Synchronization in coupled nonidentical incommensurate fractional-order systems

Physics Letters A 374 (2009) 202–207

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Physics Letters A www.elsevier.com/locate/pla

Synchronization in coupled nonidentical incommensurate fractional-order systems Jun-Wei Wang a,∗ , Yan-Bin Zhang b a b

School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510006, PR China School of Computer Science, Hangzhou Dianzi University, Hangzhou 310018, PR China

a r t i c l e

i n f o

Article history: Received 8 September 2009 Received in revised form 12 October 2009 Accepted 19 October 2009 Available online 24 October 2009 Communicated by A.R. Bishop PACS: 05.45.-a 05.45.Xt 05.45.Pq

a b s t r a c t Synchronization of fractional-order nonlinear systems has received considerable attention for many research activities in recent years. In this Letter, we consider the synchronization between two nonidentical fractional-order systems. Based on the open-plus-closed-loop control method, a general coupling applied to the response system is proposed for synchronizing two nonidentical incommensurate fractional-order systems. We also derive a local stability criterion for such synchronization behavior by utilizing the stability theory of linear incommensurate fractional-order differential equations. Feasibility of the proposed coupling scheme is illustrated through numerical simulations of a limit cycle system, a chaotic system and a hyperchaotic system. © 2009 Elsevier B.V. All rights reserved.

Keywords: Fractional-order system Caputo fractional derivative Incommensurate fractional system Open-plus-closed-loop control Chaos synchronization

1. Introduction Synchronization, one of the most universal collective rhythms, has been extensively studied in various disciplines such as physics, chemistry, biology and neuroscience since the Dutch physicist Chistiaan Huygens first observed the anti-phase sychronization phenomenon of two pendulum clocks in the 17th century. In particular, inspired by the pioneering work of Pecora and Carroll [1,2], chaos synchronization of coupled nonlinear systems has become an active research subject due to its potential applications for secure communications and control. Depending upon the coupling topology and the strength of coupling, such systems are capable of entering into a state of complete, phase, lag or generalized synchronization [3,4]. On the other hand, recent years have seen efforts to explore synchronization behavior of coupled fractional-order systems in which at least one of the differential operators is of fractionalorder [5–13]. Using the frequency-domain method, Li et al. [5] numerically investigated the master–slave synchronization of fractional Chua and Rössler systems which represents the first report on the synchronization of fractional-order dynamical systems. In

*

Corresponding author. Tel.: +86 20 39328577. E-mail address: [email protected] (J.-W. Wang).

0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.10.051

Ref. [8], chaos synchronization of fractional Duffing, Lorenz and Rössler systems was studied with three coupling methods, i.e., combination of active–passive decomposition and one-way coupling method, Pecora–Carroll method and bidirectional coupling method. A nonlinear controller for synchronizing integer-order differential systems has been successfully extended to fractional Chen system to achieve complete synchronization [10]. In Ref. [11], via a scalar transmitted signal, the author presented a drive–response synchronization method with linear output error feedback for a class of fractional chaotic systems. Using the pole placement technique, a nonlinear state observer has been designed for synchronizing a class of nonlinear fractional-order systems [13]. It is no doubt that these aforementioned contributions are indeed of theoretical and experimental importance. It should be noted that most of previous works focus on synchronization between identical fractional-order systems. However, in practice most systems are nonidentical and parameter mismatches are inevitable because of noise or other uncertain factors. For example, recent experiments have shown that only 30% of SCN neurons produce stable rhythms in the absence of VIP signaling and that these intrinsic oscillators are nonidentical, exhibiting a wide distribution of free-running periods of between 20 and 28 h [14,15]. Moreover, parameter mismatches have become a common and serious problem in modern web applications. In fact, it has been reported that parameter mismatches are one

J.-W. Wang, Y.-B. Zhang / Physics Letters A 374 (2009) 202–207

of the most frequent types of errors made by end-user web application developers [16]. Therefore, it is very necessary to synchronize coupled nonidentical fractional-order nonlinear systems. Since nonidentical fractional-order systems have different nonlinear functions, even different dynamical behaviors, synchronization of nonidentical fractional-order systems is difficult to achieve and hence has received less attention [12,13]. Thus, this Letter is devoted to a study of how synchronizing nonidentical incommensurate fractional-order differential systems. To the best of our knowledge, this problem has not yet been addressed previously in the literatures. Specifically, we will present a coupling scheme for the master– slave synchronization of two nonidentical incommensurate fractional-order systems with parameter mismatches. After a brief overview of fractional derivatives, a general coupling, based on the open-plus-close-loop (OPCL) control, is proposed for synchronization of two nonidentical fractional-order systems. Then by utilizing the stability theory of linear incommensurate fractional differential equations, we derive a local stability criterion which guarantees, if satisfied, that two coupled nonidentical systems reach complete synchronization. Interestingly, this method enables synchronization of both periodic and chaotic fractional-order systems be achieved in a systematic way. Finally, we provide numerical simulations to illustrate the effectiveness of the proposed coupling approach. 2. Preliminaries and notations 2.1. Fractional derivatives As an extension of ordinary integration and differentiation, fractional calculus has been known since the development of the classic calculus [17,18]. Nowadays, fractional calculus has been found many applications in physics and engineering. Many systems in interdisciplinary fields are known to display fractional-order dynamics, such as electrode–electrolyte polarization [19], viscoelastic systems [20], dielectric polarization [21], electromagnetic wave [22], and boundary layer effects in ducts [23]. Although there are several different definitions for fractional derivatives of order α (α > 0), the Grünwald–Letnikov, the Riemann–Liouville and the Caputo definitions are three most commonly used ones [17,18]. The Grünwald–Letnikov (GL) fractional derivative with fractional-order q is defined by GL D

q

f (t ) =

lim

h→0, mh=t

h−q

m 

(−1)k

k =0

  q k

f (t − kh).

For a wide class of functions, the Grünwald–Letnikov definition and the Riemann–Liouville definition are equivalent. However, when modeling real-world phenomena with fractional differential equations (FDEs), the Caputo fractional derivative is more popular than the Riemann–Liouville definition of fractional derivative. This is because the initial conditions for the FDEs with the Caputo derivative are in the same form as for integer-order derivatives which have well understood physical meanings. Hence, we choose the Caputo derivative through this Letter. For more details on the geometric and physical interpretation for FDEs of both the Riemann–Liouville and Caputo types, see [17]. Hereafter, we use dq the notation dt q to denote the Caputo fractional derivative operaq tor C D . 2.2. Numerical algorithm for the fractional differential equations The approximate numerical techniques for FDEs have been developed in the literature which are numerically stable and can be applied to both linear and nonlinear FDEs. In 2002, Diethelm et al. [24] presented a PECE (predict, evaluate, correct, evaluate) type method for numerical solution of FDEs with Caputo derivatives, which is a generalization of the classical one-step Adams–Bashforth–Moulton algorithm for first-order ordinary differential equations. Recently, Deng [25,26] also proposed an improved predictor–corrector approach in which the numerical approximation is more accurate and the computational cost is largely reduced. The fractional predictor–corrector algorithm is based on the analytical property that the following FDE:

dq y (t ) dt q

  = f t , y (t ) ,

1

dn

(n − q) dt n

t o

f (τ ) dτ , (t − τ )q−n+1

y (t ) =

m −1 

t z−1 e −t dt .

q

f (t ) =

k!

1

(n − q)

for n − 1 < q < n.

o

f (n) (τ )

(t − τ )q−n+1

dτ ,

t

1

(q)

  (t − s)q−1 f s, y (s) ds.

(4)

0

k!

n 

(q + 2)



hq

(k)

y0 +

hq

(q + 2)

p



f tn+1 , y h (tn+1 )





a j ,n+1 f t j , y h (t j ) ,

(5)

j =0

where

⎧ q +1 n − (n − q)(n + 1)q , j = 0, ⎪ ⎪ ⎨ ( n − j + 2)q+1 + (n − j )q+1 a j ,n+1 = ⎪ ⎪ − 2(n − j + 1)q+1 , 1  j  n, ⎩ 1, j = n + 1, m −1 k  t n +1 k =0

t

+

m −1 k  t n +1 k =0

p

The Caputo derivative is defined by

k

Set h = NT , tn = nh, n = 0, 1, . . . , N ∈ Z + . Then Eq. (4) can be discretized as

y h (tn+1 ) =

0

CD

y0

+

∞ ( z) =

(k) t

k =0

(1)

where n is the smallest integer larger than q, i.e., n − 1 < q < n and (·) denotes the gamma function:



k = 0, . . . , m − 1 m = q

is equivalent to the Volterra integral equation [27]:

y h (tn+1 ) =

(2)

0 t  T,



(k)

y (k) (0) = y 0 ,

The Riemann–Liouville (RL) derivative can be written as q RL D f (t ) =

203

k!

n 

1

(k)

y0 +

(q)

hq ((n q





b j ,n+1 f t j , y h (t j ) ,

j =0

in which b j ,n+1 = + 1 − j ) − (n − j )q ). Therefore, the estimation error of this approximation is

(3)

max

j =0,1,..., N

q

  y (t j ) − yh (t j ) = O h p ,

where p = min(2, 1 + q) [28].

204

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3. Synchronization of two nonidentical fractional-order systems Consider a fractional-order system described by the following nonlinear fractional differential equations:

  = f xm (t ) +  f (xm ),

dq xm dt q

(6)

where xm (t ) = (x1 (t ), x2 (t ), . . . , xn (t )) T ∈ R n is an n-dimensional state vector of the system, f : R n → R n defines a vector field in ndimensional vector space,  f (xm ) contains mismatch parameters and q = (q1 , q2 , . . . , qn ) indicates the fractional-orders. We define

dq xm (t ) dt q

 =

dq1 x1 (t ) dq2 x2 (t )

,

dt q1

dt q2

,...,

dqn xn (t ) dt qn

T .

(7)

We call system (6) a commensurate fractional-order system if q1 = q2 = · · · = qn , otherwise a incommensurate fractional-order system [29]. Eq. (6) represents the master system. The controller u (t ) ∈ R 3 is added into the slave system, so it is described by

dq xs dt q

= f (xs ) + u (t ).

(8)

Note that the subscripts m and s stand for the master (or drive) and the slave (or response) systems, respectively. Synchronization of the systems means finding a controller u (t ) ∈ R 3 that makes states of the slave system (8) to evolve as the states of the master system (6). Define the synchronization error as e (t ) = xs (t ) − xm (t ), then we get the error system:

dq e (t ) dt q

=

dq xs (t ) dt q

−

dq xm (t ) (9)

The goal is to design a suitable controller u (t ) such that limt →∞ e (t ) = 0. The open-plus-closed-loop (OPCL) method proposed by Jackson and Grosu [30] is a powerful approach for controlling complex nonlinear systems. It has been widely applied in synchronization problems of integer-order dynamical systems [31–34]. This Letter presents the new applications of the OPCL control for synchronization of coupled nonidentical fractional-order systems. According to the OPCL control [30,33], we design the controller u (t ) as in the form of:

  ∂ f (xm ) u (t ) = K − xs (t ) − xm (t ) +  f (xm ), ∂ xm 

(10)

where K = (ki j )n×n is an n × n constant control matrix. Via the Taylor series expansion, we expand f (xs ) as:

f (xs , t ) = f (xm ) +

dt q

n1 1

dt qn

n2 2

(12)

nn n

where all q i (i = 1, 2, . . . , n) are rational numbers between 0 and 1. Assume M is the lowest common multiple of the denominators u i of q i , where q i = v i /u i , (u i , v i ) = 1, u i , v i ∈ Z + , for i = 1, 2, . . . , n. Define

⎞ −a12 ··· −a1n λMq1 − a11 Mq2 λ − a22 · · · −a2n ⎟ ⎜ −a21 ⎟. (λ) = ⎜ .. .. .. .. ⎠ ⎝ . . . . Mqn −an2 ··· λ − ann −an1

(13)

Then the zero solution of system (12) is globally asymptotically stable in the Lyapunov sense if all roots λ of the equation det((λ)) = 0 satisfy |arg(λ)| > π /2M. The stability region of the incommensurate fractionalorder system (12) is shown in Fig. 1. Thus, according to Lemma 1, we obtain the following theorem: Theorem 1. Assume q i ∈ (0, 1) (i = 1, 2, . . . , n) are rational numbers, M be the lowest common multiple of the denominators u i of qi , where qi = v i /u i , (u i , v i ) = 1, then the error system (11) is asymptotically stable if all roots λ of the equation









det diag λ Mq1 , λ Mq2 , . . . , λ Mqn − K = 0

arg(λ) > π , 2M

(14)

(15)

which means that the master system (6) and the slave system (8) with the OPCL controller (10) have achieved synchronization.

 ∂ f (xm ) = f (x s ) + K − e (t ) +  f (xm ) ∂ xm 

− f (xm , t ) −  f (xm ) = K e (t ).

⎧ q1 d x1 ⎪ ⎪ ⎪ q1 = a11 x1 + a12 x2 + · · · + a1n xn , ⎪ ⎪ dt ⎪ ⎪ q2 ⎪ ⎪ d x2 ⎨ = a21 x1 + a22 x2 + · · · + a2n xn , dt q2 ⎪ . ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ q ⎪ ⎪ d n xn = a x + a x + · · · + a x ⎩

satisfies the following condition:

∂ f (xm ) e (t ) + · · · . ∂ xm

The error system is then rewritten as

dq e (t )

Lemma 1. (See [35].) Consider the following n-dimensional linear fractional-order system:

⎛

dt q

= f (xs ) + u (t ) − f (xm ) −  f (xm ).

Fig. 1. Stability region of linear fractional order system (12) with order 0 < qi < 1.

(11)

Before we give our main results, the following lemma should be given first.

Remark. According to the original OPCL control method [30,31], we can simplify the control matrix K in the OPCL controller (10) as long as the condition (15) is satisfied. When [∂ f (xm )/∂ xm ]i j is a constant, we then set ki j = [∂ f (xm )/∂ xm ]i j such that [ K − ∂ f (xm )/∂ xm ]i j is zero. When [∂ f (xm )/∂ xm ]i j is a variable, we choose ki j = p i j where p i j are control parameters.

J.-W. Wang, Y.-B. Zhang / Physics Letters A 374 (2009) 202–207

205

Fig. 3. Synchronization behavior of two nonidentical fractional Van der Pol oscillators (17) and (20) with initial conditions (xm (0), ym (0), xs (0), y s (0)) = (0.7, −0.6, 0.65, 0.5). Fig. 2. Limit cycle in the fractional Van der Pol system with (q1 , q2 ) = (0.8, 0.9) and initial conditions (x(0), y (0)) = (0.7, −0.6). (a) The time evolution of state variables x and y; (b) view on (x, y ) plane.

According to the OPCL controller (10), we choose

 K=

0

1



−1 + k1 a + k2

4. Numerical examples

(19)

and then the response system can be written as: In this section, in order to show the effectiveness of the proposed scheme in preceding section, numerical examples on a limit cycle system, a chaotic system and a hyperchaotic system will be provided. When numerically solving fractional differential equations, we adopt the predictor–corrector method introduced in Section 2. So, by applying the aforementioned method, the numerical schemes for the drive–response configurations (6) and (8) can be easily constructed, and are omitted here. 4.1. Synchronization of the fractional Van der Pol oscillator The Van der Pol oscillator is a type of nonconservative oscillator with nonlinear damping described by a second-order nonlinear differential equation. It has been used to develop models in many applications such as electronics, biology or acoustics. The fractional Van der Pol oscillator has been introduced by Pereira et al. [36] and Barbosa et al. [37,38] where only one of the state variables contains the fractional-order derivative. Here we present full fractional Van der Pol system in the sense that the derivatives of both x and y are the fractional-order type:

⎧ q d 1x ⎪ ⎪ ⎨ q = y, dt

1

dt

2

(16)

  ⎪ dq 2 y ⎪ ⎩ = −x − a x2 − 1 y . q

When a = 1 and (q1 , q2 ) = (0.8, 0.9), this fractional-order system has a limit cycle, as shown in Fig. 2. To implement chaos synchronization of fractional Van der Pol systems, the drive system is defined as follows:

⎧ q d 1 xm ⎪ ⎪ ⎨ q = ym , dt 1 q ⎪ ⎪ d 2 ym

⎩

dt q2

 2   2  = −xm − a xm − 1 ym + a1 xm − 1 ym ,

(17)



0 −1 − 2axm ym

1 2 −a(xm − 1)

 .

(18)

(20)

We can present the concrete coupling terms in Eq. 20 by appropriately choosing ki (i = 1, 2). If we take k1 = −30, k2 = −10, then the corresponding eigenvalues of K are λ1,2 = −4.5 ± 3.3i. Thus the arguments of both λ1 and λ2 satisfy |arg(λ)| > π /20. According to Theorem 1, we can conclude that the master system (17) and the slave system (20) are synchronized. Fig. 3 shows the time evolution of the synchronization error. From this figure, we can conclude that the components of the error dynamics decay towards zero as t → ∞. Therefore, the designed OPCL con2 troller u (t ) = (0, (k1 + 2axm ym )(xs − xm ) + (k2 + axm )( y s − ym ) + 2 a1 (xm − 1) ym )T can effectively control the fractional Van der Pol oscillator to achieve synchronization between the systems (17) and (20). 4.2. Synchronization of the fractional unified system In 2002, Lü et al. [39] introduced the unified system which can unify the Lorenz, Chen and Lü systems. The fractional version of the unified system reads as:

⎧ q1 d x ⎪ ⎪ ⎪ q1 = (25α + 10)( y − x), ⎪ dt ⎪ ⎪ ⎨ q2 d y = (28 − 35α )x − xz + (29α − 1) y , ⎪ dt q2 ⎪ ⎪ ⎪ q 3 ⎪d z 8+α ⎪ ⎩ = xy − z. q dt

where a1 represents mismatch parameter about the standard parameter a. 2 Let f (xm ) = ( ym , −xm − a(xm − 1) ym )T , then the Jacobian matrix of f is

Df =

⎧ q1 d xs ⎪ ⎪ q = ys, ⎪ 1 ⎪ ⎪ ⎨ dt  2  dq 2 y s ⎪ dt q2 = −xs − a xs − 1 y s + (k1 + 2axm ym )(xs − xm ) ⎪ ⎪ ⎪   2   ⎪ 2 ⎩ ( y s − ym ) + a1 xm + k2 + axm − 1 ym .

3

(21)

3

Some dynamical investigations of this system have been given in Li et al. [40]. There the fractional unified system is classified as: when α ∈ [0, 0.8), system (21) belongs to fractional generalized Lorenz system; when α = 0.8, it becomes the fractional Lü system; and when α ∈ (0.8, 1], it belongs to the fractional generalized Chen system. For α = 0.8, we found that chaos exists in the fractional unified system with the fraction-order

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Fig. 5. Synchronization behavior of two nonidentical fractional unified systems (22) and (23) with initial conditions (xm (0), ym (0), zm (0), xs (0), y s (0), z s (0)) = (−2, 1, 26, −3, 2, 25).

Fig. 4. Chaotic attractor in the fractional unified system with (q1 , q2 , q3 ) = (0.97, 0.98, 0.96) and initial conditions (x(0), y (0), z(0)) = (−2, 1, 26). (a) Projection onto (x, y) plane; (b) projection onto (x, z) plane; (c) projection onto ( y , z) plane; (d) view on (x, y , z) space.

(q1 , q2 , q3 ) = (0.97, 0.98, 0.96) and the phase portraits are shown in Fig. 4. With designed OPCL coupling (10), the drive and the response fractional unified systems can be explicitly expressed as:

⎧ q1 d xm ⎪ ⎪ = (25α + 10)( ym − xm ) + 25α1 ( ym − xm ), ⎪ q1 ⎪ ⎪ ⎪ dt ⎨ q d 2 ym

= (28 − 35α )xm − xm zm + (29α − 1) ym + α1 xm , (22) ⎪ dt q2 ⎪ ⎪ ⎪ q3 ⎪ ⎪ ⎩ d zm = xm ym − 8 + α zm q dt

3

3

and

⎧ q1 d xs ⎪ ⎪ ⎪ dt q1 = (25α + 10)( y s − xs ) + 25α1 ( ym − xm ), ⎪ ⎪ ⎪ ⎪ dq 2 y ⎪ s ⎪ ⎪ = (28 − 35α )xs − xs zs + (29α − 1) y s ⎪ ⎪ q ⎪ ⎨ dt 2 + (k1 + zm )(xs − xm ) + (k2 + xm )( zs − zm ) + α1 xm , ⎪ ⎪ ⎪ ⎪ ⎪ dq 3 z s 8+α ⎪ ⎪ zs + (k3 − ym )(xs − xm ) ⎪ q 3 = xs y s − ⎪ ⎪ dt 3 ⎪ ⎪ ⎪ ⎩ + (k4 − xm )( y s − ym ), (23) respectively. Here α1 = 0.01 represents mismatch parameter about the standard parameter α . We can find the control parameters k1 , k2 , k3 and k4 such that the synchronization between systems (22) and (23) is achieved. For instance, when k1 = −25, k2 = k3 = k4 = 0, the corresponding eigenvalues of control matrix K are λ1 = −2.9, λ2,3 = −3.9 ± 8.3i. According to Theorem 1, we know that the synchronization error e (t ) = xs (t ) − xm (t ) converges to zero, thus the chaotic synchronization of the systems (22) and (23) is achieved. Numerical simulation results in Fig. 5 confirm this conclusion. 4.3. Synchronization of a novel fractional hyperchaotic system Recently, Chen et al. [41] proposed a novel hyperchaotic system which only has one unstable equilibrium and has bigger positive

Fig. 6. Chaotic attractor in the fractional hyperchaotic system with (q1 , q2 , q3 , q4 ) = (0.96, 0.97, 0.98, 0.99) and initial conditions (x(0), y (0), z(0), w (0)) = (5, 2, 4, 40). (a) Projection onto (x, y) plane; (b) projection onto (x, z) plane; (c) projection onto ( y , z) plane; (d) view on (x, y , z) space.

Lyapunov exponents than those already known hyperchaotic systems. It has been shown that this system can generate complex dynamics within wide parameter ranges, including periodic orbit, quasi-periodic orbit, chaos and hyperchaos. Now, we consider the fractional version of this novel hyperchaotic system, described by:

⎧ q1 d x ⎪ ⎪ = a( y − x) + μ yz, ⎪ ⎪ dt q1 ⎪ ⎪ ⎪ dq 2 y ⎪ ⎪ ⎪ ⎨ q = cx − dxz + y + w , dt

2

⎪ dq 3 z ⎪ ⎪ ⎪ ⎪ dt q3 = xy − bz, ⎪ ⎪ ⎪ q4 ⎪ ⎪ ⎩ d w = −v y.

(24)

dt q4

The parameters are chosen to be a = 35, b = 4, c = 25, d = 5, μ = 35, ν = 100 so that system (24) in its integer version can exhibit hyperchaotic behaviors. Through numerical simulations, we find that system (24) will exhibit hyperchaotic behavior when (q1 , q2 , q3 , q4 ) = (0.96, 0.97, 0.98, 0.99), see Fig. 6. We define the drive fractional hyperchaotic system as:

J.-W. Wang, Y.-B. Zhang / Physics Letters A 374 (2009) 202–207

207

periodic and chaotic fractional-order systems. In addition, we performed some numerical simulations of paradigmatic examples including a limit cycle, a chaotic system and a hyperchaotic system to verify the effectiveness of the proposed synchronization scheme. The results of this work indicate that the OPCL control method is a powerful tool not only for integer-order nonlinear systems [31–34], but also for fractional-order nonlinear systems. Acknowledgements

Fig. 7. Synchronization behavior of nonidentical fractional hyperchaotic systems (25) and (26) with initial conditions (xm (0), ym (0), zm (0), w m (0), xs (0), y s (0), z s (0), w s (0)) = (5, 2, 4, 40, 4, 1, 3, 38).

⎧ q1 d xm ⎪ ⎪ = a( ym − xm ) + μ ym zm + a1 ( ym − xm ), ⎪ ⎪ dt q1 ⎪ ⎪ ⎪ dq 2 y ⎪ ⎪ m ⎪ = cxm − dxm zm + ym + w m , ⎨ q dt

2

⎪ dq3 zm ⎪ ⎪ ⎪ ⎪ dt q3 = xm ym − bzm + b1 zm , ⎪ ⎪ ⎪ q4 ⎪ ⎪ ⎩ d w m = − v ym ,

References

(25)

dt q4

where a1 = 0.01 and b1 = 0.01 represent mismatch parameters about the standard parameters a and b, respectively. Based on the OPCL coupling scheme (10), the response fractional hyperchaotic system can be written as:

⎧ dq 1 x s ⎪ ⎪ ⎪ dt q1 = a( y s − xs ) + μ y s zs + (k1 − μ zm )( y s − ym ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + (k2 − μ ym )( zs − zm ) + a1 ( ym − xm ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dq 2 y s ⎪ ⎪ ⎪ = cxs − dxs zs + y s + w s + (k3 + dzm )(xs − xm ) ⎪ q2 ⎪ ⎪ ⎨ dt + (k4 + dxm )( zs − zm ), ⎪ ⎪ ⎪ q3 ⎪ d zs ⎪ ⎪ ⎪ = xs y s − bzs + (k5 − ym )(xs − xm ) ⎪ ⎪ dt q3 ⎪ ⎪ ⎪ ⎪ ⎪ + (k6 − xm )( y s − ym ) + b1 zm , ⎪ ⎪ ⎪ ⎪ ⎪ q4 ⎪ ⎪ ⎩ d w s = −v y . s q dt

The authors would like to thank the referee and the editor for their valuable comments and suggestions. This work was supported by the State Key Program of National Natural Science Foundation of China (Grant No. 60736028), the National Natural Science Foundation of China (Grant Nos. 10871074, 60704045), and Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20070558053).

(26)

4

Here we set k1 = −40, k2 = k3 = k4 = k5 = k6 = 0 and the corresponding eigenvalues of control matrix K can be calculated as λ1 = −4, λ2 = −31.5, λ3,4 = −1.25 ± 10.47i. According to Theorem 1, we know that hyperchaotic synchronization of the systems (25) and (26) has been achieved. The simulation results with the initial values (xm (0), ym (0), zm (0), w m (0)) = (5, 2, 4, 40) and (xs (0), y s (0), zs (0), w s (0)) = (4, 1, 3, 38) are illustrated in Fig. 7 which show that the synchronization error e (t ) = xs (t ) − xm (t ) indeed tends to zero. 5. Conclusions The master–slave synchronization of two nonidentical fractional-order nonlinear systems has been studied in this Letter. An OPCL-based controller applied to the response system has been presented to ensure the synchronization behavior. This coupling scheme is proposed for incommensurate fractional-order systems, but in the special case it converts to the condition for the commensurate fractional-order systems. Such an OPCL-based coupling approach is general in the sense that it can be applied to both

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