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Generalized projective synchronization between Lorenz system and Chen’s system
Chaos, Solitons and Fractals 32 (2007) 1454–1458 www.elsevier.com/locate/chaos
Generalized projective synchronization between Lorenz system and Chen’s system Guo-Hui Li
*
Department of Communication Engineering, Shanghai University, Yanchang Road 149, Shanghai 200072, China Accepted 21 November 2005
Communicated by Prof. Mohamed El Naschie
Abstract On the basis of active backstepping design, this paper presents the generalized projective synchronization between two different chaotic systems: Lorenz system and Chen’s system. The proposed method combines backstepping methods and active control without having to calculate the Lyapunov exponents and the eigenvalues of the Jacobian matrix, which makes it simple and convenient. Numerical simulations show that this method works very well. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Since the pioneering work by Pecora and Carroll [1], much attention has been devoted to research on synchronization of chaos. Amongst all kinds of chaos synchronization, projective synchronization, which was reported by Mainieri and Rehacek [2] in partially linear systems, is the most noticeable due to its proportionality between the synchronized dynamical states. Because this sort of synchronization is associated with projective synchronization and generalized one, some authors [3,4] extended it to a general class of chaotic systems and termed it as ‘‘generalized projective synchronization’’ (GPS). Complete synchronizations [1] and anti-phase synchronizations [5] are the special cases of generalized projective synchronization where the scaling factor a = 1 and a = 1, respectively. This paper considers the projective synchronization problems of two different chaotic systems. In other words, we break the limit of ‘‘controlling two identical systems’’ and apply the control techniques to synchronize two different systems. The method [6] is a combination of backstepping design [7–9] and active control technique [10]. The layout of the rest of the paper is as follows. In Section 2, a brief description of Lorenz system and Chen’s system is introduced. Section 3 investigates the design of an active backstepping controller for the projective synchronization of two different chaotic systems. Numerical simulation results are given for illustration and verification in Section 4. Finally, conclusions are drawn in Section 5.
*
Tel.: +86 21 56332993. E-mail address: [email protected]
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.11.073
G.-H. Li / Chaos, Solitons and Fractals 32 (2007) 1454–1458
1455
2. System description The famous Lorenz system, the first chaotic attractor model in a three-dimensional autonomous system, is given by 8 > < x_ ¼ rðy xÞ; ð1Þ y_ ¼ cx xz y; > : z_ ¼ xy bz; where r, c, b are parameters. When r = 10, c = 28, b = 8/3, system (1) exhibits chaotic behavior. Chen’s system is another typical chaotic model, which has a more complicated topological structure than Lorenz attractor. The dynamical equations that describe the Chen’s system are 8 > < x_ ¼ aðy xÞ; y_ ¼ xz þ ðc aÞx þ cy; ð2Þ > : z_ ¼ xy bz. This system exhibits a chaotic attractor at the parameters a = 35, b = 3, c = 28. In the next sections, we will study GPS between the chaotic dynamical systems mentioned above.
3. GPS between Lorenz system and Chen’s system The projective synchronization means that the drive and response vectors synchronize up to a scaling factor a, that is, the vectors become proportional. First, we define the GPS below. Consider the following chaotic system: x_ m ¼ f ðxm Þ; ð3Þ x_ s ¼ gðxm ; xs Þ; where n-dimensional state vector xm, xs 2 Rn. The subscripts ‘m’ and ‘s’ stand for the master and slave systems, respectively. f : Rn ! Rn and g : Rn ! Rn are vector fields in n-dimensional space. If there exists a constant a (a 5 0), such that limt!1kxm axsk = 0, then GPS of the system (3) is achieved, and we call a ‘scaling factor’. Now, we want to control two different chaos systems. Take Lorenz system and Chen’s system into consideration. In order to observe GPS between Lorenz and Chen’s systems, it is assumed that Lorenz system drives Chen’s system. Therefore, we rewrite the master and slave systems as follows: 8 > < x_ m ¼ rðy m xm Þ; y_ m ¼ cxm xm zm y m ; ð4Þ > : z_ m ¼ xm y m bzm ; and 8 > < x_ s ¼ aðy s xs Þ þ u1 ; y_ s ¼ xs zs þ ðc aÞxs þ cy s þ u2 ; > : z_ s ¼ xs y s bzs þ u3 .
ð5Þ
We have introduced three control functions u1, u2 and u3 in (5). Our goal is to determine the control functions u1, u2 and u3 such that limt!1kxi ayik = 0. In order to estimate the control functions, we subtract (5) from (4) 8 > < e_ 1 ¼ ae2 ae1 þ f1 au1 ; ð6Þ e_ 2 ¼ ðc aÞe1 þ ce2 e1 zm =a ðxm e1 Þe3 =a þ f2 au2 ; > : e_ 3 ¼ y m e1 =a ðe1 xm Þe2 =a be3 þ f3 au3 ; by defining state errors e1 = xm axs, e2 = ym ays, e3 = zm azs, where f1 = (r a)(ym xm), f2 = (c c a)xm (c + 1)ym + xmzm(1/a 1), f3 = xmym(1 1/a) + (b b)zm. The objective is to find a control law ui for stabilizing the error variables of system (6) at the origin. Now we begin to design the active controllers based on the backstepping design method as follows:
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G.-H. Li / Chaos, Solitons and Fractals 32 (2007) 1454–1458
Step 1: Let w1 = e1, then we can obtain its derivative w_ 1 ¼ e_ 1 ¼ ae2 aw1 þ f1 au1 ;
ð7Þ
where e2 = a1(w1) is regarded as a virtual controller. For the design of a1(w1) to stabilize w1-subsystem (7), we can choose the first Lyapunov function 1 v1 ¼ w21 . 2
ð8Þ
The derivative of v1 is as follows: v_ 1 ¼ aw21 þ w1 ðae2 þ f1 au1 Þ.
ð9Þ
If we set e2 = a1(w1) = 0, and au1 = f1, then V1 is negative and definite. This implies that the w1-subsystem (7) is asymptotically stable. Since the virtual control function a1 is estimative, the error between e2 and a1 is w2 ¼ e2 a1 ðw1 Þ.
ð10Þ
Study the following (w1, w2)-subsystem w_ 1 ¼ aw2 aw1 ; w_ 2 ¼ ðc aÞw1 þ cw2 w1 zm =a ðxm w1 Þe3 =a þ f2 au2 .
ð11Þ
Consider e3 = a2(w1, w2) as a virtual controller to make system (11) asymptotically stable.
3 0 e1(t)
-3
e(t)
e2(t) e3(t)
-6 -9 -12 0
(a)
1
2
3
4
5
t
0
zs
80
zm
60
40
-50
-100 -40
-20
0
xs
20
40
20
-20
(b)
-10
0 Xm
10
20
Fig. 1. GPS between Lorenz system (1) and Chen’s system (4) with the scaling factor a = 0.5. (a) The time evolution of the error vectors. (b) The projection of the synchronized attractors onto the xm zm (xs zs) plane after transient states.
G.-H. Li / Chaos, Solitons and Fractals 32 (2007) 1454–1458
1457
Step 2: In order to stabilize the (w1, w2)-subsystem (11), we can choose the second Lyapunov function defined by 1 ð12Þ v2 ¼ v1 þ w22 . 2 Its derivative is v_ 2 ¼ aw21 þ w2 ðcw1 þ cw2 w1 zm =a ðxm w1 Þa2 =a þ f2 au2 Þ ð13Þ If a2(w1, w2) = 0 and au2 = c(w1 + w2) w1zm/a + f2 + w2, then v_ 2 ¼ aw21 w22 < 0 makes subsystem (11) asymptotically stable. Similarly, assume that w3 = e3 a2(w1, w2), and then we can derive the full dimension (w1, w2, w3)-subsystem: 8 > < w_ 1 ¼ aw2 aw1 ; w_ 2 ¼ aw1 w2 ; ð14Þ > : w_ 3 ¼ y m w1 =a ðw1 xm Þw2 =a bw3 þ f3 au3 . Step 3: In order to stabilize system (14), we can choose the third Lyapunov function v3 ¼ v2 þ 12 w23 . The derivative of V3 is ð15Þ v_ 3 ¼ aw21 w22 bw23 þ w3 ðy m w1 =a þ f3 au3 Þ. 2 2 2 _ Let au3 = ymw1/a + f3, then V 3 ¼ aw1 w2 bw3 < 0, which makes the (w1, w2, w3)-system (14) asymptotically stable. Based on Lyapunov stability theory, the error system (6) is globally stable about the origin. Therefore, the system (4) is projective-synchronized with the system (5) via backstepping design. In other words, the generalized projective synchronization between (4) and (5) can be realized. 30 e1(t)
20
e2(t) e3(t)
10 e(t)
0 -10 -20 -30 0
1
2
3
(a)
4
5
t
80
25 20
zs
15
60
10
zm
5
40
0 -10
0
xs
10
20
-20
(b)
-10
0 Xm
10
20
Fig. 2. GPS between Lorenz system (1) and Chen’s system (4) with the scaling factor a = 2. (a) The time evolution of the error vectors. (b) The projection of the synchronized attractors onto the xm zm (xs zs) plane after transient states.
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4. Numerical simulation In this section, numerical experiments are conducted to verify the effectiveness of the proposed approach. In the numerical simulations, fourth-order Runge–Kutta method is used to solve the systems with a time step size of 0.001. The initial states for the master system and slave system are taken as (8, 5, 6), and (10, 10, 8), respectively. In the present paper, we consider two cases below: Case 1: The scaling factor a = 0.5. The corresponding numerical results are shown in Fig. 1. Fig. 1(a) displays the time response of the synchronization error e = [e1, e2, e3]T. As expected, one can observe that the error vector e converges to zero, which implies that all the state variables tend to be synchronized in a proportional way. Fig. 1(b) depicts the projection of the synchronized attractors onto the xm zm (xs zs) plane after transient states, where the state vectors of the master state and the slave state evolve in the opposite directions. Case 2: The scaling factor a = 2. The simulation results are also illustrated in Fig. 2. From the figure, it can be seen that the synchronization error converges to zero and two different systems are indeed achieving projective-synchronization. Meanwhile, as with the phase difference being zero, the state vectors of the systems are synchronized in the same direction all the time.
5. Conclusion In this paper, an active backstepping control method, which combines backstepping design and active control technique, is proposed for GPS in a general class of chaotic systems. It extends the control capability to achieve a full range synchronization of all state variables in a proportional way. It also allows us to arbitrarily adjust the desired scaling by controlling the slave system. It is not necessary to calculate the Lyapunov exponents and the eigenvalues of the Jacobian matrix, which makes it simple and convenient. Numerical simulations are carried out to verify the effectiveness of the proposed controller.
Acknowledgment This work was supported by the Shanghai Leading Academic Disciplines (No. T0102).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4. Mainieri R, Rehacek J. Projective synchronization in three-dimensional chaotic systems. Phys Rev Lett 1999;82:3042–5. Yan J, Li C. Generalized projective synchronization of a unified chaotic system. Chaos, Solitons & Fractals 2005;26:1119–24. Wen G, Xu D. Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems. Chaos, Solitons & Fractals 2005;26:71–7. Cao LY, Lai YC. Anti-phase synchronism in chaotic systems. Phys Rev E 1998;58:382–6. Zhang H, Ma XK, Yang Y, et al. Controlling and tracking hyperchaotic Rossler system via active backstepping design. Chaos, Solitons & Fractals 2005;26:353–61. Lu JG, Wei R, Wang XF, et al. Backstepping control of discrete-time chaotic systems with application to the Henon systems. IEEE Trans CAS 2001;48(11):1359–63. Wang C, Ge SS. Adaptive synchronization of uncertain chaotic systems via backstepping design. Chaos, Solitons & Fractals 2001;12(7):1199–206. Tan XH, Zhang JY, Yang YR. Synchronization chaotic systems using backstepping design. Chaos, Solitons & Fractals 2003;16(1):37–45. Bai E-W, Lonngsen KE. Synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 1997;8:51–8.
Generalized projective synchronization between Lorenz system and Chen’s system Guo-Hui Li
*
Department of Communication Engineering, Shanghai University, Yanchang Road 149, Shanghai 200072, China Accepted 21 November 2005
Communicated by Prof. Mohamed El Naschie
Abstract On the basis of active backstepping design, this paper presents the generalized projective synchronization between two different chaotic systems: Lorenz system and Chen’s system. The proposed method combines backstepping methods and active control without having to calculate the Lyapunov exponents and the eigenvalues of the Jacobian matrix, which makes it simple and convenient. Numerical simulations show that this method works very well. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Since the pioneering work by Pecora and Carroll [1], much attention has been devoted to research on synchronization of chaos. Amongst all kinds of chaos synchronization, projective synchronization, which was reported by Mainieri and Rehacek [2] in partially linear systems, is the most noticeable due to its proportionality between the synchronized dynamical states. Because this sort of synchronization is associated with projective synchronization and generalized one, some authors [3,4] extended it to a general class of chaotic systems and termed it as ‘‘generalized projective synchronization’’ (GPS). Complete synchronizations [1] and anti-phase synchronizations [5] are the special cases of generalized projective synchronization where the scaling factor a = 1 and a = 1, respectively. This paper considers the projective synchronization problems of two different chaotic systems. In other words, we break the limit of ‘‘controlling two identical systems’’ and apply the control techniques to synchronize two different systems. The method [6] is a combination of backstepping design [7–9] and active control technique [10]. The layout of the rest of the paper is as follows. In Section 2, a brief description of Lorenz system and Chen’s system is introduced. Section 3 investigates the design of an active backstepping controller for the projective synchronization of two different chaotic systems. Numerical simulation results are given for illustration and verification in Section 4. Finally, conclusions are drawn in Section 5.
*
Tel.: +86 21 56332993. E-mail address: [email protected]
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.11.073
G.-H. Li / Chaos, Solitons and Fractals 32 (2007) 1454–1458
1455
2. System description The famous Lorenz system, the first chaotic attractor model in a three-dimensional autonomous system, is given by 8 > < x_ ¼ rðy xÞ; ð1Þ y_ ¼ cx xz y; > : z_ ¼ xy bz; where r, c, b are parameters. When r = 10, c = 28, b = 8/3, system (1) exhibits chaotic behavior. Chen’s system is another typical chaotic model, which has a more complicated topological structure than Lorenz attractor. The dynamical equations that describe the Chen’s system are 8 > < x_ ¼ aðy xÞ; y_ ¼ xz þ ðc aÞx þ cy; ð2Þ > : z_ ¼ xy bz. This system exhibits a chaotic attractor at the parameters a = 35, b = 3, c = 28. In the next sections, we will study GPS between the chaotic dynamical systems mentioned above.
3. GPS between Lorenz system and Chen’s system The projective synchronization means that the drive and response vectors synchronize up to a scaling factor a, that is, the vectors become proportional. First, we define the GPS below. Consider the following chaotic system: x_ m ¼ f ðxm Þ; ð3Þ x_ s ¼ gðxm ; xs Þ; where n-dimensional state vector xm, xs 2 Rn. The subscripts ‘m’ and ‘s’ stand for the master and slave systems, respectively. f : Rn ! Rn and g : Rn ! Rn are vector fields in n-dimensional space. If there exists a constant a (a 5 0), such that limt!1kxm axsk = 0, then GPS of the system (3) is achieved, and we call a ‘scaling factor’. Now, we want to control two different chaos systems. Take Lorenz system and Chen’s system into consideration. In order to observe GPS between Lorenz and Chen’s systems, it is assumed that Lorenz system drives Chen’s system. Therefore, we rewrite the master and slave systems as follows: 8 > < x_ m ¼ rðy m xm Þ; y_ m ¼ cxm xm zm y m ; ð4Þ > : z_ m ¼ xm y m bzm ; and 8 > < x_ s ¼ aðy s xs Þ þ u1 ; y_ s ¼ xs zs þ ðc aÞxs þ cy s þ u2 ; > : z_ s ¼ xs y s bzs þ u3 .
ð5Þ
We have introduced three control functions u1, u2 and u3 in (5). Our goal is to determine the control functions u1, u2 and u3 such that limt!1kxi ayik = 0. In order to estimate the control functions, we subtract (5) from (4) 8 > < e_ 1 ¼ ae2 ae1 þ f1 au1 ; ð6Þ e_ 2 ¼ ðc aÞe1 þ ce2 e1 zm =a ðxm e1 Þe3 =a þ f2 au2 ; > : e_ 3 ¼ y m e1 =a ðe1 xm Þe2 =a be3 þ f3 au3 ; by defining state errors e1 = xm axs, e2 = ym ays, e3 = zm azs, where f1 = (r a)(ym xm), f2 = (c c a)xm (c + 1)ym + xmzm(1/a 1), f3 = xmym(1 1/a) + (b b)zm. The objective is to find a control law ui for stabilizing the error variables of system (6) at the origin. Now we begin to design the active controllers based on the backstepping design method as follows:
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G.-H. Li / Chaos, Solitons and Fractals 32 (2007) 1454–1458
Step 1: Let w1 = e1, then we can obtain its derivative w_ 1 ¼ e_ 1 ¼ ae2 aw1 þ f1 au1 ;
ð7Þ
where e2 = a1(w1) is regarded as a virtual controller. For the design of a1(w1) to stabilize w1-subsystem (7), we can choose the first Lyapunov function 1 v1 ¼ w21 . 2
ð8Þ
The derivative of v1 is as follows: v_ 1 ¼ aw21 þ w1 ðae2 þ f1 au1 Þ.
ð9Þ
If we set e2 = a1(w1) = 0, and au1 = f1, then V1 is negative and definite. This implies that the w1-subsystem (7) is asymptotically stable. Since the virtual control function a1 is estimative, the error between e2 and a1 is w2 ¼ e2 a1 ðw1 Þ.
ð10Þ
Study the following (w1, w2)-subsystem w_ 1 ¼ aw2 aw1 ; w_ 2 ¼ ðc aÞw1 þ cw2 w1 zm =a ðxm w1 Þe3 =a þ f2 au2 .
ð11Þ
Consider e3 = a2(w1, w2) as a virtual controller to make system (11) asymptotically stable.
3 0 e1(t)
-3
e(t)
e2(t) e3(t)
-6 -9 -12 0
(a)
1
2
3
4
5
t
0
zs
80
zm
60
40
-50
-100 -40
-20
0
xs
20
40
20
-20
(b)
-10
0 Xm
10
20
Fig. 1. GPS between Lorenz system (1) and Chen’s system (4) with the scaling factor a = 0.5. (a) The time evolution of the error vectors. (b) The projection of the synchronized attractors onto the xm zm (xs zs) plane after transient states.
G.-H. Li / Chaos, Solitons and Fractals 32 (2007) 1454–1458
1457
Step 2: In order to stabilize the (w1, w2)-subsystem (11), we can choose the second Lyapunov function defined by 1 ð12Þ v2 ¼ v1 þ w22 . 2 Its derivative is v_ 2 ¼ aw21 þ w2 ðcw1 þ cw2 w1 zm =a ðxm w1 Þa2 =a þ f2 au2 Þ ð13Þ If a2(w1, w2) = 0 and au2 = c(w1 + w2) w1zm/a + f2 + w2, then v_ 2 ¼ aw21 w22 < 0 makes subsystem (11) asymptotically stable. Similarly, assume that w3 = e3 a2(w1, w2), and then we can derive the full dimension (w1, w2, w3)-subsystem: 8 > < w_ 1 ¼ aw2 aw1 ; w_ 2 ¼ aw1 w2 ; ð14Þ > : w_ 3 ¼ y m w1 =a ðw1 xm Þw2 =a bw3 þ f3 au3 . Step 3: In order to stabilize system (14), we can choose the third Lyapunov function v3 ¼ v2 þ 12 w23 . The derivative of V3 is ð15Þ v_ 3 ¼ aw21 w22 bw23 þ w3 ðy m w1 =a þ f3 au3 Þ. 2 2 2 _ Let au3 = ymw1/a + f3, then V 3 ¼ aw1 w2 bw3 < 0, which makes the (w1, w2, w3)-system (14) asymptotically stable. Based on Lyapunov stability theory, the error system (6) is globally stable about the origin. Therefore, the system (4) is projective-synchronized with the system (5) via backstepping design. In other words, the generalized projective synchronization between (4) and (5) can be realized. 30 e1(t)
20
e2(t) e3(t)
10 e(t)
0 -10 -20 -30 0
1
2
3
(a)
4
5
t
80
25 20
zs
15
60
10
zm
5
40
0 -10
0
xs
10
20
-20
(b)
-10
0 Xm
10
20
Fig. 2. GPS between Lorenz system (1) and Chen’s system (4) with the scaling factor a = 2. (a) The time evolution of the error vectors. (b) The projection of the synchronized attractors onto the xm zm (xs zs) plane after transient states.
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G.-H. Li / Chaos, Solitons and Fractals 32 (2007) 1454–1458
4. Numerical simulation In this section, numerical experiments are conducted to verify the effectiveness of the proposed approach. In the numerical simulations, fourth-order Runge–Kutta method is used to solve the systems with a time step size of 0.001. The initial states for the master system and slave system are taken as (8, 5, 6), and (10, 10, 8), respectively. In the present paper, we consider two cases below: Case 1: The scaling factor a = 0.5. The corresponding numerical results are shown in Fig. 1. Fig. 1(a) displays the time response of the synchronization error e = [e1, e2, e3]T. As expected, one can observe that the error vector e converges to zero, which implies that all the state variables tend to be synchronized in a proportional way. Fig. 1(b) depicts the projection of the synchronized attractors onto the xm zm (xs zs) plane after transient states, where the state vectors of the master state and the slave state evolve in the opposite directions. Case 2: The scaling factor a = 2. The simulation results are also illustrated in Fig. 2. From the figure, it can be seen that the synchronization error converges to zero and two different systems are indeed achieving projective-synchronization. Meanwhile, as with the phase difference being zero, the state vectors of the systems are synchronized in the same direction all the time.
5. Conclusion In this paper, an active backstepping control method, which combines backstepping design and active control technique, is proposed for GPS in a general class of chaotic systems. It extends the control capability to achieve a full range synchronization of all state variables in a proportional way. It also allows us to arbitrarily adjust the desired scaling by controlling the slave system. It is not necessary to calculate the Lyapunov exponents and the eigenvalues of the Jacobian matrix, which makes it simple and convenient. Numerical simulations are carried out to verify the effectiveness of the proposed controller.
Acknowledgment This work was supported by the Shanghai Leading Academic Disciplines (No. T0102).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4. Mainieri R, Rehacek J. Projective synchronization in three-dimensional chaotic systems. Phys Rev Lett 1999;82:3042–5. Yan J, Li C. Generalized projective synchronization of a unified chaotic system. Chaos, Solitons & Fractals 2005;26:1119–24. Wen G, Xu D. Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems. Chaos, Solitons & Fractals 2005;26:71–7. Cao LY, Lai YC. Anti-phase synchronism in chaotic systems. Phys Rev E 1998;58:382–6. Zhang H, Ma XK, Yang Y, et al. Controlling and tracking hyperchaotic Rossler system via active backstepping design. Chaos, Solitons & Fractals 2005;26:353–61. Lu JG, Wei R, Wang XF, et al. Backstepping control of discrete-time chaotic systems with application to the Henon systems. IEEE Trans CAS 2001;48(11):1359–63. Wang C, Ge SS. Adaptive synchronization of uncertain chaotic systems via backstepping design. Chaos, Solitons & Fractals 2001;12(7):1199–206. Tan XH, Zhang JY, Yang YR. Synchronization chaotic systems using backstepping design. Chaos, Solitons & Fractals 2003;16(1):37–45. Bai E-W, Lonngsen KE. Synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 1997;8:51–8.