Applied Mathematics, 2013, 4, 1-6
Published Online November 2013 (http://www.scirp.org/journal/am)
http://dx.doi.org/10.4236/am.2013.411A2001
Open Access AM
Controlling Unstable Discrete Chaos and Hyperchaos
Systems*
Xin Li, Suping Qian
School of Mathematics and Statistics, Changshu Institute of Technology, Changshu, China
Received July 23, 2013; revised August 23, 2013; accepted August 30, 2013
Copyright © 2013 Xin Li, Suping Qian. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
A method is introduced to stabilize unstable discrete systems, which does not require any adjustable control parameters
of the system. 2-dimension discrete Fold system and 3-dimension discrete hyperchaotic system are stabilized to fixed
points respectively. Numerical simulations are then provided to show the effectiveness and feasibility of the proposed
chaos and hyperchaos controlling scheme.
Keywords: 2-Dimension Discrete Fold System; 3-Dimension Discrete Hyperchaotic System; Lyapunov Stability
Theory; Controlling Chaos
1. Introduction
In many engineering and other practical problems, chaos
is undesirable and therefore needs to be controlled. Thus,
a large number of control methods have been developed
and are being applied to real systems [1-10]. The method
given by Ott, Grebogi and Yorke (OGY) [1] is to stabi-
lize an unstable orbit in the neighborhood of a hyperbolic
fixed point by forcing the orbit onto the stable manifold.
The method proposed by Romeiras, Grebogi, Ott and
Dayawansa (RGOD) [2] is not yet suitable for control-
ling hyperchaos since the method changes the stability
property of the fixed point completely. However, the
method proposed by Yang, Liu and Jian-min Mao [11]
gives a new idea to stabilize unstable orbits even if there
is no preexisting stable manifold nearby. For a finite-
dimensional dynamical system, whose governing equa-
tions may or may not be analytically available, Yang, Liu
and Mao show how to stabilize an unstable orbit in a
neighborhood of a “fully” unstable fixed point. The ad-
vantage of this method is: only one of the unstable direc-
tions is to be stabilized via time-dependent adjustments
of control parameters. The parameter adjustments can be
optimized. Recently, Bu [12] and Li [13] stabilized un-
stable discrete systems by a method which does not re-
quire any adjustable control parameters of the system.
Consider an n-dimensional dynamical system defined
by

1, kk
x
Fx
 (1)
where n
x
Ris an n-dimensional vector, F is a nonlin-
ear vector valued function. Let xf be the fixed point of the
map (1). To stabilize a chaotic orbit to this fixed point,
we take a variable feedback control described by




1kk kk
x
Fx MFx x
 , (2)
Define an infinitesimal deviation of xk from xf as
kkf
x
xx


. Then from Equation (2), one has

1, kk k
x
JxMJ Ix  
 (3)
where

kf
kxx
JFx

  is the Jacobian matrix of the
original system F evaluated at the fixed point xf and I is
the nn

identity matrix. The goal of controlling here is
to make li
km 0
k
x 
 .

For this aim, one requires
1, kk
x
Qx




(4)
where Q is an nn

matrix and takes the form
1
2
0,
0
q
Qq




(5)
where are constants. Substituting Equation (4) and Equa-
tion (5) into Equation (3) and eliminating k
x

, choosing
one special form of the matrix one
have

1,1QqI ,q ,
*The work was supported by the Special Funds of the National Natural
Science Foundation of China (Grant No. 11141003).

X. LI, S. P. QIAN
2

1.MqIJJI 
 (6)
This needs to use numeric computation to do. There-
fore the above scheme based on the symbolic numeric
computation is summarized as follows.
Input:
1) The unstable system (1);
2) The system (2) with a variable feedback controller;
3) Choose the initial values of systems (2).
Output:
1)

;
kf
kxx
JFx

 
2) M in (6);
3) Deduce the system (2) according to the results (6);
4) Numerical simulations of the states xk when
. k
In this paper, we use the method to stabilize 2-dimen-
sion discrete Fold system [14] and 3-dimension discrete
hyperchaotic system due to Wang [15] to fixed points
respectively.
2. Stabilizing 2-Dimensional Discrete Fold
System
Using the above method, we stabilize 2-dimension dis-
crete Fold system presented as:



 
121
2
21
1,
1

x
kxkxk
xk xk


 




(7)
where
 ,  are the parameters, and we choose  = −0.1,
 = −1.7.
In the following based on the method mentioned above,
we will make the Fold system stabilize at the fixed point.
It is easy to get the two fixed points (1.965097170,
2.161606887), and (−0.8650971698, −0.9516068868) of
Equation (7). The Jacobian matrix corresponding the
fixed point


12
,
f
f
x
x is
1
1.
20
f
Jx





(8)
From (6) one can have

1
1
11
11
21
12 12
,
212
12 12
f
f
ff
ff
qx q
xx
M1f
x
qxq
xx








 






 

q
(9)
here we choose




12
,1.965097170, 2.161606887
ff
xx  as our re-
search object. Choosing the parameter 0.5,q

and
0.3q

respectively, one gets
1
2
1.1766663130.1766663133 ,
0.6943329445 1.194332944
1.2473328380.2473328387 .
0.9720661223 1.272066122
M
M










(10)
From (2), respectively substitute (10) into (7), we can
obtain
 




 
 

121 211
2
12
2
21 211
2
12
1( )1.76666313
0.1766663133 ,
1 0.6943329445
1.194332944 ,
x
k xkxkxkxkxk
xk xk
x kxkx kxkxk
xk xk




 








 









(11)
and
 







 

121 211
2
12
2
21 211
2
12
1 1.247332838
0.2473328387 ,
1 0.9720661223
1.272066122 .
x
kxkxk xkxkxk
xk xk
x kxkx kxkxk
xk xk




 








 





(12)
In the following, we give the orbit of 2-dimension dis-
crete Fold system before being stabilized in Figure 1(a).
And in Figure 1(b), three orbits starting from different
initial points are stabilized to the fixed point
(1.965097170, 2.161606887). It is shown that the unsta-
ble orbit is stabilized to the desired fixed point mono-
tonically. Then the orbits stabilized of xk and yk versus tk
are depicted contrasting with the ones before being stabi-
lized in Figures 2 and 3, respectively.
3. Stabilizing 3-Dimension Discrete
Hyperchaotic System
In this section, we consider 3-dimension discrete hyper-
haotic system c
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X. LI, S. P. QIAN 3
(a) (b)
Figure 1. (a) 2-dimension discrete Fold system; (b) Three orbits starting from different initial points are stabilized to the fixed
point (1.965097170, 2.161606887), for q = 0.5.
(a) (b)
Figure 2. (a) x1(k) versus k before being stabilized; (b) x1(k) versus k after being stabilized for q = 0.5 and q = 0.3.
(a) (b)
Figure 3. (a) x2(k) versus k before being stabilized; (b) x2(k) versus k after being stabilized for q = 0.5 and q = 0.3.
 
 
 
 
13241
223112
3562373
11,
1
11
ykaykayk
ykaykaykyk
ykaa ykykayk


 
 
 

 

,
,
(13)
which was derived from the generalized Rössler system
via a first-order difference algorithm [15].
We take the fixed point (0.09610764055,
0.4420951466, 0.9130225853) as our research object,
that is


123
,,
f
ff
yyy = (0.09610764055, 0.4420951466,
0.9130225853).
Following the procedure above, the Jacobian matrix of
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X. LI, S. P. QIAN
4
map (13) is
43
12
63 62 7
10
1
0
ff
aa
Ja a
ayay a










. (14)
1





Here we let a1 = −1.9, a2 = 0.2, a3 = 0.5, a4 = −2.3, a5
= 2, a6 = −0.6, a7 = −1.9 and 1.


From (6), choosing 0.5q

and respec-
tively, the matrix M at the fixed point (0.09610764055,
0.4420951466, 0.9130225853) is correspondingly ob-
tained as following
0.2q
0.9762747382 0.2344378413 0.02165450397
0.89086379700.07841407030.09961071835 ,
0.22538998420.2728405070 0.7942822117
M








(15)
and
0.94305937220.5626508191 0.05197080956
2.1380731131.5881937680.2390657240 .
0.54093596150.6548172166 0.562773086
M








(16)
From (2), respectively substitute (15) and (16) into (13), one can obtain
 
 
 
1324111
2311562373
223112324
1 0.02372526180.97627473820.2344378412
0.02165450397 ,
1 1.078414070.890863797
0.099610
yka ykaykykyk
ayk aykaaykyk ayk
ykaykayk ykaykayk

 
 
 

 

 
 

 
    
 
562 373
3562373
4123 113
71835 ,
1 0.205717788310.2253899842
0.2728405070.7942822117 ,
aaykykayk
1




32
y
kaaykykayka
aykayk aykyk
 
 











 


 
 
 

yk
41




32
(17)
and
  
 
 
1324111
2311562373
22311232
1 0.05694062780.94305937220.5626508191
0.05197080956 ,
1 2.5881937682.138073113
0.23906
yka ykaykykyk
ayk aykaaykyk ayk
ykayk aykykayk ayk

 

  

 


 
 


 
  
 
562373
3562373
4123 113
5724 ,
1 0.493722691410.5409359615
0.65481721660.5062773086 ,
aaykykayk
ykaa ykykaykayk
aykayk aykyk
 
 










 


 
 
 

(18)
The numerical results are shown in the followed fig-
ures. The orbit of 3-dimension discrete time hyperchaotic
system is given by Figure 4(a). In Figure 4(b), three
orbits starting from different initial points are stabilized
to the fixed point (0.09610764055, 0.4420951466,
0.9130225853).
We can also get the result that 3-dimension discrete
time hyperchaotic system is stabilized. In Figures 5-7,
the stabilized orbits of
 
123
,,
y
kykyk versus tk
are plotted contrasting with the ones before being stabi-
lized, respectively.
4. Conclusion
In summary, we have introduced a method to stabilize
unstable discrete systems, which does not require any
adjustable control parameters of the system. 2-dimension
discrete Fold system and 3-dimension discrete hypercha-
otic system are stabilized to fixed points respectively.
From the process we finish, it is shown that stabilizing
the unstable discrete systems neither requires a prior
analytical knowledge of the underlying system nor any
adjustable control parameters in advance. Numerical
imulations are then provided to show the effectiveness s
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X. LI, S. P. QIAN 5
(a) (b)
Figure 4. (a) 3-dimension discrete time hyperchaotic system; (b) Three orbits starting from different initial points are stabi-
lized to the fixed point (0.09610764055, 0.4420951466, 0.9130225853), for q = 0.5.
(a) (b)
Figure 5. (a)


y
k
1 versus k before being stabilized; (b)


y
k
1 versus k after being stabilized.
(a) (b)
Figure 6. (a)


2
y
k versus k before being stabilized; (b)


2
y
k versus k after being stabilized.
(a) (b)
Figure 7. (a)


3
yk versus k before being stabilized; (b)


3
yk versus k after being stabilized.
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X. LI, S. P. QIAN
6
and feasibility of the proposed chaos and hyperchaos
controlling Scheme.
5. Acknowledgements
The authors are grateful to the reviewers for their valu-
able comments and suggestions.
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