Chaos, Solitons and Fractals 33 (2007) 1197–1203
www.elsevier.com/locate/chaos
The synchronization for time-delay of linearly
bidirectional coupled chaotic system q
Yongguang Yu
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, PR China
Accepted 17 January 2006
Communicated by Prof. Ji-Huan He
Abstract
Based on Lyapunov stability theory and matrix measure, this paper presents a new generic criterion of global chaotic
synchronization for bidirectional coupled chaotic system with time-delay. A new chaotic system with four-scroll
attractor is chosen as an example to verify the effectiveness of the criterion. Numerical simulation are shown for
demonstration.
2006 Published by Elsevier Ltd.
1. Introduction
Chaos, a very interesting complex nonlinear phenomenon, has been intensively studied in the last four decades [1–3].
It is found to be useful and it has great potential in many technological fields, such as in information and computer
sciences, biomedical systems analysis, flow dynamics and liquid mixing, power systems protection, encryption and communications, and so on. Therefore many researchers have been involved into how to control and utilize it [1–24].
Since the discovery of chaos synchronization [4], it has been a focus of intensive research [4–24]. Many new types of
chaos synchronization have appeared in the literatures on this subject, such as phase synchronization [5,6], adaptive
synchronization [7], stochastic synchronization [8], lag synchronization [9], projective synchronization [10–12] and generalized synchronization in mismatched systems or even distinct class [14,15]. Chaos synchronization can be applied to
many fields including physical, chemical and ecological systems and secure communications, etc.
Especially, chaos synchronization of linearly coupled chaotic systems is investigated by some researchers [16–24]. So
far, there have been many specific results about determining the coupling parameters for chaotic systems. In engineering
applications, time-delay always exists, and the value of the delay is often unknown in advance. In practice, It is not
reasonable to require the response system to synchronize the drive system at the same time due to the signal propagation delays. So, chaotic synchronization for time-delay of linearly bidirectional coupled chaos systems is investigated
in this paper. How to synchronize between the state of response system at time t + s and the drive system at time t?
q
Supported by the National Nature Science Foundation of China (Grant No. 10461002) and Beijing Jiaotong University Science
Foundation of China (No. 2005sm063).
E-mail address: yyg@amss.ac.cn
0960-0779/$ - see front matter 2006 Published by Elsevier Ltd.
doi:10.1016/j.chaos.2006.01.119
1198
Y. Yu / Chaos, Solitons and Fractals 33 (2007) 1197–1203
i.e. limt!þ1 k~xðt þ sÞ xðtÞk ¼ 0, where x and ~x are the states of the drive and response systems, respectively, s(>0) is the
delay time.
The organization of this paper is as follows. In Section 2, at first, the problem statement is presented. Then based on
the Lyapunov stability theory and some matrix theory, a generic condition of global chaos synchronization for timedelay of the general chaotic systems with bidirectional linear coupling is obtained. In Section 3, a new chaotic system
with four-scroll attractor is taken as examples to demonstrate the effectiveness of the proposed scheme. In Section 4,
numerical simulations for given four-scroll chaotic system are presented. Finally, the conclusion of this paper is given
in Section 5.
2. Chaos synchronization of linearly bidirectional coupled systems with time-delay
Considering a chaotic system in the form of
X_ ðtÞ ¼ AX ðtÞ þ GðX ðtÞÞ
ð1Þ
where X(t) 2 Rn is the state vector, A 2 Rn·n and G(X(t)) is a continuous nonlinear function. Then a bidirectional
coupling scheme between identical chaotic systems with time-delay can be constructed as follows:
e ðt þ sÞ X ðtÞÞ
X_ ðtÞ ¼ AX ðtÞ þ GðX ðtÞÞ þ D1 ð X
ð2Þ
e_ ðt þ sÞ ¼ A X
e ðt þ sÞ þ GðX ðt þ sÞÞ þ D2 ðX ðtÞ X
e ðt þ sÞÞ
X
ð3Þ
and
e ðtÞ is the n-dimensional state vector of the response system, s is a finite time-delay which is an unknown conwhere X
stant, G : Rn ! Rn is a continuous function, D1 and D2 are diagonal matrices which rule the dissipative coupling.
Assume that
ðX ðt þ sÞ X ðtÞÞ
GðX ðt þ sÞÞ GðX ðtÞÞ ¼ M X ;e
X
where M X ;e
is a matrix with elements depending on X(t) and X(t + s). Obviously it is bounded.
X
Let e(t) = X(t + s) X(t), then by subtracting (2) from (3), we can obtain the error dynamical system:
e_ ðtÞ ¼ ðA þ M X ;e
ðD1 þ D2 ÞÞe
X
ð4Þ
In order to realize the chaos synchronization between two linearly bidirectional systems with time-delay, we should
choose the suitable coupled parameter matrices D1, D2 which can make
lim eðtÞ ¼ 0
n!þ1
In the following, a generic criterion is given for judging the global chaos synchronization between two coupled systems with time-delay.
Theorem 1. If there exists a positive definite symmetric matrix P and a constant > 0, such that
ðA þ M X ;e
ðD1 þ D2 ÞÞT P þ P ðA þ M X ;e
ðD1 þ D2 ÞÞ 6 I
X
X
e ðt þ sÞ, where I is an identity matrix, then the error dynamics system (4) is globally stable at
uniformly for any X(t) and X
zero, i.e. the two systems (2) and (3) are realized to globally asymptotically synchronize.
Proof 1. Choose the following Lyapunov function:
V ðtÞ ¼ eT PeðtÞ
where P is a positive definite symmetric matrix, i.e. V(t) is a positive definite function. Calculating its derivative along
the trajectory of the system (4), one have
T
dV ðtÞ
ðD1 þ D2 Þ PeðtÞ þ eT ðtÞP A þ M X ;e
ðD1 þ D2 Þ eðtÞ
¼ e_ T ðtÞPeðtÞ þ eT ðtÞP e_ ðtÞ ¼ eT A þ M X ;e
X
X
dt
T
ðD
þ
D
Þ
P
þ
P
A
þ
M
ðD
þ
D
Þ
eðtÞ
¼ eT ðtÞ A þ M X ;e
1
2
1
2
X
X ;e
X
6 eT ðtÞeðtÞ < 0
Y. Yu / Chaos, Solitons and Fractals 33 (2007) 1197–1203
1199
for all e(t) 5 0. Based on the Lyapunov stability theorem, the error dynamics system (4) is globally asymptotically stable at origin. Hence the two coupled systems with time-delay are globally asymptotically synchronized. So, the theorem
is proved. h
Remark. It can be seen from Theorem 1, when the coupled parameter matrix D1 = 0 or D2 = 0, the results can be simplified as the general synchronization criterion of unidirectional coupled system with time-delay [16].
3. Synchronization of a four-scroll chaotic attractor
To verify the result of Theorem 1, a four-scroll chaotic attractor is chosen as an example. One can easily determine
the coupling parameters for realizing to synchronize two bidirectional coupled systems with time-delay.
A new four-scroll chaotic system can be described by the following three-dimensional quadratic autonomous ODE
[22–24].
8
>
< x_ ¼ ax yz
y_ ¼ by þ xz
ð5Þ
>
:
z_ ¼ cz þ xy
where a > 0, b > 0, c > 0 and b + c > a.
The system is found to be chaotic in a wide parameter range and has many complex dynamic behaviors. Differing
from other known chaotic systems, it has five equilibria, and does not have Hopf bifurcation. At the same time, the
system (5) not only can display a two-scroll chaotic attractor when parameters a = 4.5, b = 12 and c = 5 (see
Fig. 1), but also can display a four-scroll chaotic attractor when a = 0.4, b = 12 and c = 5 (see Fig. 2).
According to Eqs. (2) and (3), two linearly bidirectionally coupled systems with time-delay can be constructed for
system (5)
8
>
< x_ 1 ðtÞ ¼ ax1 ðtÞ y 1 ðtÞz1 ðtÞ þ d 11 ðx2 ðt þ sÞ x1 ðtÞÞ
y_ 1 ðtÞ ¼ by 1 ðtÞ þ x1 ðtÞz1 ðtÞ þ d 12 ðy 2 ðt þ sÞ y 1 ðtÞÞ
ð6Þ
>
:
z_ 1 ðtÞ ¼ cz1 ðtÞ þ x1 ðtÞy 1 ðtÞ þ d 13 ðz2 ðt þ sÞ z1 ðtÞÞ
and
8
>
< x_ 2 ðt þ sÞ ¼ ax2 ðt þ sÞ y 2 ðt þ sÞz2 ðt þ sÞ þ d 21 ðx1 ðtÞ x2 ðt þ sÞÞ
y_ 2 ðt þ sÞ ¼ by 2 ðt þ sÞ þ x2 ðt þ sÞz2 ðt þ sÞ þ d 22 ðy 1 ðtÞ y 2 ðt þ sÞÞ
>
:
z_ 2 ðt þ sÞ ¼ cz2 ðt þ sÞ þ x2 ðt þ sÞy 2 ðt þ sÞ þ d 23 ðz1 ðtÞ z2 ðt þ sÞÞ
ð7Þ
where dij(i = 1, 2; j = 1, 2, 3) are coupling parameters.
Let e ¼ ðe1 ; e2 ; e3 ÞT ¼ ðx1 ðtÞ ~x2 ðt þ sÞ; y 1 ðtÞ y 2 ðt þ sÞ; z1 ðtÞ z2 ðt þ sÞÞT , the corresponding error dynamics system is
Fig. 1. The graph of chaotic system (6) with two-scroll attractor when the parameters a = 4.5, b = 12 and c = 5.
1200
Y. Yu / Chaos, Solitons and Fractals 33 (2007) 1197–1203
Fig. 2. The graph of chaotic system (6) with four-scroll attractor when the parameter a = 0.4, b = 12 and c = 5.
e_ ðtÞ ¼ ðA þ M X 1 ;X 2 ðD1 þ D2 ÞÞe
where
0
1
a 0
0
B
C
A ¼ @ 0 b 0 A;
0 0 c
and
ð8Þ
0
M X 1 ;X 2
1
0
z1 ðtÞ y 2 ðt þ sÞ
B
C
¼ @ z1 ðtÞ
0
x2 ðt þ sÞ A;
y 2 ðt þ sÞ x1 ðtÞ
0
D1 ¼ diagfd 11 ; d 12 ; d 13 g
D2 ¼ diagfd 21 ; d 22 ; d 23 g
When coupling parameter matrices D1 = D2 = 0, and chosen a very small constant s = 0.1 and same initial condition
for system (6) and (7), in Fig. 3 it can show the two systems are chaotic and do not synchronize.
Fig. 3. The graph of the error dynamical system between systems (6) and (7) when the coupling parameter is zero.
Y. Yu / Chaos, Solitons and Fractals 33 (2007) 1197–1203
1201
Choose the positive definite symmetric constant matrix P = diag {p1, p2, p3}, (pi > 0, i = 1, 2, 3) and any constant
> 0, then
ðA þ M X 1 ;X 2 ðD1 þ D2 ÞÞT P þ P ðA þ M X 1 ;X 2 ðD1 þ D2 ÞÞ þ I
0
2p1 a þ ðd 11 þ d 21 Þ 2p
ðp3 p1 Þy 2 ðt þ sÞ
1
B
B
¼B
2p2 b þ ðd 12 þ d 22 Þ 2p 2
ðp2 p1 Þz1 ðtÞ
B
@
p2 x2 ðt þ sÞ þ p3 x1 ðtÞ
ðp3 p1 Þy 2 ðt þ sÞ
1
C
C
C
P 2 x2 ðt þ sÞ þ p3 x1 ðtÞ
C
A
2p3 c þ ðd 13 þ d 23 Þ 2p
ð9Þ
3
Based on basic matrix theory, the matrix (9) is negative definite if and only if its sub-determinants satisfy
< 0;
DI ¼ 2p1 a þ ðd 11 þ d 21 Þ 2p1
2p1 a þ ðd 11 þ d 21 Þ 2p
1
D2 ¼ ðp2 p1 Þz1 ðtÞ
and
> 0
2p2 b þ ðd 12 þ d 22 Þ 2p 2
ðp2 p1 Þz1 ðtÞ
D3 ¼ det ðA þ M X 1 ;X 2 ðD1 þ D2 ÞÞT P þ P ðA þ M X 1 ;X 2 ðD1 þ D2 ÞÞ þ I
D2 þ d < 0
¼ 2p3 c þ ðd 13 þ d 23 Þ 2p2
where d ¼ ðp3 p2 Þ2 y 22 ðt þ sÞð þ 2p2 ðb þ d 12 þ d 22 ÞÞ þ ðp2 x2 ðt þ sÞ þ p3 x1 ðtÞÞ2 ð a þ d 11 þ d 21 Þ þ 2ðp2 p1 Þðp3 p1 Þ y 2 ðt þ sÞz1 ðtÞðp2 x2 ðt þ sÞ þ p3 x1 ðtÞÞ. Through calculations, it can be simplified as
Fig. 4. The graph of the errors between two coupled two-scroll chaotic systems.
1202
Y. Yu / Chaos, Solitons and Fractals 33 (2007) 1197–1203
8
d 11 þ d 21 > a þ 2p 1
>
>
>
<
ðp2 p1 Þ2 z21 ðtÞ
d 12 þ d 22 > 4p1 p2 ðaþd
b þ 2p 2
11 þd 21 2p Þ
>
1
>
>
:d þ d > d c þ 13
23
2p3 D3
2p3
ð10Þ
Due to the fact that the trajectory of the coupled chaotic systems (6) and (7) is bounded, the inequalities (10) can
easily be hold by choosing appropriately coupled parameters.
For convenience, let the matrix P = I, k > 0, i.e. p1 = p2 = p3 = 1 > 0, one have
8
d 11 þ d 21 > a þ 2
>
>
<
d 12 þ d 22 > b þ 2
ð11Þ
>
>
ðx2 ðtþsÞþx1 ðtÞÞ2
:
d 13 þ d 23 > 4ðbþd 12 þd 22 Þ c þ 2
2
4. Numerical simulation
In this section, to demonstrate the effectiveness of the proposed scheme, the following numerical simulations are
given. The fourth/fifth-order Runge–Kutta method is used to solve the chaotic systems during the simulation. Let
= 0.1. First, the parameters are chosen a = 4.5, b = 12 and c = 5 for system (5), the time-delay s = 0.1, we assume
that the initial states of system (6) are (x1(0), y1(0), z1(0)) = (5, 3, 8), when the coupled parameters are chosen to satisfy
the inequalities (13), for instance (d11, d12, d13) = (1, 1, 20),(d21, d22, d23) = (4, 1, 20), the simulations in Fig. 4 are shown
that the error system (8) converge to zero very quickly, i.e. the synchronization of the coupled systems (6) and (7) is
realized. In Fig. 5, we choose the parameters a = 0.4, b = 12 and c = 5, the time-delay s = 0.1, the initial states of system
(6) are (x1(0), y1(0), z1(0)) = (5, 3, 8), the coupled parameters (d11, d12, d13) = (1, 1, 20),(d21, d22, d23) = (4, 1, 20), it can
display the systems (6) and (7) also synchronized rapidly.
Fig. 5. The graph of the errors between two coupled four-scroll chaotic systems.
Y. Yu / Chaos, Solitons and Fractals 33 (2007) 1197–1203
1203
5. Discussion
In this paper, we investigate the synchronization for time-delay of linearly bidirectional coupled chaotic systems. A
new generic criterion is given for judging the synchronization of two coupled systems. We choose a new chaotic system
with four-scroll attractor as an example to show the effectiveness of the method. Finally, numerical simulation is provided for demonstration.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
Chen G, Dong X. From chaos to order: methodologies, perspectives and applications. Singapore: World Scientific; 1998.
Lorenz EN. Deterministic non-periodic flows. J Atmos Sci 1963;20:130–41.
Ott E, Grebogi C, Yorke J. Controlling chaos. Phys Rev Lett 1990;64:1196–9.
Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4.
Rosenblum M, Pikovsky A, Kurth J. Phase synchronization in chaotic oscillators. Phys Rev Lett 1996;76:1804–7.
Rosenblum M. A characteristic frequency of chaotic dynamical system. Chaos, Solitons & Fractals 1993;3(6):617–26.
Chen S, Lu J. Synchronization of uncertain unified chaotic system via adaptive control. Chaos, Solitons & Fractals
2002;14(4):643–7.
Alexeyev A, Shalfeev V. Chaotic synchronization of mutually-coupled generators with frequency-controlled feedback loop.
Int J Bifurcat Chaos 1995;5:551–8.
Taherion S, Lai Y. Experimental observation of lag synchronization in coupled chaotic systems. Int J Bifurcat Chaos
2000;10:2587–94.
Xu D, Li Z. Controlled projective synchronization in nonpartially-linear chaotic systems. Int J Bifurcat Chaos
2002;12(6):1395–402.
Xu D, Chee C, Li C. A necessary condition of projective synchronization in discrete-time systems of arbitrary dimensions.
Chaos, Solitons & Fractals 2004;22(1):175–80.
Li Z, Xu D. A secure communication scheme using projective chaos synchronization. Chaos, Solitons & Fractals 2004;22(2):
477–81.
Rulkov N, Sushchik K, Tsimring L, Abarbanel H. Generalized synchronization of chaos in directionally coupled chaotic systems.
Phys Rev E 1995;51:980–94.
Chen G, Liu S. On generalized synchronization of spatial chaos. Chaos, Solitons & Fractals 2003;15(2):311–8.
Yang X, Chen G. Some observer-based criteria for discrete-time generalized chaos synchronization. Chaos, Solitons & Fractals
2002;13(6):1303–8.
Jiang G, Zheng W, Chen G. Global chaos synchronization with channel time-delay. Chaos, Solitons & Fractals
2004;20(2):267–75.
Jiang GP, Tang W. A global synchronization criterion for coupled chaotic systems via unidirectional linear error feedback
approach. Int J Bifurcat Chaos 2002;12(10):2239–53.
Zhou T, Lü J, Chen G, Tang Y. Synchronization stability of three chaotic systems with linear coupling. Phys Lett A
2002;301:231–40.
Kyprianidis I, Stouboulos I. Chaotic synchronization of three coupled oscillators with ring connection. Chaos, Solitons & Fractals
2003;17:327–36.
Yu Y, Zhang S. The synchronization of linearly bidirectional coupled chaotic systems. Chaos, Solitons & Fractals 2004;22:189–97.
Yu Y, Zhang S. Global synchronization of three coupled chaotic systems with ring connection. Chaos, Solitons & Fractals
2005;24:1233–42.
Elabbasy E, Agiza H. Chaos synchronization criterion and adaptive synchronization for new chaotic system. Chaos, Solitons &
Fractals 2005;23:1299–309.
Park J. Chaos synchronization of a chaotic system via nonlinear control. Chaos, Solitons & Fractals 2005;25:579–84.
Lu J, Chen G, Cheng D. A new chaotic system and beyond: the generalized Lorenz-like system. Int J Bifurcat Chaos
2004;14:1507–37.