- Email: [email protected]
Synchronization of the unified chaotic systems via active control
Chaos, Solitons and Fractals 27 (2006) 1292–1297 www.elsevier.com/locate/chaos
Synchronization of the unified chaotic systems via active control Ahmet Uc¸ar b
a,*
, Karl E. Lonngren b, Er-Wei Bai
b
a Department of Electrical and Electronics Engineering, Firat University, Elazıg 23119, Turkey Department of Electrical and Computer Engineering, University of Iowa, Iowa City, IA 52242, USA
Accepted 21 April 2005
Abstract This paper investigates the synchronization of coupled unified chaotic systems via active control. The synchronization is given in the slave–master scheme and the controller ensures that the states of the controlled chaotic slave system exponentially synchronize with the state of the master system. Numerical simulations are provided for illustration and verification of the proposed method. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Since the introduction of the Lorenz chaotic attractor [1], there has been great interest in the study and synchronization of the Lorenz and Lorenz type systems [2–8]. There are several Lorenz type systems as discussed in Chen and Ueta [9] and Lu¨ et al. [10,11]. Recently a unified chaotic system was introduced in [8], which unified the Lorenz, Chen and Lu¨ systems by setting a single adjustable parameter instead of three parameters of the individual systems. In the present paper, we study the synchronization of coupled unified chaotic systems using the active control technique introduced in [6] and further developed in [12]. In the following section, the dynamics of a unified chaotic system is presented. In Section 3, the synchronization of the unified chaotic systems is discussed. Numerical simulations are provided in Section 4 to illustrate and verify of the method. Finally a concluding remark is given.
2. The unified chaotic systems In 1963, Lorenz developed a system and observed the first chaotic attractor [1]. In 1999, Chen found a similar system [9] that exhibits chaotic behavior with a topologically different chaotic attractor. More recently, another similar system was introduced by Lu¨ et al. [10] which produces the chaotic attractor and connects the gap between the Lorenz and *
Corresponding author. E-mail addresses: aucar1@firat.edu.tr (A. Uc¸ar), [email protected] (K.E. Lonngren), [email protected] (E.-W. Bai). 0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.104
A. Uc¸ar et al. / Chaos, Solitons and Fractals 27 (2006) 1292–1297
1293
Fig. 1. (a) Lorenz chaotic attractor; (b) Chen chaotic attractor; (c) Lu¨ chaotic attractor; (d) new chaotic attractor.
Chen systems. In [8] Li et al. introduced the system defined in (1) with the parameter a being changed to observe these three attractors. x_ 1 ¼ ð25a þ 10Þðy 1 x1 Þ y_ 1 ¼ ð28 25aÞx1 x1 z1 þ ð29a 1Þy 1 8þa z1 z_ 1 ¼ x1 y 1 3
ð1Þ
where a 2 [0, 1]. It was shown in [8] that the system exhibits chaotic behavior for any a 2 [0, 1]. When a 2 [0, 0.8), the system (1) is called a Lorenz system and exhibits chaotic behavior. For a = 0, the chaotic behavior is depicted in Fig. 1(a) which shows the steady state solution for the initial condition (x(0), y(0), z(0)) = (1, 2, 3). For a = 0.8, the system is known to be the generalized Lu¨ system and its steady state solution is in Fig. 1(b). When a 2 (0.8, 1], the system is called the Chen system as in Fig. 1(c) for a = 1. In fact, we have found the system defined in (1) also shows chaotic behavior for a = 1.1 as depicted in Fig. 1(d). Our interest is to design an active control system to synchronize two unified chaotic systems and discuss its validity in the following section.
3. Chaos synchronization of the unified chaotic systems Recently chaos synchronization of coupled chaotic system has attracted a great deal of attentions. Many techniques have been developed to synchronize coupled chaotic systems, such as linear/nonlinear feedback methods, adaptive techniques, time delay feedback approaches and so on. The unified chaotic system defined in (1) has also been studied and techniques proposed to synchronize coupled unified chaotic systems [8,13,14]. We show in this paper that the technique, namely the active control method proposed in [6] can be readily designed to synchronize two chaotic systems that also include systems with time delay if the nonlinearity in the system is known [15–17]. Consider a unified chaotic system of (1) with the subscript Ô1Õ as the master system to which the slave system needs to be synchronized. Since the parameter a has been found to be very important in determining the chaotic attractors
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A. Uc¸ar et al. / Chaos, Solitons and Fractals 27 (2006) 1292–1297
observed from (1), we were led to ask whether it would be possible to synchronize two different chaotic systems with a different attractor. Consider the second unified chaotic system with the subscript Ô2Õ that contains a different value of the system parameter ~a x_ 2 ¼ ð25~a þ 10Þðy 2 x2 Þ þ la y_ 2 ¼ ð28 25~aÞx2 x2 z2 þ ð29~a 1Þy 2 þ lb ð2Þ 8 þ ~a z_ 2 ¼ x2 y 2 z2 þ lc 3 where la(t), lb(t), and lc(t) are active control signals that are yet to be determined. Here the aim of the control signals is to force the slave system to follow the master system. Thus, one-way synchronization of the two unified chaotic systems from system 1 to system 2 will be achieved. Without the controls la(t), lb(t), and lc(t), the second system defined by (2) will have a chaotic behavior depending chosen value of ~ a. In order to obtain the active control signals, we define the error state between the dependent variables of system 2 and system 1 as ex ¼ x2 x1 ey ¼ y 2 y 1 e z ¼ z2 z1
ð3Þ
Subtracting the first system (1) from the second system (2) which includes the control signals, we obtain e_ x ¼ 10ðey ex Þ þ 25~aðy 2 x2 Þ 25aðy 1 x1 Þ þ la ðtÞ e_ y ¼ 28ex 35~ ax2 þ 35ax1 x2 z2 þ x1 z1 þ 29~ay 2 29ay 1 ey þ lb ðtÞ ~a 8 a e_ z ¼ x2 y 2 x1 y 1 ez z2 þ z1 þ lc ðtÞ 3 3 3
ð4Þ
Since the nonlinear terms in both systems are known, the control signals can be defined as la ðtÞ ¼ 25~aðy 2 x2 Þ þ 25aðy 1 x1 Þ þ u1 ðtÞ lb ðtÞ ¼ ~að35x2 29y 2 Þ þ að29y 1 35x1 Þ þ x2 z2 x1 z1 þ u2 ðtÞ ~a a lc ðtÞ ¼ x2 y 2 þ x1 y 1 þ z2 z1 þ u3 ðtÞ 3 3 The substitution of 2 3 2 e_ x 10 6 7 6 4 e_ y 5 ¼ 4 28 e_ z 0
Eq. (5) into (4) leads to 32 3 2 ex 10 0 1 76 7 6 1 0 54 ey 5 þ 4 0 ez 0 8=3 0
32 3 u1 0 0 76 7 1 0 54 u2 5 u3 0 1
ð5Þ
ð6Þ
Eq. (7) described the error dynamics and can be considered in terms of a control problem where the system to be controlled is now a linear system with the control input u(t) = [u1, u2, u3]T [18]. Since the error dynamics is full state controllable, the feedback gains can be designed to stabilize the state of the error system, [ex, ey, ez]T so that the error signals converge to zero as time t goes to infinity. This implies that the two unified chaotic systems with different values the parameter a are synchronized. There are many possible choices for the controller u(t). We choose 2 3 2 32 3 ex u1 k 11 k 12 k 12 6 7 6 76 7 u k k k e ¼ ð7Þ 4 25 4 21 22 23 54 y 5 u3 k 31 k 32 k 33 ez where the constants kijÕs are controller gains. For the proper choice of kijÕs, the closed loop system characteristic matrix must have all of the eigenvalues with negative real parts. In this case the control signal defined in (7) yields the error dynamic function of kijÕs as 2 3 2 32 3 e_ x ex 0 k 13 10 k 11 10 k 12 6 7 6 76 7 k 23 ð8Þ 4 e_ y 5 ¼ 4 28 k 21 1 k 22 54 ey 5 e_ z k 31 k 32 8=3 k 33 ez The rate of convergence is now determined by the numerical values of the parameters of the feedback gains kijÕs. For the particular choice of feedback gains;
A. Uc¸ar et al. / Chaos, Solitons and Fractals 27 (2006) 1292–1297
2
k 11 6 4 k 21
k 12 k 22
3 2 k 12 9 10 7 6 k 23 5 ¼ 4 28 0
k 31
k 32
k 33
0
0
0 0
1295
3 7 5
ð9Þ
5=3
the error system given in (8) is stable and the closed loop system has eigenvalues that are found to be 1, 1 and 1. The choice of control gains will lead to a stable error system and to synchronization of two unified chaotic systems.
4. Numerical results Here the numerical results are given to verify the proposed method. In order to demonstrate the efficacy of the procedure, we keep the master system parameter a = 0 and chose different values of the slave system parameter ~ a. First, consider the case of ~a ¼ 0.8, i.e., the Lu¨ attractor a = 0. In these numerical simulations, the fourth-order Runge-Kutta is used to solve the master and slave systems defined in (1) and (2), respectively, with time step size 103. The initial values of the master system and slave system are taken as (x1(0), y1(0), z1(0)) = (1, 2, 1) and (x2(0), y2(0), z2(0)) = (1, 2, 3), respectively. The controller gains are chosen as in (9) in order to achieve the synchronization within 5 s after the time of the activation of the controller s. When the control signal is activated at s = 0, the simulation results are illustrated in Fig. 2 for the error system of (8). Fig. 2(a) shows the error signal ex = x2 x1; Fig. 2(b) shows the error signal ey = y2 y1; and Fig. 2(c) shows the error signal ez = z2 z1. Fig. 2 shows that the error signals converge to zero and this leads to the synchronization of the Lu¨ attractor with the Lorenz attractor. To synchronize the Chen attractor with the Lorenz attractor with a = 0, let the slave system parameter value be ~ a ¼ 1. The numerical results of the error system are depicted in Fig. 3 when the control signals are applied at the time s = 50. The results of this calculation clearly indicate that the slave system follows the master system after the control is applied. Note that the speed of the convergence time can be adjusted by choosing different values of controller gain defined in the previous subsection. For example if the desired convergence times of the error dynamics are 5 s for ex, 2.5 s for ey, 1 s for ez, then the controller gains are 2 3 2 3 k 11 k 12 k 12 9.9 10 0 6 7 6 7 0.95 0 ð10Þ 4 k 21 k 22 k 23 5 ¼ 4 28 5 k 31 k 32 k 33 0 0 2.6467
Fig. 2. The time response of the error signals of coupled unified chaotic systems with active controller applied at a time s = 0. The master Lorenz system (1) parameter was a = 0 and the slave Lu¨ system (2) parameter was ~a ¼ 0.8.
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A. Uc¸ar et al. / Chaos, Solitons and Fractals 27 (2006) 1292–1297
Fig. 3. The time response of the error signals of coupled unified chaotic systems with active controller applied at a time s = 50. The master Lorenz system (1) parameter was a = 0 and the slave Chen system (2) parameter was ~a ¼ 1.
~ ¼ 1.1 with the Lorenz system with of a = 0. The For those controller gains, the results are shown in Fig. 1(d) for a error signals are depicted in Fig. 4 when the control signal is activated at s = 0, Fig. 4 shows the convergence times for ex, ey, and ez, have been achieved. In the numerical calculations, we have noted that the synchronization speed is very rapid. The evolution of the synchronization can be altered by choosing different control gains. Note that when the convergence time is reduced, the
Fig. 4. The time response of the error signals of coupled unified chaotic systems with active controller applied at a time s = 50. The master Lorenz system (1) parameter was a = 0 and the slave system (2) parameter was ~a ¼ 1.1.
A. Uc¸ar et al. / Chaos, Solitons and Fractals 27 (2006) 1292–1297
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magnitudes of the controller signals are increased. This may lead to a large controller gain and a signal saturation in practical implementations such as electronically.
5. Conclusion In this paper, we have obtained a nonlinear active controller that can be used to synchronize two coupled unified chaotic systems together such that the frequency of oscillation of the slave system will follow the master system. The desired speed of the convergence time of synchronization can be modified by the linear feedback gain. In practical applications, some caution should be exercised to balance between the convergence rate and the magnitude of the controller gains. The only limitation of the proposed method is the assumption that the nonlinearity in the systems needs to be known.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
Lorenz E. Deterministic nonperiodic flows. J Atmos Sci 1963;20:130–41. Sparrow C. The Lorenz equations: bifurcations, chaos, and strange attractors. Appl Math Sci 1982;41. Stewart HB. Visualization of the Lorenz system. Physica D: Nonlinear Phenom 1986;18:479–80. Chen CC, Tsai CH, Fu CC. Rich dynamics in self-interacting Lorenz systems. Phys Lett A 1994;194:265–71. Abdelkader MA. Generalized Lorenz-type systems. J Math Anal Appl 1996;199:1–13. Bai EW, Lonngren KE. Synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 1997;8:51–8. Liao TL. Adaptive synchronization of two Lorenz systems. Chaos, Solitons & Fractals 1998;9:1555–61. Li D, Lu J, Wu X. Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems. Chaos, Solitons & Fractals 2005;23:79–85. Chen GR, Ueta T. Yet another chaotic attractor. Int J Bifurcat Chaos 1999;9:146–56. Lu¨ J, Chen G, Cheng D. A new chaotic system and beyond: the generalized Lorenz-like system. Int J Bifurcat Chaos 2004;14:1507–37. Lu¨ J, Chen G, Cheng D. Bridge the gap between the Lorenz system and the Chen system. Int J Bifurcat Chaos 2002;12:2917–26. Bai EW, Lonngren KE. Sequential synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 2000;11:1041–4. Lu¨ J. Chaos synchronization between linearly coupled chaotic systems. Chaos, Solitons & Fractals 2002;14:529–41. Chen HK. Synchronization of two different chaotic systems: a new system and each of the dynamical systems Lorenz, Chen and Lu¨. Chaos, Solitons & Fractals. Available online at www.sciencedirect.com. Lonngren KE, Bai EW, Uc¸ar A. Dynamics and synchronization of the Hastings–Powell model of the food chain. Chaos, Solitons & Fractals 2004;20:387–93. Uc¸ar A, Lonngren KE, Bai EW. Synchronization of chaotic behavior in nonlinear Bloch equations. Phys Lett A 2004;314:96–101. Lonngren KE, Bai EW. On the Uc¸ar prototype model. Int J Eng Sci 2002;40:1855–7. Goodwin GC, Graebe SF, Salgado ME. Control system design. New Jersey: Prentice Hall; 2001.
Synchronization of the unified chaotic systems via active control Ahmet Uc¸ar b
a,*
, Karl E. Lonngren b, Er-Wei Bai
b
a Department of Electrical and Electronics Engineering, Firat University, Elazıg 23119, Turkey Department of Electrical and Computer Engineering, University of Iowa, Iowa City, IA 52242, USA
Accepted 21 April 2005
Abstract This paper investigates the synchronization of coupled unified chaotic systems via active control. The synchronization is given in the slave–master scheme and the controller ensures that the states of the controlled chaotic slave system exponentially synchronize with the state of the master system. Numerical simulations are provided for illustration and verification of the proposed method. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Since the introduction of the Lorenz chaotic attractor [1], there has been great interest in the study and synchronization of the Lorenz and Lorenz type systems [2–8]. There are several Lorenz type systems as discussed in Chen and Ueta [9] and Lu¨ et al. [10,11]. Recently a unified chaotic system was introduced in [8], which unified the Lorenz, Chen and Lu¨ systems by setting a single adjustable parameter instead of three parameters of the individual systems. In the present paper, we study the synchronization of coupled unified chaotic systems using the active control technique introduced in [6] and further developed in [12]. In the following section, the dynamics of a unified chaotic system is presented. In Section 3, the synchronization of the unified chaotic systems is discussed. Numerical simulations are provided in Section 4 to illustrate and verify of the method. Finally a concluding remark is given.
2. The unified chaotic systems In 1963, Lorenz developed a system and observed the first chaotic attractor [1]. In 1999, Chen found a similar system [9] that exhibits chaotic behavior with a topologically different chaotic attractor. More recently, another similar system was introduced by Lu¨ et al. [10] which produces the chaotic attractor and connects the gap between the Lorenz and *
Corresponding author. E-mail addresses: aucar1@firat.edu.tr (A. Uc¸ar), [email protected] (K.E. Lonngren), [email protected] (E.-W. Bai). 0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.104
A. Uc¸ar et al. / Chaos, Solitons and Fractals 27 (2006) 1292–1297
1293
Fig. 1. (a) Lorenz chaotic attractor; (b) Chen chaotic attractor; (c) Lu¨ chaotic attractor; (d) new chaotic attractor.
Chen systems. In [8] Li et al. introduced the system defined in (1) with the parameter a being changed to observe these three attractors. x_ 1 ¼ ð25a þ 10Þðy 1 x1 Þ y_ 1 ¼ ð28 25aÞx1 x1 z1 þ ð29a 1Þy 1 8þa z1 z_ 1 ¼ x1 y 1 3
ð1Þ
where a 2 [0, 1]. It was shown in [8] that the system exhibits chaotic behavior for any a 2 [0, 1]. When a 2 [0, 0.8), the system (1) is called a Lorenz system and exhibits chaotic behavior. For a = 0, the chaotic behavior is depicted in Fig. 1(a) which shows the steady state solution for the initial condition (x(0), y(0), z(0)) = (1, 2, 3). For a = 0.8, the system is known to be the generalized Lu¨ system and its steady state solution is in Fig. 1(b). When a 2 (0.8, 1], the system is called the Chen system as in Fig. 1(c) for a = 1. In fact, we have found the system defined in (1) also shows chaotic behavior for a = 1.1 as depicted in Fig. 1(d). Our interest is to design an active control system to synchronize two unified chaotic systems and discuss its validity in the following section.
3. Chaos synchronization of the unified chaotic systems Recently chaos synchronization of coupled chaotic system has attracted a great deal of attentions. Many techniques have been developed to synchronize coupled chaotic systems, such as linear/nonlinear feedback methods, adaptive techniques, time delay feedback approaches and so on. The unified chaotic system defined in (1) has also been studied and techniques proposed to synchronize coupled unified chaotic systems [8,13,14]. We show in this paper that the technique, namely the active control method proposed in [6] can be readily designed to synchronize two chaotic systems that also include systems with time delay if the nonlinearity in the system is known [15–17]. Consider a unified chaotic system of (1) with the subscript Ô1Õ as the master system to which the slave system needs to be synchronized. Since the parameter a has been found to be very important in determining the chaotic attractors
1294
A. Uc¸ar et al. / Chaos, Solitons and Fractals 27 (2006) 1292–1297
observed from (1), we were led to ask whether it would be possible to synchronize two different chaotic systems with a different attractor. Consider the second unified chaotic system with the subscript Ô2Õ that contains a different value of the system parameter ~a x_ 2 ¼ ð25~a þ 10Þðy 2 x2 Þ þ la y_ 2 ¼ ð28 25~aÞx2 x2 z2 þ ð29~a 1Þy 2 þ lb ð2Þ 8 þ ~a z_ 2 ¼ x2 y 2 z2 þ lc 3 where la(t), lb(t), and lc(t) are active control signals that are yet to be determined. Here the aim of the control signals is to force the slave system to follow the master system. Thus, one-way synchronization of the two unified chaotic systems from system 1 to system 2 will be achieved. Without the controls la(t), lb(t), and lc(t), the second system defined by (2) will have a chaotic behavior depending chosen value of ~ a. In order to obtain the active control signals, we define the error state between the dependent variables of system 2 and system 1 as ex ¼ x2 x1 ey ¼ y 2 y 1 e z ¼ z2 z1
ð3Þ
Subtracting the first system (1) from the second system (2) which includes the control signals, we obtain e_ x ¼ 10ðey ex Þ þ 25~aðy 2 x2 Þ 25aðy 1 x1 Þ þ la ðtÞ e_ y ¼ 28ex 35~ ax2 þ 35ax1 x2 z2 þ x1 z1 þ 29~ay 2 29ay 1 ey þ lb ðtÞ ~a 8 a e_ z ¼ x2 y 2 x1 y 1 ez z2 þ z1 þ lc ðtÞ 3 3 3
ð4Þ
Since the nonlinear terms in both systems are known, the control signals can be defined as la ðtÞ ¼ 25~aðy 2 x2 Þ þ 25aðy 1 x1 Þ þ u1 ðtÞ lb ðtÞ ¼ ~að35x2 29y 2 Þ þ að29y 1 35x1 Þ þ x2 z2 x1 z1 þ u2 ðtÞ ~a a lc ðtÞ ¼ x2 y 2 þ x1 y 1 þ z2 z1 þ u3 ðtÞ 3 3 The substitution of 2 3 2 e_ x 10 6 7 6 4 e_ y 5 ¼ 4 28 e_ z 0
Eq. (5) into (4) leads to 32 3 2 ex 10 0 1 76 7 6 1 0 54 ey 5 þ 4 0 ez 0 8=3 0
32 3 u1 0 0 76 7 1 0 54 u2 5 u3 0 1
ð5Þ
ð6Þ
Eq. (7) described the error dynamics and can be considered in terms of a control problem where the system to be controlled is now a linear system with the control input u(t) = [u1, u2, u3]T [18]. Since the error dynamics is full state controllable, the feedback gains can be designed to stabilize the state of the error system, [ex, ey, ez]T so that the error signals converge to zero as time t goes to infinity. This implies that the two unified chaotic systems with different values the parameter a are synchronized. There are many possible choices for the controller u(t). We choose 2 3 2 32 3 ex u1 k 11 k 12 k 12 6 7 6 76 7 u k k k e ¼ ð7Þ 4 25 4 21 22 23 54 y 5 u3 k 31 k 32 k 33 ez where the constants kijÕs are controller gains. For the proper choice of kijÕs, the closed loop system characteristic matrix must have all of the eigenvalues with negative real parts. In this case the control signal defined in (7) yields the error dynamic function of kijÕs as 2 3 2 32 3 e_ x ex 0 k 13 10 k 11 10 k 12 6 7 6 76 7 k 23 ð8Þ 4 e_ y 5 ¼ 4 28 k 21 1 k 22 54 ey 5 e_ z k 31 k 32 8=3 k 33 ez The rate of convergence is now determined by the numerical values of the parameters of the feedback gains kijÕs. For the particular choice of feedback gains;
A. Uc¸ar et al. / Chaos, Solitons and Fractals 27 (2006) 1292–1297
2
k 11 6 4 k 21
k 12 k 22
3 2 k 12 9 10 7 6 k 23 5 ¼ 4 28 0
k 31
k 32
k 33
0
0
0 0
1295
3 7 5
ð9Þ
5=3
the error system given in (8) is stable and the closed loop system has eigenvalues that are found to be 1, 1 and 1. The choice of control gains will lead to a stable error system and to synchronization of two unified chaotic systems.
4. Numerical results Here the numerical results are given to verify the proposed method. In order to demonstrate the efficacy of the procedure, we keep the master system parameter a = 0 and chose different values of the slave system parameter ~ a. First, consider the case of ~a ¼ 0.8, i.e., the Lu¨ attractor a = 0. In these numerical simulations, the fourth-order Runge-Kutta is used to solve the master and slave systems defined in (1) and (2), respectively, with time step size 103. The initial values of the master system and slave system are taken as (x1(0), y1(0), z1(0)) = (1, 2, 1) and (x2(0), y2(0), z2(0)) = (1, 2, 3), respectively. The controller gains are chosen as in (9) in order to achieve the synchronization within 5 s after the time of the activation of the controller s. When the control signal is activated at s = 0, the simulation results are illustrated in Fig. 2 for the error system of (8). Fig. 2(a) shows the error signal ex = x2 x1; Fig. 2(b) shows the error signal ey = y2 y1; and Fig. 2(c) shows the error signal ez = z2 z1. Fig. 2 shows that the error signals converge to zero and this leads to the synchronization of the Lu¨ attractor with the Lorenz attractor. To synchronize the Chen attractor with the Lorenz attractor with a = 0, let the slave system parameter value be ~ a ¼ 1. The numerical results of the error system are depicted in Fig. 3 when the control signals are applied at the time s = 50. The results of this calculation clearly indicate that the slave system follows the master system after the control is applied. Note that the speed of the convergence time can be adjusted by choosing different values of controller gain defined in the previous subsection. For example if the desired convergence times of the error dynamics are 5 s for ex, 2.5 s for ey, 1 s for ez, then the controller gains are 2 3 2 3 k 11 k 12 k 12 9.9 10 0 6 7 6 7 0.95 0 ð10Þ 4 k 21 k 22 k 23 5 ¼ 4 28 5 k 31 k 32 k 33 0 0 2.6467
Fig. 2. The time response of the error signals of coupled unified chaotic systems with active controller applied at a time s = 0. The master Lorenz system (1) parameter was a = 0 and the slave Lu¨ system (2) parameter was ~a ¼ 0.8.
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A. Uc¸ar et al. / Chaos, Solitons and Fractals 27 (2006) 1292–1297
Fig. 3. The time response of the error signals of coupled unified chaotic systems with active controller applied at a time s = 50. The master Lorenz system (1) parameter was a = 0 and the slave Chen system (2) parameter was ~a ¼ 1.
~ ¼ 1.1 with the Lorenz system with of a = 0. The For those controller gains, the results are shown in Fig. 1(d) for a error signals are depicted in Fig. 4 when the control signal is activated at s = 0, Fig. 4 shows the convergence times for ex, ey, and ez, have been achieved. In the numerical calculations, we have noted that the synchronization speed is very rapid. The evolution of the synchronization can be altered by choosing different control gains. Note that when the convergence time is reduced, the
Fig. 4. The time response of the error signals of coupled unified chaotic systems with active controller applied at a time s = 50. The master Lorenz system (1) parameter was a = 0 and the slave system (2) parameter was ~a ¼ 1.1.
A. Uc¸ar et al. / Chaos, Solitons and Fractals 27 (2006) 1292–1297
1297
magnitudes of the controller signals are increased. This may lead to a large controller gain and a signal saturation in practical implementations such as electronically.
5. Conclusion In this paper, we have obtained a nonlinear active controller that can be used to synchronize two coupled unified chaotic systems together such that the frequency of oscillation of the slave system will follow the master system. The desired speed of the convergence time of synchronization can be modified by the linear feedback gain. In practical applications, some caution should be exercised to balance between the convergence rate and the magnitude of the controller gains. The only limitation of the proposed method is the assumption that the nonlinearity in the systems needs to be known.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
Lorenz E. Deterministic nonperiodic flows. J Atmos Sci 1963;20:130–41. Sparrow C. The Lorenz equations: bifurcations, chaos, and strange attractors. Appl Math Sci 1982;41. Stewart HB. Visualization of the Lorenz system. Physica D: Nonlinear Phenom 1986;18:479–80. Chen CC, Tsai CH, Fu CC. Rich dynamics in self-interacting Lorenz systems. Phys Lett A 1994;194:265–71. Abdelkader MA. Generalized Lorenz-type systems. J Math Anal Appl 1996;199:1–13. Bai EW, Lonngren KE. Synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 1997;8:51–8. Liao TL. Adaptive synchronization of two Lorenz systems. Chaos, Solitons & Fractals 1998;9:1555–61. Li D, Lu J, Wu X. Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems. Chaos, Solitons & Fractals 2005;23:79–85. Chen GR, Ueta T. Yet another chaotic attractor. Int J Bifurcat Chaos 1999;9:146–56. Lu¨ J, Chen G, Cheng D. A new chaotic system and beyond: the generalized Lorenz-like system. Int J Bifurcat Chaos 2004;14:1507–37. Lu¨ J, Chen G, Cheng D. Bridge the gap between the Lorenz system and the Chen system. Int J Bifurcat Chaos 2002;12:2917–26. Bai EW, Lonngren KE. Sequential synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 2000;11:1041–4. Lu¨ J. Chaos synchronization between linearly coupled chaotic systems. Chaos, Solitons & Fractals 2002;14:529–41. Chen HK. Synchronization of two different chaotic systems: a new system and each of the dynamical systems Lorenz, Chen and Lu¨. Chaos, Solitons & Fractals. Available online at www.sciencedirect.com. Lonngren KE, Bai EW, Uc¸ar A. Dynamics and synchronization of the Hastings–Powell model of the food chain. Chaos, Solitons & Fractals 2004;20:387–93. Uc¸ar A, Lonngren KE, Bai EW. Synchronization of chaotic behavior in nonlinear Bloch equations. Phys Lett A 2004;314:96–101. Lonngren KE, Bai EW. On the Uc¸ar prototype model. Int J Eng Sci 2002;40:1855–7. Goodwin GC, Graebe SF, Salgado ME. Control system design. New Jersey: Prentice Hall; 2001.