WUJNS Wuhan University Journal of Natural Sciences
Vol. 10 No. 6 2005 9932996 Article ID :100721202 (2005) 0620993204
Linearly Coupled Synchronization of the New Chaotic Systems □ L U J un2an1 , ZHOU Jin1 , L I Yi2tian2 1. School of Mathematics and Statistics , Wuhan University , Wuhan 430072 , Hubei , China ; 2. School of Water Resources and Hydropower , Wuhan University , Wuhan 430072 , Hubei , China
Abstract : This paper investigates synchronization within the new systems , which we denote as Liu system in this paper. New stability criteria for synchronization of linearly coupled Liu systems are attained using the Lyapunov method. Some sufficient conditions for synchronization are concluded through rigorous mathematical theory , which can be further applied to more chaotic systems. Moreover , numerical simulations are given to show the effectiveness of our synchronization criteri2 ons. Key words : linearly synchronization ; coupled synchroniza2 tion ; chaotic system ; Lyapunov method CLC number : O 322
0 Introduction haos has been intensively studied in the last three dec2 ades[1 ] . In 1963 , Lorenz found the first canonical at2 tractor [2 ,3 ] . In 1999 , Chen , et al found another similar but topologically not equivalent chaotic attractor [4 ,5 ] . In 2002 , Lu , et al found a chaotic system[6 ] , bearing the name of the Lu system. In the second year , Chen and Lu introduced the notion of generalized Lorenz canonical form ( GLCF) , which contains the Lorenz system (τ∈( 0 , ∞) ) and Chen system (τ ∈( - 1 , 0) ) as two extremes , along with infinitely many cha2 otic systems in between including Lu system (τ= 0) [7 ] . Recently , a new chaotic system is proposed in Ref. [8 ]. It relies on one multipliers and one quadratic term to introduce the non2linearity necessary for folding trajectories. The Liu system is described by
C
・ x1 ・ x2 ・ x3
Received date : 2004212212 Foundation item : Supported by t he National Key Basic Research and Development 973 Program of China (2003CB415200) and State Key Laboratory of Water Resources and Hydropower Engineering Science (2004C011) Biography : LU J un2an (19452) , male , Professor , research direction : chaos control and synchronization , complex networks. E2mail :jalu @ whu. edu. cn
g = a( g x2 - g x1 ) g = cg x1 - k g x1 g x3 g = - bg x3 + h g x 21 here , some redundancy in the set of parameters { a, b, c, h, k} can be removed. For any a, b, c ∈R , and any h, k ∈R\ { 0} , the fol2 lowing system (1) is equivalent to the system above via the line2 ar transform of coordinates[9 ] : x1 = xg1 hk , x2 = xg2 hk , x3 = kxg3 : x1 = a( x2 - x1 ) g ( 1) xg 2 = c x 1 - x1 x3 2 xg 3 = - b x 3 + x1 let a = 10 , b = 40 , c = 2 . 5 , a chaotic attracter can be genera2 ted. Worth mentioning , it has been shown that the new sys2
W uhan Universit y J ournal of N atural Sciences Vol. 10 No. 6 2005
© 1995-2006 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
993
tem (1) is state2equivalent to the GLCF with τ= - 1. Recently , coupled synchronization has attracted a great deal of attention[10213 ] . In this paper , we first in2 vestigate linearly coupled synchronization of two systems through rigorously mathematical analysis. Then we fur2 ther extend the method to the coupled synchronization within three Liu systems. Finally , computer simulations are provided for illustration and verification.
1 Linearly Coupled Synchronization of Two Liu Systems Consider the following coupled system with two Liu systems : xg 1 = a( x2 - x1 ) + d1 ( y1 - x1 ) xg 2 = c x 1 - x1 x3 + d2 ( y2 - x2 ) x3 = - b x 3 + x21 + d3 ( y3 - x3 ) g ( 2) yg 1 = a( y2 - y1 ) + d1 ( x1 - y1 ) yg 2 = c y 1 - y1 y3 + d2 ( x2 - y2 ) 2 yg 3 = - by 3 + y1 + d3 ( x3 - y3 ) where a, b, c > 0 and di > 0 ( i = 1 , 2 , 3) are the coupling coefficients. Defining the errors as ei = x i - y i , ( i = 1 , 2 , 3) , we have the errors system : eg 1 = - ( a + 2 d1 ) e1 + ae2 ( 3) eg 2 = ( c - x3 ) e1 - 2 d2 e2 - y1 e3 eg 3 = ( x1 + y1 ) e1 - ( b + 2 d3 ) e3 Denote the coefficient matrix as A , and define B A + AT as . We can see 2 - ( a + 2 d1 ) B =
a + c - x3
a + c - x3
x1 + y1
2
2
- 2 d2
2 x1 + y1
-
2
y1
-
≤‖X( t) ‖2 ≤ ‖X(0) ‖2 exp
994
max { | x1 | } = max { | y1 | }
x1 ∈( 1)
△
M3
y1 ∈( 1)
max { | a + c - x3 | } = max { | a + c - y3 | }
x3 ∈( 1)
y3 ∈( 1)
Synchronization of two systems ( 1 ) means ei →0 ( t →∞, i = 1 , 2 , 3 ) . According to Lemma , B should be negative definite. ① - ( a + 2 d1 ) < 0 is obvious. ② The second leading principal minor determinant of B is 2 d2 ( a + 2 d1 ) -
( a + c - x3 ) 2
4
2
∫ 0
4
.
( a + c - x3 ) 2
4
1 - y1 ( x1 + y1 ) ( a + c - x3 ) 4 ( x1 + y1 ) 2 y21 + ( a + 2 d1 ) + 2 d2 4 4 = - ( b + 2 d3 ) 2 d2 ( a + 2 d1 ) -
-
( a + c - x3 ) 2
4
a + c - x3
( x1 + 2 y1 ) 2 8 ( a + c - x3 ) + 2 a + 4 d1 2 + y1 8 ( a + c - x3 ) + 4 d2 ( x1 + y1 ) 2 + 8
≤- ( b + 2 d3 ) 2 d2 ( a + 2 d1 ) -
M23
4
9 M3 2 ( M3 + 2 a + 4 d1 ) 2 + M1 + M1 8 8 4 ( M3 + 4 d2 ) 2 + M1 8
2
= - ( b + 2 d3 ) 2 d2 ( a + 2 d1 ) -
M23
4
7 M3 + a + 2 d1 + 8 d2 2 + M1 . 4 Since b + 2 d3 > 0 , we can see 2 d2 ( a + 2 d1 ) -
M23
>0 4 if det B < 0. Then , B is negative definite. As a result , we have Theorem 1 If a, b, c > 0 , di > 0 ( i = 1 , 2 , 3) , and di
t
α( s) ds
M3
≥2 d2 ( a + 2 d1 ) -
③ det B = - ( b + 2 d3 ) 2 d2 ( a + 2 d1 ) -
n
‖X(0) ‖2 exp
0
△
M1
y1
A + AT d ‖X ‖2 If X g= AX, X ∈R , = 2 XT X. 2 dt Assuming that α( t) ,β( t) be the minimum and maxi2 A + AT mum eigenvalues of , we get the following re2 2 sult [11 ] . Lemma Assuming that the differential equation ( ) X g t = AX( t) has a solution X( t) , we have
β( s) d s ∫
Therefore , if there exists ε> 0 such that β( t) < - ε, then for any initial value X(0) , with exponential rate , we will have X( t) →0 ( t →∞) . Since system (1) is chaos , the terms x1 and a + c x3 are bounded. Denote that
- ( b + 2 d3 )
2
t
satisfy the condition ( b + 2 d3 ) 2 d2 ( a + 2 d1 ) -
M23
4
LU J un2an et al :Linearly Coupled Synchronization of t he …
© 1995-2006 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
-
7 M3 + a + 2 d1 + 8 d2 2 M1 > 0 , then for any initial value 4 ( x1 ( 0) , x2 ( 0) , x3 ( 0 ) , y1 ( 0 ) , y2 ( 0 ) , y3 ( 0 ) ) , we have ei →0 ( t →∞, i = 1 , 2 , 3) ,i. e. , two coupled Liu systems approach synchronization.
2 Linearly Coupled Synchronization of Three Liu Systems Consider the following coupled system with three Liu systems :
xg 1 = a ( x2 - x1 ) + d1 ( y1 - x1 ) xg 2 = cx 1 - x1 x3 + d2 ( y2 - x2 ) x3 = - bx 3 + x21 + d3 ( y3 - x3 ) g yg 1 = a ( y2 - y1 ) + d1 ( z1 - y1 ) ( 4) yg 2 = cy 1 - y1 y3 + d2 ( z2 - y2 ) 2 ( ) yg 3 = - by 3 + y1 + d3 z3 - y3 zg 1 = a( z2 - z1 ) + d1 ( x1 - z1 ) zg 2 = cz 1 - z1 z3 + d2 ( x2 - z2 ) 2 zg 3 = - bz 3 + z1 + d3 ( x3 - z3 ) where a, b, c > 0 , and di > 0 ( i = 1 , 2 , 3) are coupled coef2 ficients. Defining the errors as ei = x i - y i , ( i = 1 , 2 , 3) ej = y j - 3 - z j - 3 , ( j = 4 , 5 , 6) ek = z k- 6 - x k- 6 , ( k = 7 , 8 , 9) we have the errors system : eg 1 = - ( a + d1 ) e1 + ae2 = d1 e4 eg 2 = ( c - x3 ) e1 - d2 e2 - y1 e3 + d2 e5 e3 = ( x1 + y1 ) e1 - ( b + d3 ) e3 + d3 e6 g ( 5) eg 4 = - d1 e1 - ( a + 2 d1 ) e4 + ae5 eg 5 = - d2 e2 + ( c - z3 ) e4 - 2 d2 e5 - y1 e6 eg 6 = - d3 e3 + ( y1 + z1 ) e4 - ( b + 2 d3 ) e6 e7 = - e1 - e4 , e8 = - e2 - e5 , e9 = - e3 - e6 g According to system ( 5) , it can be seen that ei →0 ( t →∞, i = 1 , 2 , …, 9) if and only if ei →0 ( t →∞, i = 1 , 2 , …, 6) . Defining the coefficient matrix of errors system A + AT as A , then defining B as , we have B = 2 B1 0 , 0 B2 where - ( a + d1 ) B1 =
a + c - x3
2 x1 + y1
2
a + c - x3
x1 + y1
2
2
- d2 -
y1
2
-
y1
2
- ( b + d3 )
W uhan Universit y J ournal of N atural Sciences Vol. 10 No. 6 2005
- ( a + 2 d1 ) B2 =
a + c - z3
2 y1 + z1
a + c - z3
y1 + z1
2
2
- 2 d2
-
y1
2
y1
- ( b + 2 d3 ) 2 2 Matrix B will be negative definite if and only if B1 and B2 are negative definite too. Similarly , we have the following conclusion : Theorem 2 If a, b, c > 0 and di > 0 ( i = 1 , 2 , 3) satis2 fy the conditions : -
① ( b + 2 d3 ) 2 d2 ( a + 2 d1 ) -
M23
4
7 M3 + a + 2 d1 + 8 d2 2 M1 > 0 . 4 2
② ( b + d3 ) d2 ( a + d1 ) -
M3
4 7 M3 + a + d1 + 4 d2 2 M1 > 0 . 4 then for any initial value ( x1 ( 0) , x2 ( 0 ) , x3 ( 0) , y1 ( 0) , y2 ( 0) , y3 ( 0 ) , z1 ( 0 ) , z2 ( 0 ) , z3 ( 0 ) ) , the three coupled Liu systems approach synchronization , i. e. , ei →0 ( t → ∞, i = 1 , 2 , …, 9) . ① Here , d1 can be set to 0. That is , controllers are just d2 and d3 . ② The two theorems give only sufficient conditions for synchronization of system ( 2) and ( 4) . In fact , nu2 merical simulations reveal that synchronization still be reached with coupling coefficients which can not satisfy the above conditions. ③ Difficulty of linearly coupled synchronization of these systems is the estimation of the chaotic boundary. ④ For more systems , the same work can be done.
3 Numerical Simulations Numerical simulations are provided in this part. In all simulations , we assume that a = 10 , b = 40 , c = 2 . 5. Figure 1 shows the synchronization errors of system (2) with initial value ( x1 ( 0 ) , x2 ( 0 ) , x3 ( 0 ) , y1 ( 0 ) , y2 ( 0) , y3 ( 0) ) = ( 3 , 5 , 7 , 4 , 6 , 8) . Here , we assume that d1 = d2 = d3 = 1 in Fig. 1 (a) and d1 = 0 , d2 = 2 , d3 = 1 in Fig. 1 ( b) . Figure 2 display the synchronization errors of system (4) with initial value ( x1 ( 0 ) , x2 ( 0 ) , x3 ( 0 ) , y1 ( 0 ) , y2 ( 0) , y3 ( 0) , z1 ( 0) , z2 ( 0) , z3 ( 0) ) = ( 3 , 5 , 7 , 4 , 6 , 8 , 9 , 10 , 11) . Here , d1 = d2 = d3 = 1 . 995
© 1995-2006 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.
LU J un2an et al :Linearly Coupled Synchronization of t he …
© 1995-2006 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.