Anti-Synchronization Of Four-Scroll Chaotic Systems Via Sliding Mode Control

International Journal of Information Technology, Control and Automation (IJITCA) Vol.1, No.1, October 2011
1
ANTI-SYNCHRONIZATION OF FOUR-SCROLL
CHAOTIC SYSTEMS VIA SLIDING MODE CONTROL
Sundarapandian Vaidyanathan1
1
Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
Avadi, Chennai-600 062, Tamil Nadu, INDIA
sundarvtu@gmail.com
ABSTRACT
In this paper, new results are derived for the anti-synchronization of identical Liu-Chen four-scroll
chaotic systems (Liu and Chen, 2004) and identical Lü-Chen-Cheng four-scroll chaotic systems (Lü,
Chen and Cheng, 2004) by sliding mode control. The stability results derived in this paper for the anti-
synchronization of identical four-scroll chaotic systems are established using sliding mode control and
Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the
sliding mode control method is very effective and convenient to achieve anti-synchronization of the
identical four-scroll chaotic systems. Numerical simulations are shown to illustrate and validate the anti-
synchronization schemes derived in this paper for the identical four-scroll systems.
KEYWORDS
Sliding Mode Control, Chaos, Chaotic Systems, Anti-Synchronization, Four-Scroll Systems, Liu-Chen
Systems, Lü-Chen-Cheng Systems.
1. INTRODUCTION
Chaotic systems are dynamical systems described by nonlinear differential equations, which are
strongly sensitive to initial conditions. The sensitive nature of chaotic systems is commonly
called as the butterfly effect [1]. Thus, the behaviour of a chaotic system is highly unpredictable
even if the system mathematical description is deterministic. The first three-dimensional chaotic
system was discovered by Lorenz in 1963 [2], when he was studying weather models. From
then on, many Lorenz-like chaotic systems such as Rössler system [3], Chen system [4], Lü
system [5] and Liu system [6] were reported and analyzed.
Synchronization of chaotic systems is a phenomenon which may occur when two or more
chaotic oscillators are coupled or when a chaotic oscillator drives another chaotic oscillator.
Because of the butterfly effect which causes the exponential divergence of the trajectories of
two identical chaotic systems started with nearly the same initial conditions, synchronizing two
chaotic systems is seemingly a very challenging problem.
In most of the chaos 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 the synchronization is to use the
output of the master system to control the slave system so that the output of the slave system
tracks the output of the master system asymptotically.
Since the pioneering work by Pecora and Carroll ([7], 1990), chaos synchronization problem
has been studied widely in the literature [7-27]. Chaos theory has been applied to a variety of

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fields such as physical systems [8], chemical systems [9], ecological systems [10], secure
communications [11-13], etc.
In the last two decades, various control schemes have been successfully applied for chaos
synchronization such as PC method [7], OGY method [14], active control method [15-18],
adaptive control method [19-24], time-delay feedback method [25], backstepping design
method [26], sampled-data feedback method [27], sliding mode control method [28-32], etc.
In this paper, we deploy sliding mode control method for the anti-synchronization of identical
Liu-Chen four-scroll chaotic systems (Liu and Chen, [33], 2004), identical Lü-Chen-Cheng
four-scroll systems ([34], Lü, Chen and Cheng, 2004). In robust control systems, the sliding
mode control method is often adopted due to its inherent advantages of easy realization, fast
response and good transient performance as well as its insensitivity to parameter uncertainties
and external disturbances.
This paper has been organized as follows. In Section 2, we describe the problem statement and
our methodology using sliding mode control (SMC). In Section 3, we give a description of the
four-scroll chaotic systems addressed in this paper. In Section 4, we discuss the anti-
synchronization of identical Liu-Chen four-scroll chaotic systems using sliding mode control. In
Section 5, we discuss the anti-synchronization of identical Lü-Chen-Cheng four-scroll chaotic
systems. In Section 6, we summarize the main results obtained in this paper.
2. PROBLEM STATEMENT AND OUR METHODOLOGY USING SMC
In this section, we describe the problem statement for the anti-synchronization for identical
chaotic systems and our methodology using sliding mode control (SMC).
Consider the chaotic system described by
( )x Ax f x= + (1)
where n
x∈R is the state of the system, A is the n n× matrix of the system parameters and
: n n
f →R R is the nonlinear part of the system.
We consider the system (1) as the master or drive system.
As the slave or response system, we consider the following chaotic system described by the
dynamics
( )y Ay f y u= + + (2)
where n
y ∈R is the state of the system and m
u ∈R is the nonlinear controller to be designed.
If we define the anti-synchronization error as
,e y x= + (3)
then the error dynamics is obtained as
( , ) ,e Ae x y u= + + (4)
where

3
( , ) ( ) ( ).x y f y f x = + (5)
The objective of the global chaos synchronization problem is to find a controller u such that
lim ( ) 0
t
e t
→∞
= for all (0) .n
e ∈R
To solve this problem, we first define the control u as
( , )u x y Bv= − + (6)
where B is a constant gain vector selected such that ( , )A B is controllable.
Substituting (6) into (4), the error dynamics simplifies to
e Ae Bv= + (7)
which is a linear time-invariant control system with single input .v
Thus, the original global chaos synchronization problem can be replaced by an equivalent
problem of stabilizing the zero solution 0e = of the system (7) by a suitable choice of the
sliding mode control. In the sliding mode control, we define the variable
1 1 2 2( ) n ns e Ce c e c e c e= = + + + (8)
where [ ]1 2 nC c c c=  is a constant vector to be determined.
In the sliding mode control, we constrain the motion of the system (7) to the sliding manifold
defined by
{ }| ( ) 0n
S x s e= ∈ =R
which is required to be invariant under the flow of the error dynamics (7).
When in sliding manifold ,S the system (7) satisfies the following conditions:
( ) 0s e = (9)
which is the defining equation for the manifold S and
( ) 0s e = (10)
which is the necessary condition for the state trajectory ( )e t of (7) to stay on the sliding
manifold .S
Using (7) and (8), the equation (10) can be rewritten as
[ ]( ) 0s e C Ae Bv= + = (11)

4
Solving (11) for ,v we obtain the equivalent control law
1
eq ( ) ( ) ( )v t CB CA e t−
= − (12)
where C is chosen such that 0.CB ≠
Substituting (12) into the error dynamics (7), we obtain the closed-loop dynamics as
1
( )e I B CB C Ae−
 = −  (13)
The row vector C is selected such that the system matrix of the controlled dynamics
1
( )I B CB C A−
 −  is Hurwitz, i.e. it has all eigenvalues with negative real parts. Then the
controlled system (13) is globally asymptotically stable.
To design the sliding mode controller for (7), we apply the constant plus proportional rate
reaching law
sgn( )s q s k s= − − (14)
where sgn( )⋅ denotes the sign function and the gains 0,q > 0k > are determined such that the
sliding condition is satisfied and sliding motion will occur.
From equations (11) and (14), we can obtain the control ( )v t as
[ ]1
( ) ( ) ( ) sgn( )v t CB C kI A e q s−
= − + + (15)
which yields
[ ]
[ ]
1
1
( ) ( ) , if ( ) 0
( )
( ) ( ) , if ( ) 0
CB C kI A e q s e
v t
CB C kI A e q s e
−
−
− + + >
=
− + − <



(16)
Theorem 1. The master system (1) and the slave system (2) are globally and asymptotically
anti-synchronized for all initial conditions (0), (0) n
x y R∈ by the feedback control law
( ) ( , ) ( )u t x y Bv t= − + (17)
where ( )v t is defined by (15) and B is a column vector such that ( , )A B is controllable. Also, the
sliding mode gains ,k q are positive.
Proof. First, we note that substituting (17) and (15) into the error dynamics (4), we obtain the
closed-loop error dynamics as
[ ]1
( ) ( ) sgn( )e Ae B CB C kI A e q s−
= − + + (18)

5
To prove that the error dynamics (18) is globally asymptotically stable, we consider the
candidate Lyapunov function defined by the equation
21
( ) ( )
2
V e s e= (19)
which is a positive definite function on .n
R
Differentiating V along the trajectories of (18) or the equivalent dynamics (14), we get
2
( ) ( ) ( ) sgn( )V e s e s e ks q s s= = − −  (20)
which is a negative definite function on .n
R
This calculation shows that V is a globally defined, positive definite, Lyapunov function for the
error dynamics (18), which has a globally defined, negative definite time derivative .V
Thus, by Lyapunov stability theory [22], it is immediate that the error dynamics (18) is globally
asymptotically stable for all initial conditions (0) .n
e ∈R
This means that for all initial conditions (0) ,n
e R∈ we have
lim ( ) 0
t
e t
→∞
= (21)
Hence, it follows that the master system (1) and the slave system (2) are globally and
asymptotically anti-synchronized for all initial conditions (0), (0) .n
x y ∈R
This completes the proof. 
3. SYSTEMS DESCRIPTION
In this section, we describe the four-scroll chaotic systems considered in this paper, viz. the Liu-
Chen chaotic system ([33], 2004) and the Lü-Chen-Cheng chaotic system ([34], 2004).
The Liu-Chen four-scroll chaotic system is described by the dynamics
1 1 2 3
2 2 1 3
3 3 1 2
x ax x x
x bx x x
x cx x x
= −
= − +
= − +



(22)
where 1 2 3, ,x x x are state variables and , ,a b c are positive, constant parameters of the system.
The Liu-Chen system (22) is chaotic when the parameter values are taken as
0.4, 12, 5.a b c= = =
The state orbits of the Liu-Chen four-scroll chaotic system are depicted in Figure 1.

6
Figure 1. State Orbits of the Liu-Chen Chaotic System
The Lü-Chen-Cheng four-scroll chaotic system is described by the dynamics
1 1 2 3
2 2 1 3
3 3 1 2
x x x x
x x x x
x x x x

 

= −
= − + +
= − +



(23)
where 1 2 3, ,x x x are state variables and , , ,    are positive, constant parameters of the
system.
The Lü-Chen-Cheng system (23) is chaotic when the parameter values are taken as
20 / 7, 10, 4  = = = and 5. =
The state orbits of the Lü-Chen-Cheng four-scroll chaotic system are depicted in Figure 2.

7
Figure 2. State Orbits of the Lü-Chen-Cheng Chaotic System
4. ANTI-SYNCHRONIZATION OF IDENTICAL LIU-CHEN FOUR-SCROLL
CHAOTIC SYSTEMS VIA SLIDING MODE CONTROL
4.1 Theoretical Results
In this section, we apply the sliding mode control results derived in Section 2 for the anti-
synchronization of identical Liu-Chen four-scroll chaotic systems ([33], 2004).
Thus, the master system is described by the Liu-Chen dynamics
1 1 2 3
2 2 1 3
3 3 1 2
x ax x x
x bx x x
x cx x x
= −
= − +
= − +



(24)
where 1 2 3, ,x x x are state variables and , ,a b c are positive, constant parameters of the system.

8
The slave system is also described by the controlled Liu-Chen dynamics
1 1 2 3 1
2 2 1 3 2
3 3 1 2 3
y ay y y u
y by y y u
y cy y y u
= − +
= − + +
= − + +



(25)
where 1 2 3, ,y y y are state variables and 1 2 3, ,u u u are the controllers to be designed.
The chaos anti-synchronization error is defined by
, ( 1,2,3)i i ie y x i= + = (26)
The error dynamics is easily obtained as
1 1 2 3 2 3 1
2 2 1 3 1 3 2
3 3 1 2 1 2 3
e ae y y x x u
e be y y x x u
e ce y y x x u
= − − +
= − + + +
= − + + +



(27) (27)
We write the error dynamics (24) in the matrix notation as
( , )e Ae x y u= + + (28)
where
0 0
0 0 ,
0 0
a
A b
c
 
 = − 
 − 
2 3 2 3
1 3 1 3
1 2 1 2
( , )
y y x x
x y y y x x
y y x x

− − 
 = + 
 + 
and
1
2
3
u
u u
u
 
 =  
  
. (29)
The sliding mode controller design is carried out as detailed in Section 2.
First, we set u as
( , )u x y Bv= − + (30)
where B is chosen such that ( , )A B is controllable.
We take B as
1
1 .
1
B
 
 =  
  
(31)
In the chaotic case, the parameter values are
0.4, 12, 5.a b c= = =

9
The sliding mode variable is selected as
[ ] 1 39 0 1 9s Ce e e e= = = + (32)
which makes the sliding mode state equation asymptotically stable.
We choose the sliding mode gains as 5k = and 0.1.q =
We note that a large value of k can cause chattering and an appropriate value of q is chosen to
speed up the time taken to reach the sliding manifold as well as to reduce the system chattering.
From Eq. (15), we can obtain ( )v t as
1( ) 4.86 0.01 sgn( )v t e s= − − (33)
Thus, the required sliding mode controller is obtained as
( , )u x y Bv= − + (34)
where ( , ),x y B and ( )v t are defined as in the equations (29), (31) and (33).
By Theorem 1, we obtain the following result.
Theorem 2. The identical Liu-Chen four-scroll chaotic systems (24) and (25) are globally and
asymptotically anti-synchronized for all initial conditions with the sliding mode controller
u defined by (34). 
4.2 Numerical Results
In this section For the numerical simulations, the fourth-order Runge-Kutta method with time-
step 6
10h −
= is used to solve the Liu-Chen four-scroll chaotic systems (24) and (25) with the
sliding mode controller u given by (34) using MATLAB.
In the chaotic case, the parameter values are 0.4, 12, 5.a b c= = =
The sliding mode gains are chosen as 5k = and 0.1.q =
The initial values of the master system (24) are taken as
1 2 3(0) 8, (0) 25, (0) 10x x x= = =
and the initial values of the slave system (25) are taken as
1 2 3(0) 14, (0) 17, (0) 5y y y= = =
Figure 3 illustrates the anti-synchronization of the identical Liu-Chen four-scroll chaotic
systems (24) and (25).

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Figure 3. Anti-Synchronization of Identical Liu-Chen Four-Scroll Chaotic Systems
5. ANTI-SYNCHRONIZATION OF LÜ-CHEN-CHENG FOUR-SCROLL
SYSTEMS VIA SLIDING MODE CONTROL
5.1 Theoretical Results
In this section, we apply the sliding mode control results derived in Section 2 for the global anti-
synchronization of identical Lü-Chen-Cheng four-scroll chaotic systems ([34], 2004).
Thus, the master system is described by the Lü-Chen-Cheng dynamics
1 1 2 3
2 2 1 3
3 3 1 2
x x x x
x x x x
x x x x

 

= −
= − + +
= − +



(35)
where 1 2 3, ,x x x are state variables and , , ,    are positive, constant parameters of the
system.

11
The slave system is also described by the Lü-Chen-Cheng dynamics
1 1 2 3 1
2 2 1 3 2
3 3 1 2 3
y y y y u
y y y y u
y y y y u

 

= − +
= − + + +
= − + +



(36)
where 1 2 3, ,y y y are state variables and 1 2 3, ,u u u are the controllers to be designed.
The anti-synchronization error is defined by
, ( 1,2,3)i i ie y x i= + = (37)
The error dynamics is easily obtained as
1 1 2 3 2 3 1
2 2 1 3 1 3 2
3 3 1 2 1 2 3
2
e e y y x x u
e e y y x x u
e e y y x x u

 

= − − +
= − + + + +
= − + + +



(38)
We write the error dynamics (24) in the matrix notation as
( , )e Ae x y u= + + (39)
where
0 0
0 0 ,
0 0
A



 
 = − 
 − 
2 3 2 3
1 3 1 3
1 2 1 2
( , ) 2
y y x x
x y y y x x
y y x x
 
− − 
 = + + 
 + 
and
1
2
3
u
u u
u
 
 =  
  
. (40)
The sliding mode controller design is carried out as detailed in Section 2.
First, we set u as
( , )u x y Bv= − + (41)
where B is chosen such that ( , )A B is controllable.
We take B as
1
1 .
1
B
 
 = − 
  
(42)
20 / 7, 10, 4  = = = and 5. =

12
The sliding mode variable is selected as
[ ] 1 2 39 1 1 9s Ce e e e e= = = + + (43)
which makes the sliding mode state equation asymptotically stable.
We choose the sliding mode gains as 5k = and 0.1.q =
We note that a large value of k can cause chattering and an appropriate value of q is chosen to
speed up the time taken to reach the sliding manifold as well as to reduce the system chattering.
From Eq. (15), we can obtain ( )v t as
1 2 32.1429 0.5556 0.1111 0.0111 sgn( )v e e e s= − + − − (44)
Thus, the required sliding mode controller is obtained as
( , )u x y Bv= − + (45)
where ( , ),x y B and ( )v t are defined as in the equations (40), (42) and (44).
By Theorem 1, we obtain the following result.
Theorem 3. The identical Lü-Chen-Cheng four-scroll chaotic systems (35) and (36) are
globally and asymptotically anti-synchronized for all initial conditions with the sliding mode
controller u defined by (45). 
5.2 Numerical Results
In this section For the numerical simulations, the fourth-order Runge-Kutta method with time-
step 6
10h −
= is used to solve the Liu-Chen four-scroll chaotic systems (35) and (36) with the
sliding mode controller u given by (31) using MATLAB.
20 / 7, 10, 4  = = = and 5. =
The sliding mode gains are chosen as 5k = and 0.1.q =
The initial values of the master system (21) are taken as
1 2 3(0) 12, (0) 16, (0) 7x x x= = =
and the initial values of the slave system (22) are taken as
1 2 3(0) 22, (0) 10, (0) 15y y y= = =
Figure 4 illustrates the anti-synchronization of the identical Lü-Chen-Cheng four-scroll
chaotic systems (21) and (22).

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Figure 5. Anti-Synchronization of Identical Lü-Chen-Cheng Four-Scroll Chaotic Systems
4. CONCLUSIONS
In this paper, we have derived new results using sliding mode control (SMC) to achieve anti-
synchronization for the identical Liu-Chen four-scroll chaotic systems (2004) and Lü-Chen-
Cheng four-scroll chaotic systems (2004). Our synchronization results for the identical Liu-
Chen four-scroll chaotic systems and identical Lü-Chen-Cheng four-scroll chaotic systems have
been established using Lyapunov stability theory. Since the Lyapunov exponents are not
required for these calculations, the sliding mode control method is very effective and convenient
to achieve global chaos synchronization for the identical Liu-Chen four-scroll chaotic systems
and Lü-Chen-Cheng four-scroll chaotic systems. Numerical simulations are also shown to
illustrate the effectiveness of the synchronization results derived in this paper.
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control”, International Journal of Computer Science and Engineering, Vol. 3, No. 5, pp 2163-
2169.
[33] Liu, W. & Chen, G. (2004) “Can a three-dimensional smooth autonomous quadratic chaotic
system generate a single four-scroll attractor?” International Journal of Bifurcation and Chaos,
Vol. 14, No. 4, pp 1395-1403.
[34] Lü, J., Chen, G. & Cheng, D. (2004) “A new chaotic system and beyond: The generalized
Lorenz-like system,” International Journal of Bifurcation and Chaos, Vol. 14, No. 4, pp 1507-
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[35] Hahn, W. (1967) The Stability of Motion, Springer, New York.
Author
Dr. V. Sundarapandian is a Professor (Systems
and Control Engineering), Research and
Development Centre at Vel Tech Dr. RR & Dr. SR
Technical University, Chennai, India. His current
research areas are: Linear and Nonlinear Control
Systems, Chaos Theory, Dynamical Systems and
Stability Theory, Soft Computing, Operations
Research, Numerical Analysis and Scientific
Computing, Population Biology, etc. He has
published over 180 research articles in international
journals and two text-books with Prentice-Hall of
India, New Delhi, India. He has published over 50
papers in International Conferences and 100 papers
in National Conferences. He is the Editor-in-Chief
of the AIRCC Journals – IJICS, IJCTCM and
IJITCA. He has delivered several Key Note
Lectures on Control Systems, Chaos Theory,
Scientific Computing, SCILAB, etc.

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