- Email: [email protected]
Impulsive control and synchronization of unified chaotic system
Chaos, Solitons and Fractals 20 (2004) 751–758 www.elsevier.com/locate/chaos
Impulsive control and synchronization of unified chaotic system Shihua Chen b
a,*
, Qing Yang a, Changping Wang
b
a College of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR China Department of Mathematics and Statistics, Dalhousie University, Halifax NS, Canada B3H 3J5
Accepted 11 August 2003
Abstract We investigate the issue on impulsive control and synchronization of the unified chaotic system, which unifies both the Lorenz system and the Chen system. Some new and general conditions with varying impulsive distances are obtained to guarantee the impulsive control and synchronization global asymptotical stable. Especially, in the case of equal impulsive distances, some simple and easily verified sufficient conditions are derived for stabilizing and synchronizing the unified chaotic system. An illustrative example, along with computer simulation results, is finally included to visualize the effectiveness and feasibility of the developed methods. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction Research efforts have been devoted to chaos control and chaos synchronization problems in many physical chaotic systems [1–3]. The control problem attempts to stabilize a chaotic attractor to either a periodic orbit or an equilibrium point. Recently, after the pioneering work of Ott et al. [1], several control strategies for stabilizing chaos have been proposed [4,7,12]. The concept of chaos synchronization involves making two chaotic systems oscillate in a synchronous manner. The idea of synchronizing two identical chaotic systems with different initial conditions was introduced by Pecora and Carrol [2]. Several different approaches, including some conventional linear control techniques and advanced nonlinear control schemes, are already applied to the above problem [5,6,8,9]. In practice, there exist many examples of impulsive control systems and impulsive control method has been widely used to stabilize and synchronize chaotic systems [10–12]. The importance of impulsive control lies in that, in some cases, impulse control may give an efficient method to deal with systems which cannot endure continuous disturbance. Additionally, in synchronization process, the slave system receives the information about the active variables of the master system only in the discrete times. This drastically reduces the amount of information transmitted from the master system to slave system which makes this method more efficient and thus useful in a great number of real-life applications. This paper addresses an interesting issue of using impulsive control method to suppress and synchronize the unified chaotic system. Some new and less conservative criteria for global asymptotical stability are established in the case of varying impulsive distances. Particularly, some simple and easily verified sufficient conditions are derived for stabilizing and synchronizing the unified chaotic system in the case of equal impulsive distances. An illustrative example, along with computer simulation results, is finally included to visualize the satisfactory control performance.
*
Corresponding author. E-mail address: [email protected] (S. Chen).
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.08.008
752
S. Chen et al. / Chaos, Solitons and Fractals 20 (2004) 751–758
2. Impulsive control and synchronization of chaotic systems Let the general chaotic system be in the form of x_ ¼ f ðt; xÞ;
ð1Þ n
n
n
. Suppose that a discrete instant set fti g where f : Rþ R ! R is continuous, x 2 R is the state variable, x_ , dx dt satisfies t1 < t2 < < ti < tiþ1 < ;
lim ti ¼ 1
i!1
and t1 > t0 . Let Ui ðxðti Þ ¼ Dxjt¼ti , xðtiþ Þ xðti Þ
ð2Þ
be the change of state variables at instant ti , where xðtiþ Þ ¼ limt!tiþ xðtÞ and xðti Þ ¼ limt!ti xðtÞ ði ¼ 1; 2; . . .Þ. In general, for simplicity, it is assumed that xðti Þ ¼ xðti Þ. Furthermore, Ui ðxðti ÞÞ can be chosen as Bi xðti Þ with Bi being n n constant matrices ði ¼ 1; 2; . . .Þ. Accordingly, the impulsive controlled system can be expressed as follows: 8 < x_ ¼ f ðt; xÞ; t 6¼ ti ; t ¼ ti ; Dx ¼ Bi x; ð3Þ : þ xðt0 Þ ¼ x0 ; i ¼ 1; 2; . . . ; which is also called an impulsive differential system [10]. The objective is to find some (sufficient) conditions on the control gains, Bi , and the impulsive distances si , ti ti 1 < 1 ði ¼ 1; 2; . . .Þ such that the impulsive controlled system (3) is asymptotical stable at origin, namely, the nonlinear system (3) is impulsively asymptotical stable. In the impulsive synchronization configuration, the master system is given by Eq. (1), while the slave system is given by y_ ¼ f ðt; yÞ;
ð4Þ
which has the same structure as the master system but with different initial conditions. In discrete instants, ti ði ¼ 1; 2; . . .Þ, the active state variable of the master system are transmitted to the slave system which changes its values of state variables instantaneously in these discrete instants according to synchronization errors. In this sense, the impulsive controlled slave system is modelled by the following impulsive equations: 8 < y_ ¼ f ðt; yÞ; t 6¼ ti ; t ¼ ti ; Dy ¼ Bi e; ð5Þ : þ yðt0 Þ ¼ y0 ; i ¼ 1; 2; . . . ; where e ¼ ðe1 ; e2 ; e3 ÞT ¼ ðy1 x1 ; y2 x2 ; y3 x3 ÞT is the synchronization error. Thus, the goal of impulsive synchronization is to find conditions on the control gains, Bi , and the impulsive distances si ði ¼ 1; 2; . . .Þ such that the impulsive controlled slave system (5) is global asymptotical synchronous with the master system (1).
3. Impulsive control of unified chaotic system We use a specific chaotic system as an example to describe the methodology. This specific chaotic system, referred to as a unified system by other authors [13,14], has recently been discovered. The system is described by x_ 1 ¼ ð25a þ 10Þðx2 x1 Þ; x_ 2 ¼ ð28 35aÞx1 x1 x3 þ ð29a þ 1Þx2 ; aþ8 x3 ; x_ 3 ¼ x1 x2 3
ð6Þ
where a 2 ½0; 1. System (6) has some special feature and advantages because it unifies both the Lorenz system (when a ¼ 0) and the Chen system (when a ¼ 1), where the latter is the dual system of the former in a sense defined in [15]. The Lorenz system satisfies the condition a12 a21 > 0 while the Chen system satisfies a12 a21 < 0, where A ¼ ðaij Þ33 is the
S. Chen et al. / Chaos, Solitons and Fractals 20 (2004) 751–758
753
matrix of the linear part of the chaotic systems. Furthermore, system (6) is chaotic for all parameter a in ½0; 1 and it realized the entire transition spectrum from one to the other. We are interested in stabilizing this unified system at zero equilibrium by impulsive control method. To this end, we first decompose the linear and nonlinear parts of the unified system in Eq. (6) and rewrite it as x_ ¼ Ax þ UðxÞ;
ð7Þ
T
where x ¼ ðx1 ; x2 ; x3 Þ and 0
ð25a þ 10Þ 25a þ 10 B 29a þ 1 A ¼ @ 28 35a 0
The impulsive control 8 < x_ ¼ Ax þ UðxÞ; Dx ¼ Bi x; : þ xðt0 Þ ¼ x0 ;
0
1 0 0 C a þ 8 A;
3
0
1 0 UðxÞ ¼ @ x1 x3 A: x1 x2
ð8Þ
of the unified system is then given by t 6¼ ti ; t ¼ ti ; i ¼ 1; 2; . . . ;
ð9Þ
where ti denotes the moment when impulsive control occurs. We use the following theorem to guarantee that the impulsive control system will be global asymptotical stable at origin. Theorem 1. Let bi and kðaÞ be the largest eigenvalue of ðI þ Bi ÞT ðI þ Bi Þ ði ¼ 1; 2; . . .Þ and 12 ðA þ AT Þ, respectively. If there exists a constant n > 1 such that lnðnbi Þ þ 2kðaÞsi 6 0;
i ¼ 1; 2; . . .
ð10Þ
Then the impulsive controlled unified chaotic system in Eq. (9) is global asymptotical stable at origin, where 0 < si ¼ ti ti 1 < 1 ði ¼ 1; 2; . . .Þ are impulsive distances. Proof. Let the Lyapunov function be in the form of 1 V ðxÞ ¼ xT x: 2 The time derivative of V ðxÞ along the solution of Eq. (9) is
dV ðxðtÞÞ
1 1 T 1 T T T
¼ 2 ðAx þ UðxÞÞ x þ 2 x ðAx þ UðxÞÞ ¼ 2 x ðA þ AÞx; dt ð9Þ
t 2 ðti 1 ; ti ði ¼ 1; 2; . . .Þ:
ð11Þ
Since 1
ð25a þ 10Þ 19 5a 0 1 B 29a þ 1 0 C ðA þ AT Þ ¼ @ 19 5a a þ 8 A; 2 0 0
3 0
it is easy to see that kðaÞ > 0 for a 2 ½0; 1, therefore,
dV ðxðtÞÞ
6 2kðaÞV ðxðtÞÞ; t 2 ðti 1 ; ti ði ¼ 1; 2; . . .Þ; dt ð9Þ
ð12Þ
which implies that þ ÞÞ expð2kðaÞðt ti 1 ÞÞ; V ðxðtÞÞ 6 V ðxðti 1
t 2 ðti 1 ; ti ði ¼ 1; 2; . . .Þ:
ð13Þ
On the other hand, it follows from the second equation of system (9) that 1 1 V ðxðtiþ ÞÞ ¼ ½ðI þ Bi Þxðti ÞT ðI þ Bi Þxðti Þ ¼ xT ðti ÞÞ½ðI þ Bi ÞT ðI þ Bi Þxðti Þ 2 2 1 T T 6 kmax ½ðI þ Bi Þ ðI þ Bi Þx ðti Þxðti Þ 2 ¼ bi V ðxðti ÞÞ; i ¼ 1; 2; . . .
ð14Þ
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S. Chen et al. / Chaos, Solitons and Fractals 20 (2004) 751–758
Thus, let i ¼ 1 in the inequality (13) we have for any t 2 ðt0 ; t1 V ðxðtÞÞ 6 V ðxðt0 ÞÞ expð2kðaÞðt t0 ÞÞ; which leads to V ðxðt1 ÞÞ 6 V ðxðt0 ÞÞ expð2kðaÞðt1 t0 ÞÞ and V ðxðt1þ ÞÞ 6 b1 V ðxðt1 ÞÞ 6 b1 V ðxðt0 ÞÞ expð2kðaÞðt1 t0 ÞÞ: Therefore, for t 2 ðt1 ; t2 V ðxðtÞÞ 6 V ðxðt1þ ÞÞ expð2kðaÞðt t1 ÞÞ 6 b1 V ðxðt0 ÞÞ expð2kðaÞðt t0 ÞÞ: In general, for t 2 ðti ; tiþ1 , V ðxðtÞÞ 6 V ðxðt0 ÞÞb1 b2 bi expð2kðaÞðt t0 ÞÞ:
ð15Þ
In virtue of the inequality (10) we know that bi expð2kðaÞsi Þ 6
1 ; n
i ¼ 1; 2; . . .
Thus, for t 2 ðti ; tiþ1 ði ¼ 1; 2; . . .Þ, V ðxðtÞÞ 6 V ðxðt0 ÞÞb1 b2 bi expð2kðaÞðt t0 ÞÞ ¼ V ðxðt0 ÞÞ½b1 expð2kðaÞs1 Þ ½bi expð2kðaÞsi Þ expð2kðaÞðt ti ÞÞ 1 6 V ðxðt0 ÞÞ i expð2kðaÞsiþ1 Þ; n
ð16Þ
which implies that the trivial solution of system (9) is global asymptotical stable. We finished the proof of Theorem 1. h In practice, for the sake of convenience, the gain matrices Bi are always selected as a constant matrix and the impulsive distances si are set to be a positive constant ði ¼ 1; 2; . . .Þ. Then we have the following corollary. Corollary 1. Assume si ¼ s > 0 and matrices Bi ¼ B ði ¼ 1; 2; . . .Þ. If there exists a constant n > 1 such that lnðnbÞ þ 2kðaÞs 6 0;
i ¼ 1; 2; . . .
ð17Þ
Then the impulsive controlled unified chaotic system in Eq. (9) is global asymptotical stable at origin, where b is the largest eigenvalue of ðI þ BÞT ðI þ BÞ. 4. Impulsive synchronization of unified system In this section, we study the impulsive synchronization of the unified system. From Section 2, we know the impulsive controlled slave system is modelled by the following impulsive equation: 8 < y_ ¼ Ay þ UðyÞ; t 6¼ ti ; t ¼ ti ; ð18Þ Dy ¼ Bi e; : i ¼ 1; 2; . . . ; yðt0 Þ ¼ y0 ; where y ¼ ðy1 ; y2 ; y3 ÞT , e ¼ ðe1 ; e2 ; e3 ÞT ¼ ðy1 x1 ; y2 x2 ; y3 x3 ÞT is the synchronization error. Note that there exists a positive number M for the unified chaotic system (6) such that jx2 ðtÞj 6 M and jx3 ðtÞj 6 M for all t. Then, similar to the stabilization of the unified system, we have the following result. Theorem 2. Let bi and kðaÞ be the largest eigenvalue of ðI þ Bi ÞT ðI þ Bi Þ ði ¼ 1; 2; . . .Þ and 12 ðA þ AT Þ, respectively. If there exists a constant n > 1 such that lnðnbi Þ þ 2ðkðaÞ þ MÞsi 6 0;
i ¼ 1; 2; . . .
ð19Þ
S. Chen et al. / Chaos, Solitons and Fractals 20 (2004) 751–758
755
Then the impulsive controlled unified system (18) is global asymptotical synchronous with the unified system (6), where 0 < si ¼ ti ti 1 < 1 ði ¼ 1; 2; . . .Þ are impulsive distances. Proof. Let e ¼ ðe1 ; e2 ; e3 ÞT be synchronization errors, the error system of the impulsive synchronization is given by e_ ¼ Ae þ Wðx; yÞ; t 6¼ ti ; De ¼ Bi e; t ¼ ti ;
ð20Þ
where 0
1 0 @ Wðx; yÞ ¼ UðyÞ UðxÞ ¼ y1 y3 þ x1 x3 A: y1 y2 x 1 x 2 Let the Lyapunov function be in the form of 1 V ðeÞ ¼ eT e: 2 Its time derivative along the trajectory of (20) is
dV ðeðtÞÞ
1 1 T T
¼ 2 ðAe þ Wðx; yÞÞ e þ 2 e ðAe þ Wðx; yÞÞ dt ð20Þ 1 ¼ eT ðAT þ AÞe þ x3 e1 e2 x2 e1 e3 6 2kðaÞV ðeðtÞÞ þ Mðje1 jje2 j þ je1 jje3 jÞ 2 6 2ðkðaÞ þ MÞV ðeðtÞÞ;
t 2 ðti 1 ; ti ði ¼ 1; 2; . . .Þ:
ð21Þ
In the same way as in the proof of Theorem 1, one can get that, for t 2 ðti ; tiþ1 ði ¼ 1; 2; . . .Þ V ðeðtÞÞ 6 V ðeðt0 ÞÞb1 b2 bi expð2ðkðaÞ þ MÞðt t0 ÞÞ ¼ V ðeðt0 ÞÞ½b1 expð2ðkðaÞ þ MÞs1 Þ ½bi expð2ðkðaÞ þ MÞsi Þ expð2ðkðaÞ þ MÞðt ti ÞÞ 6 V ðeðt0 ÞÞ
1 expð2ðkðaÞ þ MÞsiþ1 Þ; ni
ð22Þ
which implies that the trivial solution of system (20) is global asymptotical stable at origin. Thus, the impulsive controlled unified system (18) is global asymptotical synchronous with the unified system (6). Theorem 2 is proved as desired. h Similarly, in the case of constant gain matrix and constant impulsive distances, we have the following corollary. Corollary 2. Assume si ¼ s > 0 and matrices Bi ¼ B ði ¼ 1; 2; . . .Þ. If there exists a constant n > 1 such that lnðnbÞ þ 2ðkðaÞ þ MÞs 6 0;
i ¼ 1; 2; . . .
ð23Þ
Then the impulsive controlled unified system (18) is global asymptotical synchronous with the unified system (6). 5. Numerical simulations In order to demonstrate and verify the performance of the proposed method, some numerical simulations are presented in this section. In what follows, the fourth-order Runge–Kutta method is applied to solve the differential equations with time step size equal to 0.001 in all numerical simulations. In the first simulation, we stabilize the unified chaotic system with the parameter a ¼ 1. In this situation, we have 0 1
35 14 0 1 ðA þ AT Þ ¼ @ 14 30 0 A: 2 0 0 3
756
S. Chen et al. / Chaos, Solitons and Fractals 20 (2004) 751–758
Its eigenvalues are )37.887, )3.00 and 32.887. Then kð1Þ ¼ 12 kmax ðA þ AT Þ ¼ 32:887. We choose the gain matrices Bi ði ¼ 1; 2; . . .Þ as a constant matrix 0 1 d 0 0 B ¼ @ 0 d 0 A: 0 0 d It is easy to see that b ¼ ðd þ 1Þ2 . Then estimates of bounds of stable regions are given by 06s6
ln n þ lnðd þ 1Þ2 : 65:774
ð24Þ
Fig. 1 shows the stable region for different ns. The whole region under the curve of n ¼ 1 is the stable region. When n ! 1, the stable region approaches a vertical line d ¼ 1. Let d ¼ 0:58, select n ¼ 1:1, then 0 6 s 6 0:025. The numerical simulation results with d ¼ 0:58 and the constant impulsive distance s ¼ 0:02 are shown in Fig. 2. In the second simulation, we study impulsive synchronization of the unified chaotic system with the parameter a ¼ 1. The initial conditions of master system and slave system are (3; 4; 5), (4; 7; 10), respectively. At first, we can get from the simulation, the approximate bounded value of the system (6) with a ¼ 1 is 64. If one choose B ¼ diagðb1 ; b2 ; b3 Þ ¼ diagð 0:78; 0:84; 0:84Þ:
0.25
0.2
τ
0.15
0.1 ξ=
1
3
10
10
1
3
0.05 Stable region 0 -1.5
-1.4
-1.3
-1.2
-1.1
-1 δ
-0.9
-0.8
-0.7
-0.6
-0.5
Fig. 1. The boundaries of stable region with different ns used in simulation 1.
6
x (t)
4 2 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1 t(s)
1.2
1.4
1.6
1.8
2
8
x (t)
6 4 2 0 6
x (t)
4 2 0
Fig. 2. Impulsive stabilizing unified system with a ¼ 1 and s ¼ 0:02.
S. Chen et al. / Chaos, Solitons and Fractals 20 (2004) 751–758
757
3
e (t)
2 1 0 -1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1 t(s)
1.2
1.4
1.6
1.8
2
6
e (t)
4 2 0 -2 6
e (t)
4 2 0 -2
Fig. 3. Errors during synchronizing unified chaotic system with a ¼ 1. e1 ¼ y1 x1 , e2 ¼ y2 x2 and e3 ¼ y3 x3 .
Then b ¼ maxfð1 þ b1 Þ2 ; ð1 þ b2 Þ2 ; ð1 þ b3 Þ2 g ¼ 0:1764. If one takes s ¼ 0:015 and n ¼ 1:01, then lnðnbÞ þ 2ðk þ MÞs ¼ 3:183 þ 2:90661 < 0; which implies from Corollary 2 that, the controlled unified system is synchronous with the unified system. Simulation results are shown in Fig. 3.
6. Conclusion In this paper, we investigate the issue on the stabilizing and synchronizing the unified chaotic system via an impulsive control method. some new and less conservative conditions with varying impulsive distances are obtained to guarantee the impulsive control and synchronization global asymptotical stable. Especially, in the case of equal impulsive distances, some simple and easily verified sufficient conditions are derived for stabilizing and synchronizing the unified chaotic system. An illustrative example, along with computer simulation results, is finally included to visualize the satisfactory control performance.
Acknowledgement This work was supported by the National Nature Science Foundation of P.R. China under the grant no. 69874029.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;64:1196–9. Pecora LM, Carroll TL. Synchronization in chaotic system. Phys Rev Lett 1990;64:821–4. Chen G, Dong X. From chaos to order: methodologies, perspectives and applications. Singapore: World Scientific; 1998. Agiza HN. Controlling chaos for the dynamical system of coupled dynamos. Chaos, Solitons & Fractals 2002;13:341–52. Vaidya PG. Monitoring and speeding up chaotic synchronization. Chaos, Solitons & Fractals 2003;17:433–9. Wang C, Ge S. Adaptive synchronization of uncertain chaotic system via backstepping design. Chaos, Solitons & Fractals 2001;12:1199–206. Hendrik R. Controlling chaotic systems with multiple strange attractors. Phys Lett A 2002;300:182–8. Chen S, L€ u J. Parameters identification and synchronization of chaotic systems based upon adaptive control. Phys Lett A 2002;299:353–8. Chen S, Liu J, Feng J, L€ u J. Synchronization chaotic system in strict-feedback form using a single controller. Chin Phys Lett 2002;19:1257–9. Lakshmikantham V, Bainov DD, Simeonov PS. Theory of impulsive differential equations. Singapore: World Scientific; 1989.
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[11] Itoh M, Yang T, Chua LO. Conditions for impulsive synchronization of chaotic and hyperchaotic systems. Int J Bifurcat Chaos 2001;11:551–60. [12] Yang T, Yang L, Yang C. Impulsive control of Lorenz system. Physica D 1997;110:18–24. [13] Chen S, L€ u J. Synchronization of an uncertain unified chaotic system via adaptive control. Chaos, Solitons & Fractals 2002;14:643–7. [14] Lu J, Wu X, L€ u J. Synchronization of a unified chaotic system and the application in secure communication. Phys Lett A 2002;305:365–70. elikovsky S. Control systems: from linear analysis to synthesis of chaos. London: Prentice-Hall; 1996. [15] Vanecek A, C
Impulsive control and synchronization of unified chaotic system Shihua Chen b
a,*
, Qing Yang a, Changping Wang
b
a College of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR China Department of Mathematics and Statistics, Dalhousie University, Halifax NS, Canada B3H 3J5
Accepted 11 August 2003
Abstract We investigate the issue on impulsive control and synchronization of the unified chaotic system, which unifies both the Lorenz system and the Chen system. Some new and general conditions with varying impulsive distances are obtained to guarantee the impulsive control and synchronization global asymptotical stable. Especially, in the case of equal impulsive distances, some simple and easily verified sufficient conditions are derived for stabilizing and synchronizing the unified chaotic system. An illustrative example, along with computer simulation results, is finally included to visualize the effectiveness and feasibility of the developed methods. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction Research efforts have been devoted to chaos control and chaos synchronization problems in many physical chaotic systems [1–3]. The control problem attempts to stabilize a chaotic attractor to either a periodic orbit or an equilibrium point. Recently, after the pioneering work of Ott et al. [1], several control strategies for stabilizing chaos have been proposed [4,7,12]. The concept of chaos synchronization involves making two chaotic systems oscillate in a synchronous manner. The idea of synchronizing two identical chaotic systems with different initial conditions was introduced by Pecora and Carrol [2]. Several different approaches, including some conventional linear control techniques and advanced nonlinear control schemes, are already applied to the above problem [5,6,8,9]. In practice, there exist many examples of impulsive control systems and impulsive control method has been widely used to stabilize and synchronize chaotic systems [10–12]. The importance of impulsive control lies in that, in some cases, impulse control may give an efficient method to deal with systems which cannot endure continuous disturbance. Additionally, in synchronization process, the slave system receives the information about the active variables of the master system only in the discrete times. This drastically reduces the amount of information transmitted from the master system to slave system which makes this method more efficient and thus useful in a great number of real-life applications. This paper addresses an interesting issue of using impulsive control method to suppress and synchronize the unified chaotic system. Some new and less conservative criteria for global asymptotical stability are established in the case of varying impulsive distances. Particularly, some simple and easily verified sufficient conditions are derived for stabilizing and synchronizing the unified chaotic system in the case of equal impulsive distances. An illustrative example, along with computer simulation results, is finally included to visualize the satisfactory control performance.
*
Corresponding author. E-mail address: [email protected] (S. Chen).
0960-0779/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.08.008
752
S. Chen et al. / Chaos, Solitons and Fractals 20 (2004) 751–758
2. Impulsive control and synchronization of chaotic systems Let the general chaotic system be in the form of x_ ¼ f ðt; xÞ;
ð1Þ n
n
n
. Suppose that a discrete instant set fti g where f : Rþ R ! R is continuous, x 2 R is the state variable, x_ , dx dt satisfies t1 < t2 < < ti < tiþ1 < ;
lim ti ¼ 1
i!1
and t1 > t0 . Let Ui ðxðti Þ ¼ Dxjt¼ti , xðtiþ Þ xðti Þ
ð2Þ
be the change of state variables at instant ti , where xðtiþ Þ ¼ limt!tiþ xðtÞ and xðti Þ ¼ limt!ti xðtÞ ði ¼ 1; 2; . . .Þ. In general, for simplicity, it is assumed that xðti Þ ¼ xðti Þ. Furthermore, Ui ðxðti ÞÞ can be chosen as Bi xðti Þ with Bi being n n constant matrices ði ¼ 1; 2; . . .Þ. Accordingly, the impulsive controlled system can be expressed as follows: 8 < x_ ¼ f ðt; xÞ; t 6¼ ti ; t ¼ ti ; Dx ¼ Bi x; ð3Þ : þ xðt0 Þ ¼ x0 ; i ¼ 1; 2; . . . ; which is also called an impulsive differential system [10]. The objective is to find some (sufficient) conditions on the control gains, Bi , and the impulsive distances si , ti ti 1 < 1 ði ¼ 1; 2; . . .Þ such that the impulsive controlled system (3) is asymptotical stable at origin, namely, the nonlinear system (3) is impulsively asymptotical stable. In the impulsive synchronization configuration, the master system is given by Eq. (1), while the slave system is given by y_ ¼ f ðt; yÞ;
ð4Þ
which has the same structure as the master system but with different initial conditions. In discrete instants, ti ði ¼ 1; 2; . . .Þ, the active state variable of the master system are transmitted to the slave system which changes its values of state variables instantaneously in these discrete instants according to synchronization errors. In this sense, the impulsive controlled slave system is modelled by the following impulsive equations: 8 < y_ ¼ f ðt; yÞ; t 6¼ ti ; t ¼ ti ; Dy ¼ Bi e; ð5Þ : þ yðt0 Þ ¼ y0 ; i ¼ 1; 2; . . . ; where e ¼ ðe1 ; e2 ; e3 ÞT ¼ ðy1 x1 ; y2 x2 ; y3 x3 ÞT is the synchronization error. Thus, the goal of impulsive synchronization is to find conditions on the control gains, Bi , and the impulsive distances si ði ¼ 1; 2; . . .Þ such that the impulsive controlled slave system (5) is global asymptotical synchronous with the master system (1).
3. Impulsive control of unified chaotic system We use a specific chaotic system as an example to describe the methodology. This specific chaotic system, referred to as a unified system by other authors [13,14], has recently been discovered. The system is described by x_ 1 ¼ ð25a þ 10Þðx2 x1 Þ; x_ 2 ¼ ð28 35aÞx1 x1 x3 þ ð29a þ 1Þx2 ; aþ8 x3 ; x_ 3 ¼ x1 x2 3
ð6Þ
where a 2 ½0; 1. System (6) has some special feature and advantages because it unifies both the Lorenz system (when a ¼ 0) and the Chen system (when a ¼ 1), where the latter is the dual system of the former in a sense defined in [15]. The Lorenz system satisfies the condition a12 a21 > 0 while the Chen system satisfies a12 a21 < 0, where A ¼ ðaij Þ33 is the
S. Chen et al. / Chaos, Solitons and Fractals 20 (2004) 751–758
753
matrix of the linear part of the chaotic systems. Furthermore, system (6) is chaotic for all parameter a in ½0; 1 and it realized the entire transition spectrum from one to the other. We are interested in stabilizing this unified system at zero equilibrium by impulsive control method. To this end, we first decompose the linear and nonlinear parts of the unified system in Eq. (6) and rewrite it as x_ ¼ Ax þ UðxÞ;
ð7Þ
T
where x ¼ ðx1 ; x2 ; x3 Þ and 0
ð25a þ 10Þ 25a þ 10 B 29a þ 1 A ¼ @ 28 35a 0
The impulsive control 8 < x_ ¼ Ax þ UðxÞ; Dx ¼ Bi x; : þ xðt0 Þ ¼ x0 ;
0
1 0 0 C a þ 8 A;
3
0
1 0 UðxÞ ¼ @ x1 x3 A: x1 x2
ð8Þ
of the unified system is then given by t 6¼ ti ; t ¼ ti ; i ¼ 1; 2; . . . ;
ð9Þ
where ti denotes the moment when impulsive control occurs. We use the following theorem to guarantee that the impulsive control system will be global asymptotical stable at origin. Theorem 1. Let bi and kðaÞ be the largest eigenvalue of ðI þ Bi ÞT ðI þ Bi Þ ði ¼ 1; 2; . . .Þ and 12 ðA þ AT Þ, respectively. If there exists a constant n > 1 such that lnðnbi Þ þ 2kðaÞsi 6 0;
i ¼ 1; 2; . . .
ð10Þ
Then the impulsive controlled unified chaotic system in Eq. (9) is global asymptotical stable at origin, where 0 < si ¼ ti ti 1 < 1 ði ¼ 1; 2; . . .Þ are impulsive distances. Proof. Let the Lyapunov function be in the form of 1 V ðxÞ ¼ xT x: 2 The time derivative of V ðxÞ along the solution of Eq. (9) is
dV ðxðtÞÞ
1 1 T 1 T T T
¼ 2 ðAx þ UðxÞÞ x þ 2 x ðAx þ UðxÞÞ ¼ 2 x ðA þ AÞx; dt ð9Þ
t 2 ðti 1 ; ti ði ¼ 1; 2; . . .Þ:
ð11Þ
Since 1
ð25a þ 10Þ 19 5a 0 1 B 29a þ 1 0 C ðA þ AT Þ ¼ @ 19 5a a þ 8 A; 2 0 0
3 0
it is easy to see that kðaÞ > 0 for a 2 ½0; 1, therefore,
dV ðxðtÞÞ
6 2kðaÞV ðxðtÞÞ; t 2 ðti 1 ; ti ði ¼ 1; 2; . . .Þ; dt ð9Þ
ð12Þ
which implies that þ ÞÞ expð2kðaÞðt ti 1 ÞÞ; V ðxðtÞÞ 6 V ðxðti 1
t 2 ðti 1 ; ti ði ¼ 1; 2; . . .Þ:
ð13Þ
On the other hand, it follows from the second equation of system (9) that 1 1 V ðxðtiþ ÞÞ ¼ ½ðI þ Bi Þxðti ÞT ðI þ Bi Þxðti Þ ¼ xT ðti ÞÞ½ðI þ Bi ÞT ðI þ Bi Þxðti Þ 2 2 1 T T 6 kmax ½ðI þ Bi Þ ðI þ Bi Þx ðti Þxðti Þ 2 ¼ bi V ðxðti ÞÞ; i ¼ 1; 2; . . .
ð14Þ
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Thus, let i ¼ 1 in the inequality (13) we have for any t 2 ðt0 ; t1 V ðxðtÞÞ 6 V ðxðt0 ÞÞ expð2kðaÞðt t0 ÞÞ; which leads to V ðxðt1 ÞÞ 6 V ðxðt0 ÞÞ expð2kðaÞðt1 t0 ÞÞ and V ðxðt1þ ÞÞ 6 b1 V ðxðt1 ÞÞ 6 b1 V ðxðt0 ÞÞ expð2kðaÞðt1 t0 ÞÞ: Therefore, for t 2 ðt1 ; t2 V ðxðtÞÞ 6 V ðxðt1þ ÞÞ expð2kðaÞðt t1 ÞÞ 6 b1 V ðxðt0 ÞÞ expð2kðaÞðt t0 ÞÞ: In general, for t 2 ðti ; tiþ1 , V ðxðtÞÞ 6 V ðxðt0 ÞÞb1 b2 bi expð2kðaÞðt t0 ÞÞ:
ð15Þ
In virtue of the inequality (10) we know that bi expð2kðaÞsi Þ 6
1 ; n
i ¼ 1; 2; . . .
Thus, for t 2 ðti ; tiþ1 ði ¼ 1; 2; . . .Þ, V ðxðtÞÞ 6 V ðxðt0 ÞÞb1 b2 bi expð2kðaÞðt t0 ÞÞ ¼ V ðxðt0 ÞÞ½b1 expð2kðaÞs1 Þ ½bi expð2kðaÞsi Þ expð2kðaÞðt ti ÞÞ 1 6 V ðxðt0 ÞÞ i expð2kðaÞsiþ1 Þ; n
ð16Þ
which implies that the trivial solution of system (9) is global asymptotical stable. We finished the proof of Theorem 1. h In practice, for the sake of convenience, the gain matrices Bi are always selected as a constant matrix and the impulsive distances si are set to be a positive constant ði ¼ 1; 2; . . .Þ. Then we have the following corollary. Corollary 1. Assume si ¼ s > 0 and matrices Bi ¼ B ði ¼ 1; 2; . . .Þ. If there exists a constant n > 1 such that lnðnbÞ þ 2kðaÞs 6 0;
i ¼ 1; 2; . . .
ð17Þ
Then the impulsive controlled unified chaotic system in Eq. (9) is global asymptotical stable at origin, where b is the largest eigenvalue of ðI þ BÞT ðI þ BÞ. 4. Impulsive synchronization of unified system In this section, we study the impulsive synchronization of the unified system. From Section 2, we know the impulsive controlled slave system is modelled by the following impulsive equation: 8 < y_ ¼ Ay þ UðyÞ; t 6¼ ti ; t ¼ ti ; ð18Þ Dy ¼ Bi e; : i ¼ 1; 2; . . . ; yðt0 Þ ¼ y0 ; where y ¼ ðy1 ; y2 ; y3 ÞT , e ¼ ðe1 ; e2 ; e3 ÞT ¼ ðy1 x1 ; y2 x2 ; y3 x3 ÞT is the synchronization error. Note that there exists a positive number M for the unified chaotic system (6) such that jx2 ðtÞj 6 M and jx3 ðtÞj 6 M for all t. Then, similar to the stabilization of the unified system, we have the following result. Theorem 2. Let bi and kðaÞ be the largest eigenvalue of ðI þ Bi ÞT ðI þ Bi Þ ði ¼ 1; 2; . . .Þ and 12 ðA þ AT Þ, respectively. If there exists a constant n > 1 such that lnðnbi Þ þ 2ðkðaÞ þ MÞsi 6 0;
i ¼ 1; 2; . . .
ð19Þ
S. Chen et al. / Chaos, Solitons and Fractals 20 (2004) 751–758
755
Then the impulsive controlled unified system (18) is global asymptotical synchronous with the unified system (6), where 0 < si ¼ ti ti 1 < 1 ði ¼ 1; 2; . . .Þ are impulsive distances. Proof. Let e ¼ ðe1 ; e2 ; e3 ÞT be synchronization errors, the error system of the impulsive synchronization is given by e_ ¼ Ae þ Wðx; yÞ; t 6¼ ti ; De ¼ Bi e; t ¼ ti ;
ð20Þ
where 0
1 0 @ Wðx; yÞ ¼ UðyÞ UðxÞ ¼ y1 y3 þ x1 x3 A: y1 y2 x 1 x 2 Let the Lyapunov function be in the form of 1 V ðeÞ ¼ eT e: 2 Its time derivative along the trajectory of (20) is
dV ðeðtÞÞ
1 1 T T
¼ 2 ðAe þ Wðx; yÞÞ e þ 2 e ðAe þ Wðx; yÞÞ dt ð20Þ 1 ¼ eT ðAT þ AÞe þ x3 e1 e2 x2 e1 e3 6 2kðaÞV ðeðtÞÞ þ Mðje1 jje2 j þ je1 jje3 jÞ 2 6 2ðkðaÞ þ MÞV ðeðtÞÞ;
t 2 ðti 1 ; ti ði ¼ 1; 2; . . .Þ:
ð21Þ
In the same way as in the proof of Theorem 1, one can get that, for t 2 ðti ; tiþ1 ði ¼ 1; 2; . . .Þ V ðeðtÞÞ 6 V ðeðt0 ÞÞb1 b2 bi expð2ðkðaÞ þ MÞðt t0 ÞÞ ¼ V ðeðt0 ÞÞ½b1 expð2ðkðaÞ þ MÞs1 Þ ½bi expð2ðkðaÞ þ MÞsi Þ expð2ðkðaÞ þ MÞðt ti ÞÞ 6 V ðeðt0 ÞÞ
1 expð2ðkðaÞ þ MÞsiþ1 Þ; ni
ð22Þ
which implies that the trivial solution of system (20) is global asymptotical stable at origin. Thus, the impulsive controlled unified system (18) is global asymptotical synchronous with the unified system (6). Theorem 2 is proved as desired. h Similarly, in the case of constant gain matrix and constant impulsive distances, we have the following corollary. Corollary 2. Assume si ¼ s > 0 and matrices Bi ¼ B ði ¼ 1; 2; . . .Þ. If there exists a constant n > 1 such that lnðnbÞ þ 2ðkðaÞ þ MÞs 6 0;
i ¼ 1; 2; . . .
ð23Þ
Then the impulsive controlled unified system (18) is global asymptotical synchronous with the unified system (6). 5. Numerical simulations In order to demonstrate and verify the performance of the proposed method, some numerical simulations are presented in this section. In what follows, the fourth-order Runge–Kutta method is applied to solve the differential equations with time step size equal to 0.001 in all numerical simulations. In the first simulation, we stabilize the unified chaotic system with the parameter a ¼ 1. In this situation, we have 0 1
35 14 0 1 ðA þ AT Þ ¼ @ 14 30 0 A: 2 0 0 3
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S. Chen et al. / Chaos, Solitons and Fractals 20 (2004) 751–758
Its eigenvalues are )37.887, )3.00 and 32.887. Then kð1Þ ¼ 12 kmax ðA þ AT Þ ¼ 32:887. We choose the gain matrices Bi ði ¼ 1; 2; . . .Þ as a constant matrix 0 1 d 0 0 B ¼ @ 0 d 0 A: 0 0 d It is easy to see that b ¼ ðd þ 1Þ2 . Then estimates of bounds of stable regions are given by 06s6
ln n þ lnðd þ 1Þ2 : 65:774
ð24Þ
Fig. 1 shows the stable region for different ns. The whole region under the curve of n ¼ 1 is the stable region. When n ! 1, the stable region approaches a vertical line d ¼ 1. Let d ¼ 0:58, select n ¼ 1:1, then 0 6 s 6 0:025. The numerical simulation results with d ¼ 0:58 and the constant impulsive distance s ¼ 0:02 are shown in Fig. 2. In the second simulation, we study impulsive synchronization of the unified chaotic system with the parameter a ¼ 1. The initial conditions of master system and slave system are (3; 4; 5), (4; 7; 10), respectively. At first, we can get from the simulation, the approximate bounded value of the system (6) with a ¼ 1 is 64. If one choose B ¼ diagðb1 ; b2 ; b3 Þ ¼ diagð 0:78; 0:84; 0:84Þ:
0.25
0.2
τ
0.15
0.1 ξ=
1
3
10
10
1
3
0.05 Stable region 0 -1.5
-1.4
-1.3
-1.2
-1.1
-1 δ
-0.9
-0.8
-0.7
-0.6
-0.5
Fig. 1. The boundaries of stable region with different ns used in simulation 1.
6
x (t)
4 2 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1 t(s)
1.2
1.4
1.6
1.8
2
8
x (t)
6 4 2 0 6
x (t)
4 2 0
Fig. 2. Impulsive stabilizing unified system with a ¼ 1 and s ¼ 0:02.
S. Chen et al. / Chaos, Solitons and Fractals 20 (2004) 751–758
757
3
e (t)
2 1 0 -1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1 t(s)
1.2
1.4
1.6
1.8
2
6
e (t)
4 2 0 -2 6
e (t)
4 2 0 -2
Fig. 3. Errors during synchronizing unified chaotic system with a ¼ 1. e1 ¼ y1 x1 , e2 ¼ y2 x2 and e3 ¼ y3 x3 .
Then b ¼ maxfð1 þ b1 Þ2 ; ð1 þ b2 Þ2 ; ð1 þ b3 Þ2 g ¼ 0:1764. If one takes s ¼ 0:015 and n ¼ 1:01, then lnðnbÞ þ 2ðk þ MÞs ¼ 3:183 þ 2:90661 < 0; which implies from Corollary 2 that, the controlled unified system is synchronous with the unified system. Simulation results are shown in Fig. 3.
6. Conclusion In this paper, we investigate the issue on the stabilizing and synchronizing the unified chaotic system via an impulsive control method. some new and less conservative conditions with varying impulsive distances are obtained to guarantee the impulsive control and synchronization global asymptotical stable. Especially, in the case of equal impulsive distances, some simple and easily verified sufficient conditions are derived for stabilizing and synchronizing the unified chaotic system. An illustrative example, along with computer simulation results, is finally included to visualize the satisfactory control performance.
Acknowledgement This work was supported by the National Nature Science Foundation of P.R. China under the grant no. 69874029.
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