The synchronization of fractional-order Rössler hyperchaotic systems

Physica A 387 (2008) 1393–1403 www.elsevier.com/locate/physa

The synchronization of fractional-order R¨ossler hyperchaotic systemsI Yongguang Yu a,c,∗ , Han-Xiong Li b,c a Department of Mathematics, Beijing Jiaotong University, Beijing 100044, PR China b School of Mechanical and Electrical Engineering, Central South University, Changsha, 410083, PR China c Department of MEEM, City University of Hong Kong, Kowloon, Hong Kong, China

Received 18 July 2007; received in revised form 12 October 2007 Available online 20 November 2007

Abstract The synchronization of fractional-order hyperchaotic systems is studied, using the R¨ossler system as an example. Based on the Laplace transformation theory, sufficient conditions for global synchronization of the systems are given analytically. Also, the variational iteration method is implemented to give the approximate solution for the fractional-order error system of the two identical hyperchaotic systems, which is in good agreement with the approximate solution using the classical Laplace transformation method. Numerical methods and simulations on the master–slave systems are presented to verify the results obtained. c 2007 Elsevier B.V. All rights reserved.

PACS: 05.45.-a Keywords: Fractional-order differential equation; Variational iteration method; R¨ossler system; Hyperchaos; Synchronization

1. Introduction The chaotic dynamics of fractional-order systems began to attract much attention in recent years [1,2]. It has been shown that the fractional-order generalizations of many well-known systems can also behave chaotically [3–13]. Ref. [3] showed that the fractional-order Chua’s system with order as low as 2.7 can produce a chaotic attractor. In Ref. [4], chaotic behavior of the fractional-order Lorenz system was studied. The fractional-order Chen chaotic system was also investigated in Refs. [5,6], and Ge et al. [7–12] investigated the chaotic behaviors of fractionalorder modified Duffing systems, modified van der Pol system and nonlinear damped Mathieu system, respectively. Especially, in Ref. [13], chaos and hyperchaos in the fractional-order R¨ossler system were studied, in which chaos exists in the fractional-order system with order as low as 2.4, and hyperchaos exists in the system with order as low as 3.8. I Supported by the National Natural Science Foundation of China (Grant No. 10461002 and No. 50775244) and Beijing Jiaotong University Science Foundation of China (No. 2005sm063). ∗ Corresponding author at: Department of Mathematics, Beijing Jiaotong University, Beijing 100044, PR China. E-mail address: [email protected] (Y. Yu).

c 2007 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2007.10.052

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But, there are many material differences between the ordinary differential equation systems (integer-order) and the corresponding fractional-order differential equation systems. Most of the properties or conclusions of the integer-order system cannot be simply extended to the case of the fractional-order one. Most of the fractional-order differential equations do not have exact analytic solutions, so approximation and numerical techniques must be used. Several analytical and numerical methods have been proposed to solve the fractional-order differential equations. Besides some classical solution techniques, e.g. Laplace transform method, Fractional Green’s function method, Mellin transform method and method of orthogonal polynomial [1], there are still the most commonly used methods: adomian decomposition method (ADM) [14–16], fractional difference method (FDM) [1], variation iteration method [17–27] (VIM) and homotopy perturbation method (HPM) [28–32]. Among the above-mentioned techniques, the VIM and HPM are the most effective and convenient ones for weakly and strongly nonlinear equations. The VIM and HPM are firstly proposed by He in 1998 and 1999, respectively [19, 28]. The VIM is to construct correction functionals using the general Lagrange multipliers identified optimally via the variational theory, and initial approximations can be freely chosen with unknown constants. It is particularly valuable as tools for scientists and applied mathematicians, and can provide immediate and visible symbolic terms of analytic solutions, as well as numerical approximate solutions to differential equations without linearization or discretization. The HPM is a coupling of the traditional perturbation method and homotopy in topology, deforms continuously to a simple problem which is easily solved. It can provide an analytical approximate solution to a wide range of nonlinear problems in applied sciences, and yield a very rapid convergence of solution series in most cases, usually only a few iterations leading to very accurate solutions. In Ref. [33], Li et al. firstly investigated the synchronization of fractional-order chaotic systems. It has recently attracted increasing attention due to its potential applications in secure communication and control processing [33–39]. But in many literatures [9,10,33–39], the synchronization among the fractional-order systems is only investigated through numerical simulations. In this paper, we focus on the synchronization of fractional-order R¨ossler hyperchaotic systems. A hyperchaotic attractor is characterized as a chaotic attractor with more than one positive lyapunov exponents which can increase the randomness and higher unpredictability of the corresponding system. So the hyperchaos may be more useful in some fields such as communication, encryption etc. To the best of our knowledge, there are few papers to investigate the hyperchaos synchronization for fractional-order systems [40]. Based on the Laplace transformation theory, we obtain the sufficient conditions for realizing the synchronization between two identical hyperchaotic systems. Especially, we present the approximate solution of the fractional-order hyperchaotic systems by means of the efficient VIM. The paper is organized as follows. In Section 2, the definition of fractional-order derivatives and the overview of VIM are given. In Section 3, the fractional-order R¨ossler hyperchaotic system is introduced. The globally synchronization of two fractional-order R¨ossler hyperchaotic systems is studied in Section 4. In Section 5, the numerical solution of the master–slave systems and the approximate solution of error system using VIM are proposed, respectively. Finally, conclusions are drawn in Section 6. 2. Variational iteration method There are many definitions about fractional-order derivatives. In this paper, the Riemann–Liouville definition [1] is mainly used: D α x(t) =

1 Γ (m − α)

t

Z 0

x (m) (u) du, (t − u)α−m+1

(1)

where m = dαe, i.e. m is the first integer of not less than α. It is easy to prove that the definition is the usual derivatives definition when α = 1. The case 0 < α < 1 seems to be particularly important, but there are also some applications for α > 1. Without loss of generality, in the following we assume that 0 < α < 1. The variational iteration method was first proposed by He [17–22] and was successfully applied to various kinds of differential equations [17–27] which include nonlinear fractional-order differential equations. To give the approximate solution of nonlinear fractional-order differential equations by means of the VIM, we write the system in the form

Y. Yu, H.-X. Li / Physica A 387 (2008) 1393–1403

 α D 1 x1 (t) = f 1 (x1 , x2 , . . . , xn ) + g1 (t)     D α2 x1 (t) = f 2 (x1 , x2 , . . . , xn ) + g2 (t) ..  .    αn D x1 (t) = f n (x1 , x2 , . . . , xn ) + gn (t),

1395

(2)

where 0 < αi ≤ 1 (i = 1, 2, . . . , n), subject to the initial conditions x1 (0) = c1 ,

x2 (0) = c2 , . . . xn (0) = cn .

The corresponding correction functional for system (2) can be approximately constructed as Z t  k+1 k  λ1 (D α1 x1k (τ ) − f 1 (x˜1k (τ ), x˜2k (τ ), . . . , x˜nk (τ )) − g1 (τ ))dτ x1 (t) = x1 (t) +   0  Z  t   x k+1 (t) = x k (t) + λ2 (D α2 x2k (τ ) − f 2 (x˜1k (τ ), x˜2k (τ ), . . . , x˜nk (τ )) − g2 (τ ))dτ 2 2 0

 ..   .   Z t    x k+1 (t) = x k (t) + λn (D αn xnk (τ ) − f n (x˜1k (τ ), x˜2k (τ ), . . . , x˜nk (τ )) − gn (τ ))dτ, n n

(3)

(4)

0

where λ1 , λ2 , . . . λn are the general Lagrange multipliers which can be identified optimally via variational theory [41], and x˜1 (τ ), x˜2 (τ ), . . . x˜n (τ ) denote restricted variations. Making the above functionals stationary, we can obtain the following stationary conditions  0 λi (τ ) = 0 1 + λi (τ )|τ =t = 0 for i = 1, 2, . . . , n. Therefore, the Lagrange multipliers can be easily identified as λi = −1,

i = 1, 2, . . . , n.

Thus, we can obtain the following iteration formulas: Z t  k+1 k  (D α1 x1k (τ ) − f 1 (x1k (τ ), x2k (τ ), . . . , xnk (τ )) − g1 (τ ))dτ  x1 (t) = x1 (t) −  0  Z  t   x k+1 (t) = x k (t) − (D α2 x2k (τ ) − f 2 (x1k (τ ), x2k (τ ), . . . , xnk (τ )) − g2 (τ ))dτ 2 2 0

 ..   .   Z t    x k+1 (t) = x k (t) − (D αn xnk (τ ) − f n (x1k (τ ), x2k (τ ), . . . , xnk (τ )) − gn (τ ))dτ. n n

(5)

0

According to the results of VIM, we can completely determine the approximations x1k , x2k , . . . , xnk when the initial approximations are x10 = c1 , x2k = c2 , . . . , xnk = cn . Finally, we approximate the solution xi (t) = limk→∞ xik (t) by the N th term xiN (t), for i = 1, 2, . . . , n. 3. The fractional-order R¨ossler hyperchaos system The fractional-order R¨ossler hyperchaotic system [13] is given by  α1 d x   = −(y + z)   dt α1    dα2 y    α = x + ay + w dt 2 α3 d   z = b + xz     dt α3  α4    d w = −cz + dw, dt α4

(6)

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Fig. 1. Phase plot of R¨ossler hyperchaotic system in x–y–z space (αi = 1 (i = 1, 2, 3, 4), a = 0.25).

Fig. 2. Phase plot of R¨ossler hyperchaotic system on the y–w plane (αi = 1 (i = 1, 2, 3, 4), a = 0.25).

where b = 3, c = 0.5, d = 0.05, and a is a variable parameter. When αi = 1 (i = 1, 2, 3, 4), Eq. (6) is the classical integer-order R¨ossler hyperchaos equation, and it has a hyperchaotic attractor when a = 0.25 as depicted in Figs. 1 and 2. But the two largest Lyapunov exponents of system (6) are λ1 = 0.12 > 0 and λ2 = 0.04 > 0 when a = 0.32 and αi = 0.9–0.95 (i = 1, 2, 3, 4), i.e. it has a hyperchaotic attractor. The corresponding phase plots in x–y–z space and w–y plane are shown in Figs. 3 and 4. 4. The synchronization of fractional-order R¨ossler hyperchaos system In this section, we will focus on the hyperchaotic synchronization of two identical fractional-order R¨ossler systems with different initial values. The master and the slave R¨ossler hyperchaotic systems can be given:  α1 d xm   = −(ym + z m )   dt α1    dα2 ym    α = xm + aym + wm dt 2 (7) α3 z d  m   = b + x z m m   dt α3   α  4   d wm = −cz + dw m m dt α4

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Fig. 3. The phase plot of fractional-order R¨ossler hyperchaotic attractor in x–y–z space (αi = 0.95 (i = 1, 2, 3, 4), a = 0.32).

Fig. 4. The phase plot of fractional-order R¨ossler hyperchaotic attractor in y–w plane (αi = 0.95 (i = 1, 2, 3, 4), a = 0.32).

and  α1 d xs   = −(ys + z s ) + u 1 (t)   dt α1   α 2  d ys    α = xs + ays + ws + u 2 (t) dt 2 α3 z d  s   = b + xs z s + u 3 (t)  α 3  dt   α  4   d ws = −cz + dw + u (t), s s 4 dt α4

(8)

where U (t) = (u 1 (t), u 2 (t), u 3 (t), u 4 (t))T is the controller to be designed. To investigate the synchronization of systems (7) and (8), we define the error states e1 = x s − x m ,

e2 = ys − ym ,

e3 = z s − z m

and

e4 = ws − wm .

Then, the corresponding error dynamics system can be obtained by subtracting Eq. (7) from Eq. (8)

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 α1 d e1     dt α1   α2 e  d  2   α dt 2 dα3 e3      dt α3   α4    d e4 dt α4

= −(e2 + e3 ) + u 1 (t) = e1 + ae2 + e4 + u 2 (t) (9) = e1 e3 + xm e3 + z m e1 + u 3 (t) = −ce3 + de4 + u 4 (t).

If the zero point of system (9) is globally asymptotically stable under a suitable controller, which implies that the two systems (7) and (8) are realized to synchronization. Choose the control law: u 1 (t) = −e1 (t), u 2 (t) = 0, u 3 (t) = −e1 e3 − xm e3 − z m e1 − e3 (t),

(10)

u 4 (t) = −(1 + d)e4 (t). The error system is changed as  α1 d e1     dt α1    dα2 e2    α dt 2 α3 e d  3    α3  dt   α4    d e4 dt α4

= −(e2 − e3 ) − e1 = e1 + ae2 + e4 (11) = −e3 = −e4 . αi

Take the Laplace transformation in both sides of Eq. (11), let E i (s) = L (ei (t)) and utilize L ( ddt αei i ) = s αi E i (s) − s αi −1 ei (0), (i = 1, 2, 3, 4), one has  α1 s E 1 (s) − s α1 −1 e1 (0) = −E 1 (s) − E 2 (s) − E 3 (s)    α2 s E 2 (s) − s α2 −1 e2 (0) = −E s + a E 2 (s) + E 4 (s) (12)  s α3 E (s) − s α3 −1 e3 (0) = −E 3 (s)   α4 3 s E 4 (s) − s α4 −1 e4 (0) = −cE 3 (s) − E 4 (s). From (12), it follows that E 3 (s) =

s α3 −1 e3 (0) , s α3 + 1

(13)

E 4 (s) =

s α4 −1 e4 (0) − cE 3 (s) , s α4 + 1

(14)

E 1 (s) =

s α1 −1 (s α2 − a)e1 (0) − s α2 −1 e2 (0) − (s α2 − a)E 3 (s) − E 4 (s) , (s α2 − a)(s α1 + 1) − 1

(15)

E 2 (s) =

s α2 −1 (s α1 + 1)e2 (0) − s α1 −1 e1 (0) + E 3 (s) + (s α1 + 1)E 4 (s) . (s α2 − a)(s α1 + 1) − 1

(16)

According to the Final-value theorem of the Laplace transformation, from (13), we have lim e3 (t) = lim s E 3 (s) = lim

t→∞

s→0+

s→0+

s α3 e3 (0) = 0, s α3 + 1

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Fig. 5. The error graph of master–slave fractional-order R¨ossler hyperchaotic systems.

as a result, from Eqs. (14)–(16), lim e4 (t) = lim s E 4 (s) = lim

t→∞

s→0+

s→0+

s α4 e4 (0) − cs E 3 (s) = 0, s α4 + 1

1 lim (s E 4 (s) − as E 3 (s)) = 0, a + 1 s→0+ 1 lim (s E 3 (s) + s E 4 (s)) = 0. lim e2 (t) = lim s E 2 (s) = − t→∞ a + 1 s→0+ s→0+ lim e1 (t) = lim s E 1 (s) =

t→∞

s→0+

The above results manifest the fractional-order R¨ossler hyperchaotic systems (7) and (8) which are synchronized under the control law (10). The simulation can be depicted in Fig. 5. 5. Numerical algorithms and approximate solutions Based on the predictor–corrector scheme in Refs. [42,43], we can generalize the numerical solution about the initial-value problem of fractional-order R¨ossler hyperchaos systems (7) and (8) under the control law (10). Set h = NT , tn = nh, n = 0, 1, . . . , N ∈ Z + , let (x, y, z)T and (x, ˜ y˜ , z˜ )T represent (xm , ym , z m )T and (xs , ys , z s )T , respectively. The master and slave systems (7) and (8) can be discretized as follows:  h α1 p p q  (yn+1 + z n+1 − xn+1 ) x = x −  n+1 0   Γ (α + 2) 1     h α2 p p p q   y = y + (xn+1 + ayn+1 + wn+1 + yn+1 ) n+1 0   Γ (α + 2)  2    h α3  p p q  z n+1 = z 0 + (b + xn+1 z n+1 + z n+1 )    Γ (α3 + 2)     h α4 p p q   (−cz n+1 + dwn+1 + wn+1 ) wn+1 = w0 + Γ (α4 + 2) h α1  p p p p q   (− y˜n+1 − z˜ n+1 + xn+1 − x˜n+1 + x˜n+1 ) x˜n+1 = x˜0 +   Γ (α + 2)  1    h α2  p p p q  y˜n+1 = y˜0 + (x˜n+1 + a y˜n+1 + w˜ n+1 + y˜n+1 )    Γ (α + 2) 2     h α3 p p p p q z˜  = z ˜ + (b + xn+1 z n+1 + z n+1 − z˜ n+1 + z˜ n+1 ) n+1 0   Γ (α + 2)  3    h α4  p p p q  w˜ n+1 = w˜ 0 + (−c˜z n+1 − w˜ n+1 + (1 + d)wn+1 + w˜ n+1 ), Γ (α4 + 2)

(17)

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where  n 1 X p   aβ1, j,n+1 (−y j − z j ) xn+1 = x0 +   Γ (α1 ) j=0     n  1 X  p   β2, j,n+1 (x j + ay j + w j ) y = y + 0  n+1  Γ (α2 ) j=0     n  1 X  p   β3, j,n+1 (b + x j z j ) z n+1 = z 0 +   Γ (α3 ) j=0     n  1 X  p   w = w + β4, j,n+1 (−cz j + dw j ) 0   n+1 Γ (α4 ) j=0 n  1 X p   x ˜ = x ˜ + β1, j,n+1 (− y˜ j − z˜ j + x j − x˜ j ) 0  n+1  Γ (α1 ) j=0     n   1 X p   β2, j,n+1 (x˜ j + a y˜ j + w˜ j ) y ˜ = y ˜ + 0  n+1  Γ (α2 ) j=0     n   z˜ p = z˜ + 1 X β   3, j,n+1 (b + x j z j + z j − z˜ j ) 0  Γ (α3 ) j=0  n+1    n   1 X p   w ˜ = w ˜ + β4, j,n+1 (−c˜z j − w˜ j + (1 + d)w j ),  n+1 0 Γ (α4 ) j=0

(18)

 n X q   x = γ1, j,n+1 (−y j − z j )  n+1   j=0    n X   q   y = γ2, j,n+1 (x j + ay j + w j )  n+1   j=0    n  X  q   z n+1 = γ3, j,n+1 (b + x j z j )     j=0   n  X  q  wn+1 = γ4, j,n+1 (−cz j + dw j )   j=0

(19)

n X  q   x ˜ = γ1, j,n+1 (− y˜ j − z˜ j + x j − x˜ j )  n+1    j=0   n  X  q   y˜n+1 = γ2, j,n+1 (x˜ j + a y˜ j + w˜ j )     j=0   n  X  q   z ˜ = γ3, j,n+1 (b + x j z j + z j − z˜ j )  n+1    j=0   n  X  q   γ4, j,n+1 (−c˜z j − w˜ j + (1 + d)w j ), w˜ n+1 = j=0

and h αi ((n − j − 1)αi − (n − j)αi ), 0 ≤ j ≤ n, αi  α +1 n i − (n − αi )(n + 1)αi γi, j,n+1 = (n − j + 2)αi +1 + (n − j)αi +1 − 2(n − j + 1)αi +1  1 i = 1, 2, 3, 4. βi, j,n+1 =

j =0 1≤ j ≤n j =n+1

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However, according to the formula (5) of VIM, the iteration formulas for error system (11) can be given by Z t  k+1 k  (D α1 e1k (τ ) + e1k (τ ) + e2k (τ ) + e3k (τ ))dτ  e1 (t) = e1 (t) −  0  Z  t  k+1   (D α2 e2k (τ ) − e1k (τ ) − ae2k (τ ) − e4k (τ ))dτ e2 (t) = e2k (t) − Z0 t  k+1 k  (D α3 e3k (τ ) + e3k (τ ))dτ  e3 (t) = e3 (t) −  0  Z  t   ek+1 (t) = ek (t) − (D α4 e4k (τ ) + e4k (τ ))dτ. 4 4

(20)

0

So, for the initial values e10 = e1 (0), e20 = e2 (0), e30 = e3 (0), e40 = e4 (0), the corresponding approximations can be obtained e11 (t) = e1 (0) − (e1 (0) + e2 (0) + e3 (0))t e21 (t) = e2 (0) + (e1 (0) + ae2 (0) + e4 (0))t e31 (t) = e3 (0) − e3 (0)t e41 (t) = e4 (0) − e4 (0)t e12 (t) = e1 (0) − 2(e1 (0) + e2 (0) + e3 (0))t (a − 1)e2 (0) − 2e3 (0) + e4 (0) 2 e1 (0) + e2 (0) + e3 (0) 2−α1 − t + t 2 Γ (3 − α1 ) e22 (t) = e2 (0) + 2(e1 (0) + ae2 (0) + e4 (0))t (a − 1)e1 (0) + (a 2 − 1)e2 (0) − e3 (0) + (a − 1)e4 (0) 2 e1 (0) + ae2 (0) + e4 (0) 2−α2 t − t + 2 Γ (3 − α2 ) e3 (0) 2 e3 (0) e32 (t) = e3 (0) − 2e3 (0)t + t + t 2−α3 2 Γ (3 − α3 ) e4 (0) 2 e4 (0) e42 (t) = e4 (0) − 2e4 (0)t + t + t 2−α4 2 Γ (3 − α4 ) 3((a − 1)e2 (0) − 2e3 (0) + e4 (0)) 2 t e13 (t) = e1 (0) − 3(e1 (0) + e2 (0) + e3 (0))t − 2 2 (a − a)e2 (0) + 2e3 (0) + (a − 2)e4 (0) 3 3(e1 (0) + e2 (0) + e3 (0)) 2−α1 + t + t 2·3 Γ (3 − α1 ) e1 (0) − (a − 2)e2 (0) + 3e3 (0) − e4 (0) 3−α1 e1 (0) + e2 (0) + e3 (0) 3−2α1 − t − t Γ (4 − α1 ) Γ (4 − 2α1 ) e3 (0) e1 (0) + ae2 (0) + e4 (0) 3−α2 t − t 3−α3 + Γ (4 − α2 ) Γ (4 − α3 ) 3((a − 1)e1 (0) + (a 2 − 1)e2 (0) − e3 (0) + (a − 1)e4 (0)) 2 t e23 (t) = e2 (0) + 3(e1 (0) + ae2 (0) + e4 (0))t + 2 2 2 2 (a − a)e1 (0) + (a − 1)(a + a − 1)e2 (0) + (2 − a)e3 (0) + (a − a)e4 (0) 3 + t 2·3 3(e1 (0) + ae2 (0) + e4 (0)) 2−α2 e1 (0) + ae2 (0) + e4 (0) 3−2α2 − t + t Γ (3 − α2 ) Γ (4 − 2α2 ) 2 (2a − 1)e1 (0) + (2a − 1)e2 (0) − e3 (0) + (2a − 1)e4 (0) 3−α2 t − Γ (4 − α2 ) e1 (0) + e2 (0) + e3 (0) 3−α1 e4 (0) t + t 3−α4 + Γ (4 − α1 ) Γ (4 − α4 ) 3e3 (0) 2 e3 (0) 3 2e3 (0) 3−α3 e3 (0) 3e3 (0) 2−α3 e33 (t) = e3 (0) − 3e3 (0)t + t − t + t − t − t 3−2α3 2 2·3 Γ (3 − α3 ) Γ (4 − α3 ) Γ (4 − 2α3 ) 3e4 (0) 2 e4 (0) 3 3e4 (0) 2−α4 2e4 (0) 3−α4 e4 (0) e43 (t) = e4 (0) − 3e4 (0)t + t − t + t − t − t 3−2α4 2 2·3 Γ (3 − α4 ) Γ (4 − α4 ) Γ (4 − 2α4 ) .. .

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6. Conclusion In this paper we have mainly studied the hyperchaotic synchronization of fractional-order R¨ossler system which exists a hyperchaotic attractor when its order is as low as 3.8. According to the Laplace transformation theory, we have realized the global synchronization of the hyperchaotic systems through choosing a suitable controller. We have also obtained the approximate solutions for the error system of two synchronized hyperchaotic systems using the VIM. Finally, numerical simulation is given to verify the effectiveness of the proposed synchronization scheme. There are still many interesting problem about the fractional-order hyperchaos system deserved to study in the future, such as the phase synchronization, the projective synchronization and so on. Acknowledgements The author would like to thank the referees for their valuable suggestions and hard work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

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