ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO AND HYPERCHAOTIC XU SYSTEMS VIA ACTIVE CONTROL

International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.2, No.3, June 2012
DOI : 10.5121/ijcseit.2012.2306 81
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO
AND HYPERCHAOTIC XU SYSTEMS VIA
ACTIVE CONTROL
Sundarapandian Vaidyanathan
1
Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
Avadi, Chennai-600 062, Tamil Nadu, INDIA
sundarvtu@gmail.com
Abstract
This paper investigates the anti-synchronization of identical hyperchaotic Bao systems (Bao and Liu,
2008), identical hyperchaotic Xu systems (Xu, Cai and Zheng, 2009) and non-identical hyperchaotic Bao
and hyperchaotic Xu systems. Active nonlinear control has been deployed for the anti- synchronization of
the hyperchaotic systems addressed in this paper and the main results have been established using
Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active
nonlinear control method is very effective and convenient to achieve anti-synchronization of identical and
non-identical hyperchaotic Bao and hyperchaotic Xu systems. Numerical simulations have been provided to
validate and demonstrate the effectiveness of the anti-synchronization results for the hyperchaotic Cao and
hyperchaotic Xu systems.
KEYWORDS
Active Control, Anti-Synchronization, Hyperchaos, Hyperchaotic Bao System, Hyperchaotic Xu System.
1. INTRODUCTION
Chaos is a complex nonlinear phenomenon. The first chaotic attractor was discovered by Lorenz
([1], 1963), when he was studying the atmospheric convection in the weather models. Since then,
chaos has been received extensive investigations and chaos phenomenon has been observed in a
variety of fields including physical systems [2], chemical systems [3], ecological systems [4],
secure communications [5-6], etc.
The seminal work on chaos synchronization problem was published by Pecora and Carroll ([7],
1990). From then on, chaos synchronization has been extensively and intensively studied in the
last three decades [8-30].
In most of the anti-synchronization approaches, the master-slave or drive-response formalism is
used. If a particular chaotic system is called the master or drive system and another chaotic
system is called the slave or response system, then the idea of anti-synchronization is to use the
output of the master system to control the slave system so that the states of the slave system have
the same amplitude but opposite signs as the states of the master system asymptotically. Thus,
when anti-synchronization is achieved, the sum of the states of the master and slave systems tends
to zero asymptotically with time.

82
In the recent years, various schemes have been deployed for chaos synchronization such as PC
method [2], OGY method [8], active control [9-12], adaptive control [13-15], backstepping
design [16], sampled-data feedback [17], sliding mode control [18-20], etc. Recently, active
control method has been applied to achieve anti-synchronization of two identical chaotic systems
[21-22].
Hyperchaotic system is usually defined as a chaotic system with at least two positive Lyapunov
exponents, implying that its dynamics are expanded in several different directions simultaneously.
Thus, the hyperchaotic systems have more complex dynamical behaviour which can be used to
improve the security of a chaotic communication system. Hence, the theoretical design and circuit
realization of various hyperchaotic signals have become important research topics [23-27].
In this paper, we use active control to derive new results for the anti-synchronization of identical
hyperchaotic Bao systems ([28], 2008), identical hyperchaotic Xu systems ([29], 2009) and non-
identical hyperchaotic Bao and hyperchaotic Xu systems.
This paper is organized as follows. In Section 2, we describe the problem statement and our
methodology using Lyapunov stability theory. In Section 3, we provide a description of the
hyperchaotic systems addressed in this paper. In Section 4, we discuss the anti-synchronization of
identical hyperchaotic Bao systems (2008) using active nonlinear control. In Section 5, we
discuss the anti-synchronization of identical hyperchaotic Xu systems (2009) using active
nonlinear control. In Section 6, we discuss the anti-synchronization of hyperchaotic Bao and
hyperchaotic Xu systems using active nonlinear control. In Section 7, we summarize the main
results obtained in this paper.
2. PROBLEM STATEMENT AND OUR METHODOLOGY
As the master or drive system, we consider the chaotic system described by
( ),x Ax f x= +& (1)
where n
x∈R is the state vector, A is the n n× matrix of system parameters and : n n
f →R R is
the nonlinear part of the system.
As the slave or response system, we consider the following chaotic system described by
( ) ,y By g y u= + +& (2)
where n
y ∈R is the state of the slave system, B is the n n× matrix of system parameters,
: n n
g →R R is the nonlinear part of the system and u is the active controller to be designed.
If A B= and ,f g= then x and y are the states of two identical chaotic systems. If A B≠ or
,f g≠ then x and y are the states of two different chaotic systems.
For the anti-synchronization of the chaotic systems (1) and (2) using active control, we design a
state feedback controller ,u which anti-synchronizes the states of the master system (1) and the
slave system (2) for all initial conditions (0), (0) .n
x y ∈R
Thus, we define the anti-synchronization error as

83
,e y x= + (3)
Hence, the error dynamics is obtained as
( ) ( )e By Ax g y f x u= + + + +& (4)
Thus, the anti-synchronization problem is essentially to find a feedback controller (active
controller) u so as to stabilize the error dynamics (4) for all initial conditions, i.e.
lim ( ) 0,
t
e t
→∞
= for all (0) n
e ∈R (5)
We use the Lyapunov stability theory as our methodology.
We take as a candidate Lyapunov function
( ) ,T
V e e Pe= (6)
where P is a positive definite matrix. Note that : n
V R→R is a positive definite function by
construction.
If we find a feedback controller u so that
( ) ,T
V e e Qe= −& (7)
where Q is a positive definite matrix, then : n
V →& R R is a negative definite function.
Thus, by Lyapunov stability theory [25], the error dynamics (4) is globally exponentially stable.
Hence, the states of the master system (1) and the slave system (2) will be globally and
exponentially anti-synchronized for all values of the initial conditions (0), (0) .n
x y ∈R
3. SYSTEMS DESCRIPTION
The hyperchaotic Bao system ([28], 2008) is described by the 4D dynamics
1 2 1 4
2 2 1 3
3 1 2 3
4 1 2 3
( )x a x x x
x cx x x
x x x bx
x x dx xε
= − +
= −
= −
= +
&
&
&
&
(8)
where 1 2 3 4, , ,x x x x are the state variables and , , , ,a b c d ε are positive constants.
The four-dimensional Bao system (8) is hyperchaotic when
36, 3, 20, 0.1a b c d= = = = and 21.ε =

84
The phase portrait of the hyperchaotic Bao system is shown in Figure 1.
Figure 1. Phase Portrait of the Hyperchaotic Bao System
The hyperchaotic Xu system ([29], 2009) is described by the 4D dynamics
1 2 1 4
2 1 1 3
3 3 1 2
4 1 3 2
( )x x x x
x x rx x
x x lx x
x x x mx
α
β
γ
= − +
= +
= − −
= −
&
&
&
&
(9)
where 1 2 3 4, , ,x x x x are the state variables and , , , ,l mα β γ are positive constants.
The four-dimensional Xu system (9) is hyperchaotic when
10, 40, 2.5, 16, 1r lα β γ= = = = = and 2.m =
The phase portrait of the hyperchaotic Xu system is shown in Figure 2.

85
Figure 2. Phase Portrait of the Hyperchaotic Xu System
3. ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO SYSTEMS
In this section, we discuss the anti-synchronization of identical hyperchaotic Bao systems ([28],
2008).
As the master system, we consider the hyperchaotic Bao system described by
1 2 1 4
2 2 1 3
3 1 2 3
4 1 2 3
( )x a x x x
x cx x x
x x x bx
x x dx xε
= − +
= −
= −
= +
&
&
&
&
(10)
As the slave system, we consider the controlled hyperchaotic Bao dynamics described by
1 2 1 4 1
2 2 1 3 2
3 1 2 3 3
4 1 2 3 4
( )y a y y y u
y cy y y u
y y y by u
y y dy y uε
= − + +
= − +
= − +
= + +
&
&
&
&
(11)
where 1 2 3 4, , ,y y y y are the state variables and 1 2 3 4, , ,u u u u are the active controls.

86
The anti-synchronization error is defined as
1 1 1
2 2 2
3 3 3
4 4 4
e y x
e y x
e y x
e y x
= +
= +
= +
= +
(12)
A simple calculation gives the error dynamics
1 2 1 4 1
2 2 1 3 1 3 2
3 3 1 2 1 2 3
4 1 2 3 2 3 4
( )
( )
e a e e e u
e ce y y x x u
e be y y x x u
e e d y y x x uε
= − + +
= − − +
= − + + +
= + + +
&
&
&
&
(13)
We consider the active nonlinear controller defined by
1 2 1 4 1 1
2 2 1 3 1 3 2 2
3 3 1 2 1 2 3 3
4 1 2 3 2 3 4 4
( )
( )
u a e e e k e
u ce y y x x k e
u be y y x x k e
u e d y y x x k eε
= − − − −
= − + + −
= − − −
= − − + −
(14)
where 1 2 3 4, , ,k k k k are positive constants.
Substitution of (14) into (13) yields the linear error dynamics
1 1 1 2 2 2 3 3 3 4 4 4, , ,e k e e k e e k e e k e= − = − = − = −& & & & (15)
We consider the quadratic Lyapunov function defined by
( )2 2 2 2
1 2 3 4
1 1
( ) ,
2 2
T
V e e e e e e e= = + + + (16)
which is a positive definite function on 4
.R
Next, we establish the main result of this section using Lyapunov stability theory.
Theorem 1. The identical hyperchaotic Bao systems (10) and (11) are globally and
exponentially anti-synchronized with the active nonlinear controller (13).
Proof. Differentiating (16) along the trajectories of the error system (15), we get
2 2 2 2
1 1 2 2 3 3 4 4( ) ,V e k e k e k e k e= − − − −& (17)
which is a negative definite function on 4
R

87
Thus, by Lyapunov stability theory [30], the error dynamics (15) is globally
exponentially stable. This completes the proof.
Numerical Simulations:
For the numerical simulations, the fourth order Runge-Kutta method with initial time-step
8
10h −
= is used to solve the two systems of differential equations (10) and (11) with the
active nonlinear controller (14). We take the gains as 5ik = for 1,2,3,4.i =
The parameters of the identical hyperchaotic Bao systems (8) and (9) are selected as
36, 3, 20, 0.1a b c d= = = = and 21.ε =
The initial values for the master system (8) are taken as
1 2 3 4(0) 5, (0) 2, (0) 14, (0) 20x x x x= = − = =
and the initial values for the slave system (9) are taken as
1 2 3 4(0) 24, (0) 17, (0) 18, (0) 22y y y y= = = − = −
Figure 4 depicts the anti-synchronization of the identical hyperchaotic Bao systems (10) and (11).
Figure 4 depicts the time-history of the anti-synchronization error.
Figure 3. Anti-Synchronization of Identical Hyperchaotic Bao Systems

88
Figure 4. Time-History of Anti-Synchronization Error 1 2 3 4, , ,e e e e
4. ANTI-SYNCHRONIZATION OF HYPERCHAOTIC XU SYSTEMS
In this section, we discuss the anti-synchronization of identical hyperchaotic Xu systems ([29],
2009).
As the master system, we consider the hyperchaotic Xu system described by
1 2 1 4
2 1 1 3
3 3 1 2
4 1 3 2
( )x x x x
x x rx x
x x lx x
x x x mx
α
β
γ
= − +
= +
= − −
= −
&
&
&
&
(18)
where 1 2 3 4, , ,x x x x are the state variables and , , , , ,r l mα β γ are positive constants.
As the slave system, we consider the controlled hyperchaotic Xu dynamics described by
1 2 1 4 1
2 1 1 3 2
3 3 1 2 3
4 1 3 2 4
( )y y y y u
y y ry y u
y y ly y u
y y y my u
α
β
γ
= − + +
= + +
= − − +
= − +
(19)

89
where 1 2 3 4, , ,y y y y are the state variables and 1 2 3 4, , ,u u u u are the active controls.
1 1 1
2 2 2
3 3 3
4 4 4
e y x
e y x
e y x
e y x
= +
= +
= +
= +
(20)
1 2 1 4 1
2 1 1 3 1 3 2
3 3 1 2 1 2 3
4 2 1 3 1 3 4
( )
( )
( )
e e e e u
e e r y y x x u
e e l y y x x u
e me y y x x u
α
β
γ
= − + +
= + + +
= − − + +
= − + + +
&
&
&
&
(21)
1 2 1 4 1 1
2 1 1 3 1 3 2 2
3 3 1 2 1 2 3 3
4 2 1 3 1 3 4 4
( )
( )
( )
u e e e k e
u e r y y x x k e
u e l y y x x k e
u me y y x x k e
α
β
γ
= − − − −
= − − + −
= + + −
= − − −
(22)
where 1 2 3 4, , ,k k k k are positive constants.
1 1 1 2 2 2 3 3 3 4 4 4, , ,e k e e k e e k e e k e= − = − = − = −& & & & (23)
( )2 2 2 2
1 2 3 4
1 1
( ) ,
2 2
T
V e e e e e e e= = + + + (24)
.R
Theorem 2. The identical hyperchaotic Xu systems (18) and (19) are globally and
exponentially anti-synchronized with the active nonlinear controller (22).

90
2 2 2 2
1 1 2 2 3 3 4 4( ) ,V e k e k e k e k e= − − − −& (25)
R
exponentially stable.
Hence, the states of the identical hyperchaotic Xu systems (18) and (19) are globally and
exponentially anti-synchronized for all initial conditions (0), (0) .n
x y ∈R
This completes the proof.
Numerical Simulations:
6
10h −
active nonlinear controller (22).
We take the feedback gains as
1 2 3 45, 5, 5, 5k k k k= = = =
The parameters of the identical hyperchaotic Xu systems (18) and (19) are selected as
10, 40, 2.5, 16, 1r lα β γ= = = = = and 2.m =
1 2 3 4(0) 12, (0) 14, (0) 7, (0) 21x x x x= = − = − =
1 2 3 4(0) 26, (0) 10, (0) 18, (0) 4y y y y= = = = −
Figure 5 depicts the anti-synchronization of the identical hyperchaotic Xu systems (18) and (19).
Figure 6 depicts the time-history of the anti-synchronisation error 1 2 3 4, , , .e e e e

91
Figure 5. Anti-Synchronization of Identical Hyperchaotic Xu Systems
Figure 6. Time-History of the Anti-Synchronization Error 1 2 3 4, , ,e e e e

92
5. ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO AND
HYPERCHAOTIC XU SYSTEMS
In this section, we consider the anti-synchronization of non-identical hyperchaotic Bao system
([28], 2008) and hyperchaotic Xu system ([29], 2009).
As the master system, we consider the hyperchaotic Bao system described by
1 2 1 4
2 2 1 3
3 1 2 3
4 1 2 3
( )x a x x x
x cx x x
x x x bx
x x dx xε
= − +
= −
= −
= +
&
&
&
&
(26)
As the slave system, we consider the controlled hyperchaotic Xu dynamics described by
1 2 1 4 1
2 1 1 3 2
3 3 1 2 3
4 1 3 2 4
( )y y y y u
y y ry y u
y y ly y u
y y y my u
α
β
γ
= − + +
= + +
= − − +
= − +
(27)
where 1 2 3 4, , ,y y y y are the state variables, , , , , ,r l mα β γ are positive constants and
1 2 3 4, , ,u u u u are the active controls.
1 1 1
2 2 2
3 3 3
4 4 4
e y x
e y x
e y x
e y x
= +
= +
= +
= +
(28)
1 2 1 4 2 1 1
2 1 1 2 1 3 1 3 2
3 3 3 1 2 1 2 3
4 2 1 2 1 3 2 3 4
( ) ( )( )
( )
e e e e a x x u
e e x cx ry y x x u
e e b x ly y x x u
e me x mx y y dx x u
α α
β β
γ γ
ε
= − + − − − +
= − + + − +
= − − − − + +
= − + + + + +
&
&
&
&
(29)

93
1 2 1 4 2 1 1 1
2 1 1 2 1 3 1 3 2 2
3 3 3 1 2 1 2 3 3
4 2 1 2 1 3 2 3 4 4
( ) ( )( )
( )
u e e e a x x k e
u e x cx ry y x x k e
u e b x ly y x x k e
u me x mx y y dx x k e
α α
β β
γ γ
ε
= − − − + − − −
= − + − − + −
= + − + − −
= − − − − −
(30)
1 1 1
2 2 2
3 3 3
4 4 4
e k e
e k e
e k e
e k e
= −
= −
= −
= −
&
&
&
&
(31)
( )2 2 2 2
1 2 3 4
1 1
( ) ,
2 2
T
V e e e e e e e= = + + + (32)
.R
Theorem 3. The non-identical hyperchaotic Bao system (26) and hyperchaotic Xu system
(27) are globally and exponentially anti-synchronized with the active nonlinear
controller (29).
2 2 2 2
1 1 2 2 3 3 4 4( ) ,V e k e k e k e k e= − − − −& (33)
R since α and γ are positive constants.
exponentially stable.
Hence, it follows that the states of the hyperchaotic Bao system (26) and hyperchaotic Xu
system (27) are globally and exponentially anti-synchronized for all initial conditions
(0), (0) .n
x y ∈R
This completes the proof.

94
Numerical Simulations
6
10h −
active nonlinear controller (30).
We take the feedback gains as
1 2 35, 5, 5k k k= = = and 4 5.k =
The parameters of the identical hyperchaotic Bao system (24) are selected as
36, 3, 20, 0.1a b c d= = = = and 21.ε =
The parameters of the identical hyperchaotic Xu system (25) are selected as
10, 40, 2.5, 16, 1r lα β γ= = = = = and 2.m =
1 2 3 4(0) 25, (0) 6, (0) 9, (0) 14x x x x= = − = = −
1 2 3 4(0) 12, (0) 15, (0) 20, (0) 8y y y y= = = − = −
Figure 7 depicts the anti-synchronization of the non-identical hyperchaotic systems, viz.
hyperchaotic Bao system (26) and hyperchaotic Xu system (28).
Figure 8 depicts the time-history of the anti-synchronization error.
6. CONCLUSIONS
In this paper, using the active control method, new results have been derived for the anti-
synchronization for the following pairs of hyperchaotic systems:
(A) Identical hyperchaotic Bao systems (2008)
(B) Identical hyperchaotic Xu systems (2009)
(C) Non-identical hyperchaotic Bao and hyperchaotic Xu systems.
The anti-synchronization results derived in this paper have been proved using Lyapunov stability
theory. Since the Lyapunov exponents are not required for these calculations, the active control
method is very convenient and efficient for the anti- synchronization of identical and non-
identical hyperchaotic Bao and hyperchaotic Xu systems. Numerical simulation results have been
presented to illustrate the anti-synchronization results derived in this paper.

95
Figure 7. Anti-Synchronization of Hyperchaotic Bao and Hyperchaotic Xu Systems
Figure 8. Time-History of the Anti-Synchronization Error 1 2 3 4, , ,e e e e

96
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Author
Dr. V. Sundarapandian earned his Doctor of Science degree in Electrical and Systems
Engineering from Washington University, Saint Louis, Missouri, USA. He is currently
Professor in the Research and Development Centre at Vel Tech Dr. RR & Dr. SR
Technical University, Chennai, Tamil Nadu, India. He has published over 260
publications in refereed International journals and over 170 papers in National and
International Conferences. He is the Editor-in-Chief of the AIRCC journals -
International Journal of Instrumentation and Control Systems, International Journal of
Control Systems and Computer Modelling and International Journal of Information
Technology, Control and Automation. His research interests are Linear and Nonlinear Control Systems,
Chaos Theory and Control, Soft Computing, Optimal Control, Process Control, Operations Research,
Mathematical Modelling, Scientific Computing using SCILAB/MATLAB. He has delivered many Keynote
Lectures on Chaos Theory and Control Engineering.

ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO AND HYPERCHAOTIC XU SYSTEMS VIA ACTIVE CONTROL

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ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO AND HYPERCHAOTIC XU SYSTEMS VIA ACTIVE CONTROL