Secure communication based on a four-wing chaotic system subject to disturbance inputs

Optik 125 (2014) 5920–5925

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Secure communication based on a four-wing chaotic system subject to disturbance inputs Fei Yu a,∗ , Chunhua Wang b a b

School of Computer and Communication Engineering, Changsha University of Science and Technology, Changsha 410014, China College of Information Science and Engineering, Hunan University, Changsha 410082, China

a r t i c l e

i n f o

Article history: Received 10 September 2013 Accepted 10 May 2014 PACS: 05.45.−a 05.45.Gg 05.45.Jn 05.45.Pq 05.45.Xt 05.45.Vx

a b s t r a c t In this paper, a two-input two-output secure communication scheme based on a four-wing fourdimensional chaotic system with disturbance inputs is discussed. Based on parameter modulation theory and Lyapunov stability theory, synchronization and secure communication between transmitter and receiver are achieved and two message signals are recovered via a convenient robust high-order sliding mode adaptative controller. In addition, the gains of the receiver system can be adjusted continually, the unknown parameters can be identified precisely and the disturbance inputs can be suppressed simultaneously by the proposed adaptative controller. Synchronization under the effect of noise is also considered. Computer simulations are done to verify the proposed methods and the numerical results show that the obtained theoretic results are feasible and efficient. © 2014 Elsevier GmbH. All rights reserved.

Keywords: Four-wing chaotic system Secure communication High-order sliding mode adaptative controller Parameter modulation Disturbance inputs

1. Introduction Since the pioneering work by Pecora and Carrol [1], synchronization of chaotic systems and its potential application to secure communication have received a lot of attentions. The idea of synchronization is to use the output of the drive system to control the response system so that the output of the response system follows the output of the drive system asymptotically [2]. In the past decades, a variety of many methods and techniques for handling chaos control and synchronization of various chaotic systems have been developed, such as OGY method [3], backstepping technique [4], active control method [5], sliding mode control method [6–8], adaptive control method [9,10], linear and nonlinear feedback control method [11,12], inverse control method [13], H∞ technique [14], etc. Recently, a new chaos synchronization

∗ Corresponding author. Tel.: +86 0731 85258462; fax: +86 0731 85258217. E-mail address: [email protected] (F. Yu). http://dx.doi.org/10.1016/j.ijleo.2014.08.001 0030-4026/© 2014 Elsevier GmbH. All rights reserved.

control method, high-order sliding mode technique, which has been actively researched [15–17], consider a fractional power of the absolute value of the tracking error coupled with the sign function. The signal encryption scheme by using this technique lends itself to cheap implementation and can therefore be used effectively for ensuring security and privacy in commercial consumer electronics products [17]. Compared with the general kind of sliding-mode structures, this structure provides several advantages such as simplification of the control law, higher accuracy and chattering prevention [15]. However, most of these research results are realized without any disturbance inputs [15–17]. It is well known that the noise disturbance is inevitable in the practical situations. So synchronization and secure communication of concrete models is unavoidably subject to internal and external noise disturbance. And a major problem in designing chaos-based secure communication systems can be stated as how to send a secret message from the transmitter to the receiver in the practical situations while achieving security, maintaining privacy, and providing good noise rejection [18]. Therefore investigation of synchronization and

F. Yu, C. Wang / Optik 125 (2014) 5920–5925

secure communication for the chaotic systems by the impact of disturbance inputs and channel noise has become an important research topic. Meanwhile, many different types of chaos-based secure communication schemes have been proposed and generally categorized into four different generations. For a recent survey one can refer to [19]. The first generation generally called chaotic masking [20], an addition method was based on simply adding the secret message to one of the chaotic states of the transmitter and then the composite signal was sent to the receiver, provided that the message strength was much weaker than that of the chaotic state [21]. Due to its sensitivity to channel noise and parameters mismatch between the transmitter and the receiver, this technique was impractical and have been proved to have poor security [22]. In order to improve the security, a multistage chaos synchronized system that was applied to secure communication through chaotic masking was discussed in [23]. Another method that was aimed at digital information signals, called chaotic switching or chaotic shift keying, was developed a shift keying technique for the coding/decoding of binary signals where the carrier signal was chaotic [24]. Although chaotic switching was more robust against noise than chaotic masking, it suffered from a lower information transmission rate. Chaotic parameter modulation [25] was called the second generation, where the information signal was used to modulate one of the parameters of the chaotic transmitter. At the receiver, some adaptive control methods were applied to synchronize the system in the receiver with the chaotic system in the transmitter and hence to recover the message from the adaptation rules [26]. In [27], an observer was presented to identify the unknown parameter of Liu chaotic system. Recently, another variant to this method, based on generalized function projective synchronization, was introduced in [28]. In this method, the responses of the synchronized dynamical states synchronize up to a function matrix. The unpredictability of the scaling functions can additionally strengthen the security of communication, which could be employed to get more secure communications. Combined the advantages of chaos and cryptography, a new chaos encryption technology called chaotic cryptosystems [29], was belonged to the third generation, which has very high safety performance. In this method, many nonlinear encryption methods were used to scramble the secure message at the transmitter side, while using an inverse operation at the receiver side that can effectively recover the original message, provided that synchronization was achieved [18,21]. A versatile combination of the parameter modulation technique and cryptography was proposed in [18]. Based on the impulsive synchronization, new techniques of chaotic communication were belonged to the fourth generation [30]. These systems have the advantage of reducing the information redundancy in the transmitted signal as only synchronization impulses sent to the driven system, which could increase the bandwidth utilization. Inspired by the above discussions, in this paper, we consider a two-input two-output secure communication scheme based on a four-wing four-dimensional (4D) chaotic system via a robust high order sliding mode adaptative controller, and investigate the identification of the four-wing system with partly uncertain parameters and disturbance inputs, and an application in secure communications via chaotic parameter modulation. By utilizing parameter modulation theory and Lyapunov stability theory, synchronization and secure communication between transmitter and receiver is achieved and two message signals are recovered accurately. In addition, the gains of the receiver system can be adjusted continually, the unknown parameters can be identified precisely and the disturbance inputs can be suppressed simultaneously by the adaptative controller. The corresponding theoretical proofs and numerical simulations are performed to validate the effectiveness and feasibility of the presented secure communication scheme.

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2. Four-wing 4D chaotic system and its dynamics Consider a four-wing 4D chaotic system:

⎧ x˙ = −ax + yz, ⎪ ⎪ ⎨ y˙ = by − xz,

z˙ = xy − cz + dw, ⎪ ⎪ ⎩

(1)

w˙ = xy − ew,

where a, b, c, d, e are system parameters, x, y, z, w are the state variables. When a = 10, b = 12, c = 60, d = 2 and e = 3, the Lyapunov exponents of system (1) are found to be l1 = 2.6692, l2 = 0, l3 = −3.7703 and l4 = −57.7174. It can be seen that there is a larger positive Lyapunov exponent that means system (1) can exhibit extremely rich dynamics. A four-wing chaotic attractor from system (1) is shown in Fig. 1 while the arbitrary initial conditions are selected as [2, 1, 2, 2]T . From Fig. 1, we can see that in any three-dimensional (3D) spaces, the chaotic attractor can display a four-wing type. Obviously that the system is asymmetric about (x, y, z, w) → (±x, ∓y, −z, −w), which persists for all values of the sys˙ ∂z) + ˙ ∂x) + (∂y/ ˙ ∂y) + (∂z/ tem parameters. Note that ∇ V = (∂x/ ˙ ∂w) = −a + b − c − e, so the system is dissipative as long as (∂w/ −a + b − c − e < 0. That means a volume element V0 is contracted by the flow into a volume element V0 −a+b−c−e in time t. When the parameters a = 10, b = 12, c = 60, e = 3 are fixed while d varies on the closed interval [− 5, 15]. Figs. 2 and 3 show the bifurcation diagram of the state X and the corresponding Lyapunov exponent spectrum versus increasing d respectively. From Figs. 2 and 3, it can be seen that the bifurcation diagram well coincides with the spectrum of the Lyapunov exponents. Interestingly, this chaotic system can display one-, two-, three- and four-wing attractors by varying one single parameter d while the others are fixed. The corresponding

Fig. 1. Four-wing chaotic attractors of system (1) in 3D spaces.

Fig. 2. Bifurcation diagram for increasing parameter d.

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3. A two-input two-output secure communication scheme In this section, it starts with introduction of some definitions, then we propose a fast mechanism for synchronizing the proposed four-wing chaotic system based on the drive-response approach. By using a convenient high-order sliding mode adaptative controller, a two-input two-output secure communication scheme is studied. 3.1. Some definitions Definition 2. [17] Consider a smooth nonlinear chaotic system in the form of: X˙ = f (X, P), Fig. 3. Lyapunov exponent spectrum for increasing parameter d.

Y˙ = h(X),

(2) ]T

⊂ Rn

is the state vector, Y = [x1 , x2 , . . ., where X = [x1 , x2 , . . ., xn xm ]T ⊂ Rm is the output state vector with m ≤ n. f(◦) and h(◦) are smooth vector functions. P ⊂ Rl is a parameter vector with l ≤ n. Let D(j) denote the jth time derivative of the vector D. We say that the vector state X is algebraically observable, if it can be uniquely expressed as: X=

(D, D(1) , . . ., D(j) )

T

where j is an integer and

(3) is smooth function.

Definition 3. [17,31] If P satisfies the following relation under the same conditions as in Definition 1: ϕ1 (D, D(1) , . . ., D(j) ) = ϕ2 (D, D(1) , . . ., D(j) )P,

(4)

where ϕ1 (◦) and ϕ2 (◦) are n × 1 and n × n smooth matrices, respectively, then P is said to be algebraically identifiable with respect to the output vector D. Now, we rewrite the second differential equation of system (1) as: Fig. 4. Multi-wing chaotic attractors of system (1) projection on x − z plane. (a) one-wing attractor with d = −1.8, (b) two-wing attractor with d = 0.2, (c) three-wing attractor with d = −0.8, and (d) four-wing attractor with d = 2.

z=

by − y˙ , x

(5)

then substituting Eq. (5) into the first differential equation of system (1), we have: chaotic attractors are depicted in Fig. 4. Fig. 5 shows the behavior of the whole state of the four-wing system. It can be seen that every variable of all orbits can freely move across the boundary to the opposite side. Remark 1. From Figs. 1 and 5, it is evident that the four states of system (1) are bounded when a = 10, b = 12, c = 60, d = 2, e = 3 and the initial conditions as [2, 1, 2, 2]T . In fact, for a large set of initial conditions and for a large set of system parameters, in most case all the states of system (1) are bounded.

xx˙ + yy˙ = by2 − ax2 .

(6)

According to the Definitions, it is obvious that system (1) is algebraically observable with respect to the two outputs x and y. From Eq. (6), it is further indicates that system (1) vector of parameters P = [a, b]T is algebraically identifiable with respect to the two available outputs. Hence, the non-available state z, w and the vector of parameters P can be simultaneously recovered from the two available outputs. 3.2. Design of the transmitter At the transmitter side, we consider using the parameter a, b to transmit two message signals respectively and define the following master system with disturbance inputs:

⎧ x˙ 1 = −a(t)x1 + y1 z1 + d1 , ⎪ ⎪ ⎪ ⎪ ⎨ y˙ 1 = b(t)y1 − x1 z1 + d2 ,

⎪ z˙ 1 = x1 y1 − 60z1 + 2w1 + d3 , ⎪ ⎪ ⎪ ⎩

(7)

w˙ 1 = x1 y1 − 3w1 + d4 ,

where x1 , y1 , z1 , w1 are system state variables of the four-wing system with the modulation rules: a(t) = a + sa (t), Fig. 5. Time sequence of system (1). (a) t − x wave, (b) t − y wave, (c) t − z wave, and (d) t − w wave.

b(t) = b + sb (t),

(8)

here, sa (t) and sb (t) are two input message signals which are used to transmit. D = [d1 , d2 , d3 , d4 ]T represent the external uncertainties,

F. Yu, C. Wang / Optik 125 (2014) 5920–5925

respectively, and the boundaries are satisfied by |di | ≤ i (i = 1, 2, 3, 4) and assume that  = [1 , 2 , 3 , 4 ]T and i (i = 1, 2, 3, 4) ≥ 0 are given.

At the receiver side, we define the slave system to be the partly uncertain four-wing system (1) with the available output states x1 and y1 . Therefore, the slave system with controllers is defined as follows:

⎧ x˙ 2 = −ˆa (t) x1 + y1 z2 + u1 , ⎪ ⎪ ⎪ ⎪ ⎨ y˙ = bˆ (t) y − x z + u , 1

1 2

⎡

where x2 , y2 , z2 , w2 are system state variables of the four-wing system with the modulation rules: ˆ b(t) = bˆ + sˆb (t),

(10)

here, aˆ and bˆ are the estimates of the unknown parameters a and b, respectively; sˆa (t) and sˆb (t) are the two recovered message signals. U = [u1 , u2 , u3 , u4 ]T is the controller to be designed such that systems (7) and (9) can be synchronized.

a˜ b˜˙

Proof.

⎢ ⎢ e˙ y e˙ = ⎢ ⎢ e˙ ⎣ z

−˜ax1 − s˜a (t)x1 + y1 ez + d1 − u1

⎥ ⎢˜ ⎥ ⎢ by1 + s˜b (t)y1 − x1 ez + d2 − u2 ⎥=⎢ ⎥ ⎢ −60e + 2e + d − u z w 3 3 ⎦ ⎣

⎤

⎡

x1 − x2

⎢ ⎥ ⎢ ⎢ ey ⎥ ⎢ y1 − y2 ⎥ ⎢ e=⎢ ⎢ e ⎥ = ⎢z −z z ⎣ ⎦ ⎣ 1 2 ew



, s˜˙ (t) =

−y1 ey

s˜˙ a (t) s˜˙ b (t)





=

x1 ex



−y1 ey

.

(13)

Consider the following Lyapunov function candidate: (14)

= ex e˙ x + ey e˙ y + ez e˙ z + ew e˙ w + a˜ a˜˙ + b˜ b˜˙ + s˜a (t)s˜˙ a (t) + s˜b (t)s˜˙ b (t)

˜ 1 + ey s˜b (t)y1 − ey x1 ez + ey d2 − ey k2 sign(ey )e +ey by y −2 ey sign(ey ) − 60ez2 + 2ez ew + ez d3 − ez y1 ex + x1 ez ey + 60ez2 −3 ez sign(ez ) −

⎤ ⎥ ⎥ ⎥, ⎥ ⎦

2 3ew

+ ew d4 − 2ew ez +

32w

(15)

− 4 ew sign(ew )

˜ 1 ey + x1 ex s˜a (t) − y1 ey s˜b (t) +˜ax1 ex − by = −k1 ex sign(ex )ex − k2 ey sign(ey )ey + eT D − eT ,

(11) where

= sign[diag(ex , ey , ez , ew )]. eT D,

ex



= −ex a˜ x1 − ex s˜a (t)x1 + ex y1 ez + ex d1 − ex k1 sign(ex )ex − 1 ex sign(ex )

where

⎡

=

x1 ex

By calculating the derivative of V(t) along the trajectories of the error system (11), and using Eqs. (12) and (13), we have:

−3ew + d4 − u4

e˙ w



 1 T ˜ + s˜ (t)T s˜ (t) . ˜Tp V(t) = e e+p 2

Subtracting the master system (7) from the slave system (9), we obtain the following error system:

⎡

(12)

Then the response system (9) can synchronize the drive system (7) globally and asymptotically with disturbance inputs and any strictly positive constants k1 , k2 and any even integer . Furthermore, the receiver system (9) can recover the message signals sa (t) and sb (t) which are embedded in the chaotic transmitter (7) via the modulation rules (8) and (10), respectively.

3.4. Design of the error system

⎤

⎤

k1 sign(ex )ex + 11 sign(ex )

2ez − 3ew + 14 sign(ew )

˙

˙ V(t)

e˙ x

⎡

where k = [k1 , k2 ]T is the control gain and  ∈ Z+ ,  > 1 being  an even number, sign(◦) denotes the sign function, and the adaptive laws of parameters and message signals are taken respectively as:

(9)

w˙ 2 = x1 y1 − 3w2 + u4 ,

⎡

⎤

u1

u4

˜˙ = p

2

⎪ z˙ 2 = x1 y1 − 60z2 + 2w2 + u3 , ⎪ ⎪ ⎪ ⎩

aˆ (t) = aˆ + sˆa (t),

Theorem 4. If the high-order sliding mode adaptative controller U is designed as follows:

⎢ ⎥ ⎢ ⎥ ⎢ u2 ⎥ ⎢ k2 sign(ey )ey + 12 sign(ey ) ⎥ ⎢ ⎥ ⎢ ⎥, U=⎢ ⎥=⎢ ⎥ u e − x e − 60e +  sign(e ) y x y z z 3 1 13 ⎣ ⎦ ⎣ 1 ⎦

3.3. Design of the receiver

2

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⎤

eT

(16)

eT

Here ∈ R and ≥ 0. According to the definition and assumption of D and , it is guaranteed that eT D ≤ eT , therefore it

    ⎥

s˜ (t) sa (t) − sˆa (t) ⎥ ⎥ , p˜ = a˜ = a − aˆ , s˜ (t) = a = . ⎥ b˜ b − bˆ s˜b (t) sb (t) − sˆb (t) ⎦

w1 − w2

As we can see, we know that the synchronization between two chaotic systems with unknown parameters is turned into a problem to choose a controller U and a corresponding parameter estimate law to make error system asymptotically converge to zero.

3.5. Design of the high-order sliding mode adaptative controller High-order sliding control technique provides several advantages, such as the upper bound of the convergence time is known and can be adjusted in advance, the condition on the gain implies that its tuning is constructive, and the structure of the controller is well-adapted to practical implementations [32,33]. For a detailed theoretical analysis one can refer to [34]. Under this technique, the following theorem gives the results of this paper.

can be obtained that: ˙ V(t)

= −[k1 ex sign(ex )ex + k2 ey sign(ey )ey ] + eT D − eT ≤ −[k1 ex sign(ex )ex + k2 ey sign(ey )ey ] =

−(k1 |ex |ex

(17)

+ k2 |ey |ey ).

˙ is a negative definite in the neighborhood of zero Therefore, V(t) ˙ solution of the error system (11). In fact, as V(t) < 0, then ex , ey ∈ L∞ . From the error system (11), we have e˙ x , e˙ y ∈ L∞ . Integrating both sides of (17), we have:



t

(k1 |ex (t)|ex (t) + k2 |ey (t)|ey (t))dt ≤ V (0).

(18)

0

According to Barbalat’s Lemma, we get e˙ → ∞ as t→ ∞. Therefore, synchronization between the transmitter system (7) and the

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F. Yu, C. Wang / Optik 125 (2014) 5920–5925

Fig. 6. Synchronization errors when the message signals are trigonometric signals.

Fig. 7. Estimation of uncertain parameters (a), (b) and the recovered signals errors (c), (d) when the message signals are trigonometric signals.

receiver system (9) subject to disturbance inputs is realized. This completes the proof.  in Eq. (16) is the compensator that is introduced to Remark 5. eliminate the influence of the external uncertainties. From Remark 1, it follows that the steady states x1 and y1 oscillate around zero. From the above discussion, we also get that e˙ is bounded which means that e is uniformly continuous. By employing Barbalat’s Lemma, it implies that e→ ∞ as t→ ∞. Differentiating (11) and using the same methods, we also obtain that e¨ is bounded and e˙ → ∞ as t→ ∞. Since V(t) converges, as t→ ∞, then we have ˜ and two signal errors converge s˜ (t) that the two parameter errors p ˜˙ and s˜˙ (t) converge to zero, as as t→ ∞. From (13), it follows that p t→ ∞. Therefore, the uncertain parameters a(t) and b (t) are identified in the receiver end and the receiver system (9) can recover the message signals sa (t) and sb (t) simultaneously. 4. Numerical simulations In this section, we are confirmed our analytical studies by numerical simulations. The fourth-order Runge–Kutta method is used to solve the nonlinear system with time step size 0.001. At the transmitter system side, the system parameters are chosen as a = 10, b = 12 while the arbitrary initial conditions are set as x1 (0) = 1, y1 (0) = 2, z1 (0) = 3, w1 (0) = 4. At the receiver system side, we fix the arbitrary initial conditions as x2 (0) = T ˆ ˆ 5, y2 (0) = 6, z2 (0) = 7, w2 (0) = 8, p(0) = [ˆa(0), b(0)] = [0.1, 0.1]T T

T

and sˆ (0) = [ˆsa (0), sˆb (0)] = [0.1, 0.1] . To consider the robustness of the scheme against channel noise, the additive white Gaussian noises with the strength 5 percent of x1 and y1 are added to time series in y1 and x1 respectively, and the corresponding numerical simulations are given as follows:

Fig. 8. Synchronization errors when the message signals are square signals.

unknown parameters converge to a stable value asymptotically. Fig. 7(c) and (d) displays the error signal between the original message signal and the recovered one. It is easy to see that the message signals sa (t) and sb (t) are recovered accurately. (ii) Suppose the message signals sa (t) and sb (t) are square signals with frequency of 100 Hz and 90 Hz, respectively, i.e.:



sa (t) = 0.06square (200t) ,

(21)

sb (t) = 0.08square (180t) . And the disturbance inputs are set as: D(t) = [d1 , d2 , d3 , d4 ]

T

= [4 sin(10t), −3.3 cos(20t), −2.5 cos(30t), 3 sin(10t)]T . (22) ]T

(i) Suppose the message signals sa (t) and sb (t) are trigonometric signals, here we chose one of sinusoidal signal with frequency of 90 Hz, while the other was cosine signal with frequency of 100 Hz, such as:



sa (t) = 0.08 sin(180t), sb (t) = 0.06 cos(200t).

1]T

The receiver system gains are set as k = [k1 , k2 = [1, and  = 6. Fig. 8 displays the synchronization errors between systems (7) and (9). It is indicated that synchronized errors between systems (7) and (9) converge to zero asymptotically as

(19)

And the disturbance inputs are set as: D(t) = [d1 , d2 , d3 , d4 ]

T

= [−3 cos(20t), 2 sin(10t), − sin(10t), 2 cos(20t)]T .

(20)

Meanwhile, we fix the receiver system gains as k = [k1 , k2 ]T = [0.6, 0.8]T and  = 4. Fig. 6 displays the synchronization errors between systems (7) and (9). It is shown that synchronized errors between systems (7) and (9) converge to zero asymptotically. Fig. 7(a) and (b) shows the estimations of the

Fig. 9. Estimation of uncertain parameters (a), (b) and the recovered signals errors (c), (d) when the message signals are square signals.

F. Yu, C. Wang / Optik 125 (2014) 5920–5925

t→ ∞. Fig. 9(a) and (b) shows the estimations of the unknown parameters converge to a stable value asymptotically. The convergence behavior of the message recovery errors are plotted in Fig. 9(c) and (d). It is easy to find that the message signals sa (t) and sb (t) are recovered accurately after a short transient. 5. Conclusion In this paper, a four-wing four-dimensional chaos and its basic dynamics are firstly presented. Then we propose a two-input two-output secure communication scheme based on the partly unknown parameters of the four-wing system. Via a convenient robust high order sliding mode adaptative controller, synchronization and secure communication between transmitter and receiver is achieved, the parameters of the receiver system are identified and the disturbance inputs are suppressed simultaneously. Meanwhile, two information signals in the receiver can be recovered successfully on the basis of the estimated parameters. As typical message signals, trigonometric signals and square signals are used as examples in numerical simulations and the results show the effectiveness and feasibility of the proposed secure communication scheme. References [1] L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990) 821–824. [2] N. Cai, Y. Jing, S. Zhang, Modified projective synchronization of chaotic systems with disturbances via active sliding mode control, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1613–1620. [3] E. Ott, C. Grebogi, J.A. Yorke, Controlling chaos, Phys. Rev. Lett. 64 (1990) 1196–1199. [4] H.T. Yu, J. Wang, Chaos synchronization of FitzHugh-Nagumo neurons via backstepping and adaptive dynamical sliding mode control, Acta Phys. Sin. 62 (2013) 170511. [5] Y. Lan, Q. Li, Chaos synchronization of a new hyperchaotic system, Appl. Math. Comput. 217 (2010) 2125–2132. [6] D.F. Wang, J.Y. Zhang, X.Y. Wang, Synchronization of uncertain fractional-order chaotic systems with disturbance based on a fractional terminal sliding mode controller, Chin. Phys. B 22 (2013) 040507. [7] M. Haeri, A.A. Emadzadeh, Synchronizing different chaotic systems using active sliding mode control, Chaos Solitons Fractals 31 (2007) 119–129. [8] M.S. Tavazoei, M. Haeri, Synchronization of chaotic fractional-order systems via active sliding mode controller, Physica A 387 (2008) 57–70. [9] T. Liang, Q. He, H. Li, L. He, A novel adaptive synchronization algorithm for intermediate frequency architecture CO-OFDM system, Optik 124 (2013) 3406–3411. [10] D.W. Zhu, L.L. Tu, Adaptive synchronization and parameter identification for Lorenz chaotic system with stochastic perturbations, Acta Phys. Sin. 62 (2013) 050508. [11] C. Li, K. Su, J. Zhang, D. Wei, Robust control for fractional-order four-wing hyperchaotic system using LMI, Optik 124 (2013) 5807–5810.

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