2004 8th International Conference on Control, Automation. Robotics and Wslon Kunming, China, 6-9th December 2004

Generalized Synchronization of Unified Chaotic System

and the Research of CSK SHAN Liang, LI Jun, WANG Zhi-quan

Department of Automation, Nanjing University of Science & Technology Nanjing, Jiangsu, P.R China 210094

Email: slshan@sina.com, whzwzq@,mi I .njust. edu. cn

Abstract

Unified chaotic system is a newly-suggested chaotic

system. By means of generalized synchronization (GS), the synchronized control of unified system is studied. For unified system has good properties of auto-correlation and cross-correlation, it takes advantage of chaotic shift

keying (CSK) scheme to make experiments. In this paper, the approach of GS control has been presented.

Then the entire scheme of CSK based on unified chaotic system has been detailed. Finally the results o f numerical simuhtions prove the effectiveness about GS of unified chaotic system. The approach i s suitable for the research situation with parameter mismatch and channel distortion.

1 Introduction

In 1963, American meteorologist Lorenz found the first chaotic attractor. Lorenz attractor provided the first

model in chaotic research. In 1999, Chen found another chaotic amactor [ 11, which has similarly simple structures with Lorenz system but displays even more

complicated dynamical behavior [2]. In 2002, LU and Chen found a new chaotic system-Unified Chaotic System [2,6], which united Lorenz system and Chen system. The new unified chaotic system is described by:[2]

(1) X = ( 2 5 ~ + 1O)(y - X)

y,= (28 - 350)~ + ( 2 9 ~ - l ) ~ - xz 1 i = xy - -(8 + a ) z 1 3

where E [OJ] , the system is chaotic. When D E [ 0 , 0 . 8 ) , system ( 1 ) belongs to the generalized Lorenz system; when 0 E (0.8,1] , it belongs to the

0-7803-8653-1/041$20.00 Q 2004 IEEE

generalized Chen system; when a = 0 . 8 , it belongs to the generalized LU system, which plays an important role in connecting Lorenz system with Chen system. Chaotic synchronization is an important subject in chaotic researches, and many papers about it have been

published. In recent years, generalized synchronization is brought forward The method has been used in some low dimensional chaotic systems [2-4]. Inspirited by the idea of generalized synchronization, in this paper, a control method for synchronizing unified chaotic system has been presented.

Since 1992, a number of chaotic synchronization and modulation schemes have been proposed. Chaos shift

keying (CSK) is a digital modulation scheme where each symbol is mapped to a different chaotic attractor. The information to be transmitted is modulated not by the shape of the sampIe function but by the attractor, which

produces the sample function [7,8J. In this paper based on unified chaotic system, the experiment of CSK is studied. The following of the paper is arranged as below. In section II, the definition and principle of GS are introduced. In section III, we analyze and construct the generalized synchronization of unified chaotic system.

The entire scheme of CSK based on unified chaotic system is detailed in section TV. Section V proves the effectiveness and the feasibility of the generalized synchronization method via numerical simulation. The last section, section VI, is the concluding remarks.

2 Definition and Principle of GS

Two continuous-time dynamical systems:

x = f ( x ) ( 2 - 1 ) j l = g ( Y ) ( 2 - 2 )

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Identical synchronization is achieved if

lim 1 1 y ( t ) - X ( t ) 0 for any initia1 states of x(0)

and y(0). In secure communication, system (2-1) is generally considered as driving system, while system (2-2) is seen as receiving system. Synchronization means that the states of receiving system are identical with those of the driving one. But in practice it is difficult to synchronize the two different systems thoroughly, due to the existence of parameter mismatch and channel distortion. Thereby in recent years generalized synchronization methods emerge. If there is a h c t i o n of M making

lim 11 y ( t ) - ~ ( ~ ( t ) ) I/= 0 for any initial states of n(0) and

y(O), system (2-1) is called generalized synchronization with system (2-2) [4,5,9]. Lemma[3]: Consider the driving and receiving system as

followed:

I+ -

I +-

x = Ax + O(x) ( 3 - 1)

( 3 - 2)

If all the real parts of eigenvalues in matrix A are

negative and RA = AR , then generalized

synchronization of system (3-1) and (3-2) is achieved when t + 00, that is y=M(x)=&. Prove: Let A = - fix, then

{ p = Ay + RO (x )

A = j - a i = Ay +R@((x) -R[Ax + @ ( x ) ]

= A y - Q A x = A A

If all the real parts of eigenvalues in matrix A are negative, it is concluded that A -, oand error system is asymptotic stable. a So the key issue of realizing the control of GS is to find proper parameters A and ll .

3 Generalized Synchronization of Unified Chaotic System

-(25a+10) (25a+10) 0 0 [!]=I p 0 29a-1 0 - -(8 1 + U )

3

r xi

Firstly, matrix A must satisfy the conditions in lemma.

A + (250 + 10) - ( 2 5 0 + 10) i + 1 - 29a

1 3

0 h +-(8+ a)

I 3

= [ A + - ( 8 + a ) ] [ A ' + E d + C ]

( 5 ) where B = 11 - 4 a (6)

C = -725 a' - 265 a f 10'- 25ap - l o p (7)

Obiouslywhen a E [ O , t ] , B E [7,11]

a, = - i / 3 ( 8 + a) < o (8) From formula (6)(7), we get

So the condition which makes real parts of eigenvalues in matrix A negative is:

B ~ > B Z - ~ C > O or ~ ~ - 4 c < o (10) 10 J 1 , fiom formula (7) we get: that is 0 0 . When a

- 125 a - 265 a + 10 25a + 10 P <

Under the condition of formula (1 1). that is p < -28

the real parts of Ai (i = 1,2,3) are negative.

3.2 Construct Receiving System

Based on the lemma in section 11, the receiving system

according to driving system (4) is:

3.1 Construct Driving System

To unified system (l), the dTiving system is constructed

as:

-(25a+lO) (25~+10) [!]=I p 0 29a-1 0 - - @ + U ) 1 3

0

= A[x' y' z']+nq(x)

-:I Z'

(1 2)

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n s f o

Fig. 1 Block diagram of a coherent driver of CSK

In order to satisfy Q A = A Q , R = Ais chosen, such that !24 = AR= A" System (12) is transformed as:

;]+ Z' 1 (2% + 1 0)((28 - 3% - P)x - XZ)

(2% - 1)((2 8- 3 Sa - P)x - XZ)

1 1 3

-- (8 + u > v

(13) It should be mentioned that, there are more than one

forms of TZ satisfymg the lemma condition, so the receiving system (13) also has many other forms. Specially, when R = Z , problems return to the range of identical synchronization.

3.3 Relation between driving and receiving systems

Analyzing driving system (4) and receiving system (13), we can find the linear relation between the state z and t',

and the h e a r proportion 'r' changes with parameter 'a', but doesn't change with p :

In section V, simulation resuIts also show the linear

relation. In section IV, the linear relation is used in the CSK receiving system.

. (U) = z'/ z = -0.333 ~ 1 - 2,667 (14)

4 CSK Research based on GS of Unified Chaotic Systems

Chaos shift keying (CSK) [7,8] is a digital modulation

scheme where each symbol is mapped to a different, chaotic attractor. The number of attractors is equal to the size of the signal set. Chaotic signals can be generated by

completely different chaotic circuits or be produced by the same chaotic circuit for different values of a bifurcation parameter. So the information to be

transmitted is carried not by the shape of the sample function but by the attractor which produces the sample hnctiqn [X-121. In this paper, two unified chaotic systems are used as dnving systems, which are only different from the parameter 'a', shown as Fig.1. The switch changes with the binary information I(& When Z(t)=O, it connects with the driving system 1; otherwise I(f)=l, it t u m s to the driving system 2. So x(4 gets:

x , ( t > Z ( t ) = 0 I X 2 ( t ) I ( t ) = 1 x ( t ) =

In order to enhance the security, a transform step is added before x(t) is transferred to channel. The really transferred signal is m(t):

F(x) can be chosen as any limited invertible hnctions. ARer x(l) is transformed to m(t), it is difficult for attackers to reconsmtct the phase-space. Meanwhile it

adds new keys for secure communication, because without the precise form of F(x/ it is impossible to obtain

Nr). In the coherent correlation CSK receiving system, the first job is to inverse-transform the receiving signal m '(d. Then ~ ' ( t ) = F - ' ( m ' ( t ) ) is the real useful signal. The

coherent CSK receiver is shown in Fig.2. Because generalized synchronization method is used, the state of receiver is not equal to that of driver, the relation

is linear shown in formula (14). So if Z(t)=O,

s , ( t ) - ( r ( u , ) x ' ( t ) ) + 0 , that is, the correlation

between s,(t) and r(al)x'(t) is larger than that of sr(t) and r(adx'(0. If W=l, s2(t ) - (r(a2)(x'( t ) )+0 , the

situation is inversed. Then the elements of the observation vector are given by:

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I I

Fig2 Block diagram of a coherent receiver of CSK

T

Y , (0 = J;, i, (t>(r(a, )X'(t))df

Yz t u = JT s*, ( t ) ( r ( a , )x ' ( t ) )d t

(17)

(18) Ts

where E is the time for synchronization, and observation

time is (7'-E). If I(rr=O, y , ( t ) > y2( t ) ; if I(r)=I,

y , ( r ) I y , ( t ) . So after decision step, the information

is achieved:

EO

40 N

2 0

0

20

-20 -10

Fig3(a) The State ofDrive System ( ~ 0 . 2 5 )

5 Numerical SimuIations

5.1 Generatized Synchronization Simulations

To driving system (4) and receiving system (13), we do a

series of numerical simulations about generalized synchronization. Let = 0 . 2 5 , = - 30 , the initial

values of system (4) and (13) are set as (0.5;0.2;0.5) and (0.1;0.2;0.3), the synchronization results are shown in Fig. 3(a) and 3(b). The driving and receiving systems have different attracton. As formula (20), the values of system's Lyapunov exponent are achieved:

When ~ 0 . 2 5 , we obtain a chaotic sequence with 6 x 10 data. From it, the maximum Lyapunov exponent of system (4) is 1.36; that of system (4) is 1.13. And at this time, r(0.25b t'/z = -2.75, which is accorded with

formula (14). The linear correlation is also shown in Fig.

3(4.

5.2 Numerical Simulations of CSK

In the CSK research based on GS, two unified systems are chosen which are only different &om parameter 'U'.

0

-50 t,

-loo

-150 400

Fig3(b) The State of Receive System ( ~ 0 . 2 5 )

2

Y=P.?SZ*Z

t -5 t

I 0 50 100 t 150 200 250 300

-0.5 1

Fig3(c) The error behveen n and z ' ( ~ 0 . 2 5 )

The first driving system in Fig, 1 sets ~ 0 . 2 5 , and the second one sets ~ 0 . 8 5 . Accordingly, the receiving system 1 in Fig.2 gets ~ 0 . 2 5 ; while ~ 0 . 8 5 in receiving system 2. Firstly, the property of auto-correlation and cross-correlation is analyzed. With the help of MATLAB correlation functions, we obtain the curve$ of auto-correlation and cross-correlation to unified system,

which is shown in Fig.4. That is, unified sytem fits the CSK research.

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0.4 1 I ' 0 . 2

0

-0.2 500 1000 1500 2000

Fig. 4(a) Autd=orrelation of Unified System

Fig. 4(b) Cross-Correlation of Unified System

Limited of the paper's length, the binary information is

only set as Z(t)=[I 0 1 I 0 0 1 0 1 01. In the numerical simulation, each step is defined as 0.2 second. In unified system, when 4 . 2 5 the synchronization time is about 10 steps; ~ 0 . 8 5 , it is 7 steps. So TFIO steps, and T=20 steps. Fig.5(a-f) show the results of simulations. The

b.ansmitted signal is the state of z(t) in system (4), whlch carries the useful information I(t). In order to encrypt the

information, two non-linear functions are used to change x(l) into m(0:

m ( t ) = FCx(f)) = F2 (4 tW)N (211

From Fig.5(c), after transform the shape of m(r) is completely different from xii). The attackers can't reconstruct x(t) Without the form of F(x). Fig.5(d) 5@3

display the generalized error between x(0 and S(l). For example, in 20-401(,) - 0, ( r ( u l ) . r l ( t ) ) CB s l ( t ) ,

( r (a i )x ' ( r ) ) II s , ( t ) @ 0 9 r ( 0 . 2 5 ) - 0 2 . 7 5 ,

7 ( 0 . 8 5 I 1 2 . 9 5 , that is, the correlation coefficient of

r(u3 n '(0 and S,(t) is larger 6 that of r(ud x '(Q and

W.

. . . . . . . . . XI a2 pa aa I Im la 1.m Im ltn 90

Fig.5(b) Diagram of signal x(t)

Fig.S(c) Diagram of signal m[t)

Fig.S(d) Generalized Error of x(t) and &(t)

FigS(e) Generalized Error of x(t) and &(t)

0 a 4 9 83 m I 100 I?& I* 1m 110 ax Fig.S(f) Diagram of recover signal I'(t)

At last the real speech signal of 'nan jing' is used as I(t). Fig.d(a) 6(b) are the o r i p d and recovered wave signal(without noise). To validate the anti-noise ability of CSK scheme based on GS method, Gauss white noise is added into the channel. Fig.6(c) is the recovered signal under the situation with noise. The maximal SNR is 18db, and the signal can be recovered correctly (error under 5%). The experiments show that the method is suitable for the situation with parameter mismatch and channel distortion.

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Fig.6(a) Diagram of original signal

Fig.6lb) Diagram of recovered signal(without noise)

Fig.d(c) Diagram of recovered signal(SNR=l8db)

6. Conctusion

This paper has studied the synchronization of unified chaotic system. By constructing the suitable forms and parameters of driving and receiving systems, the control of generalized synchronization has been achieved. Based on it, the CSK research of unified systems has been detailed. Numerical simulations show the effectiveness of the arithmetic. The approach is very suitable for the research situation with parameter mismatch and channel distortion. However some problems should be M e r discussed

when the method is applied in the actual systems.

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