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Anti-synchronization of two different chaotic systems
Wenlin Li, Xiuqin Chen ∗ , Shen Zhiping
College of Mathematics and Information, Henan Normal University, Xinxiang 453007, PR China
Received 17 April 2007
Abstract
In this paper, the anti-synchronization of two different chaotic systems is investigated. On the basis of a nonlinear control scheme
and Lyapunov theory, sufficient conditions for the stability of the error dynamics are derived, where the controllers are designed by
using the sum of the relevant variables in chaotic systems. Numerical simulations are performed for the Genesio–Rossler system
to demonstrate the effectiveness of the proposed control strategy.
c 2008 Elsevier B.V. All rights reserved.
Keywords: Chaotic system; Anti-synchronization
1. Introduction
Since the pioneering work by Pecora and Carroll [1], chaos synchronization, as a very active topic in nonlinear
science, has received increasing attention. The concept of synchronization has been extended in scope, for example
to generalized synchronization [2,3], phase synchronization [4], lag synchronization [5], and even anti-phase
synchronization (APS) [6,7]. APS can also be interpreted as anti-synchronization (AS), which is a phenomenon in
which the state vectors of the synchronized systems have the same amplitude as but opposite signs to those of the
driving system. Therefore, the sums of two signals are expected to converge to zero when either AS or APS appears.
Recently, Ho and Huang generalized active control to AS for two identical systems [8]. Guo-Hui Li and Shi-Ping
Zhou considered AS in two different systems using active control theory [9]. However, it seems that there are no
previous results on AS in two different systems using nonlinear control methods. In this article, we will focus on the
AS of two different systems. Nonlinear control methods will be employed. We derive rigorously a sufficient condition
for the anti-synchronization. Numerical experiments on a Genesio system and a Rossler system are performed, which
demonstrate the effectiveness and feasibility of the proposed control technique.
The layout of the rest of the paper is as follows. In Section 2, brief descriptions of the Genesio system and the
Rossler system are introduced. Section 3 investigates the design of the controller for the anti-synchronization of the
two different chaotic systems. Numerical simulation results are given for illustration and verification in Section 4.
Finally, conclusions are drawn in Section 5.
∗ Corresponding author. Tel.: +86 373 3397430.
E-mail address: lcjszp@163.com (X. Chen).
0378-4371/$ - see front matter c 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.physa.2008.02.011
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doi:10.1016/j.physa.2008.02.011
W. Li, et al., Anti-synchronization of two different chaotic systems, Physica A (2008),
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Fig. 1. (a) Genesio chaotic attractor. (b) Rossler chaotic attractor.
2. System description
The Genesio system, proposed by Genesio and Tesi [10], is one of the paradigms of chaos since it captures many
features of chaotic systems. It includes a simple square part and three simple ordinary differential equations that
depend on three negative real parameters. The dynamic equations of the system are

ẋ = y
ẏ = z
(1)

ż = −ax − by − cz + x 2 .
It has a chaotic attractor as shown in Fig. 1(a) when a = 6, b = 2.92, c = 1.2.
Following the discovery of the Lorenz attractor in 1963 [11], Rossler proposed a simpler chaotic system in
1976 [12], which is now called the Rossler system, described by

ẋ = −y − z
ẏ = x + by
(2)

ż = a + z(x − c).
It has a chaotic attractor as shown in Fig. 1(b) when a = 0.2, b = 0.2, c = 5.7.
In the following sections, we will study the anti-synchronization of the two chaotic dynamical systems.
3. Synchronization of a Genesio system and a Rossler system
In order to observe the anti-synchronization behavior of Genesio and Rossler systems using a nonlinear control
scheme, we assume that Genesio system (1) drives Rossler system (2). Therefore, we define the drive and response
systems as follows:

ẋ1 = x2
ẋ2 = x3
(3)

ẋ3 = −ax1 − bx2 − cx3 + x12
and

 ẏ1 = −y2 − y3 + u 1
ẏ2 = y1 + b1 y2 + u 2

ẏ3 = a1 + y3 (y1 − c1 ) + u 3 .
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W. Li, et al., Anti-synchronization of two different chaotic systems, Physica A (2008),
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Fig. 2. Anti-synchronization of Genesio and Rossler systems. (a) Signals x1 and y1 , (b) signals x2 and y2 , (c) signals x3 and y3 , and (d) error e(t).
For AS of the above two different systems, we should add (3) to (4):

ė1 = x2 − y2 − y3 + u 1
ė2 = x3 + y1 + b1 y2 + u 2
(5)

ė3 = a1 + y3 (y1 − c1 ) − ax1 − bx2 − cx3 + x12 + u 3
where e(t) = e1 e2 e3 = (x1 + y1 , x2 + y2 , x3 + y3 ). Now we define the control functions u 1 , u 2 and u 3 as
follows:

u 1 = y3 − 2x2 + e2 − λe1 , (λ > 0)
u 2 = b1 x2 − x3 − y1 − (b1 + 1)e2
(6)

u 3 = ax1 + bx2 − a1 − x12 − y1 y3 − cy3 .
Hence the error system (5) becomes

ė1 = −λe1
ė2 = −e2

ė3 = −ce3 .
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The error system (5) is a linear system of the form v̇ = Av, where v = e1


−λ 0
0
A =  0 −1 0  ,
0
0 −c
e2
e3
T
and A is as follows:
(8)
since the eigenvalues of the system (7) are −λ, −1 and −c. Thus AS of the Genesio system and the Rossler system is
achieved.
4. Simulation results
In this section, to verify and demonstrate the effectiveness of the proposed method, we consider the following
numerical example. In the numerical simulations, the fourth-order Runge–Kutta method is used to solve the system
with time step 0.01. We select the parameters of the Genesio system as a = 6, b = 2.92, c = 1.2. and the parameters
of the Rossler system as a = 0.2, b = 0.2, c = 5.7, so that the Genesio system and the Rossler system exhibit chaotic
behavior. The initial conditions of the drive system and response system are taken as (x1 (0), x2 (0), x3 (0)) = (5, −4, 2)
and (y1 (0), y2 (0), y3 (0)) = (−5, 3, 3), respectively, and λ = 1. Thus, the initial errors are e1 (0) = 0, e2 (0) = −1 and
e3 (0) = 5. Notice that the control is switched on at t = 0.5. The simulation results are illustrated in Fig. 2. Fig. 2(a)
shows the signals x1 and y1 , while Fig. 2(b) displays x2 and y2 , and Fig. 2(c) displays x3 and y3 , in order to show the
AS error, whose time evolution is shown in Fig. 2(d). From the simulations we find that the response system starts to
trace the drive system from an opposite direction at t = 0.5, and their AS is observed when t ≥ 3, which verifies the
validity of the proposed control technique.
5. Conclusion
In this paper, we study the problem of anti-synchronization for two different chaotic systems. By a nonlinear
control technique, we have realized anti-synchronization of two different chaotic systems. Simulations on antisynchronization for Genesio–Rossler systems encourage us to believe that the technique works very well.
Acknowledgements
This work was supported by the National Natural Science Foundation of PR China (Grant No. 60643003) and by
the Natural Science Foundation of Henan Province, PR China (Grant No. 0611051200).
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Please cite this article in press as:
doi:10.1016/j.physa.2008.02.011
W. Li, et al., Anti-synchronization of two different chaotic systems, Physica A (2008),