Controlling Unstable Discrete Chaos and Hyperchaos Systems

doi:10.4236/am.2013.411A2001

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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

Email: [email protected]

Received July 23, 2013; revised August 23, 2013; accepted August 30, 2013

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



 (1)

where n

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

Fx MFx x

 , (2)

Define an infinitesimal deviation of xk from xf as

kkf





. Then from Equation (2), one has



kk k

JxMJ Ix

 

 (3)

where



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



 .



For this aim, one requires







(4)

where Q is an nn



matrix and takes the form









(5)

where are constants. Substituting Equation (4) and Equa-

tion (5) into Equation (3) and eliminating k



, 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



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:



;

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



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





x is











(8)

From (6) one can have



12 12

212

12 12

qx q

M1f

qxq

















 













 



(9)

here we choose









,1.965097170, 2.161606887

xx  as our re-

search object. Choosing the parameter 0.5,q



and

0.3q



respectively, one gets

1.1766663130.1766663133 ,

0.6943329445 1.194332944

1.2473328380.2473328387 .

0.9720661223 1.272066122





















(10)

From (2), respectively substitute (10) into (7), we can

obtain

 









 

 



121 211

21 211

1( )1.76666313

0.1766663133 ,

1 0.6943329445

1.194332944 ,

k xkxkxkxkxk

xk xk

x kxkx kxkxk

xk xk









 

















 



















(11)

and

 















 



121 211

21 211

1 1.247332838

0.2473328387 ,

1 0.9720661223

1.272066122 .

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

Open Access AM

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,

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

yyy = (0.09610764055, 0.4420951466,

0.9130225853).

Following the procedure above, the Jacobian matrix of

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X. LI, S. P. QIAN

map (13) is

63 62 7

Ja a

ayay a



















(14)







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















(15)

and

0.94305937220.5626508191 0.05197080956

2.1380731131.5881937680.2390657240 .

0.54093596150.6548172166 0.562773086















(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









kaaykykayka

aykayk aykyk

 

 















 





 

 

 











(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

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)





1 versus k before being stabilized; (b)





1 versus k after being stabilized.

(a) (b)

Figure 6. (a)





k versus k before being stabilized; (b)





k versus k after being stabilized.

(a) (b)

Figure 7. (a)





yk versus k before being stabilized; (b)





yk versus k after being stabilized.

Open Access AM

X. LI, S. P. QIAN

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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