Global synchronization for time-delay of WINDMI System

Chaos, Solitons and Fractals 30 (2006) 629–635 www.elsevier.com/locate/chaos

Global synchronization for time-delay of WINDMI System Junxa Wang *, Dianchen Lu, Lixin Tian Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China Accepted 31 March 2005

Abstract Considering a time-delay in the receiver as compared with the transmitter, we addresses a practical issue in chaos synchronization of WINDMI system which is based on the Lyapunov stabilization theory and matrix measure, such that the state of the slave system at time t is asymptotically synchronizing with the master at time t  s. The Mathematical software is used to prove the effectiveness of this method.  2005 Elsevier Ltd. All rights reserved.

1. Introduction Chaos, as a very interesting nonlinear phenomenon, has been intensively studied in the last three decades. It is found to be useful or has great potential in communications [1–3] and other engineering applications. Refs. [4,5] study the chaos synchronization, where the unavoidable signal propagation delays in the typical master-slave synchronization. In the practical environment with signal propagation delays, it is not reasonable to require the slave system to synchronize the master system at exactly the same time. It is just like the telephone communication systems, where one hears the voice at time t on the receiver side, which was said from the transmitter side some time ago, at time t  s. For this reason, in [1], chaotic synchronization is re-defined in such a way that the state of the slave system at time t is asymptotically synchronizing with the master at time t  s, namely, e ðtÞk ¼ 0; lim kX ðt  sÞ  X

t!1

e are the state of the master and slave systems, respectively. Consequently, the control signal uses the where X and X e ðtÞ. output error between the drive system at time t  s and the slave system at time t, i.e., X ðt  sÞ  X The WINDMI system is a complex driven-damped dynamical system, which exhibits a variety of dynamical states. Ref. [6] describes the WINDMI as the energy flow through the solar wind-magnelosphere-ionosphere system. In this paper, we address a scheme on chaos synchronization of WINDMI system where the unavoidable signal propagation delays. Using Lyapunov stabilization theory we study the synchronization of WINDMI system, give the numerical results. Finally, conclusion remarks are given.

*

Corresponding author.

0960-0779/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.010

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J. Wang et al. / Chaos, Solitons and Fractals 30 (2006) 629–635

2. Time-delay chaotic synchronization Consider a chaotic continuous system described by X_ ðtÞ ¼ AX ðtÞ þ gðX ðtÞÞ;

ð1Þ

n

n·n

where X(t) 2 R is the state vector, A 2 R that

is constant matrix, and g(X) is a continuous nonlinear function. Assume

e ðt2 ÞÞ ¼ M X ;X~ ðX ðt1 Þ  X e ðt2 ÞÞ gðX ðt1 ÞÞ  gð X

ð2Þ

for a bounded matrix M X ;X~ with elements depending on X(t1) and X(t2). At this point, it is important to note that most chaotic systems can be described by (1) and (2) [4]. From the typical unidirectional linear error feedback coupling approach, and taking the time-delay into account, the slave system based on the chaotic system (1) is described by (see Fig. 1) e_ ðtÞ ¼ A X e ðtÞ þ gð X e ðtÞÞ þ LðX ðt  sÞ  X e ðtÞÞ; X

ð3Þ n·n

where s is a finite time-delay which is an unknown constant, and L 2 R is the coupling matrix to be designed to achieve synchronization. e ðtÞ is used for control, where X(t  s) is sent from the master system In Fig. 1 and formula (3), the error X ðt  sÞ  X through the channel and hence it has a time delay as compared to the signal that arrives at the slave system. The value of time-delay s is not required to be known for executing the control action. Define the error signal by e ðtÞ eðtÞ ¼ X ðt  sÞ  X

ð4Þ

which is the state error between the master system at time t  s and the slave system at time t. Synchronization requires that ke(t)k ! 0 as t ! 1. Replacing t with t  s in (1) gives the master system equation at time t  s as follows: X_ ðt  sÞ ¼ AX ðt  sÞ þ gðX ðt  sÞÞ.

ð5Þ

Then, from (3) and (5), the following error system equation is obtained: e ðtÞÞ  LeðtÞ ¼ AeðtÞ þ M X ;X~ eðtÞ  LCeðtÞ ¼ ðA þ M X ;X~  LÞeðtÞ. e_ ðtÞ ¼ AeðtÞ þ gðX ðt  sÞÞ  gð X

ð6Þ

Using Lyapunov function method, the synchronization of system (3) and (5) can be realized.

3. Synchronization for time-delay of WINDMI system WINDMI system is a physical system [7], it can be described by 8 > < x_ ¼ y; y_ ¼ z; > : z_ ¼ az  y þ b  ex :

Fig. 1. The schematic of chaos synchronization with time-delay.

ð7Þ

J. Wang et al. / Chaos, Solitons and Fractals 30 (2006) 629–635

631

Fig. 2. The attractor of WINDMI system.

From [7,8] we learn: when a = 0.7, b = 2.5, the Lyapunov exponents are (0.0755, 0, 0.7755), then system (7) is chaotic (see Fig. 2). In Eq. (7), we have ex  e~x ¼ k x;~x ðx  ~xÞ;

ð8Þ

where k x;~x is dependent on x and ~x. We choose x and ~x in order to make k x;~x varies in the interval [0, 1], that is make k x;~x bounded as 0 < k x;~x < 1. The following slave system is constructed by using (3) for (7) 8 _ > < ~xðtÞ ¼ ~y ðtÞ þ l1 ðxðt  sÞ~xðtÞÞ; ð9Þ ~y_ ðtÞ ¼ ~zðtÞ þ l2 ðyðt  sÞ  ~y ðtÞÞ; > :_ ~zðtÞ ¼ a~zðtÞ  ~y ðtÞ þ b  e~xðtÞ þ l3 ðzðt  sÞ  ~zðtÞÞ: It follows from (7) that the WINDMI system equation at time t  s is 8 > < x_ ðt  sÞ ¼ yðt  sÞ; _  sÞ ¼ zðt  sÞ; yðt > : z_ ðt  sÞ ¼ azðt  sÞ  yðt  sÞ þ b  exðtsÞ :

ð10Þ

Subtracting (10) from (9), we obtain 8 > < ex ðtÞ ¼ ey ðtÞ ¼ l1 ex ðtÞ; ey ðtÞ ¼ ez ðtÞ ¼ l2 ex ðtÞ; > : ez ðtÞ ¼ aez ðtÞ ¼ ey ðtÞ  k x;~x ex ðtÞ  l3 ez ðtÞ;

ð11Þ

where exðtÞ ¼ xðt  sÞ  ~xðtÞ; ey ðtÞ ¼ yðt  sÞ  ~y ðtÞ; ez ðtÞ ¼ zðt  sÞ  ~zðtÞ, which can be rewritten as e ðtÞÞ  LeðtÞ; e_ ðtÞ ¼ AeðtÞ þ gðX ðt  sÞÞ  gð X where

2

0

1

6 A¼6 40

0

0

3

7 1 7 5; 0 1 a

2

3

0

0

6 L¼6 40

l2

7 07 5;

0

0

l3

l1

ð12Þ 3

2

6 7 7 eðtÞ ¼ 6 4 yðt  sÞ  ~y ðtÞ 5;

6 gðX Þ ¼ 6 4

2

xðt  sÞ  ~xðtÞ

zðt  sÞ  ~zðtÞ

0 0 b  ex

3 7 7 5

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J. Wang et al. / Chaos, Solitons and Fractals 30 (2006) 629–635

then

2

3 2 3 2 0 0 0 7 6 7 6 e ðtÞÞ ¼ 6 0 g ¼ ðX ðt  sÞÞ  gð X 0 4 5¼4 5¼4 0 k x;~x ex ðtÞ k x;~x b  ex  b þ e~x

3 32 0 0 xðt  sÞ  ~xðtÞ 7 76 0 0 54 yðt  sÞ  ~y ðtÞ 5 ¼ M x;~x ex ðtÞ; 0 0 zðt  sÞ  ~zðtÞ ð13Þ

where

2

M x;~x

0 6 ¼4 0 k x;~x

3 0 0 7 0 0 5. 0 0

Theorem 1. If there exists a positive definite symmetric constant matrix P such that ðA  L þ M X ;X~ ÞT P þ P ðA  L þ M X ;X~ Þ 6 lI < 0

ð14Þ

e in the phase space, where l is a negative constant and I is the identity matrix, then the error system uniformly for all X, X (12) is globally exponentially stable about zero, implying that the two systems (9) and (10) are globally asymptotically synchronized. Proof. Choose a Lyapunov function of the form V ¼ eðtÞT PeðtÞ;

ð15Þ

where P is a positive definite symmetric constant matrix. Its derivative is V_ ¼ e_ ðtÞPeðtÞ þ eðtÞT P e_ ðtÞ ¼ ½ðA  L þ M X ;X~ ÞeðtÞT PeðtÞ þ eðtÞT P ½ðA  L þ M X ;X~ ÞeðtÞ ¼ eðtÞT ½ðA  L þ M X ;X~ ÞT P þ P ðA  L þ M X ;X~ ÞeðtÞ 6 lkeðtÞk2 < 0;

ð16Þ

where kÆk denotes the Euclidean norm. Based on the Lyapunov stability theory, system (12) is globally exponentially stable about zero. Hence the two systems (9) and (10) are globally asymptotically synchronized. Theorem 1 proves the effectiveness of the method, next work is to determine L. Let Q ¼ ðA  L þ M X ;X~ ÞT P þ P ðA  L þ M X ;X~ Þ; ki ; i ¼ 1; 2; 3 are the eigenvalves of Q. If inequality (14) holds, then ki 6 l < 0;

i ¼ 1; 2; 3.

ð17Þ

So we choose L to satisfy (17).

h

Theorem 2. Choose P = diag (pl, p2, p3), and let P ðA þ M X ;X~ Þ þ ðA þ M X ;X~ ÞT P ¼ ½aij . If a suitable L is chosen such that li P

1 ðaii þ Ri  lÞ; 2pi

i ¼ 1; 2; 3;

and

Ri ¼

3 X

j aij j

ð18Þ

j¼1;j6¼i

then the two coupled chaotic systems (9) and (10) are globally synchronized. Proof Q ¼ ðA  L þ M X ;X~ ÞT P þ P ðA  L þ M X ;X~ Þ ¼ P ðA þ M X ;X~ Þ þ ðA þ M X ;X~ ÞT P  LP  PL ¼ P ðA þ M X ;X~ Þ þ ðA þ M X ;X~ ÞT P  2LP .

ð19Þ

From Gerschgorin’s theorem in matrix theory we know, every eigenvalve is in its Gerschgorin, i.e., ki  aii  2li pi 6 Ri while l P ki, it means l is out of all Gerschgoeins, so l   aii  2li pi P Ri , then we get following inequalities,

J. Wang et al. / Chaos, Solitons and Fractals 30 (2006) 629–635

li P

633

1 ðaii þ Ri  lÞ. 2pi

In particular, we choose P = I, we obtain 2 0 6 ðA þ M X ;X~ Þ þ ðA þ M X ;X~ ÞT ¼ 4 0 k x;~x

3 2 3 2 0 0 0 0 k x;~x 7 6 7 6 1 5 ¼ 4 1 0 1 5 ¼ 4 1 1 a k x;~x 0 1 a 1 0

1 0

3 k x;~x 7 0 5.

0

2a

ð20Þ

We can choose 1 l1 P ð1 þ k x;~x  lÞ; 2

1 l2 P ð1  lÞ; 2

1 l3 P ð2a þ k x;~x  lÞ. 2

ð21Þ

According to Theorem 2, the two coupled WINDMI systems (9) and (10) are globally asymptotically synchronized. Since 0 < k x;~x < 1, we can choose 1 l1 P ð2  lÞ; 2

1 l2 P ð1  lÞ; 2

1 l3 P ð2a þ 1  lÞ. 2

ð22Þ

When we choose l = 1, we can obtain l1 = 1.51, l2 = 1.1, l3 = 0.32, the inequality (21) holds. According to Theorem 1, the two chaotic systems (9) and (10) are globally asymptotically synchronized (see Fig. 3). h Theorem 3. The two chaotic systems (3) and (5) are globally asymptotically synchronized, if one condition is satisfied at least as follows: ( ) 3 X ð1Þ maxj ~ajj þ j~aij j < 0; ( ð2Þ maxi ~aii þ

i¼1;i6¼j 3 X

) j~aij j

ð23Þ < 0;

j¼1;j6¼i

a ij Þ33 . where A  L þ M X ;X~ ¼ ðe

Fig. 3. Time evolution of valuable (a) ex(t), (b) ey(t) and (c) ez(t).

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J. Wang et al. / Chaos, Solitons and Fractals 30 (2006) 629–635

Fig. 4. Time evolution of valuable (a) ex(t), (b) ey(t) and (c) ez(t).

In this paper, have A  L þ M X ;X~

2

l1 6 ¼4 0 k x;~x

1 l2 1

3 0 7 0 5. a  l3

ð24Þ

We choose l1, l2, l3, to satisfy   max l1 þ k x;~x ; l2 þ 2; a  l3 þ 1 < 0 or

  max l1 þ 1; l2 þ 1; a  l3 þ k x;~x þ 1 < 0.

Then we can determine the bound of l1, l2, l3, and realize the chaos synchronization by choosing the proper parameters (see Fig. 4).

4. Conclusions The chaos synchronization problem, with time-delay in the channel, has been studied from the typical approach of unidirectional linear error feedback using the received time-delay signal directly without requiring any knowledge of the constant unknown time-delay [9,10]. This paper address a practical issue in chaos synchronization of MINDMI system, which will be helpful for researching and predicting the influence of solar wind to celestial bodies.

Acknowledgements Research was supported by the National Nature Science Foundation of China (no. 10071033) and Nature Science Foundation of Jiangsu Province (no.: BK2002003).

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