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Full state hybrid lag projective synchronization in chaotic (hyperchaotic) systems
Physics Letters A 372 (2008) 1416–1421 www.elsevier.com/locate/pla
Full state hybrid lag projective synchronization in chaotic (hyperchaotic) systems ✩ Qunjiao Zhang, Jun-an Lu ∗ School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China Received 13 May 2007; received in revised form 18 August 2007; accepted 21 September 2007 Available online 2 October 2007 Communicated by A.P. Fordy
Abstract This Letter introduces another novel type of chaos synchronization—full state hybrid lag projective synchronization (FSHLPS), which includes complete synchronization, anti-synchronization, lag synchronization, general projective synchronization and FSHPS in [M. Hu, Z. Xu, R. Zhang, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 456; M. Hu, Z. Xu, R. Zhang, A. Hu, Phys. Lett. A 361 (2007) 231]. Furthermore, systematic FSHLPS schemes are respectively proposed for the continuous and discrete systems. Finally, some numerical simulations are given to verify the effectiveness of the developed schemes. © 2007 Elsevier B.V. All rights reserved. Keywords: FSHLPS; Active control; Chaotic (hyperchaotic) systems; Continuous; Discrete
1. Introduction Since chaos synchronization was first introduced by Pecora and Carroll [1,2], it has been an active research topic in nonlinear science, and has been extensively studied. Over the past 15 years, a variety of approaches have been proposed for chaos synchronization, such as manifold-based method [3], adaptive method [4,5], time delay feedback approach [6], backstepping method [7], nonlinear control scheme [8,9] and many others. At the same time, many different types of synchronization in chaotic (hyperchaotic) systems were presented. For example, complete synchronization, generalized synchronization, phase synchronization, anti-synchronization, general projective synchronization, lag synchronization and anticipate synchronization, and so on. Recently, Wen [10] presented a full-state projective synchronization between two dynamical systems. For two dynamical systems x(t) ˙ = F (x)—drive system, y(t) ˙ = G(x, y)—response system, where x = (x1 , x2 , . . . , xn )T , y = (y1 , y2 , . . . , yn )T ∈ R n are state variables of the drive system and the response system, respectively. If there exists a nonzero constant α such that limt→∞ y(t) − αx(t) = 0, 0 = α ∈ R, i.e., limt→∞ |yi (t) − αxi (t)| = 0 (i = 1, 2, . . . , n), then they call it as full-state projective synchronization (FSPS). Very recently, Hu [11,12] presented the definition of full-state hybrid projective synchronization (FSHPS). For the above drive system and response system, it is said that they are full state hybrid projection synchronization (FSHPS), if there exists a nonzero ✩
This work was supported by the National Natural Science Foundation of China under the grant No. 60574045.
* Corresponding author. Tel.: +86 27 68773425.
E-mail addresses: [email protected] (Q. Zhang), [email protected] (J. Lu). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.09.051
Q. Zhang, J. Lu / Physics Letters A 372 (2008) 1416–1421
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constant matrix α = diag{α1 , α2 , . . . , αn } ∈ R n×n such that limt→∞ y(t) − αx(t) = 0, i.e., limt→∞ |yi (t) − αi xi (t)| = 0 (i = 1, 2, . . . , n). However, in the practical engineering applications, time delay is inevitable. For instance, in the telephone communication system, the voice one hears on the receiver side at time t is often the voice from the transmitter side at time t − τ . So, in many cases, it is more reasonable to require the slave system to synchronize the master system with a time-delay τ . Therefore, motivated by the existing works and take into account of the time-delay, a new kind of chaos synchronization is introduced in this Letter, which is named as full state hybrid lag projective synchronization (FSHLPS). Furthermore, some unified schemes are proposed to realize the FSHLPS. They are respectively illustrated by the continuous systems of hyperchaotic Rössler system and hyperchaotic Lorenz system, also by the discrete systems of the 3D generalized Hénon map and the 3D Baier–Klein map. Meanwhile, numerical simulations are given to verify the effectiveness of the developed methods. 2. Definition of FSHLPS and the unified FSHLPS schemes First, we will give a new concept of chaos synchronization, which is described as below: Definition 1 (FSHLPS). For the above drive system and response system, it is said that they are full state hybrid lag projection synchronization (FSHLPS), if there exists a nonzero constant matrix Λ = diag(λ1 , λ2 , . . . , λn ) ∈ R n×n such that limt→∞ e(t) = 0, where e(t) = y(t) − Λx(t − τ ), τ 0. That is, limt→∞ |yi (t) − λi xi (t − τ )| = 0, i = 1, 2, . . . , n. Remark 1. Here, we use the term FSHLPS to distinguish the FSPS in Ref. [10] and the FSHPS in Refs. [11,12]. In fact, if τ = 0 and λ1 = λ2 = · · · = λn = 0, it degenerates to be FSPS. And it is simplified to be FSHPS with τ = 0. Remark 2. It is easy to see that the complete synchronization, anti-synchronization and general projective synchronization are the special cases when scaling matrix Λ equals to I, −I and γ I (γ is a nonzero constant and I is the identity matrix) with τ = 0. And it is reduced to be the lag synchronization when λ1 = λ2 = · · · = λn = 1. In what follows, a general unified FSHLPS scheme will be proposed for continuous chaotic (hyperchaotic) systems by active control. Consider a class of n-dimensional chaotic (hyperchaotic) systems in the form of x(t) ˙ = F x(t) , (1) where x(t) = (x1 (t), x2 (t), . . . , xn (t))T ∈ R n is the state vector, and F (x(t)) = (F1 (x(t)), F2 (x(t)), . . . , Fn (x(t)))T ∈ R n is a continuous nonlinear vector function. We refer to (1) as the drive system and the response system is given by y(t) ˙ = G y(t) + U (t), (2) where y(t) = (y1 (t), y2 (t), . . . , yn (t))T , G(y(t)) = (G1 (y(t)), G2 (y(t)), . . . , Gn (y(t)))T . And U (t) = (u1 (t), u2 (t), . . . , un (t))T ∈ R n is the controller vector to be determined later. Let the error vector be e(t) = y(t) − Λx(t − τ ), where Λ is a n-order diagonal matrix, i.e., Λ = diag(λ1 , λ2 , . . . , λn ), 0 = λi ∈ R, i = 1, 2, . . . , n. Thus the error dynamical system between the drive system (1) and the response system (2) is e(t) ˙ = G y(t) − ΛF x(t − τ ) + U (t). (3) According to the active control method, choose U (t) = V (e(t)) + H (x(t − τ ), y(t)), where V (e(t)) = (v1 (e(t)), v2 (e(t)), . . . , vn (e(t)))T ∈ R n is a linear vector function relevant to e(t). Usually, we take V (e(t)) = Ae(t) with A is a n × n constant matrix. Theorem 1. If the suitable vector function H (x(t − τ ), y(t)) and matrix A are designed such that all the eigenvalues of error system (3) are on the left-hand side of the complex plane, then the system (1) would be full state hybrid lag projection synchronized with system (2). Now, consider the n-dimensional discrete chaotic systems (drive and slave system) which are described by x(k + 1) = Ax(k) + F x(k) , and
y(k + 1) = By(k) + G y(k) + U,
(4)
(5)
where x(k) = (x1 (k), x2 (k), . . . , xn (k))T , y(k) = (y1 (k), y2 (k), . . . , yn (k))T ∈ R n are the state vectors, F (x(k)) and G(y(k)) are continuous nonlinear vector functions and k is the iteration index. Matrices A, B ∈ R n×n and U is the vector controller to be determined for the purpose of making the two different chaotic (hyperchaotic) systems (4) and (5) be in FSHLPS.
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Remark 3. Many chaotic and hyperchaotic maps can be written as the form of (4), such as cubic map, logistic map, Hénon map, 3D discrete-time Baier–Klein map and 3D generalized Hénon map, etc. Similar to the above, define the error vector be e(k) = y(k) − Λx(k − τ ), where Λ is a n-order diagonal matrix, i.e., Λ = diag(λ1 , λ2 , . . . , λn ), 0 = λi ∈ R, i = 1, 2, . . . , n. τ ∈ Z+ is the time delay. Then, the error dynamics between the drive system (4) and the response system (5) can be derived as e(k + 1) = y(k + 1) − Λx(k + 1 − τ ) = By(k) + G y(k) − ΛAx(k − τ ) − ΛF x(k − τ ) + U = Be(k) + (BΛ − ΛA)x(k − τ ) + G y(k) − ΛF x(k − τ ) + U. To achieve the FSHLPS between drive system (4) and response system (5), choose the vector controller U as U = Le(k) − G y(k) + ΛF x(k − τ ) + (ΛA − BΛ)x(k − τ ),
(6)
(7)
where L = (lij ) ∈ R n×n is the unknown matrix to be designed. Substitute Eq. (7) into Eq. (6), one can simplify the error system into e(k + 1) = (B + L)e(k).
(8)
Theorem 2. If the constants lij are chosen such that the maximal eigenvalue of (B + L)T (B + L) is less than 1, then system (4) would be full state hybrid lag projection synchronized with system (5). Proof. Choose the Lyapunov function V (e(k)) = e(k)T e(k), according to Eq. (8), there is V e(k) = V e(k + 1) − V e(k) = e(k)T (B + L)T (B + L)e(k) − e(k)T e(k) λmax (B + L)T (B + L) − 1 e(k)T e(k) < 0,
(9)
which implies that limk→∞ e(k) = 0, that is, limk→∞ |yi (k) − λi xi (k − τ )| = 0, i = 1, 2, . . . , n. Thus, system (4) is full state hybrid lag projection synchronuous with system (5). 2 3. Numerical examples As an application of the proposed method, the continuous systems are illustrated by the Rössler system and the hyperchaotic Lorenz system with different structures. The hyperchaotic Rössler system [13] is ⎧ x˙ = −x2 − x3 , ⎪ ⎨ 1 x˙2 = x1 + a1 x2 + x4 , (10) ⎪ ⎩ x˙3 = x1 x3 + b1 , x˙4 = −c1 x3 + d1 x4 , it is hyperchaotic with a1 = 0.25, b1 = 3, c1 = 0.5, d1 = 0.05. The response system is given by the hyperchaotic Lorenz system, which was presented by Li et al. [14]. The controlled hyperchaotic Lorenz system is described by the following equations: ⎧ y˙1 = a(y2 − y1 ) + u1 (t), ⎪ ⎨ y˙2 = by1 + cy2 − y1 y3 + y4 + u2 (t), (11) ⎪ ⎩ y˙3 = −dy3 + y1 y2 + u3 (t), y˙4 = −ry1 + u4 (t), this system is found to be hyperchaotic for the parameters a = 35, b = 7, c = 12, d = 3 and r = 5. The phase portraits of the hyperchaotic Rössler system and the hyperchaotic Lorenz system are given in Fig. 1. For the above systems (10) and (11), let the error states be ei (t) = yi (t) − λi xi (t − τ ),
i = 1, 2, 3, 4.
From Eqs. (10) and (11), then the error dynamics can be obtained as ⎧ e˙1 (t) = a(e2 (t) − e1 (t)) − aλ1 x1 (t − τ ) + (aλ2 + λ1 )x2 (t − τ ) + λ1 x3 (t − τ ) + u1 (t), ⎪ ⎪ ⎪ ⎨ e˙2 (t) = be1 (t) + ce2 (t) + e4 (t) + (bλ1 − λ2 )x1 (t − τ ) + (c − a1 )λ2 x2 (t − τ ) + (λ4 − λ2 )x4 (t − τ ) − y1 (t)y3 (t) + u2 (t), ⎪ ⎪ (t) = −de3 (t) − λ3 [dx3 (t − τ ) + x1 (t − τ )x3 (t − τ ) + b1 ] + y1 (t)y2 (t) + u3 (t), e ˙ ⎪ 3 ⎩ e˙4 (t) = −re1 (t) − rλ1 x1 (t − τ ) + c1 λ4 x3 (t − τ ) − d1 λ4 x4 (t − τ ) + u4 (t).
(12)
(13)
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Fig. 1. Simulated phase portraits of the hyperchaotic Rössler and Lorenz systems.
Choose the control function vector ⎛
⎞ aλ1 x1 (t − τ ) − (aλ2 + λ1 )x2 (t − τ ) − λ1 x3 (t − τ ) ⎜ (λ − bλ1 )x1 (t − τ ) + (a1 − c)λ2 x2 (t − τ ) + (λ2 − λ4 )x4 (t − τ ) + y1 (t)y3 (t) ⎟ H x(t − τ ), y(t) = ⎝ 2 ⎠ λ3 [dx3 (t − τ ) + x1 (t − τ )x3 (t − τ ) + b1 ] − y1 (t)y2 (t) rλ1 x1 (t − τ ) − c1 λ4 x3 (t − τ ) + d1 λ4 x4 (t − τ )
and
⎛
0 −a ⎜ 0 −1 − c A=⎝ 0 0 0 0
⎞ 0 0 0 −1 ⎟ ⎠. 0 0 0 −1
(14)
(15)
Notice that (u1 (t), u2 (t), u3 (t), u4 (t))T = A(e1 (t), e2 (t), e3 (t), e4 (t))T + H (x(t − τ ), y(t)), substitute Eqs. (14) and (15) into Eq. (13), we have the error system as below ⎧ e˙ (t) = −ae1 (t), ⎪ ⎨ 1 e˙2 (t) = be1 (t) − e2 (t), (16) ⎪ ⎩ e˙3 (t) = −de3 (t), e˙4 (t) = −re1 (t) − e4 (t), whose eigenvalues are −a, −1, −d, −1, all of them are on the left-hand side of the complex plane. This implies that ei (t) → 0 (i = 1, 2, 3, 4) as t → ∞. That’s to say, the FSHLPS between the Rössler system and the hyperchaotic Lorenz system is achieved. For the numerical simulations, the history initial states of the drive system are set to be x1 (t) = 10, x2 (t) = 20, x3 (t) = 30, x4 (t) = 40 (t 0), and the initial conditions of the response system are y1 (0) = −40, y2 (0) = −30, y3 (0) = −20, y4 (0) = −10. Without loss of generality, we set the scaling factors λ1 = 1, λ2 = −1, λ3 = 2, λ4 = −2 and time delay τ = 1. The variables’ states during the synchronizing process between the hyperchaotic Rössler system and the hyperchaotic Lorenz system are shown in Fig. 2. From it, one can see that each yi (t) (by dotted line) converges to λi xi (t − 1) (by solid line) quickly. Another example is for the discrete systems, the drive system is the 3D generalized Hénon map [9,15] ⎧ ⎨ x1 (k + 1) = bx3 (k), x2 (k + 1) = b(1 + x2 (k) − ax32 (k)), (17) ⎩ x3 (k + 1) = 1 + x1 (k) − a(1 + x2 (k) − ax32 (k))2 , where a = 1.58 and b = 0.06. The slave system is the 3D discrete-time Baier–Klein map [16], which is described as ⎧ ⎨ y1 (k + 1) = −0.1y3 (k) − y22 (k) + 1.76 + u1 , y (k + 1) = y1 (k) + u2 , ⎩ 2 y3 (k + 1) = y2 (k) + u3 . So the A, B and F (x(k)), G(y(k)) in Eqs. (4)–(5) are respectively 0 0 −0.1 0 0 b , A= 0 0 b 0 , and B = 1 0 1 −2a 0 0 1 0
(18)
(19)
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Fig. 2. Synchronizing state variables between the hyperchaotic Rössler and Lorenz system.
⎛ F x(k) = ⎝
⎞ 0 2 ⎠ b(1 − ax3 (k)) 2 4 2 2 2 1 − a(1 + x2 (k) + a x3 (k) − 2ax3 (k) − 2ax2 (k)x3 (k))
(20)
−y22 (k) + 1.76 . 0 0
(21)
and
G y(k) =
Choose the matrix L as L=
−0.5 0 0.1 −1 −0.6 0 , 0 −1 0.9
(22)
then the λmax (B + L)T (B + L) = 0.81 < 1. According to Theorem 2, the system (17) and (18) is in FSHLPS. Fig. 3 displays the simulation results with time-delay τ = 1, where the controllers are switched on at k = 5s and scale factors λ1 = 1, λ2 = −1, λ3 = 3. 4. Conclusion In this Letter, we have introduced a new concept of chaos synchronization—full state hybrid lag projective synchronization (FSHLPS) in chaos (hyperchaotic) systems, which is more general than the existing chaos synchronization and more practical in the applications. What is more, general FSHLPS schemes have been proposed for the continuous and discrete-time chaotic (hyperchaotic) systems with different structures. The Rössler system and the hyperchaotic Lorenz system are chosen to illustrate the proposed scheme for continuous systems. Further, the discrete systems are verified by the 3D generalized Hénon map and the 3D Baier–Klein map. All the numerical simulation results are in line with the theoretical analysis.
Q. Zhang, J. Lu / Physics Letters A 372 (2008) 1416–1421
Fig. 3. Synchronizing errors between 3D generalized Hénon map and the 3D Baier–Klein map.
Acknowledgements The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. 64 (1990) 821. T.L. Carroll, L.M. Pecora, IEEE Trans. Circuits Syst. I 38 (1991) 453. S. Chen, Q. Zhang, J. Xie, Chaos Solitons Fractals 20 (2004) 947. X. Chen, J. Lu, Phys. Lett. A 364 (2007) 123. J.H. Park, J. Comput. Appl. Math., in press. Q. Han, Phys. Lett. A 360 (2007) 563. G. Li, S. Zhou, K. Yang, Phys. Lett. A 355 (2006) 326. H.K. Chen, Chaos Solitons Fractals 23 (2005) 1245. Z. Yan, Phys. Lett. A 343 (2005) 423. G. Wen, D. Xu, Chaos Solitons Fractals 26 (2005) 71. M. Hu, Z. Xu, R. Zhang, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 456. M. Hu, Z. Xu, R. Zhang, A. Hu, Phys. Lett. A 361 (2007) 231. O.E. Rössler, Phys. Lett. A 71 (1979) 155. Y. Li, Wallace, S. Tang, G. Chen, Int. J. Circ. Theory Appl. 33 (2005) 235. K. Stefa´nski, Chaos Solitons Fractals 9 (1998) 83. G. Baier, M. Klein, Phys. Lett. A 151 (1990) 281.
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Full state hybrid lag projective synchronization in chaotic (hyperchaotic) systems ✩ Qunjiao Zhang, Jun-an Lu ∗ School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China Received 13 May 2007; received in revised form 18 August 2007; accepted 21 September 2007 Available online 2 October 2007 Communicated by A.P. Fordy
Abstract This Letter introduces another novel type of chaos synchronization—full state hybrid lag projective synchronization (FSHLPS), which includes complete synchronization, anti-synchronization, lag synchronization, general projective synchronization and FSHPS in [M. Hu, Z. Xu, R. Zhang, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 456; M. Hu, Z. Xu, R. Zhang, A. Hu, Phys. Lett. A 361 (2007) 231]. Furthermore, systematic FSHLPS schemes are respectively proposed for the continuous and discrete systems. Finally, some numerical simulations are given to verify the effectiveness of the developed schemes. © 2007 Elsevier B.V. All rights reserved. Keywords: FSHLPS; Active control; Chaotic (hyperchaotic) systems; Continuous; Discrete
1. Introduction Since chaos synchronization was first introduced by Pecora and Carroll [1,2], it has been an active research topic in nonlinear science, and has been extensively studied. Over the past 15 years, a variety of approaches have been proposed for chaos synchronization, such as manifold-based method [3], adaptive method [4,5], time delay feedback approach [6], backstepping method [7], nonlinear control scheme [8,9] and many others. At the same time, many different types of synchronization in chaotic (hyperchaotic) systems were presented. For example, complete synchronization, generalized synchronization, phase synchronization, anti-synchronization, general projective synchronization, lag synchronization and anticipate synchronization, and so on. Recently, Wen [10] presented a full-state projective synchronization between two dynamical systems. For two dynamical systems x(t) ˙ = F (x)—drive system, y(t) ˙ = G(x, y)—response system, where x = (x1 , x2 , . . . , xn )T , y = (y1 , y2 , . . . , yn )T ∈ R n are state variables of the drive system and the response system, respectively. If there exists a nonzero constant α such that limt→∞ y(t) − αx(t) = 0, 0 = α ∈ R, i.e., limt→∞ |yi (t) − αxi (t)| = 0 (i = 1, 2, . . . , n), then they call it as full-state projective synchronization (FSPS). Very recently, Hu [11,12] presented the definition of full-state hybrid projective synchronization (FSHPS). For the above drive system and response system, it is said that they are full state hybrid projection synchronization (FSHPS), if there exists a nonzero ✩
This work was supported by the National Natural Science Foundation of China under the grant No. 60574045.
* Corresponding author. Tel.: +86 27 68773425.
E-mail addresses: [email protected] (Q. Zhang), [email protected] (J. Lu). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.09.051
Q. Zhang, J. Lu / Physics Letters A 372 (2008) 1416–1421
1417
constant matrix α = diag{α1 , α2 , . . . , αn } ∈ R n×n such that limt→∞ y(t) − αx(t) = 0, i.e., limt→∞ |yi (t) − αi xi (t)| = 0 (i = 1, 2, . . . , n). However, in the practical engineering applications, time delay is inevitable. For instance, in the telephone communication system, the voice one hears on the receiver side at time t is often the voice from the transmitter side at time t − τ . So, in many cases, it is more reasonable to require the slave system to synchronize the master system with a time-delay τ . Therefore, motivated by the existing works and take into account of the time-delay, a new kind of chaos synchronization is introduced in this Letter, which is named as full state hybrid lag projective synchronization (FSHLPS). Furthermore, some unified schemes are proposed to realize the FSHLPS. They are respectively illustrated by the continuous systems of hyperchaotic Rössler system and hyperchaotic Lorenz system, also by the discrete systems of the 3D generalized Hénon map and the 3D Baier–Klein map. Meanwhile, numerical simulations are given to verify the effectiveness of the developed methods. 2. Definition of FSHLPS and the unified FSHLPS schemes First, we will give a new concept of chaos synchronization, which is described as below: Definition 1 (FSHLPS). For the above drive system and response system, it is said that they are full state hybrid lag projection synchronization (FSHLPS), if there exists a nonzero constant matrix Λ = diag(λ1 , λ2 , . . . , λn ) ∈ R n×n such that limt→∞ e(t) = 0, where e(t) = y(t) − Λx(t − τ ), τ 0. That is, limt→∞ |yi (t) − λi xi (t − τ )| = 0, i = 1, 2, . . . , n. Remark 1. Here, we use the term FSHLPS to distinguish the FSPS in Ref. [10] and the FSHPS in Refs. [11,12]. In fact, if τ = 0 and λ1 = λ2 = · · · = λn = 0, it degenerates to be FSPS. And it is simplified to be FSHPS with τ = 0. Remark 2. It is easy to see that the complete synchronization, anti-synchronization and general projective synchronization are the special cases when scaling matrix Λ equals to I, −I and γ I (γ is a nonzero constant and I is the identity matrix) with τ = 0. And it is reduced to be the lag synchronization when λ1 = λ2 = · · · = λn = 1. In what follows, a general unified FSHLPS scheme will be proposed for continuous chaotic (hyperchaotic) systems by active control. Consider a class of n-dimensional chaotic (hyperchaotic) systems in the form of x(t) ˙ = F x(t) , (1) where x(t) = (x1 (t), x2 (t), . . . , xn (t))T ∈ R n is the state vector, and F (x(t)) = (F1 (x(t)), F2 (x(t)), . . . , Fn (x(t)))T ∈ R n is a continuous nonlinear vector function. We refer to (1) as the drive system and the response system is given by y(t) ˙ = G y(t) + U (t), (2) where y(t) = (y1 (t), y2 (t), . . . , yn (t))T , G(y(t)) = (G1 (y(t)), G2 (y(t)), . . . , Gn (y(t)))T . And U (t) = (u1 (t), u2 (t), . . . , un (t))T ∈ R n is the controller vector to be determined later. Let the error vector be e(t) = y(t) − Λx(t − τ ), where Λ is a n-order diagonal matrix, i.e., Λ = diag(λ1 , λ2 , . . . , λn ), 0 = λi ∈ R, i = 1, 2, . . . , n. Thus the error dynamical system between the drive system (1) and the response system (2) is e(t) ˙ = G y(t) − ΛF x(t − τ ) + U (t). (3) According to the active control method, choose U (t) = V (e(t)) + H (x(t − τ ), y(t)), where V (e(t)) = (v1 (e(t)), v2 (e(t)), . . . , vn (e(t)))T ∈ R n is a linear vector function relevant to e(t). Usually, we take V (e(t)) = Ae(t) with A is a n × n constant matrix. Theorem 1. If the suitable vector function H (x(t − τ ), y(t)) and matrix A are designed such that all the eigenvalues of error system (3) are on the left-hand side of the complex plane, then the system (1) would be full state hybrid lag projection synchronized with system (2). Now, consider the n-dimensional discrete chaotic systems (drive and slave system) which are described by x(k + 1) = Ax(k) + F x(k) , and
y(k + 1) = By(k) + G y(k) + U,
(4)
(5)
where x(k) = (x1 (k), x2 (k), . . . , xn (k))T , y(k) = (y1 (k), y2 (k), . . . , yn (k))T ∈ R n are the state vectors, F (x(k)) and G(y(k)) are continuous nonlinear vector functions and k is the iteration index. Matrices A, B ∈ R n×n and U is the vector controller to be determined for the purpose of making the two different chaotic (hyperchaotic) systems (4) and (5) be in FSHLPS.
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Remark 3. Many chaotic and hyperchaotic maps can be written as the form of (4), such as cubic map, logistic map, Hénon map, 3D discrete-time Baier–Klein map and 3D generalized Hénon map, etc. Similar to the above, define the error vector be e(k) = y(k) − Λx(k − τ ), where Λ is a n-order diagonal matrix, i.e., Λ = diag(λ1 , λ2 , . . . , λn ), 0 = λi ∈ R, i = 1, 2, . . . , n. τ ∈ Z+ is the time delay. Then, the error dynamics between the drive system (4) and the response system (5) can be derived as e(k + 1) = y(k + 1) − Λx(k + 1 − τ ) = By(k) + G y(k) − ΛAx(k − τ ) − ΛF x(k − τ ) + U = Be(k) + (BΛ − ΛA)x(k − τ ) + G y(k) − ΛF x(k − τ ) + U. To achieve the FSHLPS between drive system (4) and response system (5), choose the vector controller U as U = Le(k) − G y(k) + ΛF x(k − τ ) + (ΛA − BΛ)x(k − τ ),
(6)
(7)
where L = (lij ) ∈ R n×n is the unknown matrix to be designed. Substitute Eq. (7) into Eq. (6), one can simplify the error system into e(k + 1) = (B + L)e(k).
(8)
Theorem 2. If the constants lij are chosen such that the maximal eigenvalue of (B + L)T (B + L) is less than 1, then system (4) would be full state hybrid lag projection synchronized with system (5). Proof. Choose the Lyapunov function V (e(k)) = e(k)T e(k), according to Eq. (8), there is V e(k) = V e(k + 1) − V e(k) = e(k)T (B + L)T (B + L)e(k) − e(k)T e(k) λmax (B + L)T (B + L) − 1 e(k)T e(k) < 0,
(9)
which implies that limk→∞ e(k) = 0, that is, limk→∞ |yi (k) − λi xi (k − τ )| = 0, i = 1, 2, . . . , n. Thus, system (4) is full state hybrid lag projection synchronuous with system (5). 2 3. Numerical examples As an application of the proposed method, the continuous systems are illustrated by the Rössler system and the hyperchaotic Lorenz system with different structures. The hyperchaotic Rössler system [13] is ⎧ x˙ = −x2 − x3 , ⎪ ⎨ 1 x˙2 = x1 + a1 x2 + x4 , (10) ⎪ ⎩ x˙3 = x1 x3 + b1 , x˙4 = −c1 x3 + d1 x4 , it is hyperchaotic with a1 = 0.25, b1 = 3, c1 = 0.5, d1 = 0.05. The response system is given by the hyperchaotic Lorenz system, which was presented by Li et al. [14]. The controlled hyperchaotic Lorenz system is described by the following equations: ⎧ y˙1 = a(y2 − y1 ) + u1 (t), ⎪ ⎨ y˙2 = by1 + cy2 − y1 y3 + y4 + u2 (t), (11) ⎪ ⎩ y˙3 = −dy3 + y1 y2 + u3 (t), y˙4 = −ry1 + u4 (t), this system is found to be hyperchaotic for the parameters a = 35, b = 7, c = 12, d = 3 and r = 5. The phase portraits of the hyperchaotic Rössler system and the hyperchaotic Lorenz system are given in Fig. 1. For the above systems (10) and (11), let the error states be ei (t) = yi (t) − λi xi (t − τ ),
i = 1, 2, 3, 4.
From Eqs. (10) and (11), then the error dynamics can be obtained as ⎧ e˙1 (t) = a(e2 (t) − e1 (t)) − aλ1 x1 (t − τ ) + (aλ2 + λ1 )x2 (t − τ ) + λ1 x3 (t − τ ) + u1 (t), ⎪ ⎪ ⎪ ⎨ e˙2 (t) = be1 (t) + ce2 (t) + e4 (t) + (bλ1 − λ2 )x1 (t − τ ) + (c − a1 )λ2 x2 (t − τ ) + (λ4 − λ2 )x4 (t − τ ) − y1 (t)y3 (t) + u2 (t), ⎪ ⎪ (t) = −de3 (t) − λ3 [dx3 (t − τ ) + x1 (t − τ )x3 (t − τ ) + b1 ] + y1 (t)y2 (t) + u3 (t), e ˙ ⎪ 3 ⎩ e˙4 (t) = −re1 (t) − rλ1 x1 (t − τ ) + c1 λ4 x3 (t − τ ) − d1 λ4 x4 (t − τ ) + u4 (t).
(12)
(13)
Q. Zhang, J. Lu / Physics Letters A 372 (2008) 1416–1421
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Fig. 1. Simulated phase portraits of the hyperchaotic Rössler and Lorenz systems.
Choose the control function vector ⎛
⎞ aλ1 x1 (t − τ ) − (aλ2 + λ1 )x2 (t − τ ) − λ1 x3 (t − τ ) ⎜ (λ − bλ1 )x1 (t − τ ) + (a1 − c)λ2 x2 (t − τ ) + (λ2 − λ4 )x4 (t − τ ) + y1 (t)y3 (t) ⎟ H x(t − τ ), y(t) = ⎝ 2 ⎠ λ3 [dx3 (t − τ ) + x1 (t − τ )x3 (t − τ ) + b1 ] − y1 (t)y2 (t) rλ1 x1 (t − τ ) − c1 λ4 x3 (t − τ ) + d1 λ4 x4 (t − τ )
and
⎛
0 −a ⎜ 0 −1 − c A=⎝ 0 0 0 0
⎞ 0 0 0 −1 ⎟ ⎠. 0 0 0 −1
(14)
(15)
Notice that (u1 (t), u2 (t), u3 (t), u4 (t))T = A(e1 (t), e2 (t), e3 (t), e4 (t))T + H (x(t − τ ), y(t)), substitute Eqs. (14) and (15) into Eq. (13), we have the error system as below ⎧ e˙ (t) = −ae1 (t), ⎪ ⎨ 1 e˙2 (t) = be1 (t) − e2 (t), (16) ⎪ ⎩ e˙3 (t) = −de3 (t), e˙4 (t) = −re1 (t) − e4 (t), whose eigenvalues are −a, −1, −d, −1, all of them are on the left-hand side of the complex plane. This implies that ei (t) → 0 (i = 1, 2, 3, 4) as t → ∞. That’s to say, the FSHLPS between the Rössler system and the hyperchaotic Lorenz system is achieved. For the numerical simulations, the history initial states of the drive system are set to be x1 (t) = 10, x2 (t) = 20, x3 (t) = 30, x4 (t) = 40 (t 0), and the initial conditions of the response system are y1 (0) = −40, y2 (0) = −30, y3 (0) = −20, y4 (0) = −10. Without loss of generality, we set the scaling factors λ1 = 1, λ2 = −1, λ3 = 2, λ4 = −2 and time delay τ = 1. The variables’ states during the synchronizing process between the hyperchaotic Rössler system and the hyperchaotic Lorenz system are shown in Fig. 2. From it, one can see that each yi (t) (by dotted line) converges to λi xi (t − 1) (by solid line) quickly. Another example is for the discrete systems, the drive system is the 3D generalized Hénon map [9,15] ⎧ ⎨ x1 (k + 1) = bx3 (k), x2 (k + 1) = b(1 + x2 (k) − ax32 (k)), (17) ⎩ x3 (k + 1) = 1 + x1 (k) − a(1 + x2 (k) − ax32 (k))2 , where a = 1.58 and b = 0.06. The slave system is the 3D discrete-time Baier–Klein map [16], which is described as ⎧ ⎨ y1 (k + 1) = −0.1y3 (k) − y22 (k) + 1.76 + u1 , y (k + 1) = y1 (k) + u2 , ⎩ 2 y3 (k + 1) = y2 (k) + u3 . So the A, B and F (x(k)), G(y(k)) in Eqs. (4)–(5) are respectively 0 0 −0.1 0 0 b , A= 0 0 b 0 , and B = 1 0 1 −2a 0 0 1 0
(18)
(19)
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Fig. 2. Synchronizing state variables between the hyperchaotic Rössler and Lorenz system.
⎛ F x(k) = ⎝
⎞ 0 2 ⎠ b(1 − ax3 (k)) 2 4 2 2 2 1 − a(1 + x2 (k) + a x3 (k) − 2ax3 (k) − 2ax2 (k)x3 (k))
(20)
−y22 (k) + 1.76 . 0 0
(21)
and
G y(k) =
Choose the matrix L as L=
−0.5 0 0.1 −1 −0.6 0 , 0 −1 0.9
(22)
then the λmax (B + L)T (B + L) = 0.81 < 1. According to Theorem 2, the system (17) and (18) is in FSHLPS. Fig. 3 displays the simulation results with time-delay τ = 1, where the controllers are switched on at k = 5s and scale factors λ1 = 1, λ2 = −1, λ3 = 3. 4. Conclusion In this Letter, we have introduced a new concept of chaos synchronization—full state hybrid lag projective synchronization (FSHLPS) in chaos (hyperchaotic) systems, which is more general than the existing chaos synchronization and more practical in the applications. What is more, general FSHLPS schemes have been proposed for the continuous and discrete-time chaotic (hyperchaotic) systems with different structures. The Rössler system and the hyperchaotic Lorenz system are chosen to illustrate the proposed scheme for continuous systems. Further, the discrete systems are verified by the 3D generalized Hénon map and the 3D Baier–Klein map. All the numerical simulation results are in line with the theoretical analysis.
Q. Zhang, J. Lu / Physics Letters A 372 (2008) 1416–1421
Fig. 3. Synchronizing errors between 3D generalized Hénon map and the 3D Baier–Klein map.
Acknowledgements The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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