- Email: [email protected]
Impulsive control and synchronization of the Lorenz systems family
Chaos, Solitons and Fractals 31 (2007) 631–638 www.elsevier.com/locate/chaos
Impulsive control and synchronization of the Lorenz systems family q Xiaoqun Wu b
a,b,*
, Jun-an Lu a, Chi K. Tse b, Jinjun Wang c, Jie Liu
a
a School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong, China c College of Mechanical and Energy Engineering, Zhejiang University, Hangzhou 310027, China
Accepted 5 October 2005
Abstract In this paper, impulsive control and synchronization for the newly presented Lorenz systems family are systematically investigated. Some new and more comprehensive criteria for global exponential stability and asymptotical stability of impulsively controlled Lorenz systems family are established with varying impulsive intervals. In particular, several simple and easily verified criteria are derived with equivalent impulsive intervals. An illustrative example is also provided to show the effectiveness and feasibility of the impulsive control method. 2005 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, there has been considerable interest in the control of chaos in nonlinear dynamical systems. For the past years, many different techniques have been proposed to control chaos, including OGY method [1], P–C technique [2], backstepping approach [3], and so on [4–9]. Recently, impulsive control has been widely used to stabilize and synchronize chaotic systems [10–15]. Its necessity and importance lie in that, in some cases, the system cannot be controlled by continuous control. For example, a government cannot change savings rates of its central bank every day. Additionally, impulsive control may give a more efficient method to deal with systems that cannot endure continuous disturbance. Furthermore, impulsive method can also greatly reduce the control cost. In this letter, we systematically investigate the impulsive control and synchronization for the newly presented Lorenz systems family [24]. To stabilize the Lorenz systems family to the origin and synchronize two chaotic systems, we establish some new and more comprehensive criteria for global exponential stability and asymptotical stability with varying impulsive intervals. Particularly, several simple and easily verified criteria are also derived with equivalent impulsive
q
Supported by National Key Basic Research Development 973 Program of China (Grant No. 2003CB415200) and National Natural Science Foundation of China (Grant No. 60574045). * Corresponding author. Address: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China. E-mail address: [email protected] (X. Wu). 0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.10.017
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X. Wu et al. / Chaos, Solitons and Fractals 31 (2007) 631–638
intervals. An illustrative example, along with numerical simulations is also provided to show the effectiveness and feasibility of the developed method.
2. Impulsive control of nonlinear systems Let the general chaotic system be in the form of x_ ¼ f ðt; xÞ;
ð1Þ n
n
n
where t 2 J = [t0, +1] (t0 P 0), x 2 R is the state variable, and f: J · R ! R is a continuous vector-value function. An impulsive control law of system (1) is given by a sequence, {tk, uk(x(tk))}, which has the effect of suddenly changing the state of the system at the instants tk, where t1 < t2 < < tk < , limk!1tk = 1 and t1 > t0; that is Dxjtk ¼ xðtþ k Þ xðt k Þ ¼ uk ðxðt k ÞÞ;
ð2Þ
xðtÞ and xðtk Þ ¼ limt!tk xðtÞ. In general, for simplicity, it is assumed that xðt where xðtþ k Þ ¼ limt!tþ k Þ ¼ xðtk Þ. Furtherk more, uk(x(tk)) can be chosen as Bkx(tk) with Bk being n · n matrices. Accordingly, the impulsively controlled system can be expressed as follows: 8 > < x_ ¼ f ðt; xÞ; t 6¼ tk ; Dx ¼ Bk x; t ¼ tk ; ð3Þ > : þ xðt0 Þ ¼ x0 ðk ¼ 1; 2; . . .Þ; which is also called an impulsive differential system. The objective is to find some (sufficient) conditions on the constant control gains, Bk, and the impulsive intervals sk = tk tk1 < 1 (k = 1, 2, . . .), such that the impulsively controlled system (3) is stable; namely, the nonlinear system (1) is impulsively stabilizable.
3. Control the Lorenz systems family to the origin by impulsive method The Lorenz system is known to be a simplified model of several physical systems. Originally, it was derived from a model of the earthÕs atmospheric convection flow heated from below and cooled from above [16]. Furthermore, it has been reported that Lorenz equations may describe such different systems as laser devices, disk dynamos and several problems related to convection [17]. Recently, the Lorenz attractor has just been mathematically confirmed to exist [18]. The Lorenz system is described by 8 > < x_ 1 ¼ aðx2 x1 Þ; x_ 2 ¼ cx1 x1 x3 x2 ; ð4Þ > : x_ 3 ¼ x1 x2 bx3 ; which has a chaotic attractor when a = 10, b = 8/3, c = 28. Chen system is a typical chaos anti-control model, which has a more complicated topological structure than the Lorenz attractor [19]. It has been implemented by circuitry [20] and has widely applicable prospect in secret communication. The nonlinear differential equations that describe the Chen system are 8 > < x_ 1 ¼ aðx2 x1 Þ; x_ 2 ¼ ðc aÞx1 x1 x3 þ cx2 ; ð5Þ > : x_ 3 ¼ x1 x2 bx3 ; which has a chaotic attractor when a = 35, b = 3, c = 28. Lu¨ system is a typical transition system, which connects the Lorenz and Chen attractors and represents the transition from one to the other [21]. The Lu¨ system is described by 8 > < x_ 1 ¼ aðx2 x1 Þ; x_ 2 ¼ x1 x3 þ cx2 ; ð6Þ > : x_ 3 ¼ x1 x2 bx3 ; which has a chaotic attractor when a = 36, b = 3, c = 20.
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ˇ elikovskyˇ [22]: It is noticed that these systems can be classified into three types by the definition of Vaneˇcˇek and C The Lorenz system satisfies the condition a12a21 > 0, the Chen system satisfies a12a21 < 0, and the Lu¨ system satisfies a12a21 = 0, where a12 and a21 are the corresponding elements in the linear part matrix A = (aij)3·3 of the system. Very recently, Lu¨ et al. produced a unified chaotic system [23], which contains the Lorenz and Chen systems as two extremes and the Lu¨ system as a special case. The unified chaotic system is described by 8 > < x_ ¼ ð25h þ 10Þðx2 x1 Þ; y_ ¼ ð28 35hÞx1 x1 x3 þ ð29h 1Þx2 ; ð7Þ > : hþ8 z_ ¼ x1 x2 3 x3 ; where h 2 [0, 1]. Obviously, system (7) is the original Lorenz system for h = 0 while system (7) belongs to the original Chen system for h = 1. When h = 0.8, system (7) is the critical system—the Lu¨ system. In fact, system (7) bridges the gap between the Lorenz system and Chen system. Especially, system (7) is always chaotic for the whole interval h 2 [0, 1]. According to G. Chen et al [24], the above four systems belong to the Lorenz systems family. If we rewrite the above systems into the form x_ ¼ Ax þ UðxÞ;
ð8Þ
where A is the linear part matrix of the corresponding system, and UðxÞ ¼ ð0; x1 x3 ; x1 x2 ÞT . The impulsive control of 8 > < x_ ¼ Ax þ UðxÞ; Dx ¼ Bk x; > : þ xðt0 Þ ¼ xðt0 Þ
ð9Þ
the Lorenz family is then given by t 6¼ tk t ¼ tk ðk ¼ 1; 2; . . .Þ;
ð10Þ
where tk denotes the instant when impulsive control occurs. For convenience, define the following notations: 1 k2 ðAÞ ¼ kmax ðA þ AT Þ; 2
bk ¼ kmax ½ðI þ Bk ÞT ðI þ Bk Þ;
ð11Þ
where I is the n · n identity matrix, and kmax(M) is the maximal eigenvalue of matrix M. Theorem 1 (i) If 2k2(A) = k < 0 (k is a constant) and there exists a constant 0 6 a < k, such that ln bk aðtk tk1 Þ 6 0;
k ¼ 1; 2; . . .
ð12Þ
Then the trivial solution of system (10) is globally exponentially stable. (ii) If 2k2(A) = k P 0 (k is a constant) and there exists a constant a P 1, such that lnðabk Þ þ kðtkþ1 tk Þ 6 0;
k ¼ 1; 2; . . .
ð13Þ
Then a = 1 implies that the trivial solution of system (10) is stable and a > 1 implies that the trivial solution of system (10) is globally asymptotically stable. Proof. Construct a Lyapunov function in the form of V(x) = xTx, its time derivative along system (10) is V_ ðxðtÞÞ ¼ ðAx þ UÞT x þ xT ðAx þ UÞ ¼ xT ðAT þ AÞx þ UT x þ xT U ¼ xT ðA þ AT Þx 6 2k2 ðAÞV ðxðtÞÞ; t 2 ðtk1 ; tk ; k ¼ 1; 2; . . . ;
ð14Þ
which implies that V ðxðtÞÞ 6 V ðxðtþ k1 ÞÞ expð2k2 ðAÞðt tk1 ÞÞ;
t 2 ðtk1 ; tk ; k ¼ 1; 2; . . . .
ð15Þ
On the other hand, it follows from system (10) that T V ðtþ k Þ ¼ ½ðI þ Bk Þxðt k Þ ðI þ Bk Þxðtk Þ 6 bk V ðxðt k ÞÞ;
k ¼ 1; 2; . . . .
The following results come from (15) and (16). For t 2 (t0, t1], V ðxðtÞÞ 6 V ðxðtþ 0 ÞÞ expð2k2 ðAÞðt t 0 ÞÞ;
ð16Þ
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which leads to V ðxðt1 ÞÞ 6 V ðxðtþ 0 ÞÞ expð2k2 ðAÞðt 1 t 0 ÞÞ; and þ V ðxðtþ 1 ÞÞ 6 b1 V ðxðt1 ÞÞ 6 b1 V ðxðt 0 ÞÞ expð2k2 ðAÞðt 1 t 0 ÞÞ.
Similarly, for t 2 (t1, t2], þ V ðxðtÞÞ 6 V ðxðtþ 1 ÞÞ expð2k2 ðAÞðt t 1 ÞÞ 6 b1 V ðxðt 0 ÞÞ expð2k2 ðAÞðt t 0 ÞÞ.
In general, for t 2 (tk, tk+1], V ðxðtÞÞ 6 V ðxðtþ 0 ÞÞb1 b2 bk expð2k2 ðAÞðt t 0 ÞÞ.
ð17Þ
(i) When 2k2(A) = k < 0, it follows from (12) and (17) that for t 2 (tk, tk+1], þ V ðxðtÞÞ 6 V ðxðtþ 0 ÞÞb1 b2 bk expð2k2 ðAÞðt t 0 ÞÞ ¼ V ðxðt 0 ÞÞb1 b2 bk expðkðt t 0 ÞÞ
¼ V ðxðtþ 0 ÞÞb1 b2 bk expðaðt t 0 ÞÞ expððk þ aÞðt t0 ÞÞ 6 V ðxðtþ 0 ÞÞb1 b2 bk expðaðtk t0 ÞÞ expððk þ aÞðt t 0 ÞÞ ¼ V ðxðtþ 0 ÞÞb1 expðaðt 1 t 0 ÞÞb2 expðaðt 2 t 1 ÞÞ bk expðaðt k t k1 ÞÞ expððk þ aÞðt t 0 ÞÞ 6 V ðxðtþ 0 ÞÞ expððk þ aÞðt t0 ÞÞ;
ð18Þ
namely, V ðxðtÞÞ 6 V ðxðtþ 0 ÞÞ expððk þ aÞðt t 0 ÞÞ, t P t0, which implies that the trivial solution of system (10) is globally exponentially stable. (ii) When 2k2(A) = k P 0, it follows from (13) and (17) that, for t 2 (tk, tk+1], þ V ðxðtÞÞ 6 V ðxðtþ 0 ÞÞb1 b2 bk expð2k2 ðAÞðt t 0 ÞÞ 6 V ðxðt 0 ÞÞb1 b2 bk expðkðtkþ1 t 0 ÞÞ
6 V ðxðtþ 0 ÞÞb1 expðkðt 2 t 1 ÞÞb2 expðkðt 3 t 2 ÞÞ bk expðkðt kþ1 tk ÞÞ expðkðt 1 t 0 ÞÞ 6 V ðxðtþ 0 ÞÞ
1 expðkðt1 t0 ÞÞ; ak
ð19Þ
which implies that the conclusion (ii) of Theorem 1 holds. This completes the proof.
h
Remark 1. Theorem 1 gives sufficient conditions for the global exponential stability and global asymptotical stability for controlling the systems to the origin. The results are new and comprehensive for the impulsive control of the Lorenz systems family. Also, the conditions imply that the impulsive intervals may not be equidistant. Moreover, by condition (12), we do not require that bk 6 1, as required in many previous works. In practice, for convenience, the gain matrices Bk are always selected as a constant matrix and the impulsive intervals sk = tk tk1 (k = 1, 2, . . .) are set to be a positive constant. Thus we have the following corollary. Corollary 1. Assume sk = s > 0 and matrices Bk = B (k = 1,2,. . .) (i) If 2k2(A) = k < 0 (k is a constant) and there exists a constant 0 6 a < k, such that ln b as 6 0, then the trivial solution of system (10) is globally exponentially stable. (ii) If 2k2(A) = k P 0 (k is a constant) and there exists a constant a P 1, such that ln(ab) + ks 6 0, then the conclusion (ii) of Theorem 1 holds. 4. Impulsive synchronization of the Lorenz systems family In this section, we will study the impulsive synchronization of two chaotic systems. Let system (8) be the drive system, and the response system is modeled by the following impulsive equation: 8 > < y_ ¼ Ay þ UðyÞ; t 6¼ tk ; Dy ¼ Bk e; t ¼ tk ; ð20Þ > : þ yðt0 Þ ¼ yðt0 Þ ðk ¼ 1; 2; . . .Þ;
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where y = (y1, y2, y3)T. Let e = (e1, e2, e3)T = (y1 x1, y2 x2, y3 x3)T be the synchronization error. If we define 0
1 0 B C Wðx; yÞ ¼ UðyÞ UðxÞ ¼ @ y 1 y 3 þ x1 x3 A; y 1 y 2 x1 x2 then the error system of the impulsive synchronization is given by 8 t 6¼ tk ; > < e_ ¼ Ae þ Wðx; yÞ; De ¼ Bk e; t ¼ tk ; > : þ eðt0 Þ ¼ yðt0 Þ xðt0 Þ ðk ¼ 1; 2; . . .Þ.
ð21Þ
Note that there exists a positive constant M for the chaotic system (8) that jx(t)j 6 M for all t. For convenience, define the following notations: 1 k2 ðAÞ ¼ kmax ðA þ AT Þ; 2
bk ¼ kmax ½ðI þ Bk ÞT ðI þ Bk Þ.
ð22Þ
Similar to the stabilization of the Lu¨ system, we have the following results. Theorem 2 (i) If 2k2(A) + M = k < 0 (k is a constant) and there exists a constant 0 6 a < k, such that ln bk a(tk tk1) 6 0, k = 1, 2, . . .. Then the trivial solution of system (21) is globally exponentially stable, that is, system (20) is globally exponentially synchronous with system (8). (ii) If 2k2(A) + M = k P 0 (k is a constant) and there exists a constant a P 1, such that ln(abk) + k(tk+1 tk) 6 0, k = 1, 2, . . .. Then system (20) is globally asymptotically synchronous with system (8). Proof. Construct a Lyapunov function in the form of V(e) = eTe, its time derivative along system (21) is V_ ðeðtÞÞ ¼ ðAe þ WðeÞÞT e þ eT ðAe þ WðeÞÞ ¼ eT ðA þ AT Þe þ ðx2 e1 e3 x3 e1 e2 Þ 6 2k2 ðAÞeT e þ Mðje1 ke2 j þ je1 ke3 j 6 ð2k2 ðAÞ þ MÞV ðeðtÞÞ;
t 2 ðtk1 ; tk ; k ¼ 1; 2; . . .
The remaining reasoning is similar to that of Theorem 1, so details are omitted. This completes the proof. h The following results easily follow from Theorem 2. Corollary 2. Assume sk = s > 0 and matrices Bk = B (k = 1, 2, . . .). (i) If 2k2(A) + M = k < 0 and there exists a constant 0 6 a < k, such that ln b as 6 0, then system (20) is globally exponentially synchronous with system (8). (ii) If 2k2(A) + M = k P 0 (k is a constant) and there exists a constant a P 1, such that ln(a b) + ks 6 0, then system (20) is globally asymptotically synchronous with system (8).
5. An illustrative example Take the Lu¨ system (6) as an illustrative example, where a = 36, b = 3, c = 20. Firstly, we control system(6) to its equilibrium point (0, 0, 0)T, thus 0
72 B A þ AT ¼ @ 36 0
36 40 0
1 0 C 0 A. 6
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Its eigenvalues are 82.5733, 6.0000 and 50.5733. Then 2k2(A) = k = 50.5733 > 0. If we choose B = diag (b1, b2, b3) = (0.58, 0.68, 0.78), then b ¼ maxfð1 þ b1 Þ2 ; ð1 þ b2 Þ2 ; ð1 þ b3 Þ2 g ¼ 0:1764. If s < lnkab, from Theorem 1, we have that the equilibrium point of the impulsively controlled system is asymptotically stable. Take a = 1.01, thus if s < 0.0341, system (6) will be stabilized at the origin. Let s = 0.025 and s = 0.01 respectively, the simulation results are shown in Figs. 1 and 2, where the initial conditions are (3.0, 4.0, 5.0)T. From the figures we have that the state variables quickly tend to the origin under impulsive control. What is more, the settling time gets shorter as the impulse intervals get smaller. If the impulse intervals are too large, as proved previously, the impulsively controlled system cannot be stabilized, as shown in Fig. 3 with s = 0.2. In the synchronization of two Lu¨ systems, the initial conditions for the drive and response systems are chosen as (3, 4, 5)T and (7, 8, 9)T. We can get from the simulation that the approximate bounds M of system (6) is 40. Thus k = 2k2(A0) + M = 90.5733. Choose B = diag(b1, b2, b3) = (0.58, 0.68, 0.78), then b = 0.1764. Take a = 1.01, thus if s < lnkab ¼ 0:019, system (20) is globally asymptotically synchronous with system (6). Fig. 4 shows the results when s = 0.025. It is obvious that with this impulsive control, two chaotic systems synchronize very fast.
Fig. 1. Impulsively control Lu¨ system to the origin with s = 0.025.
Fig. 2. Impulsively control Lu¨ system to the origin with s = 0.01.
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Fig. 3. The impulsively controlled Lu¨ system cannot be stabilized with s = 0.2.
Fig. 4. Synchronization errors of two Lu¨ systems with s = 0.025.
6. Conclusions In this paper, we investigate the impulsive control and synchronization of the Lorenz systems family. Some new and more comprehensive criteria for the global exponential stability and asymptotical stability of impulsively controlled Lorenz systems family are obtained with varying impulsive intervals. In particular, some simple and easily verified criteria are established with equivalent impulsive intervals. An illustrative example, along with numerical simulations is presented to prove the effectiveness and feasibility of the developed method.
References [1] Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;64:1196–9. [2] Pecora L, Carroll T. Synchronization in chaotic system. Phys Rev Lett 1990;64:821–4. [3] Wu X, Lu J. Parameter identification and backstepping control of uncertain Lu¨ system. Chaos, Solitons & Fractals 2003;18:721–9.
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[4] Chen G, Dong X. From chaos to order: Perspectives, methodologies and applications. Singapore: World Scientific; 1998. [5] Hu G, Xiao J, Zheng Z. Chaos control. China: Shanghai Scientific and Technological Education Publishing House; 2000 [in Chinese]. [6] Wu X, Lu J. Adaptive control of uncertain Lu system. Chaos, Solitons & Fractals 2004;22:375–81. [7] Lu¨ J, Zhou T, Zhang S. Chaos synchronization between linearly coupled chaotic system. Chaos, Solitons & Fractals 2002;14:529–41. [8] Guan Z. Decentralized stabilization for impulsive large scale systems with delays. Dynam Continuous Discrete Impuls Syst 1999;6:369–79. [9] Lu J, Wu X, Lii J. Synchronization of a unified system and the application in secure communication. Phys Lett A 2002;305:365–70. [10] Lakshmikantham V, Bainov D, Simeonov P. Theory of impulsive differential equations. Singapore: World Scientific; 1989. [11] Yang T, Yang L, Yang C. Impulsive control of Lorenz system. Physica D 1997;110:18–24. [12] Yang T. Impulsive control. IEEE Trans Automat Control 1999;44:1081–4. [13] Li Z, Wen C, Soh Y. Analysis and design of impulsive control systems. IEEE Trans Automat Control 2001;46:894–7. [14] Sun J, Zhang Y. Impulsive control of Ro¨ssler systems. Phys Lett A 2003;306:306–12. [15] Chen S, Yang Q, Wang C. Impulsive control and synchronization of unified chaotic system. Chaos, Solitons & Fractals 2004;20:751–8. [16] Lorenz EN. Deterministic non-periodic flows. J Atmospheric Sci 1963;20:130–41. [17] Richter H. Controlling the Lorenz system: combining global and local schemes. Chaos, Solitons & Fractals 2001;12:2375–80. [18] Stewart I. The Lorenz attractor exists. Nature 2000;406:948–9. [19] Chen G, Ueta T. Yet another chaotic attractor. Int J Bifurcat Chaos 1999;9:1465–6. [20] Zhong G, Tang K. Circuitry implementation and synchronization of ChenÕs attractor. Int J Bifurcat Chaos 2000;10:549–70. [21] Lu¨ J, Chen G, Zhang S. Dynamical analysis of a new chaotic attractor. Int J Bifurcat Chaos 2002;12:1001–15. ˇ elikovskyˇ S. Control systems: From linear analysis to synthesis of chaos. London: Prentice-Hall; 1996. [22] Vaneˇcˇek A, C ˇ elikovskyˇ S. Bridge the gap between the Lorenz system and the Chen system. Int J Bifurcat Chaos [23] Lu¨ J, Chen G, Cheng D, C 2002;12:2917–26. [24] Chen G, Lu J. Dynamics of the Lorenz systems family: Analysis, control and synchronization. Beijing, China: Science Press; 2003.
Impulsive control and synchronization of the Lorenz systems family q Xiaoqun Wu b
a,b,*
, Jun-an Lu a, Chi K. Tse b, Jinjun Wang c, Jie Liu
a
a School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong, China c College of Mechanical and Energy Engineering, Zhejiang University, Hangzhou 310027, China
Accepted 5 October 2005
Abstract In this paper, impulsive control and synchronization for the newly presented Lorenz systems family are systematically investigated. Some new and more comprehensive criteria for global exponential stability and asymptotical stability of impulsively controlled Lorenz systems family are established with varying impulsive intervals. In particular, several simple and easily verified criteria are derived with equivalent impulsive intervals. An illustrative example is also provided to show the effectiveness and feasibility of the impulsive control method. 2005 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, there has been considerable interest in the control of chaos in nonlinear dynamical systems. For the past years, many different techniques have been proposed to control chaos, including OGY method [1], P–C technique [2], backstepping approach [3], and so on [4–9]. Recently, impulsive control has been widely used to stabilize and synchronize chaotic systems [10–15]. Its necessity and importance lie in that, in some cases, the system cannot be controlled by continuous control. For example, a government cannot change savings rates of its central bank every day. Additionally, impulsive control may give a more efficient method to deal with systems that cannot endure continuous disturbance. Furthermore, impulsive method can also greatly reduce the control cost. In this letter, we systematically investigate the impulsive control and synchronization for the newly presented Lorenz systems family [24]. To stabilize the Lorenz systems family to the origin and synchronize two chaotic systems, we establish some new and more comprehensive criteria for global exponential stability and asymptotical stability with varying impulsive intervals. Particularly, several simple and easily verified criteria are also derived with equivalent impulsive
q
Supported by National Key Basic Research Development 973 Program of China (Grant No. 2003CB415200) and National Natural Science Foundation of China (Grant No. 60574045). * Corresponding author. Address: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China. E-mail address: [email protected] (X. Wu). 0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.10.017
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X. Wu et al. / Chaos, Solitons and Fractals 31 (2007) 631–638
intervals. An illustrative example, along with numerical simulations is also provided to show the effectiveness and feasibility of the developed method.
2. Impulsive control of nonlinear systems Let the general chaotic system be in the form of x_ ¼ f ðt; xÞ;
ð1Þ n
n
n
where t 2 J = [t0, +1] (t0 P 0), x 2 R is the state variable, and f: J · R ! R is a continuous vector-value function. An impulsive control law of system (1) is given by a sequence, {tk, uk(x(tk))}, which has the effect of suddenly changing the state of the system at the instants tk, where t1 < t2 < < tk < , limk!1tk = 1 and t1 > t0; that is Dxjtk ¼ xðtþ k Þ xðt k Þ ¼ uk ðxðt k ÞÞ;
ð2Þ
xðtÞ and xðtk Þ ¼ limt!tk xðtÞ. In general, for simplicity, it is assumed that xðt where xðtþ k Þ ¼ limt!tþ k Þ ¼ xðtk Þ. Furtherk more, uk(x(tk)) can be chosen as Bkx(tk) with Bk being n · n matrices. Accordingly, the impulsively controlled system can be expressed as follows: 8 > < x_ ¼ f ðt; xÞ; t 6¼ tk ; Dx ¼ Bk x; t ¼ tk ; ð3Þ > : þ xðt0 Þ ¼ x0 ðk ¼ 1; 2; . . .Þ; which is also called an impulsive differential system. The objective is to find some (sufficient) conditions on the constant control gains, Bk, and the impulsive intervals sk = tk tk1 < 1 (k = 1, 2, . . .), such that the impulsively controlled system (3) is stable; namely, the nonlinear system (1) is impulsively stabilizable.
3. Control the Lorenz systems family to the origin by impulsive method The Lorenz system is known to be a simplified model of several physical systems. Originally, it was derived from a model of the earthÕs atmospheric convection flow heated from below and cooled from above [16]. Furthermore, it has been reported that Lorenz equations may describe such different systems as laser devices, disk dynamos and several problems related to convection [17]. Recently, the Lorenz attractor has just been mathematically confirmed to exist [18]. The Lorenz system is described by 8 > < x_ 1 ¼ aðx2 x1 Þ; x_ 2 ¼ cx1 x1 x3 x2 ; ð4Þ > : x_ 3 ¼ x1 x2 bx3 ; which has a chaotic attractor when a = 10, b = 8/3, c = 28. Chen system is a typical chaos anti-control model, which has a more complicated topological structure than the Lorenz attractor [19]. It has been implemented by circuitry [20] and has widely applicable prospect in secret communication. The nonlinear differential equations that describe the Chen system are 8 > < x_ 1 ¼ aðx2 x1 Þ; x_ 2 ¼ ðc aÞx1 x1 x3 þ cx2 ; ð5Þ > : x_ 3 ¼ x1 x2 bx3 ; which has a chaotic attractor when a = 35, b = 3, c = 28. Lu¨ system is a typical transition system, which connects the Lorenz and Chen attractors and represents the transition from one to the other [21]. The Lu¨ system is described by 8 > < x_ 1 ¼ aðx2 x1 Þ; x_ 2 ¼ x1 x3 þ cx2 ; ð6Þ > : x_ 3 ¼ x1 x2 bx3 ; which has a chaotic attractor when a = 36, b = 3, c = 20.
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ˇ elikovskyˇ [22]: It is noticed that these systems can be classified into three types by the definition of Vaneˇcˇek and C The Lorenz system satisfies the condition a12a21 > 0, the Chen system satisfies a12a21 < 0, and the Lu¨ system satisfies a12a21 = 0, where a12 and a21 are the corresponding elements in the linear part matrix A = (aij)3·3 of the system. Very recently, Lu¨ et al. produced a unified chaotic system [23], which contains the Lorenz and Chen systems as two extremes and the Lu¨ system as a special case. The unified chaotic system is described by 8 > < x_ ¼ ð25h þ 10Þðx2 x1 Þ; y_ ¼ ð28 35hÞx1 x1 x3 þ ð29h 1Þx2 ; ð7Þ > : hþ8 z_ ¼ x1 x2 3 x3 ; where h 2 [0, 1]. Obviously, system (7) is the original Lorenz system for h = 0 while system (7) belongs to the original Chen system for h = 1. When h = 0.8, system (7) is the critical system—the Lu¨ system. In fact, system (7) bridges the gap between the Lorenz system and Chen system. Especially, system (7) is always chaotic for the whole interval h 2 [0, 1]. According to G. Chen et al [24], the above four systems belong to the Lorenz systems family. If we rewrite the above systems into the form x_ ¼ Ax þ UðxÞ;
ð8Þ
where A is the linear part matrix of the corresponding system, and UðxÞ ¼ ð0; x1 x3 ; x1 x2 ÞT . The impulsive control of 8 > < x_ ¼ Ax þ UðxÞ; Dx ¼ Bk x; > : þ xðt0 Þ ¼ xðt0 Þ
ð9Þ
the Lorenz family is then given by t 6¼ tk t ¼ tk ðk ¼ 1; 2; . . .Þ;
ð10Þ
where tk denotes the instant when impulsive control occurs. For convenience, define the following notations: 1 k2 ðAÞ ¼ kmax ðA þ AT Þ; 2
bk ¼ kmax ½ðI þ Bk ÞT ðI þ Bk Þ;
ð11Þ
where I is the n · n identity matrix, and kmax(M) is the maximal eigenvalue of matrix M. Theorem 1 (i) If 2k2(A) = k < 0 (k is a constant) and there exists a constant 0 6 a < k, such that ln bk aðtk tk1 Þ 6 0;
k ¼ 1; 2; . . .
ð12Þ
Then the trivial solution of system (10) is globally exponentially stable. (ii) If 2k2(A) = k P 0 (k is a constant) and there exists a constant a P 1, such that lnðabk Þ þ kðtkþ1 tk Þ 6 0;
k ¼ 1; 2; . . .
ð13Þ
Then a = 1 implies that the trivial solution of system (10) is stable and a > 1 implies that the trivial solution of system (10) is globally asymptotically stable. Proof. Construct a Lyapunov function in the form of V(x) = xTx, its time derivative along system (10) is V_ ðxðtÞÞ ¼ ðAx þ UÞT x þ xT ðAx þ UÞ ¼ xT ðAT þ AÞx þ UT x þ xT U ¼ xT ðA þ AT Þx 6 2k2 ðAÞV ðxðtÞÞ; t 2 ðtk1 ; tk ; k ¼ 1; 2; . . . ;
ð14Þ
which implies that V ðxðtÞÞ 6 V ðxðtþ k1 ÞÞ expð2k2 ðAÞðt tk1 ÞÞ;
t 2 ðtk1 ; tk ; k ¼ 1; 2; . . . .
ð15Þ
On the other hand, it follows from system (10) that T V ðtþ k Þ ¼ ½ðI þ Bk Þxðt k Þ ðI þ Bk Þxðtk Þ 6 bk V ðxðt k ÞÞ;
k ¼ 1; 2; . . . .
The following results come from (15) and (16). For t 2 (t0, t1], V ðxðtÞÞ 6 V ðxðtþ 0 ÞÞ expð2k2 ðAÞðt t 0 ÞÞ;
ð16Þ
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which leads to V ðxðt1 ÞÞ 6 V ðxðtþ 0 ÞÞ expð2k2 ðAÞðt 1 t 0 ÞÞ; and þ V ðxðtþ 1 ÞÞ 6 b1 V ðxðt1 ÞÞ 6 b1 V ðxðt 0 ÞÞ expð2k2 ðAÞðt 1 t 0 ÞÞ.
Similarly, for t 2 (t1, t2], þ V ðxðtÞÞ 6 V ðxðtþ 1 ÞÞ expð2k2 ðAÞðt t 1 ÞÞ 6 b1 V ðxðt 0 ÞÞ expð2k2 ðAÞðt t 0 ÞÞ.
In general, for t 2 (tk, tk+1], V ðxðtÞÞ 6 V ðxðtþ 0 ÞÞb1 b2 bk expð2k2 ðAÞðt t 0 ÞÞ.
ð17Þ
(i) When 2k2(A) = k < 0, it follows from (12) and (17) that for t 2 (tk, tk+1], þ V ðxðtÞÞ 6 V ðxðtþ 0 ÞÞb1 b2 bk expð2k2 ðAÞðt t 0 ÞÞ ¼ V ðxðt 0 ÞÞb1 b2 bk expðkðt t 0 ÞÞ
¼ V ðxðtþ 0 ÞÞb1 b2 bk expðaðt t 0 ÞÞ expððk þ aÞðt t0 ÞÞ 6 V ðxðtþ 0 ÞÞb1 b2 bk expðaðtk t0 ÞÞ expððk þ aÞðt t 0 ÞÞ ¼ V ðxðtþ 0 ÞÞb1 expðaðt 1 t 0 ÞÞb2 expðaðt 2 t 1 ÞÞ bk expðaðt k t k1 ÞÞ expððk þ aÞðt t 0 ÞÞ 6 V ðxðtþ 0 ÞÞ expððk þ aÞðt t0 ÞÞ;
ð18Þ
namely, V ðxðtÞÞ 6 V ðxðtþ 0 ÞÞ expððk þ aÞðt t 0 ÞÞ, t P t0, which implies that the trivial solution of system (10) is globally exponentially stable. (ii) When 2k2(A) = k P 0, it follows from (13) and (17) that, for t 2 (tk, tk+1], þ V ðxðtÞÞ 6 V ðxðtþ 0 ÞÞb1 b2 bk expð2k2 ðAÞðt t 0 ÞÞ 6 V ðxðt 0 ÞÞb1 b2 bk expðkðtkþ1 t 0 ÞÞ
6 V ðxðtþ 0 ÞÞb1 expðkðt 2 t 1 ÞÞb2 expðkðt 3 t 2 ÞÞ bk expðkðt kþ1 tk ÞÞ expðkðt 1 t 0 ÞÞ 6 V ðxðtþ 0 ÞÞ
1 expðkðt1 t0 ÞÞ; ak
ð19Þ
which implies that the conclusion (ii) of Theorem 1 holds. This completes the proof.
h
Remark 1. Theorem 1 gives sufficient conditions for the global exponential stability and global asymptotical stability for controlling the systems to the origin. The results are new and comprehensive for the impulsive control of the Lorenz systems family. Also, the conditions imply that the impulsive intervals may not be equidistant. Moreover, by condition (12), we do not require that bk 6 1, as required in many previous works. In practice, for convenience, the gain matrices Bk are always selected as a constant matrix and the impulsive intervals sk = tk tk1 (k = 1, 2, . . .) are set to be a positive constant. Thus we have the following corollary. Corollary 1. Assume sk = s > 0 and matrices Bk = B (k = 1,2,. . .) (i) If 2k2(A) = k < 0 (k is a constant) and there exists a constant 0 6 a < k, such that ln b as 6 0, then the trivial solution of system (10) is globally exponentially stable. (ii) If 2k2(A) = k P 0 (k is a constant) and there exists a constant a P 1, such that ln(ab) + ks 6 0, then the conclusion (ii) of Theorem 1 holds. 4. Impulsive synchronization of the Lorenz systems family In this section, we will study the impulsive synchronization of two chaotic systems. Let system (8) be the drive system, and the response system is modeled by the following impulsive equation: 8 > < y_ ¼ Ay þ UðyÞ; t 6¼ tk ; Dy ¼ Bk e; t ¼ tk ; ð20Þ > : þ yðt0 Þ ¼ yðt0 Þ ðk ¼ 1; 2; . . .Þ;
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where y = (y1, y2, y3)T. Let e = (e1, e2, e3)T = (y1 x1, y2 x2, y3 x3)T be the synchronization error. If we define 0
1 0 B C Wðx; yÞ ¼ UðyÞ UðxÞ ¼ @ y 1 y 3 þ x1 x3 A; y 1 y 2 x1 x2 then the error system of the impulsive synchronization is given by 8 t 6¼ tk ; > < e_ ¼ Ae þ Wðx; yÞ; De ¼ Bk e; t ¼ tk ; > : þ eðt0 Þ ¼ yðt0 Þ xðt0 Þ ðk ¼ 1; 2; . . .Þ.
ð21Þ
Note that there exists a positive constant M for the chaotic system (8) that jx(t)j 6 M for all t. For convenience, define the following notations: 1 k2 ðAÞ ¼ kmax ðA þ AT Þ; 2
bk ¼ kmax ½ðI þ Bk ÞT ðI þ Bk Þ.
ð22Þ
Similar to the stabilization of the Lu¨ system, we have the following results. Theorem 2 (i) If 2k2(A) + M = k < 0 (k is a constant) and there exists a constant 0 6 a < k, such that ln bk a(tk tk1) 6 0, k = 1, 2, . . .. Then the trivial solution of system (21) is globally exponentially stable, that is, system (20) is globally exponentially synchronous with system (8). (ii) If 2k2(A) + M = k P 0 (k is a constant) and there exists a constant a P 1, such that ln(abk) + k(tk+1 tk) 6 0, k = 1, 2, . . .. Then system (20) is globally asymptotically synchronous with system (8). Proof. Construct a Lyapunov function in the form of V(e) = eTe, its time derivative along system (21) is V_ ðeðtÞÞ ¼ ðAe þ WðeÞÞT e þ eT ðAe þ WðeÞÞ ¼ eT ðA þ AT Þe þ ðx2 e1 e3 x3 e1 e2 Þ 6 2k2 ðAÞeT e þ Mðje1 ke2 j þ je1 ke3 j 6 ð2k2 ðAÞ þ MÞV ðeðtÞÞ;
t 2 ðtk1 ; tk ; k ¼ 1; 2; . . .
The remaining reasoning is similar to that of Theorem 1, so details are omitted. This completes the proof. h The following results easily follow from Theorem 2. Corollary 2. Assume sk = s > 0 and matrices Bk = B (k = 1, 2, . . .). (i) If 2k2(A) + M = k < 0 and there exists a constant 0 6 a < k, such that ln b as 6 0, then system (20) is globally exponentially synchronous with system (8). (ii) If 2k2(A) + M = k P 0 (k is a constant) and there exists a constant a P 1, such that ln(a b) + ks 6 0, then system (20) is globally asymptotically synchronous with system (8).
5. An illustrative example Take the Lu¨ system (6) as an illustrative example, where a = 36, b = 3, c = 20. Firstly, we control system(6) to its equilibrium point (0, 0, 0)T, thus 0
72 B A þ AT ¼ @ 36 0
36 40 0
1 0 C 0 A. 6
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Its eigenvalues are 82.5733, 6.0000 and 50.5733. Then 2k2(A) = k = 50.5733 > 0. If we choose B = diag (b1, b2, b3) = (0.58, 0.68, 0.78), then b ¼ maxfð1 þ b1 Þ2 ; ð1 þ b2 Þ2 ; ð1 þ b3 Þ2 g ¼ 0:1764. If s < lnkab, from Theorem 1, we have that the equilibrium point of the impulsively controlled system is asymptotically stable. Take a = 1.01, thus if s < 0.0341, system (6) will be stabilized at the origin. Let s = 0.025 and s = 0.01 respectively, the simulation results are shown in Figs. 1 and 2, where the initial conditions are (3.0, 4.0, 5.0)T. From the figures we have that the state variables quickly tend to the origin under impulsive control. What is more, the settling time gets shorter as the impulse intervals get smaller. If the impulse intervals are too large, as proved previously, the impulsively controlled system cannot be stabilized, as shown in Fig. 3 with s = 0.2. In the synchronization of two Lu¨ systems, the initial conditions for the drive and response systems are chosen as (3, 4, 5)T and (7, 8, 9)T. We can get from the simulation that the approximate bounds M of system (6) is 40. Thus k = 2k2(A0) + M = 90.5733. Choose B = diag(b1, b2, b3) = (0.58, 0.68, 0.78), then b = 0.1764. Take a = 1.01, thus if s < lnkab ¼ 0:019, system (20) is globally asymptotically synchronous with system (6). Fig. 4 shows the results when s = 0.025. It is obvious that with this impulsive control, two chaotic systems synchronize very fast.
Fig. 1. Impulsively control Lu¨ system to the origin with s = 0.025.
Fig. 2. Impulsively control Lu¨ system to the origin with s = 0.01.
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Fig. 3. The impulsively controlled Lu¨ system cannot be stabilized with s = 0.2.
Fig. 4. Synchronization errors of two Lu¨ systems with s = 0.025.
6. Conclusions In this paper, we investigate the impulsive control and synchronization of the Lorenz systems family. Some new and more comprehensive criteria for the global exponential stability and asymptotical stability of impulsively controlled Lorenz systems family are obtained with varying impulsive intervals. In particular, some simple and easily verified criteria are established with equivalent impulsive intervals. An illustrative example, along with numerical simulations is presented to prove the effectiveness and feasibility of the developed method.
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