Journal of Software Engineering and Applications, 20 12, 5, 23-25
doi:10.4236/jsea.2012.512b005 Published Online December 2012 (http://www.SciRP.org/journal/jsea)
23
Impulsive Control for Synchronization of Lo renz Chaotic
System
Wenxiang Zhang, Zhanji Gui, Kaihua Wang*
Scho ol of Mathematics and S tatistics, Hainan N ormal University, Haiko u, Hainan, Ch in a.
Received 2012
ABSTRACT
Chaotic synchronization is the key technology of secure communication. In this paper, an impulsive control method
for chaotic synchronization of two coupled Lorenz chaotic system was proposed. The global asymptotic synchroniza-
tion of two Lorenz systems was realized by using the linear error feedback of the state variables of the drive system and
the response system as i mpulsive contro l signal. B ased on s tabilit y theory of i mpulsive di fferential eq uation, conditions
were obtained to guarantee the global asymptotic synchronization of two Lorenz systems. The theory analysis and
computer simula tion result s validated its effectiveness.
Keywords: Lorenz System; Impulsi ve Co ntrol; Synchro nization
1. Introduction
As the ke y technology of secure communication, chaotic
synchronization has been widely development since Pe-
cora and Carroll [1] proposed the principle of chaos syn-
chronization and realized it in the circuit in 1990. Several
synchronization me tho d s ha ve b ee n pro p o se d so far , s uch
as drive-response synchronization, coupling synchronizatio n,
feedback-perturb sync hr oniz atio n, self-adapt synchroniza-
tion, impulse synchronization, and so on [2-4]. Impulse
synchroniz ati on has been widel y appreciated b y researcher
and made some good progress [5-8]. However, many of
the impulsive control methods for synchronization are
subject to c ertain restrictio ns. Pap er [9] studied impulsi ve
synchronization for Rössler chaotic system, and paper
[10] researched impulsive control for synchronization of
a class of chaotic system. In their papers the drive signal
is generated by the impulsive signal and continuous signal
of the system variables, so the controller is very compli-
cated. In this paper, we use an impulsive control method,
and design the controller for Lorenz chaotic system. De-
signed controller is simple and easy to be realized.
2. Problem Formulation
In this section, we study the impulsive control of Lorenz
chaotic system [11] described by the following differential
equation:
1 12
21 2 13
33 12
x xx
xx xxx
xx xx
σσ
ρ
β
=−+

= −−

=−+




(1)
where
σ
,
ρ
,
β
are system’s positive real number
parameters. We choose the parameters
10 σ =
,
28 ρ =
,
83 β =
, the initial condition is give n by
123
( (0),(0),(0))(0,0,1)
TT
xxx =
. The uncontrolled tra-
jectories are shown in Figure 1, which is the notable
Lorenz attractor.
System (1) can b e r ewritten in to the following form:
( )
xAxg x= +

(2)
where
123
(, , )
T
x xxx=
is state variab le ,
13
12
00
10 ,().
00
Ag xxx
xx
σσ
ρ
β
−
 
 
=−=−
 
 
−
 
Figure 1. Lorenz attractor.
*Corresponding author.

Impulsive Control for Synchronization of Lorenz Chaotic System
24
Regarding (2) as a drive system, the response system
can be described as:
()
yAyg y= +
 (3)
where
123
(,, )
T
y yyy=
is state variable of response
system. Using linear feedback of synchronization error a s
impulsive signal, we can obtain the following impulsive
response s y s te m:
00
(),1, 2,
()
()
k
kk
yAyg yttk
yByxt t
yt y
+
=+ ≠=
∆= −=

=


(4)
where
33
(1, 2,)
k
BRk
×
∈=
are constant matrices describing
the li near na ture o f t he d rivi ng i mpul ses,
()
kk
y ytt
+−
∆−
,
( )()
kk
yt yt
−

.
From system (2) and system (4), the error dynamics is
given as followi ng:
00
(,)(),1, 2,
()
k
kk
e AeMxygettk
eBe tt
et e
+
=+ ≠=
∆= =

=


(5)
here
1 12233
(,,)
T
ey xyxyxyx=−=−−−
is synchroni-
zation error,
31
21
000
(, )0
0
Mxy yx
yx


=−−



3. Synchronization of Lorenz Chaotic
System
Theorem: Denote
k
β
,
A
λ
and
M
λ be the largest
eigenvalue of
()()
T
kk
IB IB
++
(1, 2,)k=
,
T
AA
+,
(,) (,)
T
M yxMyx+
respectively, if there exists a constant
1 α >
such that
ln() ()0
kA Mk
αβλλ τ
++ ≤
,
1, 2,k=
,
then sy stem (4) and (2) are global Asymptotic synchro-
nization. Here
1
k kk
tt τ
−
= −
is pulse interva l.
Proof: Choose the Lyapunov function
()
T
Ve ee=
,
then for
1
(,],(1,2,)
kk
ttt k
−
∈=
we have
()[(, )][(, )]
[][ (,)(,)]
()()()
TT TT
T TTT
T
AM AM
VeeeeeAeM yxeeeAeMyxe
eAAeeMyxMyx e
ee Ve λλ λλ
=+=++ +
=++ +
≤+ =+

So
11
( ())( ())exp[() ()]
kAM k
VetVett t λλ
+
−−
≤+ ⋅−
(6)
On the other hand, when ,(1,2,)
k
tt k==, we get
( ())[() ()] () ()
()( )()()
( ())
T
kkk kk
TT
k kkk
kk
VetIB etIB et
etIB IBet
Vet β
+
=++
= ++
≤
(7)
According to the inequality (6) and (7), we can get
0 120
01 1
22
1
0
( ())( ())exp[()()]
(()){exp[()]}
{exp[()]}
{exp[()()]}
(())exp[()()]/
k AM
AM
AM
kAM k
k
kAM k
VetVett t
Vet
tt
Vett t
βββλλ
βλ λτ
βλ λτ
β λλ
β λλα
−
≤ +−
= +
+
+−
≤ +−

 (8)
Considering 0 ( ())Vetand 1 exp[()( )]
2
AM k
tt λλ
+−
are both bounded, so
lim( )0
t
Ve
→∞
=
. Then the tr ivial solu-
tion o f s yste m (5) is global a symptotic al ly stab le. T hat is,
system (4) and (2) are global asymptotic synchronization.
Now, according matrix theor y, we have
||( ,)( ,)||
T
M
M yxMyx λ ≤+
, where
⋅
denotes any
kinds of norms of a matrix. We can obtain the follow-
ing corollary:
Corollary: if there exists a constant
1 α >
such that
ln()(||( ,)( ,)||)0,
1, 2,
T
kA k
M yxMyx
k
αβ λτ ++ +≤
=
(9)
then system (4) and (2) are global asymptotic synchroni-
zation.
4. Numerical Simulation
In this section, we present some numerica l simulation s to
demonstrate our results. We choose the parameters of
Lorenz system (1) as
10 σ =
,
28 ρ =
,
83
β =
, then
we can calculate that
28.05
A
λ
=
. By ob serving the Lo-
renz attractor (Figure 1), we can get state variable’s val-
ue ranges:
1
20 20x−≤≤
,
2
0 50x≤≤
,
3
20 20x−≤≤
,
and work out
140
M
λ ≤
. Take the equal impulsive in-
tervals and the equal impulsive feedback gain matrix,
that is, set k
ττ
=，
(,,)
k
BBdiag
µµµ = =
,
1, 2,k=
,
then
2
( 1)
k
βµ = +
. According to the corollary in the
previous section, if
2
0[lnln(1) ]/168.05 τ αµ < ≤−++
,
then system (4) and (2) are global asymptotically synchro-
nization. Let
0.5 µ = −
,
1.1 α =
, then
0 0.0077 τ <≤
.
Figure 2 shows the simulation results in which initial
Figure 2. Synchronization error of Lorenz chaotic system.

Impulsive Control for Synchronization of Lorenz Chaotic System
25
value of drive sys tem (2) is
( 2,0.1,0.1)
T
while initial valu e
of response system (4) is (0.1,5,0.5),
T and
0.5, µ = −
0.007 τ =
. We can see that synchronization error con-
verges to zero quickly.
5. Acknowledgements
This work is supported jointly by the Natural Sciences
Foundation of China under Grant No.60963025, Natural
Scienc es Fo undat ion of Ha ina n Pro vince und er Gr ant No.
110007 and the Start-up fund of Haina n Normal Univer-
sity under Project No. 00203020201.
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