- Email: [email protected]
Synchronization and anti-synchronization of a hyperchaotic Chen system
Chaos, Solitons and Fractals 39 (2009) 1790–1797 www.elsevier.com/locate/chaos
Synchronization and anti-synchronization of a hyperchaotic Chen system M.M. El-Dessoky Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Accepted 18 June 2007
Abstract Based on the active control theory, synchronization and anti-synchronization between two identical chaotic systems is investigated. Anti-synchronization can be characterized by the vanishing of the sum of relevant variables. Through rigorous mathematical theory, the sufficient condition is drawn for the stability of the error dynamics, where the controllers are designed by using the sum of the relevant variables in chaotic systems. Numerical simulations are performed for Chen hyperchaotic dynamical system to demonstrate the effectiveness of the proposed control strategy. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Chaos synchronization has been investigated extensively in the last decades and many effective methods have been suggested since the pioneering work developed by Ott et al. [1]. The aim of synchronization is to use the output of a master system to control the slave system so that its output of the slave system follows that of the master system asymptotically. The original synchronization technique developed by Pecorra and Carroll [2,3] is relevant to complete synchronization. Recently, the concept of synchronization has been extended to the scope, such as generalized synchronization [3–5], phase synchronization [6,7], lag synchronization [8–12], and anti-synchronization (AS) [13–19], in which the state vectors of synchronized systems have the same absolute values but opposite signs. Therefore, the sum of two signals can converge to zero when AS appears. The aim of this work is to further develop the state observer method for constructing anti-synchronized slave system. Finally, simulations on Chen hyperchaotic dynamical system are performed to verify the effectiveness and feasibility of the proposed control technique.
2. Chaotic synchronization We consider a chaotic continuous system described by x_ ¼ f ðxðtÞ; tÞ E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.053
ð1Þ
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1791
where x 2 Rn is a n-dimensional state vector of the system, and f : Rn ! Rn defines a vector field in n-dimensional space. We decompose the function f(x(t), t) as f ðxðtÞ; tÞ ¼ gðxðtÞ; tÞ þ hðxðtÞ; tÞ
ð2Þ
where g(x(t)) is the linear part of f(x(t), t) described by gðxðtÞÞ ¼ AxðtÞ
ð3Þ
where A is a full rank constant matrix and all eigenvalues of A have negative real parts, hðxðtÞ; tÞ ¼ f ðxðtÞ; tÞ gðxðtÞ; tÞ is the nonlinear part of f(x(t), t). Then the system (1) can be written as x_ ¼ gðxðtÞ; tÞ þ hðxðtÞ; tÞ
ð4Þ
2.1. Synchronization The chaotic synchronization discussed in this Letter is defined as the complete synchronization, which means that the difference of the states of two relevant systems converges to zero. For the given chaotic system (4), we construct a new system as follows: _ wðtÞ ¼ gðwðtÞ; tÞ þ hðxðtÞ; tÞ
ð5Þ
n
where wðtÞ 2 R is the state vector of the new system. The synchronization error between system (1) and system (5) is defined as eðtÞ ¼ xðtÞ wðtÞ, the evolution of which is determined by the following equation: e_ ðtÞ ¼ x_ ðtÞ wðtÞ _ ¼ AðxðtÞ wðtÞÞ ¼ AeðtÞ
ð6Þ
obviously that the zero point of e(t) is its equilibrium point. Since all eigenvalues of the matrix A have negative real parts, according to the stability criterion of linear system, the zero point of synchronization error is asymptotically stable and e(t) tends to zero when t ! 1. Then the state vectors x(t) and w(t) of different systems (1) and (5) are synchronized. 2.2. Anti-synchronization Anti-synchronization is a phenomenon that the state vectors of synchronized systems have the same absolute values but opposite signs. We say that anti-synchronization of two system S1 and S2 is achieved if the following equation holds: lim kx1 ðtÞ þ x2 ðtÞk ¼ 0
t!1
ð7Þ
where xi ðtÞ, ði ¼ 1; 2Þ is the state vector of the system SiðtÞ, ði ¼ 1; 2Þ. We construct a new system described by _ wðtÞ ¼ gðwðtÞ; tÞ hðxðtÞ; tÞ
ð8Þ
The synchronization error between system (1) and system (8) is defined as eðtÞ ¼ xðtÞ þ wðtÞ, the evolution of which is determined by the following equation: _ e_ ðtÞ ¼ x_ ðtÞ þ wðtÞ ¼ AðxðtÞ þ wðtÞÞ ¼ AeðtÞ
ð9Þ
Since all eigenvalues of the matrix A have negative real parts, the zero point of synchronization error is asymptotically stable and e(t) tends to zero when t ! 1. Then the state vectors x(t) and w(t) of different systems (1) and (8) can be anti-synchronized. The simplest configuration of matrix A ¼ diagðk1 ; k2 ; . . . ; kn Þ through separation of systems can be used for the adjustment of parameters to satisfy the stability criterion of linear system that all eigenvalues have negative real.
3. Synchronization of Chen hyperchaotic system The differential equations of Chen hyperchaotic system [20,21] is described by x_ ¼ aðy xÞ þ w y_ ¼ dx xz þ cy z_ ¼ xy bz w_ ¼ yz þ rw
ð10Þ
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where a, b, c, d and r are positive parameters. If a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and 0 6 r 6 0:085, then the system (10) is chaotic; while if a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and 0:085 6 r 6 0:798, then the system (10) is hyperchaotic; and if a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and 0:798 6 r 6 0:9 system (10) is periodic. In particular, these projections of hyperchaotic attractor of Chen hyperchaotic system (10) with a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r ¼ 0:5 are displayed in Fig. 1a–d. In order to observe the synchronization behavior in the Chen hyperchaotic system, we have two Chen hyperchaotic systems where the drive system with four state variables denoted by the subscript 1 drives the response system having identical equations denoted by the subscript 2. However, the initial condition on the drive system is different from that of the response system. The two Chen systems are described, respectively, by the following equations: x_ 1 ¼ aðy 1 x1 Þ þ w1 y_ 1 ¼ dx1 x1 z1 þ cy 1 z_ 1 ¼ x1 y 1 bz1 w_ 1 ¼ y 1 z1 þ rw1
ð11Þ
x_ 2 ¼ aðy 2 x2 Þ þ w2 þ u1 y_ 2 ¼ dx2 x2 z2 þ cy 2 þ u2 z_ 2 ¼ x2 y 2 bz2 þ u3 w_ 2 ¼ y 2 z2 þ rw2 þ u4
ð12Þ
and
where U ¼ ½u1 ; u2 ; u3 ; u4 T is the active control function to be designed. The aim of this section is to determine the controller U for the chaos synchronization of two Chen hyperchaotic dynamical system with the same unknown parameters and different initial conditions. For this purpose, the error dynamical system between the drive system (11) and response system (12) can be expressed by
Fig. 1. Shows the chaotic attractor of Chen hyperchaotic system at: (a) a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5 in x; y; z subspace; (b) a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5 in x; y; w subspace; (c) a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5 in x; z; w subspace; (d) a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5 in y; z; w subspace.
M.M. El-Dessoky / Chaos, Solitons and Fractals 39 (2009) 1790–1797
e_ x ¼ aðey ex Þ þ ew þ u1 e_ y ¼ dex þ x1 z1 x2 z2 þ cey þ u2 e_ 2 ¼ x2 y 2 x1 y 1 bez þ u3 e_ w ¼ y 2 z2 y 1 z1 þ rew þ u4
1793
ð13Þ
where ex ¼ x2 x1 , ey ¼ y 2 y 1 , ez ¼ z2 z1 and ew ¼ w2 w1 . The active control function U ¼ ½u1 ; u2 ; u3 ; u4 T u1 u2 u3 u4
¼ v1 ¼ x2 z2 x1 z1 þ v2 ¼ x1 y 1 x2 y 2 þ v3 ¼ y 1 z1 y 2 z2 þ v4
ð14Þ
This leads to e_ x ¼ aðey ex Þ þ ew þ v1 e_ y ¼ dex þ cey þ v2 e_ 2 ¼ bez þ v3 e_ w ¼ rew þ v4
ð15Þ
Eq. (15) describe the error dynamics and can be considered in terms of a control problem where the system to be controlled is a linear system with a control input v1 ðtÞ, v2 ðtÞ, v3 ðtÞ and v4 ðtÞ as function of ex , ey , ez and ew . As long as these feedbacks stabilize the system, ex , ey , ez and ew converge to zero as time t goes to infinity. This implies that two Chen hyperchaotic systems are anti-synchronized with feedback control. There are many possible choices for the control v1 ðtÞ, v2 ðtÞ, v3 ðtÞ and v4 ðtÞ. We choose v1 v2 v3 v4
¼ aey ew ¼ dex ð1 þ cÞey ¼0 ¼ ð1 þ rÞew
ð16Þ
then the error dynamical system is e_ x ¼ aex e_ y ¼ ey e_ 2 ¼ bez e_ w ¼ ew Eq. (17) describes the error dynamics. Now we define the Laypunov function for the system (17) as follows: 1 2 V ðex ; ey ; ez ; ew Þ ¼ ex þ e2y þ e2z þ e2w 2
ð17Þ
ð18Þ
This function is positive definite and equal zero at the equilibrium of the system (17). Moreover, the derivative of the Laypunov function (18) has the form h i V_ ¼ ae2x þ e2y þ be2z þ e2w which is negative definite. From Lyapunov direct method, we have that the zero solution of the system (17) is asymptotically stable. This implies that the two Chen hyperchaotic systems are synchronized. 3.1. Numerical results By using MAPLE to solve the systems of differential Eqs. (11)–(13). The parameters are chosen to be a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r ¼ 0:5 in all simulations so that the Chen hyperchaotic system exhibits a chaotic behavior if no control is applied.
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Fig. 2. (a) Shows the behavior of the trajectory ex of the error system tends to zero as t tends to 1 when the parameter values are a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5. (b) Shows the behavior of the trajectory ey of the error system tends to zero as t tends to 7 when the parameter values are a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5. (c) Shows the behavior of the trajectory ez of the error system tends to zero as t tends to 3 when the parameter values are a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5. (d) Shows the behavior of the trajectory ew of the error system tends to zero as t tends to 6 when the parameter values are a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5.
The initial states of the drive system are x1 ð0Þ ¼ 0:1, y 1 ð0Þ ¼ 0:2, z1 ð0Þ ¼ 0:6 and w1 ð0Þ ¼ 0:4 and initial states of the response system are x2 ð0Þ ¼ 1, y 2 ð0Þ ¼ 0:4, z2 ð0Þ ¼ 0:2 and w2 ð0Þ ¼ 1, hence the error system has the initial values ex ð0Þ ¼ 0:9, ey ð0Þ ¼ 0:2, ez ð0Þ ¼ 0:4 and ew ð0Þ ¼ 0:6. The results of the two identical Chen hyperchaotic systems with active control are shown in Fig. 2a–c displays the trajectories of ex , ey , ez and ew of the error system tended to zero.
4. Anti-synchronization of Chen hyperchaotic system In this section to study the anti-synchronization problem of Chen hyperchaotic system. In order to observe the antisynchronization behavior in the Chen hyperchaotic system, we have two Chen hyperchaotic systems where the drive system with four state variables denoted by the subscript 1 drives the response system having identical equations denoted by the subscript 2. However, the initial condition on the drive system is different from that of the response system. The two Chen systems are described, respectively, by the following equations: x_ 1 ¼ aðy 1 x1 Þ þ w1 y_ 1 ¼ dx1 x1 z1 þ cy 1 z_ 1 ¼ x1 y 1 bz1 w_ 1 ¼ y 1 z1 þ rw1 and
ð19Þ
M.M. El-Dessoky / Chaos, Solitons and Fractals 39 (2009) 1790–1797
x_ 2 ¼ aðy 2 x2 Þ þ w2 þ u1 y_ 2 ¼ dx2 x2 z2 þ cy 2 þ u2 z_ 2 ¼ x2 y 2 bz2 þ u3 w_ 2 ¼ y 2 z2 þ rw2 þ u4
1795
ð20Þ
be the response system where we have introduced four control functions u1 ðtÞ, u2 ðtÞ, u3 ðtÞ and u4 ðtÞ in Eq. (20). These functions are to be determined for the purpose of anti-synchronizing the two Chen hyperchaotic dynamical system with the same unknown parameters and different initial conditions. Let us define the state errors between the response system that is to be controlled and the controlling derive system as ex ¼ x2 þ x1 , ey ¼ y 2 þ y 1 , ez ¼ z2 þ z1 and ew ¼ w2 þ w1 . By adding Eq. (19) from Eq. (20) yields the error dynamical system between Eqs. (19) and (20) e_ x ¼ aðey ex Þ þ ew þ u1 e_ y ¼ dex x1 z1 x2 z2 þ cey þ u2 e_ 2 ¼ x2 y 2 þ x1 y 1 bez þ u3 e_ w ¼ y 2 z2 þ y 1 z1 þ rew þ u4
ð21Þ
The anti-synchronization problem for Chen hyperchaotic dynamical system is to achieve the asymptotic stability of the zero solution of the error system (21). To this end we take the active control functions u1 ðtÞ, u2 ðtÞ, u3 ðtÞ and u4 ðtÞ as follows: u1 ¼ v1 u2 ¼ x2 z2 þ x1 z1 þ v2 u3 ¼ x1 y 1 x2 y 2 þ v3 u4 ¼ y 1 z1 y 2 z2 þ v4
ð22Þ
This leads to e_ x ¼ aðey ex Þ þ ew þ v1 e_ y ¼ dex þ cey þ v2 e_ 2 ¼ bez þ v3 e_ w ¼ rew þ v4
ð23Þ
Eq. (23) describe the error dynamics and can be considered in terms of a control problem where the system to be controlled is a linear system with a control input v1 ðtÞ, v2 ðtÞ, v3 ðtÞ and v4 ðtÞ as function of ex , ey , ez and ew . As long as these feedbacks stabilize the system, ex , ey , ez and ew converge to zero as time t goes to infinity. This implies that two Chen hyperchaotic systems are anti-synchronized with feedback control. There are many possible choices for the control v1 ðtÞ, v2 ðtÞ, v3 ðtÞ and v4 ðtÞ. We choose v1 ¼ aey ew v2 ¼ dex ð1 þ cÞey v3 ¼ 0 v4 ¼ ð1 þ rÞew
ð24Þ
then the error dynamical system is e_ x ¼ aex e_ y ¼ ey e_ 2 ¼ bez e_ w ¼ ew Eq. (25) describes the error dynamics. Now we define the Laypunov function for the system (25) as follows: 1 2 V ðex ; ey ; ez ; ew Þ ¼ ex þ e2y þ e2z þ e2w 2
ð25Þ
ð26Þ
This function is positive definite and equal zero at the equilibrium of the system (25). Moreover, the derivative of the Laypunov function (26) has the form
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h i V_ ¼ ae2x þ e2y þ be2z þ e2w which is negative definite. From Lyapunov direct method, we have that the zero solution of the system (25) is asymptotically stable. This implies that the two Chen hyperchaotic systems are anti-synchronized. 4.1. Numerical results By using MAPLE to solve the systems of differential Eqs. (19)–(21). The parameters are chosen to be a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5 in all simulations so that the Chen hyperchaotic system exhibits a chaotic behavior if no control is applied. The initial states of the drive system are x1 ð0Þ ¼ 0:1, y 1 ð0Þ ¼ 0:2, z1 ð0Þ ¼ 0:6 and w1 ð0Þ ¼ 0:4 and initial states of the response system are x2 ð0Þ ¼ 1, y 2 ð0Þ ¼ 0:4, z2 ð0Þ ¼ 0:2 and w2 ð0Þ ¼ 1, hence the error system has the initial values ex ð0Þ ¼ 1:1, ey ð0Þ ¼ 0:6, ez ð0Þ ¼ 0:8 and ew ð0Þ ¼ 1:4. The results of the two identical Chen hyperchaotic systems
a 40.00
b
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-40.00
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-40.00
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0.00
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Fig. 3. Shows that the time response of states for drive system ðx1 ; y 1 ; z1 Þ and response system ðx2 ; y 2 ; z2 Þ with active control law activated. (a) Signals x1 ðtÞ and x2 ðtÞ; (b) signals y 1 ðtÞ and y 2 ðtÞ; (c) signals z1 ðtÞ and z2 ðtÞ; (d) signals w1 ðtÞ and w2 ðtÞ.
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with active control are shown in Fig. 3: (a) displays the trajectories x1 ðtÞ and x2 ðtÞ, (b) displays the trajectories y 1 ðtÞ and y 2 ðtÞ, (c) displays the trajectories z1 ðtÞ and z2 ðtÞ and (d) displays the trajectories w1 ðtÞ and w2 ðtÞ.
5. Conclusion In this paper, we study the problem of chaos synchronization and anti synchronization between two identical Chen hyperchaotic dynamical systems. Using the active control technique, we have realized the synchronization and anti-synchronization between two identical Chen hyperchaotic dynamical systems. It has found that one can use active control theory to synchronize and anti-synchronization chaotic systems. Numerical simulations are used to verify the effectiveness of the proposed control techniques.
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Ott E, Grebogi C, Yorke JA. Controlling chaos [J]. Phys Rev Lett 1990;64:1196–9. Pecora LM, Carroll TM. Synchronization of chaotic systems. Phys Rev Lett 1990;64(8):821–30. Carroll TL, Pecora LM. Synchronizing a chaotic systems. IEEE Trans Circuit Syst 1991;38:453–6. Zhang Gang, Liu Zengrong, Ma Zhongjun. Generalized synchronization of different dimensional chaotic dynamical systems. Chaos, Solitons & Fractals 2007;32(2):773–9. Li Guo-Hui, Zhou Shi-Ping, Yang Kui. Generalized projective synchronization between two different chaotic systems using active backstepping control. Phys Lett A 2006;355(4–5):326–30. Li Guo-Hui. Generalized projective synchronization of two chaotic systems by using active control. Chaos, Solitons & Fractals 2006;30(1):77–82. Winterhalder Matthias, Schelter Bjo¨rn, Kurths Ju¨rgen, Schulze-Bonhage Andreas, Timmer Jens. Sensitivity and specificity of coherence and phase synchronization analysis. Phys Lett A 2006;356(1):26–34. Rosenblum Michael G, Pikovsky Arkady S, Kurths Ju¨rgen. Phase synchronization of chaotic oscillators. Phys Rev Lett 1996;76:1804–7. Liu Zengrong, Luo Jigui. From lag synchronization to pattern formation in one-dimensional open flow models. Chaos, Solitons & Fractals 2006;30(5):1198–205. Taherionl S, Lai YC. Observability of lag synchronization of coupled chaotic oscillators. Phys Rev E 1999;59:6247–50. Zhang Xiaohong, Liao Xiaofeng, Li Chuandong. Impulsive control, complete and lag synchronization of unified chaotic system with continuous periodic switch. Chaos, Solitons & Fractals 2005;26(3):845–54. Li Chuandong, Liao Xiaofeng. Lag synchronization of Rossler system and Chua circuit via a scalar signal. Phys Lett A 2004;329(4–5):301–8. Li Guo-Hui, Zhou Shi-Ping. Anti-synchronization in different chaotic systems. Chaos, Solitons & Fractals 2007;32(2):516–20. Li Guo-Hui, Zhou Shi-Ping. An observer-based anti-synchronization. Chaos, Solitons & Fractals 2006;29(2):495–8. Li Guo-Hui. Synchronization and anti-synchronization of Colpitts oscillators using active control. Chaos, Solitons & Fractals 2005;26(1):87–93. Hu Jia, Chen Shihua, Chen Li. Adaptive control for anti-synchronization of Chua’s chaotic system. Phys Lett A 2005;339(6):455–60. Zhang Yinping, Sun Jitao. Chaotic synchronization and anti-synchronization based on suitable separation. Phys Lett A 2004;330(6):442–7. Kim Chil-Min, Rim Sunghwan, Kye Won-Ho, Ryu Jung-Wan, Park Young-Jai. Anti-synchronization of chaotic oscillators. Phys Lett A 2003;320(1):39–46. Yu Hongjie, Liu Yanzhu. Chaotic synchronization based on stability criterion of linear systems. Phys Lett A 2003;314:292–8. Li Y, Tang Wallack S, Chen Guanrong. Generating hyperchaos via state feedback control. Int J Bifurcation Chaos 2005;5(10):3367–75. Yan Zhenya. Controlling hyperchaos in the new hyperchaotic Chen system. Appl Math Comput 2005;68(2):1239–50.
Synchronization and anti-synchronization of a hyperchaotic Chen system M.M. El-Dessoky Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Accepted 18 June 2007
Abstract Based on the active control theory, synchronization and anti-synchronization between two identical chaotic systems is investigated. Anti-synchronization can be characterized by the vanishing of the sum of relevant variables. Through rigorous mathematical theory, the sufficient condition is drawn for the stability of the error dynamics, where the controllers are designed by using the sum of the relevant variables in chaotic systems. Numerical simulations are performed for Chen hyperchaotic dynamical system to demonstrate the effectiveness of the proposed control strategy. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Chaos synchronization has been investigated extensively in the last decades and many effective methods have been suggested since the pioneering work developed by Ott et al. [1]. The aim of synchronization is to use the output of a master system to control the slave system so that its output of the slave system follows that of the master system asymptotically. The original synchronization technique developed by Pecorra and Carroll [2,3] is relevant to complete synchronization. Recently, the concept of synchronization has been extended to the scope, such as generalized synchronization [3–5], phase synchronization [6,7], lag synchronization [8–12], and anti-synchronization (AS) [13–19], in which the state vectors of synchronized systems have the same absolute values but opposite signs. Therefore, the sum of two signals can converge to zero when AS appears. The aim of this work is to further develop the state observer method for constructing anti-synchronized slave system. Finally, simulations on Chen hyperchaotic dynamical system are performed to verify the effectiveness and feasibility of the proposed control technique.
2. Chaotic synchronization We consider a chaotic continuous system described by x_ ¼ f ðxðtÞ; tÞ E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.053
ð1Þ
M.M. El-Dessoky / Chaos, Solitons and Fractals 39 (2009) 1790–1797
1791
where x 2 Rn is a n-dimensional state vector of the system, and f : Rn ! Rn defines a vector field in n-dimensional space. We decompose the function f(x(t), t) as f ðxðtÞ; tÞ ¼ gðxðtÞ; tÞ þ hðxðtÞ; tÞ
ð2Þ
where g(x(t)) is the linear part of f(x(t), t) described by gðxðtÞÞ ¼ AxðtÞ
ð3Þ
where A is a full rank constant matrix and all eigenvalues of A have negative real parts, hðxðtÞ; tÞ ¼ f ðxðtÞ; tÞ gðxðtÞ; tÞ is the nonlinear part of f(x(t), t). Then the system (1) can be written as x_ ¼ gðxðtÞ; tÞ þ hðxðtÞ; tÞ
ð4Þ
2.1. Synchronization The chaotic synchronization discussed in this Letter is defined as the complete synchronization, which means that the difference of the states of two relevant systems converges to zero. For the given chaotic system (4), we construct a new system as follows: _ wðtÞ ¼ gðwðtÞ; tÞ þ hðxðtÞ; tÞ
ð5Þ
n
where wðtÞ 2 R is the state vector of the new system. The synchronization error between system (1) and system (5) is defined as eðtÞ ¼ xðtÞ wðtÞ, the evolution of which is determined by the following equation: e_ ðtÞ ¼ x_ ðtÞ wðtÞ _ ¼ AðxðtÞ wðtÞÞ ¼ AeðtÞ
ð6Þ
obviously that the zero point of e(t) is its equilibrium point. Since all eigenvalues of the matrix A have negative real parts, according to the stability criterion of linear system, the zero point of synchronization error is asymptotically stable and e(t) tends to zero when t ! 1. Then the state vectors x(t) and w(t) of different systems (1) and (5) are synchronized. 2.2. Anti-synchronization Anti-synchronization is a phenomenon that the state vectors of synchronized systems have the same absolute values but opposite signs. We say that anti-synchronization of two system S1 and S2 is achieved if the following equation holds: lim kx1 ðtÞ þ x2 ðtÞk ¼ 0
t!1
ð7Þ
where xi ðtÞ, ði ¼ 1; 2Þ is the state vector of the system SiðtÞ, ði ¼ 1; 2Þ. We construct a new system described by _ wðtÞ ¼ gðwðtÞ; tÞ hðxðtÞ; tÞ
ð8Þ
The synchronization error between system (1) and system (8) is defined as eðtÞ ¼ xðtÞ þ wðtÞ, the evolution of which is determined by the following equation: _ e_ ðtÞ ¼ x_ ðtÞ þ wðtÞ ¼ AðxðtÞ þ wðtÞÞ ¼ AeðtÞ
ð9Þ
Since all eigenvalues of the matrix A have negative real parts, the zero point of synchronization error is asymptotically stable and e(t) tends to zero when t ! 1. Then the state vectors x(t) and w(t) of different systems (1) and (8) can be anti-synchronized. The simplest configuration of matrix A ¼ diagðk1 ; k2 ; . . . ; kn Þ through separation of systems can be used for the adjustment of parameters to satisfy the stability criterion of linear system that all eigenvalues have negative real.
3. Synchronization of Chen hyperchaotic system The differential equations of Chen hyperchaotic system [20,21] is described by x_ ¼ aðy xÞ þ w y_ ¼ dx xz þ cy z_ ¼ xy bz w_ ¼ yz þ rw
ð10Þ
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where a, b, c, d and r are positive parameters. If a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and 0 6 r 6 0:085, then the system (10) is chaotic; while if a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and 0:085 6 r 6 0:798, then the system (10) is hyperchaotic; and if a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and 0:798 6 r 6 0:9 system (10) is periodic. In particular, these projections of hyperchaotic attractor of Chen hyperchaotic system (10) with a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r ¼ 0:5 are displayed in Fig. 1a–d. In order to observe the synchronization behavior in the Chen hyperchaotic system, we have two Chen hyperchaotic systems where the drive system with four state variables denoted by the subscript 1 drives the response system having identical equations denoted by the subscript 2. However, the initial condition on the drive system is different from that of the response system. The two Chen systems are described, respectively, by the following equations: x_ 1 ¼ aðy 1 x1 Þ þ w1 y_ 1 ¼ dx1 x1 z1 þ cy 1 z_ 1 ¼ x1 y 1 bz1 w_ 1 ¼ y 1 z1 þ rw1
ð11Þ
x_ 2 ¼ aðy 2 x2 Þ þ w2 þ u1 y_ 2 ¼ dx2 x2 z2 þ cy 2 þ u2 z_ 2 ¼ x2 y 2 bz2 þ u3 w_ 2 ¼ y 2 z2 þ rw2 þ u4
ð12Þ
and
where U ¼ ½u1 ; u2 ; u3 ; u4 T is the active control function to be designed. The aim of this section is to determine the controller U for the chaos synchronization of two Chen hyperchaotic dynamical system with the same unknown parameters and different initial conditions. For this purpose, the error dynamical system between the drive system (11) and response system (12) can be expressed by
Fig. 1. Shows the chaotic attractor of Chen hyperchaotic system at: (a) a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5 in x; y; z subspace; (b) a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5 in x; y; w subspace; (c) a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5 in x; z; w subspace; (d) a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5 in y; z; w subspace.
M.M. El-Dessoky / Chaos, Solitons and Fractals 39 (2009) 1790–1797
e_ x ¼ aðey ex Þ þ ew þ u1 e_ y ¼ dex þ x1 z1 x2 z2 þ cey þ u2 e_ 2 ¼ x2 y 2 x1 y 1 bez þ u3 e_ w ¼ y 2 z2 y 1 z1 þ rew þ u4
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ð13Þ
where ex ¼ x2 x1 , ey ¼ y 2 y 1 , ez ¼ z2 z1 and ew ¼ w2 w1 . The active control function U ¼ ½u1 ; u2 ; u3 ; u4 T u1 u2 u3 u4
¼ v1 ¼ x2 z2 x1 z1 þ v2 ¼ x1 y 1 x2 y 2 þ v3 ¼ y 1 z1 y 2 z2 þ v4
ð14Þ
This leads to e_ x ¼ aðey ex Þ þ ew þ v1 e_ y ¼ dex þ cey þ v2 e_ 2 ¼ bez þ v3 e_ w ¼ rew þ v4
ð15Þ
Eq. (15) describe the error dynamics and can be considered in terms of a control problem where the system to be controlled is a linear system with a control input v1 ðtÞ, v2 ðtÞ, v3 ðtÞ and v4 ðtÞ as function of ex , ey , ez and ew . As long as these feedbacks stabilize the system, ex , ey , ez and ew converge to zero as time t goes to infinity. This implies that two Chen hyperchaotic systems are anti-synchronized with feedback control. There are many possible choices for the control v1 ðtÞ, v2 ðtÞ, v3 ðtÞ and v4 ðtÞ. We choose v1 v2 v3 v4
¼ aey ew ¼ dex ð1 þ cÞey ¼0 ¼ ð1 þ rÞew
ð16Þ
then the error dynamical system is e_ x ¼ aex e_ y ¼ ey e_ 2 ¼ bez e_ w ¼ ew Eq. (17) describes the error dynamics. Now we define the Laypunov function for the system (17) as follows: 1 2 V ðex ; ey ; ez ; ew Þ ¼ ex þ e2y þ e2z þ e2w 2
ð17Þ
ð18Þ
This function is positive definite and equal zero at the equilibrium of the system (17). Moreover, the derivative of the Laypunov function (18) has the form h i V_ ¼ ae2x þ e2y þ be2z þ e2w which is negative definite. From Lyapunov direct method, we have that the zero solution of the system (17) is asymptotically stable. This implies that the two Chen hyperchaotic systems are synchronized. 3.1. Numerical results By using MAPLE to solve the systems of differential Eqs. (11)–(13). The parameters are chosen to be a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r ¼ 0:5 in all simulations so that the Chen hyperchaotic system exhibits a chaotic behavior if no control is applied.
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Fig. 2. (a) Shows the behavior of the trajectory ex of the error system tends to zero as t tends to 1 when the parameter values are a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5. (b) Shows the behavior of the trajectory ey of the error system tends to zero as t tends to 7 when the parameter values are a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5. (c) Shows the behavior of the trajectory ez of the error system tends to zero as t tends to 3 when the parameter values are a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5. (d) Shows the behavior of the trajectory ew of the error system tends to zero as t tends to 6 when the parameter values are a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5.
The initial states of the drive system are x1 ð0Þ ¼ 0:1, y 1 ð0Þ ¼ 0:2, z1 ð0Þ ¼ 0:6 and w1 ð0Þ ¼ 0:4 and initial states of the response system are x2 ð0Þ ¼ 1, y 2 ð0Þ ¼ 0:4, z2 ð0Þ ¼ 0:2 and w2 ð0Þ ¼ 1, hence the error system has the initial values ex ð0Þ ¼ 0:9, ey ð0Þ ¼ 0:2, ez ð0Þ ¼ 0:4 and ew ð0Þ ¼ 0:6. The results of the two identical Chen hyperchaotic systems with active control are shown in Fig. 2a–c displays the trajectories of ex , ey , ez and ew of the error system tended to zero.
4. Anti-synchronization of Chen hyperchaotic system In this section to study the anti-synchronization problem of Chen hyperchaotic system. In order to observe the antisynchronization behavior in the Chen hyperchaotic system, we have two Chen hyperchaotic systems where the drive system with four state variables denoted by the subscript 1 drives the response system having identical equations denoted by the subscript 2. However, the initial condition on the drive system is different from that of the response system. The two Chen systems are described, respectively, by the following equations: x_ 1 ¼ aðy 1 x1 Þ þ w1 y_ 1 ¼ dx1 x1 z1 þ cy 1 z_ 1 ¼ x1 y 1 bz1 w_ 1 ¼ y 1 z1 þ rw1 and
ð19Þ
M.M. El-Dessoky / Chaos, Solitons and Fractals 39 (2009) 1790–1797
x_ 2 ¼ aðy 2 x2 Þ þ w2 þ u1 y_ 2 ¼ dx2 x2 z2 þ cy 2 þ u2 z_ 2 ¼ x2 y 2 bz2 þ u3 w_ 2 ¼ y 2 z2 þ rw2 þ u4
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ð20Þ
be the response system where we have introduced four control functions u1 ðtÞ, u2 ðtÞ, u3 ðtÞ and u4 ðtÞ in Eq. (20). These functions are to be determined for the purpose of anti-synchronizing the two Chen hyperchaotic dynamical system with the same unknown parameters and different initial conditions. Let us define the state errors between the response system that is to be controlled and the controlling derive system as ex ¼ x2 þ x1 , ey ¼ y 2 þ y 1 , ez ¼ z2 þ z1 and ew ¼ w2 þ w1 . By adding Eq. (19) from Eq. (20) yields the error dynamical system between Eqs. (19) and (20) e_ x ¼ aðey ex Þ þ ew þ u1 e_ y ¼ dex x1 z1 x2 z2 þ cey þ u2 e_ 2 ¼ x2 y 2 þ x1 y 1 bez þ u3 e_ w ¼ y 2 z2 þ y 1 z1 þ rew þ u4
ð21Þ
The anti-synchronization problem for Chen hyperchaotic dynamical system is to achieve the asymptotic stability of the zero solution of the error system (21). To this end we take the active control functions u1 ðtÞ, u2 ðtÞ, u3 ðtÞ and u4 ðtÞ as follows: u1 ¼ v1 u2 ¼ x2 z2 þ x1 z1 þ v2 u3 ¼ x1 y 1 x2 y 2 þ v3 u4 ¼ y 1 z1 y 2 z2 þ v4
ð22Þ
This leads to e_ x ¼ aðey ex Þ þ ew þ v1 e_ y ¼ dex þ cey þ v2 e_ 2 ¼ bez þ v3 e_ w ¼ rew þ v4
ð23Þ
Eq. (23) describe the error dynamics and can be considered in terms of a control problem where the system to be controlled is a linear system with a control input v1 ðtÞ, v2 ðtÞ, v3 ðtÞ and v4 ðtÞ as function of ex , ey , ez and ew . As long as these feedbacks stabilize the system, ex , ey , ez and ew converge to zero as time t goes to infinity. This implies that two Chen hyperchaotic systems are anti-synchronized with feedback control. There are many possible choices for the control v1 ðtÞ, v2 ðtÞ, v3 ðtÞ and v4 ðtÞ. We choose v1 ¼ aey ew v2 ¼ dex ð1 þ cÞey v3 ¼ 0 v4 ¼ ð1 þ rÞew
ð24Þ
then the error dynamical system is e_ x ¼ aex e_ y ¼ ey e_ 2 ¼ bez e_ w ¼ ew Eq. (25) describes the error dynamics. Now we define the Laypunov function for the system (25) as follows: 1 2 V ðex ; ey ; ez ; ew Þ ¼ ex þ e2y þ e2z þ e2w 2
ð25Þ
ð26Þ
This function is positive definite and equal zero at the equilibrium of the system (25). Moreover, the derivative of the Laypunov function (26) has the form
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h i V_ ¼ ae2x þ e2y þ be2z þ e2w which is negative definite. From Lyapunov direct method, we have that the zero solution of the system (25) is asymptotically stable. This implies that the two Chen hyperchaotic systems are anti-synchronized. 4.1. Numerical results By using MAPLE to solve the systems of differential Eqs. (19)–(21). The parameters are chosen to be a ¼ 35, b ¼ 3, c ¼ 12, d ¼ 7 and r = 0.5 in all simulations so that the Chen hyperchaotic system exhibits a chaotic behavior if no control is applied. The initial states of the drive system are x1 ð0Þ ¼ 0:1, y 1 ð0Þ ¼ 0:2, z1 ð0Þ ¼ 0:6 and w1 ð0Þ ¼ 0:4 and initial states of the response system are x2 ð0Þ ¼ 1, y 2 ð0Þ ¼ 0:4, z2 ð0Þ ¼ 0:2 and w2 ð0Þ ¼ 1, hence the error system has the initial values ex ð0Þ ¼ 1:1, ey ð0Þ ¼ 0:6, ez ð0Þ ¼ 0:8 and ew ð0Þ ¼ 1:4. The results of the two identical Chen hyperchaotic systems
a 40.00
b
40.00
y1 20.00
20.00
x1
0.00
0.00
-20.00
-20.00
x2
-40.00
c
-40.00
t
0.00
2.00
4.00
6.00
8.00
10.00
40.00
y2
t
0.00
d
2.00
4.00
6.00
8.00
10.00
6.00
8.00
10.00
200.00
z1 20.00
100.00
w1 0.00
0.00
w2 -20.00
-100.00
z2 -40.00 0.00
t 2.00
4.00
6.00
8.00
10.00
-200.00 0.00
t 2.00
4.00
Fig. 3. Shows that the time response of states for drive system ðx1 ; y 1 ; z1 Þ and response system ðx2 ; y 2 ; z2 Þ with active control law activated. (a) Signals x1 ðtÞ and x2 ðtÞ; (b) signals y 1 ðtÞ and y 2 ðtÞ; (c) signals z1 ðtÞ and z2 ðtÞ; (d) signals w1 ðtÞ and w2 ðtÞ.
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with active control are shown in Fig. 3: (a) displays the trajectories x1 ðtÞ and x2 ðtÞ, (b) displays the trajectories y 1 ðtÞ and y 2 ðtÞ, (c) displays the trajectories z1 ðtÞ and z2 ðtÞ and (d) displays the trajectories w1 ðtÞ and w2 ðtÞ.
5. Conclusion In this paper, we study the problem of chaos synchronization and anti synchronization between two identical Chen hyperchaotic dynamical systems. Using the active control technique, we have realized the synchronization and anti-synchronization between two identical Chen hyperchaotic dynamical systems. It has found that one can use active control theory to synchronize and anti-synchronization chaotic systems. Numerical simulations are used to verify the effectiveness of the proposed control techniques.
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