Synchronization of two different chaotic systems with unknown parameters

Physics Letters A 361 (2007) 98–102 www.elsevier.com/locate/pla

Synchronization of two different chaotic systems with unknown parameters Shuang Li a,∗ , Wei Xu a , Ruihong Li b,a a Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, China b Department of Applied Mathematics, Xidian University, Xi’an 710071, China

Received 28 April 2006; received in revised form 1 September 2006; accepted 6 September 2006 Available online 3 October 2006 Communicated by A.P. Fordy

Abstract In this Letter, a general method is proposed for the synchronization between two different chaotic systems with unknown parameters. This approach is based on the invariance principle of differential equations, and employs a combination of feedback control and adaptive control. With this method one can achieve synchronization between almost all chaotic systems, e.g., Lorenz system, Chen system, Liu system, Rössler system, Chua’s system, etc. Although the models used in the research are the systems with unknown parameters, the method is also applicable to the systems with known parameters. Numerical simulations results are presented to demonstrate the effectiveness of the method. © 2006 Elsevier B.V. All rights reserved. Keywords: Synchronization; Feedback control; Adaptive control; Different chaotic systems

1. Introduction Since the idea of synchronizing two identical chaotic systems from different initial conditions was first introduced by Pecora and Carroll in 1990 [1], chaos synchronization has gained a lot of attention among scientists due to its important application in secure communication, chemical and biological systems, human heartbeat regulation, etc. Until now, a wide variety of approaches have been proposed for the synchronization of chaotic systems which include adaptive control [2], observerbased control [3], variable structure control [4], backstepping control [5], active control [6], nonlinear control [7], and so on. However, most of the methods mentioned above are designed to synchronize two identical chaotic systems. In fact, in many practical world such as laser array, biological systems and cognitive processes, it is hardly the case that every component can be assumed to be identical. Also, more and more applications of chaos synchronization in secure communication make it much more important to synchronize two different chaotic systems [8,9].

A chaotic system is extremely sensitive to tiny variations of parameters. But in practical situation, some systems’ parameters cannot be exactly known in priori, the effect of these uncertainties will destroy the synchronization and even break it. Therefore, it is important and necessary to study the synchronization in such systems with unknown parameters [10,11]. In this Letter, we present a general method to synchronize two different chaotic systems with unknown parameters. Based on the invariance principle of differential equations, we prove that the suggested approach can realize chaos synchronization globally and asymptotically. Although the models used in the research are the systems with unknown parameters, the result is also applicable to the systems with known parameters. Simulation results validate the effectiveness and universality of the proposed synchronization method. 2. Design of synchronization method Let a chaotic system be given as x˙ = f (x) + F (x)θˆ ,

* Corresponding author.

E-mail address: [email protected] (S. Li). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.09.068

(1)

is the state vector, f (x) ∈ F (x) ∈ where x ∈ θˆ ∈ R m is the unknown parameter vector. And let Ω ⊂ R n be a bounded closed set which contains the whole attractor of Rn

Rn ,

R n×m ,

S. Li et al. / Physics Letters A 361 (2007) 98–102

Eq. (1). Relation (1) represents the drive system, and the response system with a controller u ∈ R n is introduced as follows: y˙ = g(y) + G(y)δˆ + u,

(2)

where y ∈ g(y) ∈ G(y) ∈ δˆ ∈ R p is the unknown parameter vector. Suppose y is defined on a set S, and S ⊃ Ω. The purpose of chaos synchronization is how to design the controller u, which is able to synchronize the states of both the drive and the response systems. If we define the error vector as e = y − x, the dynamic equation of synchronization error can be expressed as Rn ,

Rn ,

R n×p ,

99

can get V˙ = e˙T e + θ T θ˙ + δ T δ˙ −

n  (εi + L)ei2 i=1



= g(y) − g(x) + F (x)θ − G(y)δ + εe  T  T − θ T F (x) e + δ T G(y) e − 

T = g(y) − g(x) e − LeT e =

n  

T

e

n 

(εi + L)ei2

i=1

 gi (y) − gi (x) ei − LeT e

i=1

e˙ = g(y) − f (x) − F (x)θˆ + G(y)δˆ + u.

(3)

Hence, the objective of synchronization is to make limt→∞ e(t) = 0. Assumption 1. For any y = (y1 , y2 , . . . , yn )T , z = (z1 , z2 , . . . , zn )T ∈ S, there exists a positive constant K such that   gi (y) − gi (z)  Ky − z, i = 1, 2, . . . , n. (4) Remark 1. We call the above condition as the uniform Lipschitz condition, and K refers to the uniform Lipschitz constant. Note that the condition (4) will hold as long as ∂gi /∂yj (i, j = 1, 2, . . . , n) are bounded on the set S. Therefore the class of systems satisfying the above condition include almost all well-known chaotic systems, such as Rössler system, Chua’s system, Lorenz system, Chen system, etc. Theorem 1. If the controller is selected as u = f (x) − g(x) + F (x)θ˜ − G(y)δ˜ + εe

(5)

and θ˜ ∈ R m , δ˜ ∈ R p , ε = diag(εi ) ∈ R n×n are updated according to the following laws  T θ˜˙ = − F (x) e, ε˙ i = −βi ei2 ,

 T δ˜˙ = G(y) e,

βi > 0, i = 1, 2, . . . , n

(6)

then the error dynamical system (3) is globally asymptotically stable at the origin, thus implying that the drive system (1) and response system (2) are globally synchronized asymptotically. ˆ by (5) the error system (3) can Proof. Let θ = θ˜ − θˆ , δ = δ˜ − δ, be rewritten as e˙ = g(y) − g(x) + F (x)θ − G(y)δ + εe.

(7)

Construct a non-negative function in the form of  1 1 1 T (εi + L)2 , e e + θ T θ + δT δ + 2 2 βi n

V=

(8)

i=1

where L is a constant bigger than nK, i.e. L > nK. Differentiating V with respect to time t along the trajectories of (7), we



n 

Ke|ei | − LeT e

(9)

i=1

 (nK − L)eT e  0.

(10)

In the inequality (9), we have used the uniform Lipschitz condition (4). It is obvious that V˙ = 0 if and only if ei = 0, i = 1, 2, . . . , n. According to the well-known invariance principle of differential equations [12], the trajectories of the error dynamical system, starting with arbitrary initial values, converge asymptotically to the largest invariant set E = {(e, θ, δ, ε): e = 0, θ = θ 0 , δ = δ 0 , ε = ε 0 } contained in V˙ = 0 as t → ∞, which implies that the two systems (1) and (2) are globally synchronized asymptotically. 2 Remark 2. Note that in the proposed method, some adaptive parameters εi may be redundant to achieve synchronization, because we find from the above proof that one may set εi ≡ 0 if |ei |  γ |ej | or ei = m(ej ), where γ > 0 and m(·) is a continuously differentiable function satisfying m(0) = 0. Although we cannot give a general criteria to determine which adaptive parameters εi may be omitted, for some concrete systems with known parameters one may determine it by the technique used in the first example. Of course for the low-dimensional systems one may determine it by directly testing again and again. Remark 3. Although the models used in the research are the systems with unknown parameters, the result is also applicable to the systems with known parameters because the latter can be viewed as a special case of the former, for example, if the parameters of the system (1) have been known exactly, one can take θˆ = 0, F (x) = 0. 3. Examples In this section, three examples and corresponding numerical simulations are given to illustrate the validity of the proposed method. In Example 1, we consider the application of the method to synchronization of two different chaotic systems with known parameters, and discuss how to omit some redundant εi . In Examples 2 and 3, we consider the synchronization of drive systems with unknown parameters, while response systems with known parameters and unknown parameters, respectively.

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S. Li et al. / Physics Letters A 361 (2007) 98–102

Example 1. Consider the synchronization between Rössler system ⎧ ⎨ x˙1 = −x2 − x3 , (11) x˙ = x1 + 0.2x2 , ⎩ 2 x˙3 = x1 x3 − 5.7x3 + 0.2, and Shimizu–Morioka model [13] ⎧ ⎪ ⎨ y˙1 = y2 + u1 , y˙2 = y1 − 0.85y2 − y1 y3 + u2 , ⎪ ⎩ y˙ = −0.5y + y 2 + u , 3

3

1

(12)

3

where x = (x1 , x2 , x3 )T , y = (y1 , y2 , y3 )T are the state vectors of drive system (11) and response system (12), respectively, u = (u1 , u2 , u3 )T is the controller. Compare systems (11) and (12) with Eqs. (1) and (2), we know that Fig. 1. Time evolution of synchronization errors.

θˆ = 0,

⎛

F (x) = 0,

δˆ = 0, ⎞

G(y) = 0,

−x2 − x3 ⎠, x1 + 0.2x2 x1 x3 − 5.7x3 + 0.2 ⎛ ⎞ y2 g(y) = ⎝ y1 − 0.85y2 − y1 y3 ⎠ .

f (x) = ⎝

(13)

−0.5y3 + y12 According to Theorem 1, the controller is taken as u = f (x) − g(x) + εe ⎛ ⎞ ⎛ ⎞ ε1 e1 −(2x2 + x3 ) ⎠ + ⎝ ε2 e2 ⎠ =⎝ 1.05x2 + x1 x3 −(x12 + 5.2x3 − x1 x3 − 0.2) ε3 e3

Fig. 2. Time evolution of adaptive parameter ε1 .

(14)

and furthermore, subtracting Eq. (11) from Eq. (12), we get the error system ⎧ e˙1 = e2 + (2x2 + x3 ) + u1 , ⎪ ⎪ ⎪ ⎪ ⎪ e ⎨ ˙2 = e1 − 0.85e2 − (x1 e3 + e1 x3 + e1 e3 ) − (1.05x2 + x1 x3 ) + u2 , ⎪ ⎪ ⎪ e˙3 = −0.5e3 + (2x1 e1 + e12 ) ⎪ ⎪ ⎩ + (x12 + 5.2x3 − x1 x3 − 0.2) + u3 ,

(15)

by (15) we know that if u3 = −(x12 + 5.2x3 − x1 x3 − 0.2), e1 → 0, then e3 → 0; if u2 = 1.05x2 + x1 x3 , e1 → 0, e3 → 0, then e2 → 0, thus we can set ε2 ≡ 0, ε3 ≡ 0, while ε˙ 1 = −β1 e12 , β1 > 0. RK4 method is used to all of our simulations with time step size 0.001. The initial conditions are selected as x(0) = (−1, 1, 0)T , y(0) = (3.1, 0.2, 0.3)T , ε1 (0) = 0, β1 = 1. The simulation results are shown in Figs. 1 and 2. From Fig. 1, one can see the errors e1 , e2 , e3 converge to zero as time increases. It shows that the trajectories of the response system asymptotically converge to those of the drive system. Fig. 2 shows that the adaptive parameter ε1 tends to a constant as t goes to infinity.

Example 2. Consider Lorenz system with unknown parameters as the drive system ⎧ ⎪ ⎨ x˙1 = θˆ1 (x2 − x1 ), (16) x˙2 = θˆ2 x1 − x2 − x1 x3 , ⎪ ⎩ x˙3 = −θˆ3 x3 + x1 x2 , and the response system is Liu system [14] ⎧ ⎪ ⎨ y˙1 = 10(y2 − y1 ) + u1 , y˙2 = 40y1 − y1 y3 + u2 , ⎪ ⎩ y˙ = −2.5y + 4y 2 + u . 3 3 3 1

(17)

Rewrite system (16) and system (17) in the form of Eqs. (1) and (2) as follows x˙ = f (x) + F (x)θˆ ,   x −x 0  0 2 1 0 x1 −x −x x where f (x) = 2 1 3 , F (x) = x1 x2

0 0 0 −x3

0

(θˆ1 , θˆ2 , θˆ3 )T is the unknown parameter vector. y˙ = g(y) + G(y)δˆ + u, where G(y) = 0, δˆ = 0, g(y) = u3 )T is the controller.

(18) 

, θˆ =

(19) 

10(y2 −y1 ) 40y1 −y1 y3 −2.5y3 +4y12

 , u = (u1 , u2 ,

S. Li et al. / Physics Letters A 361 (2007) 98–102

101

Fig. 5. Time evolution of parameters θi = θ˜i − θˆi (i = 1, 2, 3).

Fig. 3. Time evolution of synchronization errors.

and the response system is a new chaotic system [15] with unknown parameters ⎧ ⎪ ⎨ y˙1 = δˆ1 y1 − y2 y3 + u1 , (23) y˙2 = −δˆ2 y2 + y1 y3 + u2 , ⎪ ⎩ y˙3 = −δˆ3 y3 + y1 y2 + u3 . Rewrite system (22) and system (23) in the form of Eqs. (1) and (2) as follows Fig. 4. Time evolution of adaptive parameter ε2 .

According to Theorem 1, the controller is taken as u = f (x) − g(x) + F (x)θ˜ + εe ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ (x2 − x1 )θ˜1 ε1 e1 10(x1 − x2 ) ⎠+⎝ ⎠ + ⎝ ε2 e2 ⎠ =⎝ x1 θ˜2 −(40x1 + x2 ) 2 −x3 θ˜3 ε3 e3 −4x1 + 2.5x3 + x1 x2 (20) and T   T θ˜˙ = − F (x) e = (x1 − x2 )e1 , −x1 e2 , x3 e3 .

(21)

In the simulations, we choose the unknown parameter vector θˆ = (10, 28, 83 )T , the initial conditions x(0) = (−1, 1, 5)T , y(0) = (0.1, 0.2, 1.3)T , θ˜ (0) = (0, 0, 0)T . Let ε1 ≡ 0, ε3 ≡ 0, ε˙ 2 = −β2 e22 , β2 = 1, ε2 (0) = 0, the numerical results are illustrated in Figs. 3–5. Fig. 3 shows the errors converge to zero as time increases, which implies that the synchronization can be realized. Figs. 4 and 5 display the evolution of ε2 , θ1 , θ2 , and θ3 . Obviously, with time passing, these parameters tend to some constants. Example 3. Consider Chen system with unknown parameters as the drive system ⎧ ⎪ ⎨ x˙1 = θˆ1 (x2 − x1 ), (22) x˙2 = (θˆ3 − θˆ1 )x1 − x1 x3 + θˆ3 x2 , ⎪ ⎩ ˆ x˙3 = x1 x2 − θ2 x3 ,

x˙ = f (x) + F (x)θˆ ,   x −x  0 2 1 −x1 where f (x) = −x1 x3 , F (x) = x1 x2

0

0 0 0 x1 +x2 −x3 0



(24) , θˆ =

(θˆ1 , θˆ2 , θˆ3 )T is the unknown parameter vector. y˙ = g(y) + G(y)δˆ + u,  −y y  y 0 1 2 3 where g(y) = y1 y3 , G(y) = 0 −y2 y1 y2

0

0 0 0 −y3



(25) , δˆ = (δˆ1 , δˆ2 ,

δˆ3 )T is the unknown parameter vector, u = (u1 , u2 , u3 )T is the controller. According to Theorem 1, the controller is taken as u = f (x) − g(x) + F (x)θ˜ − G(y)δ˜ + εe ⎞ ⎛ ⎞ ⎛ (x2 − x1 )θ˜1 x2 x3 = ⎝ −2x1 x3 ⎠ + ⎝ −x1 θ˜1 + (x1 + x2 )θ˜3 ⎠ 0 −x3 θ˜2 ⎞ ⎛ ⎞ ˜ ε1 e1 y1 δ1 ⎝ ⎠ ⎝ ˜ − −y2 δ2 + ε2 e2 ⎠ −y3 δ˜3 ε3 e3 ⎛

(26)

and ⎧ ˙ ⎪ ⎨ θ˜ = −[F (x)]T e = ((x2 − x1 )e1 − x1 e2 , −x3 e3 , (x1 + x2 )e2 )T , ⎪ ⎩˙ δ˜ = [G(y)]T e = (y e , −y e , −y e )T . 1 1

2 2

(27)

3 3

In the simulations, we choose the unknown parameter vectors θˆ = (35, 3, 28)T , δˆ = (0.4, 12, 5)T , the initial conditions x(0) = (−3, 1, 1)T , y(0) = (−1, 11, 3)T , θ˜ (0) = (25, 0,

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S. Li et al. / Physics Letters A 361 (2007) 98–102

Fig. 6. Time evolution of synchronization errors.

Fig. 9. Time evolution of parameters δi = δ˜i − δˆi (i = 1, 2, 3).

4. Conclusion

Fig. 7. Time evolution of adaptive parameter ε2 .

This work presents a general method for the synchronization of different chaotic systems with unknown parameters. The controller used in this synchronization consists of feedback and adaptive control. Based on the invariance principle of differential equations, we prove that the controller can realize chaos synchronization globally and asymptotically. Although the models used in the research are the systems with unknown parameters, the result is also applicable to the systems with known parameters. Three illustrative examples are given to demonstrate the validity of this technique, and numerical simulations are also given to show the effectiveness of the method. Acknowledgements The authors are grateful for the support to the National Natural Science Foundation of China (Grant Nos. 10472091, 10332030 and 10502042). References

Fig. 8. Time evolution of parameters θi = θ˜i − θˆi (i = 1, 2, 3).

˜ 25)T , δ(0) = (0, 0, 0)T . Let ε1 ≡ 0, ε3 ≡ 0, ε˙ 2 = −β2 e22 , β2 = 1, ε2 (0) = 0, the numerical results are illustrated in Figs. 6–9. Fig. 6 shows the errors converge to zero as time increases, which implies that the synchronization can be realized. Figs. 7–9 display the evolution of ε2 , θ1 , θ2 , θ3 , δ1 , δ2 , and δ3 . Obviously, with time passing, these parameters tend to some constants.

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