General hybrid projective complete dislocated synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems

Accepted Manuscript General hybrid projective complete dislocated synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems Junwei Sun, Yanfeng Wang, Lina Yao, Yi Shen, Guangzhao Cui PII: DOI: Reference:

S0307-904X(15)00052-9 http://dx.doi.org/10.1016/j.apm.2015.01.049 APM 10400

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Appl. Math. Modelling

Please cite this article as: J. Sun, Y. Wang, L. Yao, Y. Shen, G. Cui, General hybrid projective complete dislocated synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems, Appl. Math. Modelling (2015), doi: http://dx.doi.org/10.1016/j.apm.2015.01.049

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General hybrid projective complete dislocated synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems Junwei Sun1,2 , Yanfeng Wang1,2 , Lina Yao1,2 , Yi Shen3 , Guangzhao Cui1,2,∗ College of Electric and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou, 450002, China Henan Key Lab of Information-based Electrical Appliances, Zhengzhou University of Light Industry, Zhengzhou 450002, China School of Automation, Huazhong University of Science and Technology, Wuhan, 430074, China

Abstract The synchronization of a class of chaotic real nonlinear systems and the synchronization of a class of chaotic complex nonlinear systems have been widely reported in the previous studies, respectively. In the paper, we investigate the general hybrid projective complete dislocated synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems for the first time. Based on the Lyapunov stability theory, an feedback control scheme has been designed to realize the general hybrid projective complete dislocated synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems with different initial conditions. The general hybrid projective complete dislocated synchronization between the real Lorenz system and the complex Lorenz system, the hyperchaotic real L¨u system and the hyperchaotic complex L¨u system are presented as two examples to demonstrate the validity and feasibility of the presented idea. Keywords: General hybrid projective complete dislocated synchronization; complex Lorenz system; hyperchaotic real L¨u system; hyperchaotic complex L¨u system.

1. Introduction Since Pecora and Carrol has presented the groundbreaking chaos synchronization work in 1990 [1], chaos synchronization has attracted a great deal of attention due to its potential application in many scientific and engineering fields in recent years. Many different types of synchronization phenomenon have been intensively investigated and a lot of theoretical results have been obtained in the past 20 years, such as complete synchronization [1], anti-synchronization [2], generalized synchronization [3, 4], phase synchronization [5], anti-phase synchronization [6], lag synchronization [7], partial synchronization [8], projective synchronization [9, 10, 11, 12, 13], time scale synchronization [14], combination synchronization [15, 16], compound synchronization [17] etc. During the past decades, there exist the following methods to realize chaos synchronization such as OGY method [19], feedback control method [20, 21, 22], H∞ control method [23], optimal control method [24], PID control method [25], active control method [26], passive control method [27], backstepping method [28], adaptive control method [29, 30], sliding mode control method [31], impulsive control method [32], coupling control method [33, 34, 35] etc. For these above many types of synchronization, projective synchronization has been paid lots of attention because of its proportionality between the synchronized dynamical systems. The response system could be achieved synchronization with the drive system by a scaling factor in a class of chaotic systems. There are two special conditions of projective synchronization, where a scaling factor 1 (or -1) is complete synchronization (or anti-synchronization), respectively. In the paper [36], a novel kind of chaos synchronization, called mismatch synchronization, has been investigated under the scheme of drive-response systems. Motivated by previous works, Kim et al. has proposed a ∗

E-mail address: [email protected](J. Sun), [email protected] (G. Cui).

Preprint submitted to Elsevier

January 28, 2015

new approach of full state hybrid projective synchronization between two identical (different) nonlinear chaotic systems, which complete synchronization, anti-synchronization and projective synchronization are included as its special items [37]. Based on the Lyapunov stability theorem, a new type of general hybrid projective complete dislocated synchronization has been designed for the structure of drive systems and response systems [38]. The synchronization of a large number of real dynamical systems [39] has been investigated by the above researches. However, there are also many cases involving complex variables in the real world. Fowler et al. are the first to construct the complex Lorenz equations [40]. Researchers have devoted much time and efforts to study dynamical properties, synchronization and control of many chaotic complex systems in the recent years. In 2007, Mahmoud et al. have studied the complex Chen and complex L¨u systems and investigated their dynamical properties such as chaotic attractors and the stability properties of their fixed points [41]. A new hyperchaotic complex L¨u system has been studied and investigated by adding a state feedback controller and using complex periodic forcing [42]. Another new chaotic complex nonlinear continuous system has been generated from the real Lorenz system, and its dynamical properties has been also analyzed and studied [43]. According to Lyapunov stability theory, an active control scheme has been designed to realize phase and antiphase synchronization of two identical n-dimensional hyperchaotic complex Lorenz systems [44]. What is more, two identical n-dimensional chaotic complex nonlinear systems have been achieved the synchronization by an adaptive control scheme [45]. It is known to all that the complex chaotic system also has much wider applications in many important fields of nonlinear sciences. For example, the adoption of complex chaotic systems has been given for secure communication and the complex variables can increase the contents and security of the transmitted information [40]. However, the synchronization of a class of chaotic real nonlinear systems, or the synchronization of a class of chaotic complex nonlinear systems, has been widely reported in the previous studies, respectively. The studies of chaos synchronization have mostly been limited to within a class of chaotic real nonlinear systems, or a class of chaotic complex nonlinear systems. Thus, an interesting and meaningful problem is: can the synchronization be realized between one real system and one complex system? To the best of our knowledge, none of the previous studies is investigated on the synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems. Obviously, the synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems seems to be more difficult and challenging than the synchronization of a class of chaotic real nonlinear systems and the synchronization of a class of chaotic complex nonlinear systems. Motivated by the above discussions, the general hybrid projective complete dislocated synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems is investigated according to Lyapunov stability theory. An feedback control scheme has been designed to realize the GHPCDS between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems with different initial conditions. The GHPCDS between the real Lorenz system and the complex Lorenz system, the hyperchaotic real L¨u system and the hyperchaotic complex L¨u system are presented as two examples to demonstrate the validity and feasibility of the presented control technique. Compared with prior work, our idea can break barriers between chaotic real nonlinear systems and chaotic complex nonlinear systems, and combine chaotic real nonlinear systems and chaotic complex nonlinear systems to realize synchronization. Our idea from numerical computations and theoretical analysis is a wonderful bridge between chaotic real nonlinear systems and chaotic complex nonlinear systems. What is more, the synchronization of a class of chaotic real nonlinear systems, or the synchronization of a class of chaotic complex nonlinear systems, has been selected in the previous study. Now, the synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems can provide a novel choice for secure communication. More choices of chaotic systems are given to realize secure communication by chaotic synchronization, more stronger anti-attack ability and anti-translated capacity are strengthened for our method to a certain degree. It seems to be potentially useful for our idea, a more bright future will be waiting for secure communication and information processing. This paper is organized as follows: in Section 2, the GHPCDS scheme between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems is discussed, followed by, in Section 3, the GHPCDS scheme between the real Lorenz system and the complex Lorenz system is realized by the proposed control scheme. In Section 4, the GHPCDS scheme between the hyperchaotic real L¨u system and the hyperchaotic complex L¨u system is realized by the proposed control method. Finally, our concluding remarks are presented in Section 5.

2

2. General hybrid projective complete dislocated synchronization scheme A drive chaotic real system is given in the form of x˙r = f r (xr ) + F r (xr )θ,

(1)

where xr = (u1 , u2 , · · · , un )T is the state real vector of the drive system (1), f r = ( f1 (xr ), f2 (xr ), · · · , fn (xr ))T is a vector of real continuous nonlinear function, F r = (F1r (x), F2r (x), · · · , Fnr (x))T , Fir (x) is the ith row of an n × n matrix (F(xr )) whose elements are real continuous nonlinear functions, θ = (θ1 ,θ2 , · · · ,θn )T is an n × 1 parameter real vector of the drive system (1). The drive n-dimensional chaotic real system is described by   u˙ 1 (t) = f1r (u1 , u2 , · · · , un ) + F1r (u1 , u2 , · · · , un )θ,    r r     u˙ 2 (t) = f2 (u1 , u2 , · · · , un ) + F2 (u1 , u2 , · · · , un )θ, (2)  ..    .     u˙ (t) = f r (u , u , · · · , u ) + F r (u , u , · · · , u )θ. n

1

n

2

n

n

1

2

n

The response chaotic complex system is written by y˙ c = gc (yc ) + Gc (yc )ψ + v,

(3)

where y = (yc1 , yc2 , · · · , ycn )T is a state complex vector of the response system (3), and yc = yr + jyi . Define yc1 = u′1 + ju′n+1 , yc2 = u′2 + ju′n+2 , · · · , ycn = u′n + ju′2n , then yr = (u′1 , u′2 , · · · , u′n )T , yi = (u′n+1 , u′n+2 , · · · , u′2n )T . gc = (gc1 , gc2 , · · · , gcn )T c r i r i c

is a state complex vector of nonlinear functions, and g = g + jg , g and g are two real and complex vectors of nonlinear functions, respectively. Gc = (Gc1 , Gc2 , · · · , Gcn )T , Gci is the ith row of an n × n matrix (G(yc )) whose elements are complex continuous nonlinear functions, and Gc = Gr + jGi , the elements of matrices Gr and Gi are two real and complex continuous nonlinear functions, respectively. ψ = (ψ1 ,ψ2 , · · · ,ψn )T is an n × 1 parameter vector of the response system (3). The designed controller is v = vr + jvi , v1 = v1 + jvn+1 , v2 = v2 + jvn+2 , · · · , vn = vn + jv2n , where vr = (v1 , v2 , · · · , vn )T , and vi = (vn+1 , vn+2 , · · · , v2n )T . The response chaotic complex system (3) is further written in the form of y˙ c = gr (yc ) + Gr (yc )ψ + j[gi (yc ) + Gi (yc )ψ] + vr + jvi ,

(4)

the response n-dimensional chaotic complex system is gained by  ′  u˙ 1 (t) + j˙u′n+1 (t) = gr1 (u′1 , u′2 , · · · , u′2n ) + Gr1 (u′1 , u′2 , · · · , u′2n )ψ + j[gi1 (u′1 , u′2 , · · · , u′2n )      + Gi1 (u′1 , u′2 , · · · , u′2n )ψ] + v1 (t) + jvn+1 (t),     ′ ′ r ′ ′ ′ r ′ ′ ′ i ′ ′ ′  u ˙ (t) + j˙ u (t) = g   2 n+2 2 (u1 , u2 , · · · , u2n ) + G 2 (u1 , u2 , · · · , u2n )ψ + j[g2 (u1 , u2 , · · · , u2n )   i ′ ′ ′  + G (u , u , · · · , u )ψ] + v (t) + jv (t), 2 n+2 2n 2 1 2    ..     .      u˙ ′n (t) + j˙u′2n (t) = grn (u′1 , u′2 , · · · , u′2n ) + Grn (u′1 , u′2 , · · · , u′2n )ψ + j[gin (u′1 , u′2 , · · · , u′2n )     + Gi (u′ , u′ , · · · , u′ )ψ] + v (t) + jv (t). n

1

2

n

2n

(5)

2n

Definition 1: Consider the drive real chaotic systems and the response complex chaotic systems described by Eqs. (2) and (5), respectively. If there exists a non-zero constant di j (i = 1, 2, · · · , n, j = 1, 2, · · · , n) such that the GHPCDS error vector e(t) tends to zero as t → ∞, namely, lim ∥ei (t)∥ = lim ∥

t→∞

t→∞

n ∑

di j ycj + xir ∥ = 0.

(6)

i=1,i, j

In other words, the GHPCDS between the drive real chaotic system (2) and the response complex chaotic system (5) is achieved, therefore, the GHPCDS between the drive real chaotic system (1) and the response complex chaotic system (3) is also achieved. n ∑ Remark 1: According to Definition 1, the error system of GHPCDS is chosen as ei (t) = di j y j + xi . In other i=1,i, j

words, the error dynamics system is written by 3

                

e˙ 1 (t) = x˙1 + d12 y˙ 2 + d13 y˙ 3 + · · · + d1n y˙ n , e˙ 2 (t) = x˙2 + d21 y˙ 1 + d23 y˙ 3 + · · · + d2n y˙ n , .. .

(7)

e˙ n (t) = x˙n + dn1 y˙ 1 + dn2 y˙ 2 + · · · + dnn−1 y˙ n−1 ,

if there exist a suitable feedback controller v and the constants di j , then the following condition lim ∥e(t)∥ = 0 holds t→∞ true, the GHPCDS will be achieved. To solve the GHPCDS problem, the error between the drive system (2) and the response system (5) can be defined n ∑ as ei (t) = di j y j + xi . Therefore, the error dynamical system is obtained as follows: i=1,i, j

                                                                              

e˙ 1 (t) = f1r (u1 , u2 , · · · , un ) + F1r (u1 , u2 , · · · , un )θ + d12 {gr2 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )ψ + j[gi2 (u′1 , u′2 , · · · , u′2n ) + Gi2 (u′1 , u′2 , · · · , u′2n )ψ] + v2 (t) + jvn+2 (t)} + d13 {gr3 (u′1 , u′2 , · · · , u′2n ) + Gr3 (u′1 , u′2 , · · · , u′2n )ψ + j[gi3 (u′1 , u′2 , · · · , u′2n ) + Gi3 (u′1 , u′2 , · · · , u′2n )ψ] + v3 (t) + jvn+3 (t)} + · · · + d1n {grn (u′1 , u′2 , · · · , u′2n ) + Grn (u′1 , u′2 , · · · , u′2n )ψ + j[gin (u′1 , u′2 , · · · , u′2n ) + Gin (u′1 , u′2 , · · · , u′2n )ψ] + vn (t) + jv2n (t)}, e˙ 2 (t) = f2r (u1 , u2 , · · · , un ) + F2r (u1 , u2 , · · · , un )θ + d21 {gr1 (u′1 , u′2 , · · · , u′2n ) + Gr1 (u′1 , u′2 , · · · , u′2n )ψ + j[gi1 (u′1 , u′2 , · · · , u′2n ) + Gi1 (u′1 , u′2 , · · · , u′2n )ψ] + v1 (t) + jvn+1 (t)} + d23 {gr3 (u′1 , u′2 , · · · , u′2n ) + Gr3 (u′1 , u′2 , · · · , u′2n )ψ + j[gi3 (u′1 , u′2 , · · · , u′2n ) + Gi3 (u′1 , u′2 , · · · , u′2n )ψ] + v3 (t) + jvn+3 (t)} + · · · + d2n {grn (u′1 , u′2 , · · · , u′2n ) + Grn (u′1 , u′2 , · · · , u′2n )ψ + j[gin (u′1 , u′2 , · · · , u′2n ) + Gin (u′1 , u′2 , · · · , u′2n )ψ] + vn (t) + jv2n (t)}, .. .

(8)

e˙ n (t) = fnr (u1 , u2 , · · · , un ) + Fnr (u1 , u2 , · · · , un )θ + dn1 {gr1 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )ψ + j[gi1 (u′1 , u′2 , · · · , u′2n ) + Gi1 (u′1 , u′2 , · · · , u′2n )ψ] + v1 (t) + jvn+1 (t)} + dn2 {gr2 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )ψ + j[gi2 (u′1 , u′2 , · · · , u′2n ) + Gi2 (u′1 , u′2 , · · · , u′2n )ψ] + v2 (t) + jvn+2 (t)} + · · · + dnn−1 {grn−1 (u′1 , u′2 , · · · , u′2n ) + Grn−1 (u′1 , u′2 , · · · , u′2n )ψ + j[gin−1 (u′1 , u′2 , · · · , u′2n ) + Gin−1 (u′1 , u′2 , · · · , u′2n )ψ] + vn−1 (t) + jv2n−1 (t)}.

The objective of this paper is to design a suitable feedback control law v(t) such that for any given drive real chaotic system (2) and the response complex chaotic system (5), the stability of the resulting error system (8) can be achieved in the sense of Definition 1.   A1 = − f1r (u1 , u2 , · · · , un ) − F1r (u1 , u2 , · · · , un )θ − d12 [gr2 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )]ψ      − d13 [gr3 (u′1 , u′2 , · · · , u′2n ) + Gr3 (u′1 , u′2 , · · · , u′2n )]ψ − · · · − d1n [grn (u′1 , u′2 , · · · , u′2n )      + Grn (u′1 , u′2 , · · · , u′2n )ψ] − k1 (u1 + d12 u′2 + d13 u′3 + · · · + d1n u′n ),       A2 = − f2r (u1 , u2 , · · · , un ) − F2r (u1 , u2 , · · · , un )θ − d21 [gr1 (u′1 , u′2 , · · · , u′2n ) + Gr1 (u′1 , u′2 , · · · , u′2n )ψ]      − d23 [gr3 (u′1 , u′2 , · · · , u′2n ) + Gr3 (u′1 , u′2 , · · · , u′2n )ψ] − · · · − d2n [grn (u′1 , u′2 , · · · , u′2n )   (9)  + Grn (u′1 , u′2 , · · · , u′2n )ψ] − k2 (u2 + d21 u′1 + d23 u′3 + · · · + d2n u′n ),      .   ..       An = − fnr (u1 , u2 , · · · , un ) − Fnr (u1 , u2 , · · · , un )θ − dn1 [gr1 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )ψ]       − dn2 [gr2 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )ψ] − · · · − dnn−1 [grn−1 (u′1 , u′2 , · · · , u′2n )     + Grn−1 (u′1 , u′2 , · · · , u′2n )ψ] − kn (un + dn1 u′1 + dn2 u′2 + · · · + dnn−1 u′n−1 ), where ki (i = 1, 2, · · · , n) are the positive constants. Let

4

Ξ =

0

d12

d13

···

d1n−1

d21

0

d23

···

d2n−1

.. .

.. .

.. .

.. .

.. .

dn1

dn2

dn3

···

dnn−1

A1

d12

d13

···

d1n−1

A2

0

d23

···

d2n−1

.. .

.. .

.. .

.. .

.. .

An

dn2

dn3

···

dnn−1

0

A1

d13

···

d1n−1

d21

A2

d23

···

d2n−1

.. .

.. .

.. .

.. .

.. .

dn1

An

dn3

···

dnn−1

0

d12

d13

· · · d1n−1

d21

0

d23

· · · d2n−1

.. .

.. .

.. .

dn1

dn2

dn3

d1n d2n , .. . 0

and Ω1 = Ω2 =

d1n d2n , .. . 0 d1n d2n , .. . 0

··· Ωn =

.. .

A1 A2 . .. . An

.. .

· · · dnn−1

The feedback control laws are designed as follows:   v1 = Ω1 /Ξ,        v2 = Ω2 /Ξ,  ..    .     v = Ω /Ξ, n n   vn+1 = −gi1 (u′1 , u′2 , · · · , u′2n ) − Gi1 (u′1 , u′2 , · · · , u′2n )ψ,    i ′ ′ ′ i ′ ′ ′     vn+2 = −g2 (u1 , u2 , · · · , u2n ) − G2 (u1 , u2 , · · · , u2n )ψ,  ..    .     v = −gi (u′ , u′ , · · · , u′ ) − Gi (u′ , u′ , · · · , u′ )ψ. 2n

2n

1

2

2n

2n

1

2

(10)

(11)

2n

Theorem 1: If the error system (8) is controlled with the control laws (10) and (11), then the GHPCDS between the response complex system (5) and the drive real system (2) is achieved asymptotically, where ki > 0(i = 1, 2, · · · , n) are the constants. Proof. Choose a positive definite function in the form of 5

V(t) =

1 2 (e + e22 + · · · + e2n ). 2 1

(12)

Then time derivative of V along the trajectory of the error system (6) is as follows ˙ V(t) = e1 e˙ 1 + e2 e˙ 2 + · · · + en e˙ n . (13) Substituting e˙ k , the control laws (10) and (11) into (13), one obtains

˙ V(t) =

n ∑

e1 { f1r (u1 , u2 , · · · , un ) + F1r (u1 , u2 , · · · , un ) + d12 {gr2 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )ψ

i=1

+ j[gi2 (u′1 , u′2 , · · · , u′2n ) + Gi2 (u′1 , u′2 , · · · , u′2n )ψ] + v2 (t) + jvn+2 (t)} + d13 {gr3 (u′1 , u′2 , · · · , u′2n ) +Gr3 (u′1 , u′2 , · · · , u′2n )ψ + j[gi3 (u′1 , u′2 , · · · , u′2n ) + Gi3 (u′1 , u′2 , · · · , u′2n )ψ] + v3 (t) + jvn+3 (t)} + · · · + d1n {grn (u′1 , u′2 , · · · , u′2n ) + Grn (u′1 , u′2 , · · · , u′2n )ψ + j[gin (u′1 , u′2 , · · · , u′2n )

+Gin (u′1 , u′2 , · · · , u′2n )ψ] + vn (t) + jv2n (t)}} + e2 { f2r (u1 , u2 , · · · , un ) + F2r (u1 , u2 , · · · , un )θ +d21 {gr1 (u′1 , u′2 , · · · , u′2n ) + Gr1 (u′1 , u′2 , · · · , u′2n )ψ + j[gi1 (u′1 , u′2 , · · · , u′2n ) + Gi1 (u′1 , u′2 , · · · , u′2n )ψ] +v1 (t) + jvn+1 (t)} + d23 {gr3 (u′1 , u′2 , · · · , u′2n ) + Gr3 (u′1 , u′2 , · · · , u′2n )ψ + j[gi3 (u′1 , u′2 , · · · , u′2n ) +Gi3 (u′1 , u′2 , · · · , u′2n )ψ] + v3 (t) + jvn+3 (t)} + · · · + d2n {grn (u′1 , u′2 , · · · , u′2n ) +Grn (u′1 , u′2 , · · · , u′2n )ψ + j[gin (u′1 , u′2 , · · · , u′2n ) + Gin (u′1 , u′2 , · · · , u′2n )ψ] + vn (t) + jv2n (t)}}

+ · · · + en { fnr (u1 , u2 , · · · , un ) + Fnr (u1 , u2 , · · · , un )θ + dn1 {gr1 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )ψ + j[gi1 (u′1 , u′2 , · · · , u′2n ) + Gi1 (u′1 , u′2 , · · · , u′2n )ψ] + v1 (t) + jvn+1 (t)} + dn2 {gr2 (u′1 , u′2 , · · · , u′2n ) +Gr2 (u′1 , u′2 , · · · , u′2n )ψ + j[gi2 (u′1 , u′2 , · · · , u′2n ) + Gi2 (u′1 , u′2 , · · · , u′2n )ψ] + v2 (t) + jvn+2 (t)} + · · · + dnn−1 {grn−1 (u′1 , u′2 , · · · , u′2n ) + Grn−1 (u′1 , u′2 , · · · , u′2n )ψ + j[gin−1 (u′1 , u′2 , · · · , u′2n ) +Gin−1 (u′1 , u′2 , · · · , u′2n )ψ] + vn−1 (t) + jv2n−1 (t)}}.

(14) Introducing v into the above equation (14) and simplifying it, this yields ˙ V(t) =

e1 { f1r (u1 , u2 , · · · , un ) + F1r (u1 , u2 , · · · , un ) + d12 [gr2 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )ψ] +d13 [gr3 (u′1 , u′2 , · · · , u′2n ) + Gr3 (u′1 , u′2 , · · · , u′2n )ψ] + · · · + d1n [grn (u′1 , u′2 , · · · , u′2n ) +Grn (u′1 , u′2 , · · · , u′2n )ψ] + d12 v2 (t) + d13 v3 (t) + · · · + d1n vn (t)}

+e2 { f2r (u1 , u2 , · · · , un ) + F2r (u1 , u2 , · · · , un )θ + d21 [gr1 (u′1 , u′2 , · · · , u′2n ) + Gr1 (u′1 , u′2 , · · · , u′2n )ψ] +d23 [gr3 (u′1 , u′2 , · · · , u′2n ) + Gr3 (u′1 , u′2 , · · · , u′2n )ψ] + · · · + d2n [grn (u′1 , u′2 , · · · , u′2n ) +Grn (u′1 , u′2 , · · · , u′2n )]ψ + d21 v1 (t) + d23 v3 (t) + · · · + d2n vn (t)} + · · · + en { fnr (u1 , u2 , · · · , un ) +Fnr (u1 , u2 , · · · , un )θ + dn1 [gr1 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )ψ] +dn2 [gr2 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )ψ] + · · · + dnn−1 [grn−1 (u′1 , u′2 , · · · , u′2n ) +Grn−1 (u′1 , u′2 , · · · , u′2n )ψ] + dn1 v1 (t) + dn2 v2 (t) + · · · + dnn−1 vn−1 (t)}.

(15) In this case, equation (15) can be reduced to

6

˙ V(t) =

e1 { f1r (u1 , u2 , · · · , un ) + F1r (u1 , u2 , · · · , un ) + d12 [gr2 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )ψ]

+d13 [gr3 (u′1 , u′2 , · · · , u′2n ) + Gr3 (u′1 , u′2 , · · · , u′2n )ψ] + · · · + d1n [grn (u′1 , u′2 , · · · , u′2n ) +Grn (u′1 , u′2 , · · · , u′2n )ψ] + A1 } + e2 { f2r (u1 , u2 , · · · , un ) + F2r (u1 , u2 , · · · , un )θ +d21 [gr1 (u′1 , u′2 , · · · , u′2n ) + Gr1 (u′1 , u′2 , · · · , u′2n )ψ] + d23 [gr3 (u′1 , u′2 , · · · , u′2n ) + Gr3 (u′1 , u′2 , · · · , u′2n )ψ] + · · · + d2n [grn (u′1 , u′2 , · · · , u′2n ) + Grn (u′1 , u′2 , · · · , u′2n )]ψ + A2 } + · · · + en { fnr (u1 , u2 , · · · , un ) +Fnr (u1 , u2 , · · · , un )θ + dn1 [gr1 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )ψ] + dn2 [gr2 (u′1 , u′2 , · · · , u′2n ) +Gr2 (u′1 , u′2 , · · · , u′2n )ψ] + · · · + dnn−1 [grn−1 (u′1 , u′2 , · · · , u′2n ) + Grn−1 (u′1 , u′2 , · · · , u′2n )ψ] + An }.

(16) Substituting equation (9) into equation (16), the equation (16) can be further rewritten as ˙ V(t) =

e1 { f1r (u1 , u2 , · · · , un ) + F1r (u1 , u2 , · · · , un ) + d12 [gr2 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )ψ] +d13 [gr3 (u′1 , u′2 , · · · , u′2n ) + Gr3 (u′1 , u′2 , · · · , u′2n )ψ] + · · · + d1n [grn (u′1 , u′2 , · · · , u′2n ) +Grn (u′1 , u′2 , · · · , u′2n )ψ] − f1r (u1 , u2 , · · · , un ) − F1r (u1 , u2 , · · · , un )θ − d12 [gr2 (u′1 , u′2 , · · · , u′2n ) +Gr2 (u′1 , u′2 , · · · , u′2n )]ψ − d13 [gr3 (u′1 , u′2 , · · · , u′2n ) + Gr3 (u′1 , u′2 , · · · , u′2n )]ψ − · · · − d1n [grn (u′1 , u′2 , · · · , u′2n ) + Grn (u′1 , u′2 , · · · , u′2n )ψ] − k1 (u1 +d12 u′2 + d13 u′3 + · · · + d1n u′n )} + e2 { f2r (u1 , u2 , · · · , un ) + F2r (u1 , u2 , · · · , un )θ +d21 [gr1 (u′1 , u′2 , · · · , u′2n ) + Gr1 (u′1 , u′2 , · · · , u′2n )ψ] + d23 [gr3 (u′1 , u′2 , · · · , u′2n )

+Gr3 (u′1 , u′2 , · · · , u′2n )ψ] + · · · + d1n [grn (u′1 , u′2 , · · · , u′2n ) + Grn (u′1 , u′2 , · · · , u′2n )]ψ − f2r (u1 , u2 , · · · , un ) − F2r (u1 , u2 , · · · , un )θ − d21 [gr1 (u′1 , u′2 , · · · , u′2n ) + Gr1 (u′1 , u′2 , · · · , u′2n )ψ] −d23 [gr3 (u′1 , u′2 , · · · , u′2n ) + Gr3 (u′1 , u′2 , · · · , u′2n )ψ] − · · · − d2n [grn (u′1 , u′2 , · · · , u′2n )

+Grn (u′1 , u′2 , · · · , u′2n )ψ] − k2 (u2 + d21 u′1 + d23 u′3 + · · · + d2n u′n )} + · · · + en { fnr (u1 , u2 , · · · , un ) +Fnr (u1 , u2 , · · · , un )θ + dn1 [gr1 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )ψ] +dn2 [gr2 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )ψ] + · · · + d1n−1 [grn−1 (u′1 , u′2 , · · · , u′2n ) +Grn−1 (u′1 , u′2 , · · · , u′2n )ψ] − fnr (u1 , u2 , · · · , un ) − Fnr (u1 , u2 , · · · , un )θ −dn1 [gr1 (u′1 , u′2 , · · · , u′2n ) + Gr2 (u′1 , u′2 , · · · , u′2n )ψ] − dn2 [gr2 (u′1 , u′2 , · · · , u′2n )

+Gr2 (u′1 , u′2 , · · · , u′2n )ψ] − · · · − dnn−1 [grn−1 (u′1 , u′2 , · · · , u′2n ) + Grn−1 (u′1 , u′2 , · · · , u′2n )ψ] − kn (un +dn1 u′1 + dn2 u′2 + · · · + dnn−1 u′n−1 )}. (17) After the computations, we can obtain ˙ V(t) = −k1 e21 − k2 e22 − · · · − kn e2n ≤ 0.

(18)

Since V is positive definite, and V˙ is negative semi-definite. According to the Lyapunov theorem, we know ei → 0 (i = 1, 2, 3, · · · , n), which means that the GHPCDS between the response complex system (5) and the drive real system (2) is achieved asymptotically. Therefore, the response complex system (5) can be achieved the GHPCDS with the drive real system (2) asymptotically with the control laws (10) and (11). This completes the proof. Remark 2: In the simulation, di j are chosen to make Ξ , 0 hold true. The control input v is too complex to achieve the GHPCDS problem, which is not only the key to our method, but the difficult point. 3. General hybrid projective complete dislocated synchronization of real and complex Lorenz systems In order to study the GHPCDS between real and complex Lorenz chaotic systems, assume that we have real Lorenz system where the drive real system is denoted with the real variable u(t) and the complex response Lorenz system is 7

denoted with the complex variable w(t). Therefore, we define the drive real Lorenz system [46] in the form of   u˙ = 10(u2 − u1 ),    1 u˙ 2 = 28u1 − u2 − u1 u3 ,     u˙ = u u − 8/3u , 3

1 2

(19)

3

where the initial state of the drive system is given by (0.3,0.2,0.1), the chaotic attractor is shown in FIG. 1. Furthermore, we rewrite the real Lorenz system (19) as follows:         u2 − u1 0 0 0   10        0 u1 0   28  . u˙ =  −u1 u3 − u2  +       u1 u2 0 0 −u3 8/3 | {z } | {z } | {z } θ

F r (u)

f r (u)

(20)

The response complex Lorenz system [43] is described by   w˙ = 10(w2 − w1 ) + v1 + jv4 ,    1 w˙ 2 = 28w1 − w2 − w1 w3 + v2 + jv5 ,     w˙ = −8/3w + 1/2(w¯ w + w w¯ ) + v , 3 3 1 2 1 2 3

(21)

where w1 = u′1 + ju′4 , w2 = u′2 + ju′5 are complex variables, w3 = u′3 is a real variable, u′1 , u′2 , u′3 , u′4 , u′5 are real variables √ and j = −1. The response complex Lorenz system is further written as follows:  ′  u˙ + j˙u′4 = 10[u′2 − u′1 + j(u′5 − u′4 )] + v1 + jv4 ,    1′ u˙ + j˙u′5 = 28u′1 − u′2 − u′1 u′3 + j(28u′4 − u′5 − u′3 u′4 ) + v2 + jv5 , (22)     u˙ 2′ = −8/3u ′ + u′ u′ + u′ u′ + v , 3

3

1 2

4 5

3

50

50

45

45

40

40

35

35

30

30 u3

u3

where the initial state of the response system is given by (3+3j,2+2j,1), the chaotic attractor is shown in FIG. 2. One can rewrite the response complex Lorenz system in the form of Eq. (4) as follows:

25

25

20

20

15

15

10

10

5

5

0 −30

−20

−10

0 u2

10

20

0 −30

30

Figure 1: Chaotic attractors of the real Lorenz system.

−20

−10

0 u2

10

20

30

Figure 2: Chaotic attractors of the complex Lorenz system.

8

u

′

  0  =  −u′1 u′3 − u′2  ′ ′ u u + u′ u′ | 1 2{z 4 5 gr (u′ )

  v1 + jv4  +  v2 + jv5  v3 | {z

  ′   u2 − u′1 0   0 u′1  +  0 0 } | {z Gr (u′ )

    . }

     0   10   0    0   28  + j  −u′3 u′4 − u′5     −u′3 8/3 0 } | {z } | {z ψ

gr (u′ )

  ′   u5 − u′4 0   0 u′4  +  0 0 } | {z Gr (u′ )

    0   10    0   28    0 8/3  } | {z } ψ

(23)

v

Therefore, the synchronization error is obtained as follows:   e = u1 + d12 w2 + d13 w3 ,    1 e2 = u2 + d21 w1 + d23 w3 ,     e =u +d w +d w . 3

3

31

1

32

(24)

2

Assume   A = −k1 (u1 + d12 u′2 + d13 u′3 ) − 10(u2 − u1 ) − d12 (28u′1 − u′2 − u′1 u′3 ) − d13 (−8/3u′3 + u′1 u′2 + u′4 u′5 ),    1 A2 = −k2 (u2 + d21 u′ + d23 u′ ) − (28u1 − u2 − u1 u3 ) − 10d21 (u′ − u′ ) − d23 (−8/3u′3 + u′1 u′2 + u′4 u′5 ),     A = −k (u + d u1′ + d u3′ ) − (u u − 8/3u ) − 10d (u′ − u2 ′ ) −1d (28u′ − u′ − u′1 u′3 ). 3 3 3 31 1 32 2 1 2 3 31 2 32 1 1 2 Let the feedback control law be as follows:   v1 = −d23 d32 A1 + d13 d32 A2 + d12 d23 A3 /d12 d23 d31 + d13 d21 d32 ,         v2 = d23 d31 A1 − d13 d31 A2 + d13 d21 A3 /d12 d23 d31 + d13 d21 d32 ,          v = d d A + d d A − d d A /d d d + d d d , 3

21 32 1

12 31 2

{

12 21 3

12 23 31

v4 = −10(u′5 − u′4 ), v5 = −(28u′4 − u′5 − u′3 u′4 ).

(25)

(26)

13 21 32

(27)

To verify and demonstrate the validity and feasibility of the proposed idea, the simulation results of the GHPCDS between real Lorenz chaotic system (20) and complex Lorenz chaotic system (23) are studied. For the numerical simulations, fourth-order Runge-Kutta method is used to our simulation with time step 0.001. The initial states for the drive system and the response system are given by (0.3,0.2,0.1) and (-0.4+3j,-0.3+2j,-0.2), respectively. The parameters are chosen as d12 = 1, d13 = 2, d21 = 1, d23 = 2, d31 = 1, d32 = 2, respectively. What is more, the constants are taken as k1 = 0.5, k2 = 0.4 and k3 = 0.3. The GHPCDS between response real Lorenz chaotic system (20) and the drive√complex Lorenz chaotic system (23) is achieved in FIG. 3. In order to show the GHPCDS error, we define e(t) = e21 (t) + e22 (t) + e23 (t), whose time evolution is shown in FIG. 4. From the simulations we find every state variable of the drive system can equal other ones of the response system while evolving in time, which verifies the validity of the proposed control technique. Finally, FIG. 5 depicts the time histories of the applied control inputs of complex Lorenz chaotic system (23). 4. General hybrid projective complete dislocated synchronization of hyperchaotic real and complex Lu¨ systems To investigate the GHPCDS between hyperchaotic real L¨u system and hyperchaotic complex L¨u system, assume that we have hyperchaotic real L¨u system where the drive real system is denoted with the real variable u(t) and the response hyperchaotic complex L¨u system is denoted with the complex variable w(t). Therefore, we define the drive hyperchaotic real L¨u system [47] in the form of 9

20

30

15 20

10 10

5

w2+2w3 w1+2w3

0

0 −5

−10 u2

−10 u1

−20

−15 −20

0

5

10

15

20 t

25

30

35

−30

40

0

5

10

15

20 t

25

30

35

40

50 40 30 u3

20 10 0 −10 −20 −30

w1+2w2 −40 −50

0

5

10

15

20 t

25

30

35

40

Figure 3: Synchronization between real Lorenz chaotic system (20) and complex Lorenz chaotic system (23).

              

u˙ 1 u˙ 2 u˙ 3 u˙ 4

= 36(u2 − u1 ) + u4 , = 20u2 − u1 u3 , = u1 u2 − 3u3 , = u1 u3 + u4 ,

(28)

where the initial states for the drive system are given by (1,2,3,4), the chaotic attractor is shown in FIG. 6. What is more, the hyperchaotic real L¨u system (28) is rewritten by   u4  −u u 1 3 u˙ =   u1 u2 u1 u3 | {z

    u2 − u1   0  +    0   0 } |

0 0 u2 0 0 −u3 0 0 {z

0 0 0 u4

    36    20    .   3  1 } | {z }

F r (u)

f r (u)

The hyperchaotic complex L¨u system is described as response system   w˙ 1 = 36(w2 − w1 ) + w4 + v1 + jv5 ,       w˙ 2 = 20w2 − w1 w3 + w4 + v2 + jv6 ,    w˙ 3 = −3w3 + 1/2(w¯ 1 w2 + w1 w¯ 2 ) + v3 ,     w˙ 4 = −w4 + 1/2(w¯ 1 w2 + w1 w¯ 2 ) + v4 , 10

(29)

θ

(30)

0.25

0.2

e(t)

0.15

0.1

0.05

0

0

5

10

15

20 t

25

30

35

40

Figure 4: Synchronization error e(t) between real Lorenz chaotic system (20) and complex Lorenz chaotic system (23).

where w1 = u′1 + ju′5 , w2 = u′2 + ju′6 are complex variables, w3 = u′3 and w4 = u′4 are real variables, u′1 , u′2 , u′3 , u′4 , u′5 , √ u′6 are real variables and j = −1. The response hyperchaotic complex L¨u system [42] is further written as follows:  ′  u˙ 1 + j˙u′5 = 36(u′2 − u′1 ) + u′4 + j36(u′5 − u′4 ) + v1 + jv5 ,       u˙ ′2 + j˙u′6 = 20u′2 − u′1 u′3 + u′4 + j(20u′6 − u′3 u′5 ) + v2 + jv6 , (31)    u˙ ′3 = −3u′3 + u′1 u′2 + u′5 u′6 + v3 ,     u˙ ′ = −u′ + u′ u′ + u′ u′ + v4 , 4

4

1 2

5 6

where the initial states for the response system are given by (1+2j,3+4j,5,6), the chaotic attractor is shown in FIG. 7. One can rewrite the response hyperchaotic complex L¨u system in the form of Eq. (4) as follows:

u′

=

 u′   −u′ u′4 + u′  1 3 4  u′ u′ + u′ u′ 1 2 5 6  ′ ′ u1 u2 + u′5 u′6 | {z gr (u′ )

  v1 + jv5  v + jv 6 +  2 v3  v4 | {z

  ′   u2 − u′1   0  +  0   0 } |

    .  }

0 0 u′2 0 0 −u′3 0 0 {z Gr (u′ )

0 0 0 −u′4

     36     0   20   −u′ u′     3 5   3  + j  0     1 0 } | {z } | {z ψ

gr (u′ )

  ′   u6 − u′5 0   0 u′6  +  0 0   0 0 } | {z

Gr (u′ )

    0 0   36    0 0   20     0 0   3  0 0 1  } | {z } ψ

(32)

v

Therefore, the synchronization error is obtained as follows:   e1 = u1 + d12 w2 + d13 w3 + d14 w4 ,       e2 = u2 + d21 w1 + d23 w3 + d24 w4 ,    e3 = u3 + d31 w1 + d32 w2 + d34 w4 ,     e4 = u4 + d41 w1 + d42 w2 + d43 w3 . Assume

11

(33)

100 v1 0 −100

0

5

10

15

20 t

25

30

35

40

0

5

10

15

20 t

25

30

35

40

0

5

10

15

20 t

25

30

35

40

0

5

10

15

20 t

25

30

35

40

0

5

10

15

20 t

25

30

35

40

100 v2 0 −100 50 v30 −50 20 v0 4 −20 20 v50 −20

Figure 5: Time histories of the applied control inputs of complex Lorenz chaotic system (23).

                                  

A1 = −k1 (u1 + d12 u′2 + d13 u′3 + d14 u′4 ) − [36(u2 − u1 ) + u4 ] − d12 (20u′2 − u′1 u′3 + u′4 ) − d13 (−3u′3 + u′1 u′2 + u′5 u′6 ) − d14 (−u′4 + u′1 u′2 + u′5 u′6 ), A2 = −k2 (u2 + d21 u′1 + d23 u′3 + d24 u′4 ) − (20u2 − u1 u3 ) − d21 [36(u′2 − u′1 ) + u′4 ] − d23 (−3u′3 + u′1 u′2 + u′5 u′6 ) − d24 (−u′4 + u′1 u′2 + u′5 u′6 ), A3 = −k3 (u3 + d31 u′1 + d32 u′2 + d34 u′4 ) − (u1 u2 − 3u3 ) − d31 [36(u′2 − u′1 ) + u′4 ] − d32 (20u′2 − u′1 u′3 + u′4 ) − d34 (−u′4 + u′1 u′2 + u′5 u′6 ), A4 = −k4 (u4 + d41 u′1 + d42 u′2 + d43 u′3 ) − (u1 u3 + u4 ) − d41 [36(u′2 − u′1 ) + u′4 ] − d42 (20u′2 − u′1 u′3 + u′4 ) − d43 (−3u′3 + u′1 u′2 + u′5 u′6 ).

(34)

Ξ = d13 d24 d31 d42 − d12 d24 d31 d43 − d14 d23 d31 d42 − d13 d21 d34 d42 − d14 d21 d32 d43 + d12 d21 d34 d43 − d13 d24 d32 d41 + d14 d23 d32 d41 − d12 d23 d34 d41 ,

(35)

Ω1 = A1 (d23 d34 d42 + d32 d24 d43 ) + A2 (d12 d34 d43 − d14 d32 d43 − d13 d34 d42 ) + A3 (d13 d24 d42 − d12 d24 d43 − d14 d23 d42 ) + A4 (d14 d23 d32 − d12 d23 d34 − d13 d24 d32 ), Ω2 = A1 (d21 d34 d43 − d31 d24 d43 − d23 d34 d41 ) + A2 (d13 d34 d41 + d14 d31 d43 ) + A3 (d14 d41 d23 − d14 d21 d43 − d13 d24 d41 ) + A4 (d13 d24 d31 − d13 d21 d34 − d14 d23 d31 ), Ω3 = −A1 (d24 d32 d41 + d21 d34 d42 − d24 d31 d42 ) − A2 (d14 d31 d42 + d12 d34 d41 − d14 d32 d41 ) + A3 (d12 d24 d41 + d14 d21 d42 ) − A4 (d12 d24 d31 + d14 d21 d32 − d12 d21 d34 ), Ω4 = A1 (−d21 d32 d43 − d23 d31 d42 + d23 d32 d41 ) + A2 (−d32 d13 d41 − d12 d31 d43 + d13 d31 d42 ) + A3 (−d13 d21 d42 − d12 d23 d41 + d12 d21 d43 ) + A4 (d13 d21 d32 + d12 d23 d31 ).

(36)

Let

                                  

The control inputs can be obtained as follows:

12

50

60

45 50

40 35

40

u’3

u3

30 25

30

20 20

15 10

10

5 0 −30

−20

−10

0 u1

10

20

0 −30

30

Figure 6: Chaotic attractors of the hyperchaotic real L¨u system.

               {

v˙ 1 v˙ 2 v˙ 3 v˙ 4

−20

−10

0 u’1

10

20

30

Figure 7: Chaotic attractors of the hyperchaotic complex L¨u system.

= Ω1 /Ξ, = Ω2 /Ξ, = Ω3 /Ξ, = Ω4 /Ξ,

v˙ 5 = −36(u′5 − u′4 ), v˙ 6 = −(20u′6 − u′3 u′5 ).

(37)

(38)

In order to verify and demonstrate the feasibility of the proposed scheme, the simulation results of the GHPCDS between hyperchaotic real L¨u system (29) and hyperchaotic complex L¨u system (32) are investigated. For the numerical simulations, fourth-order Runge-Kutta method is used to our simulation with time step 0.001. In the simulation process, the initial states for the drive system and the response system are given by (1,2,3,4) and (1+2j,3+4j,5,6), respectively. The parameters are chosen as d12 = 1, d13 = 1, d14 = 1, d21 = 1, d23 = 1, d24 = 1, d31 = 1, d32 = 1, d34 = 1, d41 = 1, d42 = 1, d43 = 1, respectively. What is more, the constants are taken as k1 = 0.7, k2 = 0.6, k3 = 0.5 and k4 = 0.6. The GHPCDS between hyperchaotic real L¨u system (29) and√ hyperchaotic complex L¨u system (32) is achieved in FIG. 8. In order to show the GHPCDS error, we define e(t) = e21 (t) + e22 (t) + e23 (t) + e24 (t), whose time evolution is shown in FIG. 9. From the simulations we find every state variable of the drive system can equal other ones of the response system while evolving in time, which verifies the validity of the proposed control technique. Finally, FIG. 10 depicts the time histories of the applied control inputs of hyperchaotic complex L¨u system (32). 5. Conclusion In this paper, we have studied the GHPCDS between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems. The GHPCDS includes complete synchronization, anti-synchronization and projective synchronization as its special items. Every state variable of the drive real system does not equal the corresponding state variable, but equal other ones of the response complex system while evolving in time. Based on Lyapunov stability theory, an adaptive control scheme has been constructed to realize the GHPCDS between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems with different initial conditions. The GHPCDS between the real Lorenz system and the complex Lorenz system, hyperchaotic real L¨u system and hyperchaotic complex L¨u system are presented as two examples to demonstrate the validity and feasibility of the proposed control technique. These theoretical and numerical results can provide a bridge between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems. More and better methods for the synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems should be investigated and 13

25

30 u1

20 u

2

20

15 10

10

5 0

0

−5 −10

−10 −15

−20 w1+w3+w4

−20 −25

w2+w3+w4 0

5

10 t

15

−30

20

50

0

5

10 t

15

20

15

20

150

40

w1+w2+w3

u3

100

30 20

50

10 0

0 −10

−50

−20 −30

−100

w +w +w 1

2

4

u4

−40 −50

0

5

10 t

15

−150

20

0

5

10 t

Figure 8: Synchronization between hyperchaotic real L¨u system (29) and hyperchaotic complex L¨u system (32).

developed. What is more, the idea should be applied in engineering fields such as secure communications and that will be an interesting and challenging research subject of our further work in the future. Acknowledgments The authors thank the editor and the anonymous reviewers for their resourceful and valuable comments and constructive suggestions. Project is supported by the State Key Program of the National Natural Science Foundation of China (Grant No. 61134012), the National Natural Science Foundation of China (Grant Nos. 11271146, 61070238 and 61472371), and the Science and Technology Program of Wuhan (Grant No. 20130105010117). References References [1] L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64(8)(1990) 821-824. [2] H.T. Yau, Synchronization and anti-synchronization coexist in two-degree-of-freedom dissipative gyroscope with nonlinear inputs, Nonlinear Ana. Real World Appl. 9(5)(2008)2253-2261. [3] A.A. Koronovskii, O.I. Moskalenko, A.E. Hramov, Hidden data transmission using generalized synchronization in the presence of noise, Tech. Phys. 55(4)(2010)435-441. [4] P.K. Roy, C. Hens, I. Grosu, S.K. Dana, Engineering generalized synchronization in chaotic oscillators, Chaos 21(1)(2011)013106-1-0131067.

14

0.7

0.6

0.5

e(t)

0.4

0.3

0.2

0.1

0

0

5

10 t

15

20

Figure 9: Synchronization error e(t) between hyperchaotic real L¨u system (29) and hyperchaotic complex L¨u system (32).

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0 v

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0

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10 t

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0 v2−20 −40 0 v−20 3 −40 100 v4 0 −100 50 v5 0 −50 50 v6 0 −50

Figure 10: Time histories of the applied control inputs of hyperchaotic complex L¨u system (32).

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