- Email: [email protected]

Synchronization of chaotic systems via nonlinear control Lilian Huang ∗ , Rupeng Feng, Mao Wang Inertial Navigation Test Equipment Research Center, Harbin Institute of Technology, Harbin 150001, PR China Received 15 July 2003; received in revised form 31 October 2003; accepted 6 November 2003 Communicated by C.R. Doering

Abstract The Letter introduces nonlinear control method and based on Lyapunov stability theory to design controller to synchronize two identical chaotic systems or two different chaotic systems. The technique is applied to two identical Lü systems and two different chaotic systems. Numerical simulations are shown for demonstration. 2003 Elsevier B.V. All rights reserved. PACS: 05.45.+b Keywords: Nonlinear control; Lyapunov stability theory; Chaotic synchronization

1. Introduction Since Pecora and Carroll introduced a method [1] to synchronize two identical chaotic systems with different initial conditions, chaos synchronization, as a very important topic in the nonlinear science, has been developed extensively in the last few years [2–6]. Many scientists in various fields have been attracted to investigate chaos synchronization due to its applications in a variety of fields including secure communications, optics, chemical and biological systems, neural networks and so on. So various synchronization schemes, such as variable structure control [7], parameters adaptive control [8], observer based control [9, 10], active control [11–17], and so on, have been successfully applied to the chaos synchronization. * Corresponding author.

E-mail address: [email protected] (L. Huang). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.11.027

Most of the methods mentioned above synchronize two identical chaotic systems. However, the method of the synchronization of two different chaotic systems is far from being straightforward. There is little work about this challenging problem because it consists of different structures and parameter mismatch of the two chaotic systems. In fact, in systems such as laser array, biological systems to cognitive processes, it is hardly the case that every component can be assumed to be identical. Consequently, in these years, more and more applications of chaos synchronization in secure communications make it much more important to synchronize two different chaotic systems. In this Letter, we use nonlinear control method to synchronize two identical or different chaotic systems, and determine the controller based on Lyapunov stability theory. Then we simulate the method by using two identical Lü systems and two different chaotic systems.

272

L. Huang et al. / Physics Letters A 320 (2004) 271–275

where x = [x1 , x2 , x3 ]T is the state variable and a, b, c are three known positive constant parameters. Eq. (4) is a drive system and the controlled response system is given by

2. Design of controller via nonlinear control method Consider the following system described by x˙ = Ax + Bf (x),

(1)

where x ∈ R n is the state vector, A ∈ R n×n , B ∈ R n are metrices and vectors of system parameters, and f : R n → R n is a nonlinear function. Eq. (1) is considered as a drive system. Inject an additive controller U ∈ R n , then the controlled response system is given by y˙ = A1 y + B1 g(y) + U,

(2)

∈ Rn

y˙1 = a(y2 − y1 ) + u1 , y˙2 = −y1y3 + cy2 + u2 , y˙3 = y1 y2 − by3 + u3 ,

where U = [u1 , u2 , u3 ]T is the controller to be designed. We subtract (4) from (5) and get the error equations as follows: e˙1 = a(e2 − e1 ) + u1 ,

denotes the state vector of the response where y system, A1 ∈ R n×n , B1 ∈ R n are metrices and vectors of this controlled response system parameters, and g : R n → R n is a nonlinear function. A = A1 , B = B1 , for two identical chaotic systems, A = A1 , B = B1 , for two different chaotic systems. The synchronization problem is to design a controller U which synchronizes the states of both the drive and response systems. We subtract (1) from (2) and get

e˙2 = x1 x3 − y1 y3 + ce2 + u2 ,

e˙ = A1 y + B1 g(y) − Ax − Bf (x) + U,

e˙1 = a(e2 − e1 ) + u1 ,

(3)

where e = y − x. The aim of synchronization is to make lim e(t) = 0.

t →∞

Then let Lyapunov error function be V (e) = where V (e) is a positive definite function. Assuming that the parameters of the drive and response systems are known and the states of both systems are measurable, we may achieve the synchronization by selecting the controller U to make the first derivative of V (e), i.e., V˙ (e) < 0. Then the states of response system and drive system are synchronized asymptotically globally. 1 T 2 e e,

(5)

e˙3 = −x1 x2 + y1 y2 − be3 + u3 ,

(6)

where e1 = y1 − x1 , e2 = y2 − x2 , e3 = y3 − x3 . In order to determine the controller, first let u2 = u2a + u2b ,

where u2a = −x1 x3 + y1 y3 ,

u3 = u3a + u3b ,

where u3a = x1 x2 − y1 y2 .

Then we rewrite (6) in the following form

e˙2 = ce2 + u2b , e˙3 = −be3 + u3b .

(7)

Based on Lyapunov stability theory, when controller satisfies the assumption with V (e) = 12 eT e, a positive definite function and the first derivative of V (e), V˙ (e) < 0, the synchronization of two identical Lü systems from different initial conditions is achieved. Take a Lyapunov function for Eq. 7 into consideration 1 V (e) = eT e, 2 we get the first derivative of V (e): V˙ (e) = ae1 e2 − ae12 + e1 u1 + ce22 + e2 u2b − be32 + e3 u3b .

3. Synchronization of two identical Lü systems

Therefore, if we choose U as follows,

The Lü systems is considered in the form of x˙1 = a(x2 − x1 ),

u1 = 0,

x˙2 = −x1 x3 + cx2 ,

u2b = −ae1 − 2ce2 ,

x˙3 = x1 x2 − bx3 ,

(8)

(4)

u3b = 0,

(9)

L. Huang et al. / Physics Letters A 320 (2004) 271–275

then V˙ (e) = −ae12 − ce22 − be32.

(10)

V˙ (e) < 0 is satisfied. Since V˙ (e) is a negative-definite function, the error states lim e(t) = 0.

t →∞

Therefore, the states of controlled response system and drive system are globally synchronized asymptotically.

273

considered as drive system. The other is chaotic Genesio system considered as controlled response system. Our aim is to design a controller and make the controlled response system trace the drive system and become the same finally. Lur’e-like system, as drive system, is considered as follows: x˙1 = x2 , x˙2 = x3 , x˙3 = a1 x1 + a2 x2 + a3 x3 + 12f (x1 ),

3.1. Simulation results In this section, numerical simulations are given to verify the method proposed. In these numerical simulations, the fourth-order Runge–Kutta method is used to solve Lü system, with time step size 0.001. The parameters are selected as follows a = 36, c = 20, b = 3, with initial values x1 (0) = 0.5, x2 (0) = 1, x3 (0) = 1, y1 (0) = 5, y2 (0) = 2, y3 (0) = 2. The simulation results are illustrated in Fig. 1. Fig. 1(a) shows e1 = y1 − x1 , Fig. 1(b) shows e2 = y2 − x2 , and Fig. 1(c) shows e3 = y3 − x3 . From the figures, we can see that the synchronization error will converge to zero finally and two identical Lü systems from different initial values are indeed achieving chaos synchronization.

4. Synchronization of two different chaotic systems We use the proposed method to synchronize two different chaotic systems. One is Lur’e-like system

where

f (x1 ) = kx1 , sign(x1 ),

(11)

if |x1 | < k1 , otherwise,

and x = [x1 , x2 , x3 ]T is the state variable and a1 , a2 , a3 are parameters of the system and k = 1.5 the system shows a chaotic behavior. The chaotic Genesio system is described by the set of ordinary 3-order differential equations y˙1 = y2 + u1 , y˙2 = y3 + u2 , y˙3 = −b1y1 − b2 y2 − b3 y3 + y12 + u3 ,

(12)

where y = [y1 , y2 , y3 ]T is the state variable and the parameters b1 , b2 , b3 are taken in a range to ensure the chaotic behaviour of (12). U = [u1 , u2 , u3 ]T is the controller to be designed. We subtract (11) from (12) and get the error equation as follows: e˙1 = e2 + u1 , e˙2 = e3 + u2 ,

Fig. 1. Dynamics of synchronization errors for two identical Lü systems (drive system and response system), (a) e1 = y1 − x1 , (b) e2 = y2 − x2 , and (c) e3 = y3 − x3 .

274

L. Huang et al. / Physics Letters A 320 (2004) 271–275

e˙3 = −b1 y1 − b2 y2 − b3 y3 + y12 − a1 x1 − a2 x2 − a3 x3 − 12f (x1 ) + u3 ,

(13)

where e1 = y1 − x1 , e2 = y2 − x2 , e3 = y3 − x3 . In order to determine the controller, first let u3 = u3a + u3b , where u3a = −y12 + 12f (x1 ). Then we rewrite (13) in the following form: e˙1 = e2 + u1 , e˙2 = e3 + u2 , e˙3 = −b1 y1 − b2 y2 − b3 y3 − a1 x1 − a2 x2 − a3 x3 + u3b .

(14)

Take a Lyapunov function for Eq. (14) into consideration 1 V (e) = eT e, 2

V˙ (e) < 0 is satisfied. Since V˙ (e) is a negative-definite function, the error states lim e(t) = 0.

t →∞

Therefore, this choice will lead the error states e1 , e2 , e3 to converge to zero as time t tends to infinity and hence the synchronization of two different chaotic systems is achieved. For the two different chaotic systems, which contain different structures and parameter mismatch, the controller proposed can synchronize the states of the drive system and the response system. This shows that the control method is more robust to unintentional mismatch in the transmitter and receiver than the single-variable coupling between the systems that is currently used in experimental demonstrations of chaotic communications.

we get the first derivative of V (e). 4.1. Simulation results

V˙ (e) = e1 (e2 + u1 ) + e2 (e3 + u2 ) + e3 (−b1 y1 − b2 y2 − b3 y3 − a1 x1 − a2 x2 − a3 x3 + u3b ).

(15)

Therefore, if we choose as follows: u1 = −e1 − e2 , u2 = −e2 − e3 , u3b = −e3 + b1 y1 + b2 y2 + b3 y3 + a1 x1 + a2 x2 + a3 x3 ,

(16)

then V˙ (e) = −e12 − e22 − e32 .

(17)

In this section, numerical simulations are given to verify the method proposed. In these numerical simulations, the fourth-order Runge–Kutta method is used to solve two systems of differential equations (11) and (12), with time step size 0.001. The parameters are selected as follows: a1 = −6.8, a2 = −3.9, a3 = −1, b1 = 6, b2 = 2.92, b3 = 1.2, with initial values x1 (0) = 1, x2 (0) = 4, x3 (0) = 10, y1 (0) = 2, y2 (0) = 1, y3 (0) = 0.1. The simulation results are illustrated in Fig. 2. Fig. 2(a) shows e1 = y1 − x1 , Fig. 2(b) shows e2 = y2 − x2 , and Fig. 2(c) shows e3 = y3 − x3 . From the figure, we can see that the synchronization error will converge to zero and two

Fig. 2. Dynamics of synchronization errors for two different chaotic systems (drive system and response system), (a) e1 = y1 − x1 , (b) e2 = y2 − x2 , and (c) e3 = y3 − x3 .

L. Huang et al. / Physics Letters A 320 (2004) 271–275

different chaotic systems are indeed achieving chaos synchronization.

5. Conclusion Based on Lyapunov stability theory, we propose a nonlinear control method to synchronize two identical chaotic systems and two different chaotic systems. The simulation results show that the states of two identical Lü system are synchronized asymptotically globally. For two different chaotic systems, the Genesio system is controlled to trace the Lur’e-like system and the states of two systems become the same finally. This shows that the method proposed has strong robustness. The simulation results are presented to show the effectiveness of this approach.

References [1] L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. 64 (8) (1990) 821.

[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

275

L.M. Pecora, T.L. Carroll, Phys. Rev. A 44 (4) (1991) 2374. L. Kocarev, U. Parlitz, Phys. Rev. Lett. 74 (1995) 5028. K. Pyragas, Phys. Lett. A 181 (1993) 203. M. Ding, E. Ott, Phys. Rev. E 49 (1994) R945. T.L. Carroll, J.F. Heagy, L.M. Pecora, Phys. Rev. E 54 (1996) 4676. T.L. Liao, N.S. Huang, Phys. Lett. A 234 (1997) 262. D. Vassiliadis, Physica D 71 (1994) 319. E. Solak, O. Morgul, U. Ersoy, Phys. Lett. A 279 (2001) 47. J.A. Ramirez, H. Puebla, H. Cervantes, Phys. Lett. A 289 (2001) 193. E.-W. Bai, K.E. Lonngren, Chaos Solitons Fractals 10 (9) (1999) 1571. E.-W. Bai, K.E. Lonngren, Chaos Solitons Fractals 11 (7) (2000) 1041. H.N. Agiza, M.T. Yassen, Phys. Lett. A 278 (2001) 191. S.H. Chen, J.H. Lü, Phys. Lett. A 299 (2002) 353. Z. Li, C.Z. Han, S.J. Shi, Phys. Lett. A 301 (2002) 224. M.C. Ho, Y.C. Hung, C.H. Chou, Phys. Lett. A 296 (2002) 43. M.C. Ho, Y.C. Hung, Phys. Lett. A 301 (2002) 424.

Copyright © 2022 COEK.INFO. All rights reserved.