system of fractional order

Dα∗ x = Ax, (14)

with x(0) = x0, where x ∈ Rn is the state vector, we have [31]

Lemma 1. System (14) is

i. asymptotically stable if and only if | arg(λi (A))| > απ/2, i = 1, 2, . . . , n, where arg(λi (A)) denotes the argumentof the eigenvalue λi of A. In this case, the component of the state decay towards 0 like t−α;

ii. stable if and only if either it is asymptotically stable or those critical eigenvalues which satisfy | arg(λi (A))| =

απ/2 have geometric multiplicity one.

Now, according to Lemma 1, we can obtain the following results

Proposition 2. Given fractional order drive system (10) and response system (12), there exists a feedback gain matrixK such that a generalized projective synchronization between (10) and (12) can be achieved if and only if all theuncontrollable eigenvalues λi of A − BK satisfy | arg(λi )| > απ/2.

Proof. For the fractional order linear system (13), there exists a proper coordinate transformation

e = Pε = (P1, P2) ε, (15)

where the columns of P1 form a set of basis vectors for the controllable state subspace and the columns of P2 areorthogonal to these [32]. Since P is an orthogonal matrix, system (13) can be transformed to the following Kalmancontrollable canonical form [32](

Dα∗ εc

Dα∗ εuc

)=

(PT

1 AP1 PT1 AP2

0 PT2 AP2

)(εcεuc

)+

(PT

1 B0

)(K e)

=

(Ac PT

1 AP20 Auc

)(εcεuc

)+

(PT

1 B0

)u,

where the eigenvalues of Ac are controllable, whereas the eigenvalues of Auc are uncontrollable. By introducing aproper state feedback u, we can get any eigenvalues of the controllable part without affecting the eigenvalues of Auc.Therefore, from Lemma 1 and (15), we can conclude that system (13) is stable if and only if the eigenvalues of Aucsatisfy | arg(λi (Auc))| > απ/2, i.e., the uncontrollable eigenvalues λi of A − BK satisfy | arg(λi )| > απ/2. Thisfinishes the proof. �

3. Numerical algorithms

The numerical calculation of a fractional differential equation is not as simple as that of an ordinary differentialequation. Here we choose the Caputo version and use a predictor-corrector algorithm for fractional order differentialequations [33–35], which is the generalization of Adams–Bashforth–Moulton one. When α > 0, the algorithm isuniversal. The following is a brief introduction of the algorithm.

The differential equation{Dα

∗ y(t) = f (t, y(t)), 0 ≤ t ≤ T,

y(k)(0) = y(k)0 , k = 0, 1, . . . , m − 1,

(16)

is equivalent to the Volterra integral equation

y(t) =

dαe−1∑k=0

tk

k!y(k)

0 +1

Γ (α)

∫ t

0(t − τ)α−1 f (τ, y(τ ))dτ. (17)

Set h =TN , N ∈ Z , tn = nh, n = 0, 1, . . . , N . Then (17) can be discreted as follows:

yh(tn+1) =

dαe−1∑k=0

tkn+1

k!y(k)

0 +hα

Γ (α + 2)f (tn+1, y p

h (tn+1)) +hα

Γ (α + 2)

n∑j=0

a j,n+1 f (t j , yh(t j )), (18)

3742 G. Peng et al. / Physica A 387 (2008) 3738–3746

where

a j,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,

y ph (tn+1) =

dαe−1∑k=0

tkn+1

k!y(k)

0 +1

Γ (α)

n∑j=0

b j,n+1 f (t j , yh(t j )),

b j,n+1 =hα

α((n + 1 − j)α − (n − j)α).

The error estimate is

maxj=0,1,...,N

|y(t j ) − yh(t j )| = O(h p), where p = min(2, 1 + α).

4. Two examples

First, consider the fractional order Arneodo’s system [27] which can be written in the form of (10) asDα∗ x1

Dα∗ x2

Dα∗ x3

=

0 1 00 0 1a b c

x1x2x3

+

00d

x31 . (19)

When α = 1, and (a, b, c, d) = (11/2, −7/2, −1, −1), the integral order Arneodo’s system can produce chaoticsolutions. Here the transmitted synchronizing signal (11) is a scalar and has the following form

G(x) = x31 + K

x1x2x3

, (20)

where x = (x1, x2, x3)T and K = (k1, k2, k3). The response system (12) is constructed asDα

∗ y1Dα

∗ y2Dα

∗ y3

=

0 1 00 0 1a b c

y1y2y3

+

00

dσ

x31 +

00d

K

σ x1 − y1σ x2 − y2σ x3 − y3

. (21)

Using the error as defined in Proposition 1, we have

e1 = σ x1 − y1, e2 = σ x2 − y2, e3 = σ x3 − y3, (22)

and get the error systemDα∗ e1

Dα∗ e2

Dα∗ e3

=

0 1 00 0 1a b c

−

00d

K

e1e2e3

. (23)

Since the controllability matrix of A − BK in (23) is full rank, from Proposition 2 we know that there exists a gainvector K such that the systems (19) and (21) are synchronized. For example, when α = 0.9, and (a, b, c, d) =

(11/2, −7/2, −2/5, −1), the drive system (19) is chaotic. We can set the eigenvalues λi = −1 of A − BK withK = (−13/2, 1/2, −13/5) and λi = −2 with K = (−27/2, −17/2, −28/5), which satisfy | arg(λi )| > 0.9π/2,i = 1, 2, 3. Setting K = (−13/2, 1/2, −13/5), the phase diagrams of (19) and (21) are plotted together in Fig. 1(a)and (b) with scaling factors σ = 2 and 4, respectively. For displaying clearly, we move the phase diagrams of (19)along the positive direction of z1 by 30 units and remove the first several points. The curves of synchronization errorare shown in Fig. 2(a) and (b), with σ = 2, K = (−13/2, 1/2, −13/5) and σ = 2, K = (−27/2, −17/2, −28/5),respectively, which indicate that the chaos synchronization between (19) and (21) is achieved.

G. Peng et al. / Physica A 387 (2008) 3738–3746 3743

Fig. 1. The attractors of drive system (19) and response system (21) with α = 0.9, (a, b, c, d) = (11/2, −7/2, −2/5, −1), and (a): σ = 2,K = (−13/2, 1/2, −13/5), (b): σ = 4, K = (−13/2, 1/2, −13/5).

Fig. 2. Synchronization errors of drive system (19) and response system (21) with α = 0.9, (a, b, c, d) = (11/2, −7/2, −2/5, −1), and (a): σ = 2,K = (−13/2, 1/2, −13/5), (b): σ = 2, K = (−27/2, −17/2, −28/5).

Next, consider the fractional order Chen’s system [28], which can be expressed in the form of (10) asDα∗ x1

Dα∗ x2

Dα∗ x3

=

−a a 0c − a c 0

0 0 −b

x1x2x3

+

0 0−1 00 1

(x1x3x1x2

). (24)

When α = 1, and (a, b, c) = (35, 3, 27), the integral order Chen’s system exhibits chaotic behavior. Here thetransmitted synchronizing signal (11) is a vector and has the following form:

G(x) =

(x1x3x1x2

)+ K

x1x2x3

, (25)

where

x = (x1, x2, x3)T and K =

(k1 k2 k3k4 k5 k6

).

3744 G. Peng et al. / Physica A 387 (2008) 3738–3746

Fig. 3. The attractors of the drive system (24) and response system (26) with α = 0.85, (a, b, c) = (35, 3, 27), and (a): σ = −2, K = K ,(b): σ = −4, K = K .

We construct the response system (12) asDα∗ y1

Dα∗ y2

Dα∗ y3

=

−a a 0c − a c 0

0 0 −b

y1y2y3

+

0 0−σ 00 σ

(x1x3x1x2

)+

0 0−1 00 1

K

σ x1 − y1σ x2 − y2σ x3 − y3

(26)

and use the error definition of Proposition 1. Subsequently, we get the error system between the system (24) and (26)Dα∗ e1

Dα∗ e2

Dα∗ e3

=

−a a 0c − a c 0

0 0 −b

−

0 0−1 00 1

K

e1e2e3

. (27)

Since the controllability matrix of A − BK in (27) is full rank, from Proposition 2 we know that there exists a gainmatrix K such that the synchronization of system (24) and (26) can be realized. For example, when α = 0.85 and(a, b, c) = (35, 3, 27), the drive system (24) is chaotic. We set the eigenvalues λi = −4 of A − BK which satisfy| arg(λi )| > 0.85π/2, i = 1, 2, 3, by an easy calculation, and know that the gain matrix K has the following properties

k4 = (−1085k1 − 280k2 + 35k1k2 − 21111)/(35k3),

k5 = (35k1 + 31k2 + k22 + 681)/k3,

k6 = k2 + 1.

For simplicity, choosing (k1, k2, k3) = (0, 0, 681), we get a gain matrix

K =

(0 0 681

−3135

1 1

).

When we set the eigenvalues λi = −2 of A − BK , i = 1, 2, 3, similarly, we can get another gain matrix

K =

(0 0 693

−2557724255

1 −5

).

Setting K = K , the phase diagrams of (24) and (26) are plotted together in Fig. 3(a) and (b) with scaling factor σ = −2and −4, respectively. Similar to the above, we move the phase diagrams of (24) along the positive direction of z1 with30 units and remove the first several points. Choosing σ = −2, the curves of synchronization error function ln(‖e‖2)

are shown in Fig. 4(a) and (b) with K = K and K = K , respectively, which indicate that the chaos synchronizationbetween (24) and (26) is achieved.

G. Peng et al. / Physica A 387 (2008) 3738–3746 3745

Fig. 4. Synchronization errors of the drive system (24) and response system (26) with α = 0.85, (a, b, c) = (35, 3, 27), and (a): σ = −2, K = K ,(b): σ = −2, K = K .

5. Conclusions

In this paper, we investigated the generalized projective synchronization of fractional order chaotic systems anddeveloped a new synchronization scheme for a class of fractional order chaotic systems based on the idea of a nonlinearobserver. By utilizing the stability criterion of linear fractional systems, the proposed synchronization approachis simple and theoretically rigorous. Two examples, i.e., the Arneodo and Chen systems, are used to illustrate theeffectiveness of the proposed synchronization method.

Acknowledgement

This research was supported by the National Nature Science Foundation of PR China under Grant No. 10771168and by the National Key Basic Research Program of PR China (973 program) under Grant No. 2005CB321701.

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