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Chaos synchronization in a Josephson junction system via active sliding mode control
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Chaos, Solitons and Fractals 41 (2009) 60–66 www.elsevier.com/locate/chaos
Chaos synchronization in a Josephson junction system via active sliding mode control Yang Zhao a,*, Wei Wang b a
Department of Mathematics, School of Science, Beijing Institute of Technology, Beijing 100081, China b Ministry of Information Industry Software and Integrated Circuit Promotion Center, China Accepted 1 November 2007
Abstract In this letter, two types of active siding control methods are proposed and applied to achieve chaotic synchronization in a Josephson junction system. Numerical simulations are used to verify the proposed control techniques. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Chaos synchronization has gained intense interests from large varieties of disciplines, due to its useful applications in many engineering areas such as secure communication, digital communication, power electronic devices and power quality, biological systems, chemical reaction analysis and design, and information processing. A large variety of synchronization techniques have been obtained to achieve stable synchronous state between identical and non-identical chaotic systems. Among them the active control [1–8] and the sliding mode control [9,10] have been widely recognized as two powerful design methods to synchronize chaotic systems. As a combination of the two control approaches, the newest method is the active sliding mode control [11,12], which has successfully applied to several systems [13,14]. This technique is a discontinuous control strategy that requires two design stages. The first stage is to select an appropriate active controller in order to facilitate the design of the sequent sliding mode controller. The second stage is to design a sliding mode controller to achieve the synchronization. In addition, the resistively shunted Josephson junction driven by a DC and/or an AC current is of fundamental interest as a basic example of nonlinear dynamical systems. One of the Josephson junction systems can be modeled by the following equation in the dimensionless form €x þ a_x þ sin x ¼ b þ A sin xt;
ð1Þ
where a is the damping coefficient, b > 0 is the constant driving force, A and x represent the amplitude and the frequency of the external driving force, respectively. This kind of equation is also commonly used to describe the motion of a forced under-damped pendulum. When the system parameters change, the system (1) may exhibit a wide variety of chaotic phenomena. *
Corresponding author. E-mail address: [email protected] (Y. Zhao).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.11.010
Y. Zhao, W. Wang / Chaos, Solitons and Fractals 41 (2009) 60–66
61
In this letter, we will generalize the skills of active sliding mode control and apply them to realize chaotic synchronization in the system (1). In the following section, the dynamics of the system (1) is presented. Section 3 introduces and analyzes two types of active sliding mode control techniques, which are applied to obtain chaos synchronization for the system (1). Numerical simulations are given in Section 4 to illustrate the effectiveness of these methods. Finally, our findings are summarized in the Conclusion.
2. The dynamics of the system (1) Taking a new variables y the system (1) is equivalent to the following equations x_ ¼ y; y_ ¼ ay sin x þ b þ A sin xt;
ð2Þ
whose dynamical behavior is rather complicated with different parameters. For a > 2; b P 0, and A P 0, or 0 < a < 2, and b sufficiently large no chaotic motion takes place in this system [15]. For 0 < a < 2, and moderate b, the chaotic behavior can be observed by numerical computation [16]. This system exhibits two different chaotic attractors at the parameter values a ¼ 0:7; b ¼ 0:905; A ¼ 0:4; x ¼ 0:25 (see Fig. 1) and a ¼ 0:5; b ¼ 0:89; A ¼ 0:4; x ¼ 0:46 (see Fig. 2), respectively.
2.5 2 1.5
y
1 0.5 0 −0.5 −1 −1
−0.5
0
0.5
1
sinx Fig. 1. The chaotic attractor with a ¼ 0:7; b ¼ 0:905; A ¼ 0:4; x ¼ 0:25.
3 2.5
y
2 1.5 1 0.5 0 −1
−0.5
0
0.5
1
sinx Fig. 2. The chaotic attractor with a ¼ 0:5; b ¼ 0:89; A ¼ 0:4; x ¼ 0:46.
62
Y. Zhao, W. Wang / Chaos, Solitons and Fractals 41 (2009) 60–66
3. Active sliding mode controller design and analysis The active sliding mode controller is a combination of two design procedure: the active controller and the sliding mode controller, which are given, respectively. 3.1. Active controller design Assume that we have two above-mentioned systems and that the master system with the subscript 1 is to control the slave system with the subscript 2. The master system is given by x_1 ¼ y 1 ; ð3Þ y_1 ¼ a1 y 1 sin x1 þ b1 þ A1 sin x1 t: The control function uðtÞ is added into the slave system, which is given by x_2 ¼ y 2 ; y_2 ¼ a2 y 2 sin x2 þ b2 þ A2 sin x2 t þ uðtÞ:
ð4Þ
Thus, the control problem considered is to design an appropriate active sliding mode controller u, which can make two systems to become synchronized, in the sense that lim jy 2 y 1 j ¼ 0; lim jx2 x1 j ¼ 0:
t!1
t!1
In order to obtain the active control function, defining the synchronization error as e ¼ ðe1 ; e2 Þ ¼ ðx2 x1 ; y 2 y 1 Þ and subtracting the first system (3) from the second system (4), the error dynamics is determined by e_1 ¼ e2 ; ð5Þ e_2 ¼ þa1 y 1 a2 y 2 þ b2 b1 þ sin x1 sin x2 þ A2 sin x2 t A1 sin x1 t þ uðtÞ: We redefine the control function uðtÞ to eliminate nonlinear term in (5) as follows: uðtÞ ¼ e2 a1 y 1 þ a2 y 2 b2 þ b1 sin x1 þ sin x2 A2 sin x2 t þ A1 sin x1 t þ wðtÞ: Substituting (6) in (5) we have e_1 ¼ e2 ; e_2 ¼ e2 þ wðtÞ:
ð6Þ
ð7Þ
Eq. (7) describes the error dynamics with a newly defined control input wðtÞ. There are many possible choices for the control wðtÞ. We choose the sliding mode control law as follows: þ w ðtÞ; sðeÞ > 0 ð8Þ wðtÞ ¼ w ðtÞ; sðeÞ < 0; and s ¼ sðeÞ is a switching surface which prescribes the desired dynamics. 3.2. Sliding surface design The sliding surface can be defined as follows: sðeÞ ¼ c1 e1 þ e2 ;
ð9Þ
where c1 is a constant. Based on the equivalent control approach [17], when in sliding mode, the controlled system satisfies the following conditions sðeÞ ¼ 0
ð10aÞ
s_ ðeÞ ¼ 0:
ð10bÞ
and Combining (7), (9) with (10b), one can obtain s_ ðeÞ ¼ c1 e_1 þ e_2 ¼ c1 e2 e2 þ wðtÞ ¼ 0;
ð11Þ
which yields the equivalent control weq ðtÞ weq ðtÞ ¼ e2 c1 e2 :
ð12Þ
Y. Zhao, W. Wang / Chaos, Solitons and Fractals 41 (2009) 60–66
Replacing wðtÞ in Eq. (7) by weq ðtÞ in (12), the state equation in the sliding mode is determined by e_1 ¼ e2 ; e_2 ¼ c1 e2 ;
63
ð13Þ
which can be reduced to one-dimensional system using Eq. (10a) e_1 ¼ c1 e1 : It can be easily seen that e1 ¼ Cec1 t : Therefore as long as c1 > 0, then the system behavior in the sliding mode is asymptotically stable. 3.3. Design of the sliding mode controller Now it is necessary to choose a sliding mode controller to drive all error trajectories onto the sliding surface, that is, the reaching condition s_s < 0 must be satisfied. Two types of controllers are provided in the following text. Case i. Limiting the controller in the following form 1; sðeÞ > 0; wðtÞ ¼ sgnðc1 e1 þ e2 Þ ¼ 1; sðeÞ < 0;
ð14Þ
where sgn(s) denotes the sign function. Hence the only remaining is to choose appropriate constant c1 2 Rþ such that s_s < 0. It induces that c1 e2 e2 1 < 0; sðeÞ > 0; s_ ðeÞ ¼ c1 e_1 þ e_2 ¼ c1 e2 e2 þ wðtÞ ¼ ð15Þ c1 e2 e2 þ 1 > 0; sðeÞ < 0; which implies 1 < ðc1 1Þe2 < 1: It can be easily found that s_s < 0 is maintained on the whole plane when c1 ¼ 1, which means that all error trajectories will converge to the sliding surface sðeÞ ¼ 0. The corresponding error dynamics is e_1 ¼ e2 ð16Þ e_2 ¼ e2 sgnðe1 þ e2 Þ: Case ii. Here the constant plus proportional rate reaching law is applied [17]. Assume that s_ ¼ esgnðsÞ rs;
ð17Þ
where e > 0 and r > 0 are constants. This choice not only guarantees that the reaching condition s_s < 0 is satisfied on the whole plane, but also assures that the converging velocity is exponential and r controls the converging rate. One can enhance converging rate by increasing r and reduce the chattering phenomenon by decreasing e. From (7) and (9), it can be found that s_ ðeÞ ¼ c1 e_ 1 þ e_ 2 ¼ c1 e2 e2 þ wðtÞ:
ð18Þ
Using (17) and (18), the control input is determined as wðtÞ ¼ e2 c1 e2 esgnðsÞ rs: The corresponding error dynamics is e_1 ¼ e2 ; e_2 ¼ c1 e2 esgnðsÞ rs:
ð19Þ
ð20Þ
In this case, there is no request for choice of c1 . as long as the stability condition c1 > 0 is achieved, the sliding motion can occur, 3.4. Numerical results In all the numerical simulations, the standard fourth-order Runge–Kutta algorithm is used to solve the above system with a time grid of 0.001. We choose the system parameters: a1 ¼ 0:7; b1 ¼ 0:905; A1 ¼ 0:4; x1 ¼ 0:25 and a2 ¼ 0:5; b2 ¼ 0:89; A2 ¼ 0:4; x2 ¼ 0:46, which places the master system and the slave system in the chaotic state, respectively. The same initial conditions are x1 ð0Þ ¼ 0; y 1 ð0Þ ¼ 0; x2 ð0Þ ¼ 0; y 2 ð0Þ ¼ 0.
64
Y. Zhao, W. Wang / Chaos, Solitons and Fractals 41 (2009) 60–66
The simulation results of the error states e1 ; e2 without active sliding mode control are shown in Fig. 3. In the Case i, when the control function is activated at the time t ¼ 50, the simulation results are illustrated in Fig. 4. Meanwhile Fig. 5 plots the phase portraits for the error system (16). In the Case ii, with the parameters c1 ¼ 1; e ¼ 1 and r ¼ 10, the simulation results are illustrated in Fig. 6 as the control function is added on at the time t ¼ 50. Fig. 7 also plots the phase portraits for the error system (20). In two cases all the error states quickly converge to zero. However the converging velocity in the Case ii is more faster than that in the Case i. 200
3 2
150
1
e2
e
1
100 50
−1
0
−2 50
100
150
−3
200
0
50
100
t
150
t
Fig. 3. Time evolution of the error state with the active sliding mode control deactivated.
40
20
e
1
1
e ,e
2
30
10 e
2
0 −10
0
50
100
150
t Fig. 4. Time evolution of the error states for the Case i.
2
1
2
0
e
−50
0
0
−1 −2 −2
−1
0
1
e
1
Fig. 5. The phase portraits for the error system (16).
2
200
Y. Zhao, W. Wang / Chaos, Solitons and Fractals 41 (2009) 60–66
65
40 30
2
0
e ,e
10
1
20 e
1
−10
e2
−20 −30 0
50
100
150
t Fig. 6. Time evolution of the error states for the Case ii.
2
e2
1
0
−1 −2
−1
0
e1
1
2
Fig. 7. The phase portraits for the error system (20).
4. Conclusion In this letter, two types of active siding control methods are provided and are successfully used to synchronize chaos motions in the J–J system. For the Case i, the range of applicability of the proposed method may be narrow owing to strong restricting of the controller form. Whereas no superabundant restrictive assumption is imposed for the Case ii, so this method can be generalized to other systems.
Acknowledgement This work was supported by the National Natural Science Foundation of China (under Grant No. 10401007).
References [1] Bai EW, Lonngren KE. Synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 1997;8:51–8. [2] Bai EW, Lonngren KE. Sequential synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 2000;11:1041–4. [3] Agiza HN, Yassen MT. Synchronization of Rossler and Chen chaotic dynamical systems using active control. Phys Lett A 2001;278:191–7. [4] Ho MC, Hung YC, Chou CH. Phase and anti-phase synchronization of two chaotic systems by using active control. Phys Lett A 2002;296:43–8.
66
Y. Zhao, W. Wang / Chaos, Solitons and Fractals 41 (2009) 60–66
[5] Ho MCh, Hung YCh. Synchronization of two different systems by using generalized active control. Phys Lett A 2002;301:424–8. [6] Codreanu S. Synchronization of spatiotemporal nonlinear dynamical systems by an active control. Chaos, Solitons & Fractals 2003;15:507–10. [7] Vincent UE. Synchronization of Rikitake chaotic attractor using active control. Phys Lett A 2005;343:133–8. [8] Ucar A, Lonngren KE, Bai EW. Synchronization of the unified chaotic systems via active control. Chaos, Solitons & Fractals 2006;27:1292–7. [9] Yau HT. Design of adaptive sliding mode controller for chaos synchronization with uncertainties. Chaos, Solitons & Fractals 2004;22:341–7. [10] Lin JSh, Yan JJ, Liao TL. Chaotic synchronization via adaptive sliding mode observers subject to input nonlinearity. Chaos, Solitons & Fractals 2005;24:371–81. [11] Zhang H, Ma XK, Liu WZ. Synchronization of chaotic systems with parametric uncertainty using active sliding mode control. Chaos, Solitons & Fractals 2004;21:1249–57. [12] Haeri M, Emadzadeh AA. Synchronizing different chaotic systems using active sliding mode control. Chaos, Solitons & Fractals 2007;31:119–29. [13] Tavazoei MS, Haeri M. Determination of active sliding mode controller parameters in synchronizing different chaotic systems. Chaos, Solitons & Fractals 2007;32:583–91. [14] Haeri M, Tavazoei MS, Naseh MR. Synchronization of uncertain chaotic systems using active sliding mode control. Chaos, Solitons & Fractals 2007;33:1230–9. [15] Qian M, Shen WX, Zhang JY. Global behavior in the dynamical equation of J–J type. J Diff Eqs 1988;71:315–33. [16] Shen WenXian. Numerical analysis of the dynamical behavior of the equation of J–J type. Acta Math Appl Sinica 1989;5:242–52. [17] Hu YM. Variable structure control theory and its applications. Beijing: Science Press; 2003.
Chaos, Solitons and Fractals 41 (2009) 60–66 www.elsevier.com/locate/chaos
Chaos synchronization in a Josephson junction system via active sliding mode control Yang Zhao a,*, Wei Wang b a
Department of Mathematics, School of Science, Beijing Institute of Technology, Beijing 100081, China b Ministry of Information Industry Software and Integrated Circuit Promotion Center, China Accepted 1 November 2007
Abstract In this letter, two types of active siding control methods are proposed and applied to achieve chaotic synchronization in a Josephson junction system. Numerical simulations are used to verify the proposed control techniques. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Chaos synchronization has gained intense interests from large varieties of disciplines, due to its useful applications in many engineering areas such as secure communication, digital communication, power electronic devices and power quality, biological systems, chemical reaction analysis and design, and information processing. A large variety of synchronization techniques have been obtained to achieve stable synchronous state between identical and non-identical chaotic systems. Among them the active control [1–8] and the sliding mode control [9,10] have been widely recognized as two powerful design methods to synchronize chaotic systems. As a combination of the two control approaches, the newest method is the active sliding mode control [11,12], which has successfully applied to several systems [13,14]. This technique is a discontinuous control strategy that requires two design stages. The first stage is to select an appropriate active controller in order to facilitate the design of the sequent sliding mode controller. The second stage is to design a sliding mode controller to achieve the synchronization. In addition, the resistively shunted Josephson junction driven by a DC and/or an AC current is of fundamental interest as a basic example of nonlinear dynamical systems. One of the Josephson junction systems can be modeled by the following equation in the dimensionless form €x þ a_x þ sin x ¼ b þ A sin xt;
ð1Þ
where a is the damping coefficient, b > 0 is the constant driving force, A and x represent the amplitude and the frequency of the external driving force, respectively. This kind of equation is also commonly used to describe the motion of a forced under-damped pendulum. When the system parameters change, the system (1) may exhibit a wide variety of chaotic phenomena. *
Corresponding author. E-mail address: [email protected] (Y. Zhao).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.11.010
Y. Zhao, W. Wang / Chaos, Solitons and Fractals 41 (2009) 60–66
61
In this letter, we will generalize the skills of active sliding mode control and apply them to realize chaotic synchronization in the system (1). In the following section, the dynamics of the system (1) is presented. Section 3 introduces and analyzes two types of active sliding mode control techniques, which are applied to obtain chaos synchronization for the system (1). Numerical simulations are given in Section 4 to illustrate the effectiveness of these methods. Finally, our findings are summarized in the Conclusion.
2. The dynamics of the system (1) Taking a new variables y the system (1) is equivalent to the following equations x_ ¼ y; y_ ¼ ay sin x þ b þ A sin xt;
ð2Þ
whose dynamical behavior is rather complicated with different parameters. For a > 2; b P 0, and A P 0, or 0 < a < 2, and b sufficiently large no chaotic motion takes place in this system [15]. For 0 < a < 2, and moderate b, the chaotic behavior can be observed by numerical computation [16]. This system exhibits two different chaotic attractors at the parameter values a ¼ 0:7; b ¼ 0:905; A ¼ 0:4; x ¼ 0:25 (see Fig. 1) and a ¼ 0:5; b ¼ 0:89; A ¼ 0:4; x ¼ 0:46 (see Fig. 2), respectively.
2.5 2 1.5
y
1 0.5 0 −0.5 −1 −1
−0.5
0
0.5
1
sinx Fig. 1. The chaotic attractor with a ¼ 0:7; b ¼ 0:905; A ¼ 0:4; x ¼ 0:25.
3 2.5
y
2 1.5 1 0.5 0 −1
−0.5
0
0.5
1
sinx Fig. 2. The chaotic attractor with a ¼ 0:5; b ¼ 0:89; A ¼ 0:4; x ¼ 0:46.
62
Y. Zhao, W. Wang / Chaos, Solitons and Fractals 41 (2009) 60–66
3. Active sliding mode controller design and analysis The active sliding mode controller is a combination of two design procedure: the active controller and the sliding mode controller, which are given, respectively. 3.1. Active controller design Assume that we have two above-mentioned systems and that the master system with the subscript 1 is to control the slave system with the subscript 2. The master system is given by x_1 ¼ y 1 ; ð3Þ y_1 ¼ a1 y 1 sin x1 þ b1 þ A1 sin x1 t: The control function uðtÞ is added into the slave system, which is given by x_2 ¼ y 2 ; y_2 ¼ a2 y 2 sin x2 þ b2 þ A2 sin x2 t þ uðtÞ:
ð4Þ
Thus, the control problem considered is to design an appropriate active sliding mode controller u, which can make two systems to become synchronized, in the sense that lim jy 2 y 1 j ¼ 0; lim jx2 x1 j ¼ 0:
t!1
t!1
In order to obtain the active control function, defining the synchronization error as e ¼ ðe1 ; e2 Þ ¼ ðx2 x1 ; y 2 y 1 Þ and subtracting the first system (3) from the second system (4), the error dynamics is determined by e_1 ¼ e2 ; ð5Þ e_2 ¼ þa1 y 1 a2 y 2 þ b2 b1 þ sin x1 sin x2 þ A2 sin x2 t A1 sin x1 t þ uðtÞ: We redefine the control function uðtÞ to eliminate nonlinear term in (5) as follows: uðtÞ ¼ e2 a1 y 1 þ a2 y 2 b2 þ b1 sin x1 þ sin x2 A2 sin x2 t þ A1 sin x1 t þ wðtÞ: Substituting (6) in (5) we have e_1 ¼ e2 ; e_2 ¼ e2 þ wðtÞ:
ð6Þ
ð7Þ
Eq. (7) describes the error dynamics with a newly defined control input wðtÞ. There are many possible choices for the control wðtÞ. We choose the sliding mode control law as follows: þ w ðtÞ; sðeÞ > 0 ð8Þ wðtÞ ¼ w ðtÞ; sðeÞ < 0; and s ¼ sðeÞ is a switching surface which prescribes the desired dynamics. 3.2. Sliding surface design The sliding surface can be defined as follows: sðeÞ ¼ c1 e1 þ e2 ;
ð9Þ
where c1 is a constant. Based on the equivalent control approach [17], when in sliding mode, the controlled system satisfies the following conditions sðeÞ ¼ 0
ð10aÞ
s_ ðeÞ ¼ 0:
ð10bÞ
and Combining (7), (9) with (10b), one can obtain s_ ðeÞ ¼ c1 e_1 þ e_2 ¼ c1 e2 e2 þ wðtÞ ¼ 0;
ð11Þ
which yields the equivalent control weq ðtÞ weq ðtÞ ¼ e2 c1 e2 :
ð12Þ
Y. Zhao, W. Wang / Chaos, Solitons and Fractals 41 (2009) 60–66
Replacing wðtÞ in Eq. (7) by weq ðtÞ in (12), the state equation in the sliding mode is determined by e_1 ¼ e2 ; e_2 ¼ c1 e2 ;
63
ð13Þ
which can be reduced to one-dimensional system using Eq. (10a) e_1 ¼ c1 e1 : It can be easily seen that e1 ¼ Cec1 t : Therefore as long as c1 > 0, then the system behavior in the sliding mode is asymptotically stable. 3.3. Design of the sliding mode controller Now it is necessary to choose a sliding mode controller to drive all error trajectories onto the sliding surface, that is, the reaching condition s_s < 0 must be satisfied. Two types of controllers are provided in the following text. Case i. Limiting the controller in the following form 1; sðeÞ > 0; wðtÞ ¼ sgnðc1 e1 þ e2 Þ ¼ 1; sðeÞ < 0;
ð14Þ
where sgn(s) denotes the sign function. Hence the only remaining is to choose appropriate constant c1 2 Rþ such that s_s < 0. It induces that c1 e2 e2 1 < 0; sðeÞ > 0; s_ ðeÞ ¼ c1 e_1 þ e_2 ¼ c1 e2 e2 þ wðtÞ ¼ ð15Þ c1 e2 e2 þ 1 > 0; sðeÞ < 0; which implies 1 < ðc1 1Þe2 < 1: It can be easily found that s_s < 0 is maintained on the whole plane when c1 ¼ 1, which means that all error trajectories will converge to the sliding surface sðeÞ ¼ 0. The corresponding error dynamics is e_1 ¼ e2 ð16Þ e_2 ¼ e2 sgnðe1 þ e2 Þ: Case ii. Here the constant plus proportional rate reaching law is applied [17]. Assume that s_ ¼ esgnðsÞ rs;
ð17Þ
where e > 0 and r > 0 are constants. This choice not only guarantees that the reaching condition s_s < 0 is satisfied on the whole plane, but also assures that the converging velocity is exponential and r controls the converging rate. One can enhance converging rate by increasing r and reduce the chattering phenomenon by decreasing e. From (7) and (9), it can be found that s_ ðeÞ ¼ c1 e_ 1 þ e_ 2 ¼ c1 e2 e2 þ wðtÞ:
ð18Þ
Using (17) and (18), the control input is determined as wðtÞ ¼ e2 c1 e2 esgnðsÞ rs: The corresponding error dynamics is e_1 ¼ e2 ; e_2 ¼ c1 e2 esgnðsÞ rs:
ð19Þ
ð20Þ
In this case, there is no request for choice of c1 . as long as the stability condition c1 > 0 is achieved, the sliding motion can occur, 3.4. Numerical results In all the numerical simulations, the standard fourth-order Runge–Kutta algorithm is used to solve the above system with a time grid of 0.001. We choose the system parameters: a1 ¼ 0:7; b1 ¼ 0:905; A1 ¼ 0:4; x1 ¼ 0:25 and a2 ¼ 0:5; b2 ¼ 0:89; A2 ¼ 0:4; x2 ¼ 0:46, which places the master system and the slave system in the chaotic state, respectively. The same initial conditions are x1 ð0Þ ¼ 0; y 1 ð0Þ ¼ 0; x2 ð0Þ ¼ 0; y 2 ð0Þ ¼ 0.
64
Y. Zhao, W. Wang / Chaos, Solitons and Fractals 41 (2009) 60–66
The simulation results of the error states e1 ; e2 without active sliding mode control are shown in Fig. 3. In the Case i, when the control function is activated at the time t ¼ 50, the simulation results are illustrated in Fig. 4. Meanwhile Fig. 5 plots the phase portraits for the error system (16). In the Case ii, with the parameters c1 ¼ 1; e ¼ 1 and r ¼ 10, the simulation results are illustrated in Fig. 6 as the control function is added on at the time t ¼ 50. Fig. 7 also plots the phase portraits for the error system (20). In two cases all the error states quickly converge to zero. However the converging velocity in the Case ii is more faster than that in the Case i. 200
3 2
150
1
e2
e
1
100 50
−1
0
−2 50
100
150
−3
200
0
50
100
t
150
t
Fig. 3. Time evolution of the error state with the active sliding mode control deactivated.
40
20
e
1
1
e ,e
2
30
10 e
2
0 −10
0
50
100
150
t Fig. 4. Time evolution of the error states for the Case i.
2
1
2
0
e
−50
0
0
−1 −2 −2
−1
0
1
e
1
Fig. 5. The phase portraits for the error system (16).
2
200
Y. Zhao, W. Wang / Chaos, Solitons and Fractals 41 (2009) 60–66
65
40 30
2
0
e ,e
10
1
20 e
1
−10
e2
−20 −30 0
50
100
150
t Fig. 6. Time evolution of the error states for the Case ii.
2
e2
1
0
−1 −2
−1
0
e1
1
2
Fig. 7. The phase portraits for the error system (20).
4. Conclusion In this letter, two types of active siding control methods are provided and are successfully used to synchronize chaos motions in the J–J system. For the Case i, the range of applicability of the proposed method may be narrow owing to strong restricting of the controller form. Whereas no superabundant restrictive assumption is imposed for the Case ii, so this method can be generalized to other systems.
Acknowledgement This work was supported by the National Natural Science Foundation of China (under Grant No. 10401007).
References [1] Bai EW, Lonngren KE. Synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 1997;8:51–8. [2] Bai EW, Lonngren KE. Sequential synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 2000;11:1041–4. [3] Agiza HN, Yassen MT. Synchronization of Rossler and Chen chaotic dynamical systems using active control. Phys Lett A 2001;278:191–7. [4] Ho MC, Hung YC, Chou CH. Phase and anti-phase synchronization of two chaotic systems by using active control. Phys Lett A 2002;296:43–8.
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