ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTOR

International Journal of Information Sciences and Techniques (IJIST) Vol.1, No.2, September 2011
DOI : 10.5121/ijist.2011.1201 1
ADAPTIVE CONTROL AND SYNCHRONIZATION OF A
HIGHLY CHAOTIC ATTRACTOR
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
Recently, a novel three-dimensional highly chaotic attractor has been discovered by Srisuchinwong and
Munmuangsaen (2010). This paper investigates the adaptive control and synchronization of this highly
chaotic attractor with unknown parameters. First, adaptive control laws are designed to stabilize the
highly chaotic system to its unstable equilibrium point at the origin based on the adaptive control theory
and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos
synchronization of identical highly chaotic systems with unknown parameters. Numerical simulations are
shown to demonstrate the effectiveness of the proposed adaptive control and synchronization schemes.
KEYWORDS
Adaptive Control, Stabilization, Chaos Synchronization, highly chaotic system, Srisuchinwong system.
1. INTRODUCTION
Chaotic systems are nonlinear dynamical systems, which are highly sensitive to initial conditions.
The sensitive nature of chaotic systems is usually called as the butterfly effect [1]. In 1963,
Lorenz first observed the chaos phenomenon in weather models. Since then, a large number of
chaos phenomena and chaos behaviour have been discovered in physical, social, economical,
biological and electrical systems.
The control of chaotic systems is to design state feedback control laws that stabilizes the chaotic
systems around the unstable equilibrium points. Active control technique is used when the system
parameters are known and adaptive control technique is used when the system parameters are
unknown [2-4].
Chaos synchronization is a phenomenon that 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 the chaos literature [5-16].
In 1990, Pecora and Carroll [5] introduced a method to synchronize two identical chaotic systems
and showed that it was possible for some chaotic systems to be completely synchronized. From
then on, chaos synchronization has been widely explored in a variety of fields including physical
systems [6], chemical systems [7], ecological systems [8], secure communications [9-10], etc.
In most of the chaos synchronization approaches, the master-slave or drive-response formalism
has been used. If a particular chaotic system is called the master or drive system and another

2
chaotic system is called the slave or response system, then the idea of 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 seminal work by Pecora and Carroll [5], a variety of impressive approaches have been
proposed for the synchronization of chaotic systems such as the OGY method [11], active control
method [12-16], adaptive control method [17-21], sampled-data feedback synchronization
method [22], time-delay feedback method [23], backstepping method [24], sliding mode control
method [25-28], etc.
In this paper, we investigate the adaptive control and synchronization of an uncertain novel three-
dimensional highly chaotic attractor discovered by B. Srisuchinwong and B. Munmuangsaen
([29], 2010). First, we devise adaptive stabilization scheme using state feedback control for the
highly chaotic system about its unstable equilibrium at the origin. Then, we devise adaptive
synchronization scheme for identical highly chaotic systems with unknown parameters. The
stability results derived in this paper are established using Lyapunov stability theory.
This paper is organized as follows. In Section 2, we give a system description of the highly
chaotic system (Srisuchinwong and Munmuangsaen, 2010). In Section 3, we derive results for the
adaptive stabilization of the highly chaotic system with unknown parameters. In Section 4, we
derive results for the adaptive synchronization of identical highly chaotic systems with unknown
parameters. In Section 5, we summarize the main results obtained in this paper.
2. SYSTEM DESCRIPTION
The highly chaotic system ([29], 2010) is a one-parameter family of three-dimensional chaotic
systems, which is described by the dynamics
1 2 1
2 1 1 3
3 1 2 3
10( )
40
10
x x x
x ax x x
x x x x
= −
= −
= −
&
&
&
(1)
where , ( 1,2,3)
i
x i = are the state variables and a is a constant positive parameter of the system.
The system (1) is highly chaotic when the parameter value is taken as
296.5
a = (2)
The state orbits of the highly chaotic system (2) are described in Figure 1. In [29], it has been
shown that when a is near 296.5, the maximum Lyapunov exponent (max) 2.6148
L = and the
maximum Kaplan-Yorke dimension (max) 2.1921.
KY
D =

3
Figure 1. State Orbits of the Highly Chaotic System
When the parameter values are taken as in (2), the system (1) is highly chaotic and the system
linearization matrix at the equilibrium point 0 (0,0,0)
E = is given by
10 10 0
296.5 0 0
0 0 1
A
−
 
 
=  
 
−
 
which has the eigenvalues
1 2
1, 59.6809
λ λ
= − = − and 3 49.6809
λ = (4)
Since 3
λ is a positive eigenvalue, it is immediate from Lyapunov stability theory [30] that the
system (1) is unstable at the equilibrium point 0 (0,0,0).
E =
3. ADAPTIVE CONTROL OF THE HIGHLY CHAOTIC SYSTEM
3.1 Theoretical Results
In this section, we design adaptive control law for globally stabilizing the highly chaotic system
(1) when the parameter value is unknown.
Thus, we consider the controlled highly chaotic system as follows.
1 2 1 1
2 1 1 3 2
3 1 2 3 3
10( )
40
10
x x x u
x ax x x u
x x x x u
= − +
= − +
= − +
&
&
&
(5)
where 1 2
,
u u and 3
u are feedback controllers to be designed using the states and estimates of the
unknown parameter of the system.
In order to ensure that the controlled system (5) globally converges to the origin asymptotically,
we consider the following adaptive control functions

4
1 2 1 1 1
2 1 1 3 2 2
3 1 2 3 3 3
ˆ( )
ˆ 40
10
u a x x k x
u ax x x k x
u x x x k x
= − − −
= − + −
= − + −
(6)
where â is the estimate of the parameter a and ,( 1,2,3)
i
k i = are positive constants.
Substituting the control law (6) into the highly chaotic dynamics (5), we obtain
1 1 1
2 1 2 2
3 3 3
ˆ
( )
x k x
x a a x k x
x k x
= −
= − −
= −
&
&
&
(7)
Let us now define the parameter estimation error as
ˆ
a
e a a
= − (8)
Using (8), the closed-loop dynamics (7) can be written compactly as
1 1 1
2 1 2 2
3 3 3
a
x k x
x e x k x
x k x
= −
= −
= −
&
&
&
(9)
For the derivation of the update law for adjusting the parameter estimate ˆ,
a the Lyapunov
approach is used.
Consider the quadratic Lyapunov function
( )
2 2 2 2 2
1 2 3 4
1
,
2
a
V x x x x e
= + + + + (10)
which is a positive definite function on 5
.
R
Note also that
ˆ
a
e a
= − &
& (11)
Differentiating V along the trajectories of (9) and using (11), we obtain
2 2 2
1 1 2 2 3 3 1 2
ˆ
a
V k x k x k x e x x a
 
= − − − + −
 
&
& (12)
In view of Eq. (12), the estimated parameters are updated by the following law:
1 2 4
ˆ a
a x x k e
= +
& (13)
where 4
k is a positive constant.
Substituting (13) into (12), we get
2 2 2 2
1 1 2 2 3 3 4 a
V k x k x k x k e
= − − − −
& (14)
which is a negative definite function on 4
.
R
Thus, by Lyapunov stability theory [27], we obtain the following result.

5
Theorem 1. The highly chaotic system (5) with unknown parameters is globally and
exponentially stabilized for all initial conditions 3
(0)
x R
∈ by the adaptive control law (6), where
the update law for the parameter is given by (13) and , ( 1, ,4)
i
k i = K are positive constants.
2.2 Numerical Results
For the numerical simulations, the fourth order Runge-Kutta method is used to solve the highly
chaotic system (5) with the adaptive control law (6) and the parameter update law (13).
The parameter a of the highly chaotic system (5) is selected as
296.5
a =
For the adaptive and update laws, we take 4, ( 1,2,3,4).
i
k i
= =
Suppose that the initial value of the estimated parameter is taken as ˆ(0) 7.
a =
The initial values of the highly chaotic system (5) are taken as (0) (14,15,22).
x =
When the adaptive control law (6) and the parameter update law (13) are used, the controlled
highly chaotic system (5) converges to the equilibrium 0 (0,0,0)
E = exponentially as shown in
Figure 2. The parameter estimate â is shown in Figure 3, which converges to 296.5
a =
Figure 2. Time Responses of the Controlled Highly Chaotic System

6
Figure 3. Parameter Estimate ˆ( )
a t
4. ADAPTIVE SYNCHRONIZATION OF IDENTICAL HIGHLY CHAOTIC
SYSTEMS
4.1 Theoretical Results
In this section, we discuss the adaptive synchronization of identical highly chaotic systems with
unknown parameter.
As the master system, we consider the highly chaotic dynamics described by
1 2 1
2 1 1 3
3 1 2 3
10( )
40
10
x x x
x ax x x
x x x x
= −
= −
= −

(15)
where , ( 1,2,3)
i
x i = are the state variables and a is the unknown system parameter.
As the slave system, we consider the controlled highly chaotic dynamics described by
1 2 1 1
2 1 1 3 2
3 1 2 3 3
10( )
40
10
y y y u
y ay y y u
y y y y u
= − +
= − +
= − +

(16)
where , ( 1,2,3)
i
y i = are the state variables and , ( 1,2,3)
i
u i = are the nonlinear controllers to be
designed.

7
The synchronization error is defined by
, ( 1,2,3)
i i i
e y x i
= − = (17)
Then the error dynamics is obtained as
1 2 1 1
2 1 1 3 1 3 2
3 1 2 1 2 3 3
10( )
40( )
10( )
e e e u
e ae y y x x u
e y y x x e u
= − +
= − − +
= − − +

(18)
Let us now define the adaptive control functions 1 2 3
( ), ( ), ( )
u t u t u t as
1 2 1 1 1
2 1 1 3 1 3 2 2
3 1 2 1 2 3 3 3
10( )
ˆ 40( )
10( )
u e e k e
u ae y y x x k e
u y y x x e k e
= − − −
= − + − −
= − − + −
(19)
where â is the estimate of the parameter ,
a and 1,
k 2 ,
k 3
k are positive constants.
Substituting the control law (19) into (18), we obtain the error dynamics as
1 1 1
2 1 2 2
3 3 3
ˆ
( )
e k e
e a a e k e
e k e
= −
= − −
= −

(20)
Let us now define the parameter estimation error as
ˆ
a
e a a
= − (21)
Substituting (21) into (20), the error dynamics simplifies to
1 1 1
2 1 2 2
3 3 3
a
e k e
e e e k e
e k e
= −
= −
= −

(22)
For the derivation of the update law for adjusting the estimate of the parameter, the Lyapunov
approach is used.
Consider the quadratic Lyapunov function
( )
2 2 2 2
1 2 3
1
2
a
V e e e e
= + + + (23)
which is a positive definite function on 4
.
R
Note also that
ˆ
a
e a
= −
(24)
Differentiating V along the trajectories of (22) and using (24), we obtain
2 2 2
1 1 2 2 3 3 1 2
ˆ
a
V k e k e k e e e e a
 
= − − − + −
 

(25)
In view of Eq. (25), the estimated parameter is updated by the following law:

8
1 2 4
ˆ a
a e e k e
= +
(26)
where 4
k is a positive constants.
Substituting (24) into (23), we get
2 2 2 2
1 1 2 2 3 3 4 ,
a
V k e k e k e k e
= − − − −
(27)
which is a negative definite function on 4
.
R
Thus, by Lyapunov stability theory [30], it is immediate that the synchronization error and the
parameter error decay to zero exponentially with time for all initial conditions.
Hence, we have proved the following result.
Theorem 2. The identical highly chaotic systems (15) and (16) with unknown parameters are
globally and exponentially synchronized for all initial conditions by the adaptive control law
(19), where the update law for parameter is given by (26) and ,( 1,2,3,4)
i
k i = are positive
constants.
3.2 Numerical Results
For the numerical simulations, the fourth order Runge-Kutta method is used to solve the two
systems of differential equations (15) and (16) with the adaptive control law (19) and the
parameter update law (26).
Here, we take the parameter value as 296.5
a = and the gains as 4
i
k = for 1,2,3,4.
i =
We take the initial value of the estimated parameter as ˆ(0) 10.
a = We take the initial state of the
master system (15) as (0) (2,15,10)
x = and the slave system (16) as (0) (18,6,4).
y =
Figure 4 shows the adaptive chaos synchronization of the identical highly chaotic systems. Figure
5 shows that the estimated value â converges to the system parameter 296.5.
a =
Figure 4. Adaptive Synchronization of the Highly Chaotic Systems

9
Figure 5. Parameter Estimate ˆ( )
a t
5. CONCLUSIONS
In this paper, we applied adaptive control theory for the stabilization and synchronization of the
highly chaotic system (Srisuchinwong and Munmuangsaen, 2010) with unknown system
parameters. First, we designed adaptive control laws to stabilize the highly chaotic system to its
equilibrium point at the origin based on the adaptive control theory and Lyapunov stability
theory. Then we derived adaptive synchronization scheme and update law for the estimation of
system parameters for identical highly chaotic systems with unknown parameters. Our
synchronization schemes were established using Lyapunov stability theory. Since the Lyapunov
exponents are not required for these calculations, the proposed adaptive control method is very
effective and convenient to achieve chaos control and synchronization of the highly chaotic
system. Numerical simulations are shown to validate and illustrate the effectiveness of the
adaptive stabilization and synchronization schemes derived in this paper.
REFERENCES
[1] Alligood, K.T., Sauer, T. Yorke, J.A. (1997) Chaos: An Introduction to Dynamical Systems,
Springer, New York.
[2] Ge, S.S., Wang, C. Lee, T.H. (2000) “Adaptive backstepping control of a class of chaotic systems,”
Internat. J. Bifur. Chaos, Vol. 10, pp 1149-1156.
[3] Wang, X., Tian, L. Yu, L. (2006) “Adaptive control and slow manifold analysis of a new chaotic
system,” Internat. J. Nonlinear Science, Vol. 21, pp 43-49.
[4] Sun, M., Tian, L., Jiang, S. Xun, J. (2007) “Feedback control and adaptive control of the energy
resource chaotic system,” Chaos, Solitons Fractals, Vol. 32, pp 168-180.
[5] Pecora, L.M. Carroll, T.L. (1990) “Synchronization in chaotic systems”, Phys. Rev. Lett., Vol. 64,
pp 821-824.
[6] Lakshmanan, M. Murali, K. (1996) Nonlinear Oscillators: Controlling and Synchronization, World
Scientific, Singapore.

10
[7] Han, S.K., Kerrer, C. Kuramoto, Y. (1995) “Dephasing and bursting in coupled neural oscillators”,
Phys. Rev. Lett., Vol. 75, pp 3190-3193.
[8] Blasius, B., Huppert, A. Stone, L. (1999) “Complex dynamics and phase synchronization in
spatially extended ecological system”, Nature, Vol. 399, pp 354-359.
[9] Feki, M. (2003) “An adaptive chaos synchronization scheme applied to secure communication”,
Chaos, Solitons and Fractals, Vol. 18, pp 141-148.
[10] Murali, K. Lakshmanan, M. (1998) “Secure communication using a compound signal from
generalized synchronizable chaotic systems”, Phys. Rev. Lett. A, Vol. 241, pp 303-310.
[11] Ott, E., Grebogi, C. Yorke, J.A. (1990) “Controlling chaos”, Phys. Rev. Lett., Vol. 64, pp 1196-
1199.
[12] Wu, Y., Zhou, X. Chen, J. (2009) “Chaos synchronization of a new chaotic system,” Chaos,
Solitons Fractals, Vol. 42, pp 1812-1819.
[13] Sundarapandian, V. Karthikeyan, R. (2011), “Global chaos synchronization of four-scroll chaotic
systems by active nonlinear control,” International Journal of Control Theory and Applications, Vol.
4, No. 1, pp 73-83.
[14] Sundarapandian, V. (2011) “Hybrid chaos synchronization of hyperchaotic Liu and hyperchaotic
Chen systems by active nonlinear control,” International Journal of Computer Science, Engineering
and Information Technology, Vol. 1, No. 2, pp 1-14.
[15] Sundarapandian, V. Suresh, R. (2011) “Global chaos synchronization of hyperchaotic Qi and Jia
systems by nonlinear control,” International Journal of Distributed and Parallel Systems, Vol. 2, No.
2, pp 83-94.
[16] Sundarapandian, V. (2011) “Global chaos synchronization of Shimizu-Morioka and Liu-Chen chaotic
systems by active nonlinear control,” International Journal of Advances in Science and Technology,
Vol. 2, No. 4, pp. 11-20.
[17] Samuel, B. (2007) “Adaptive synchronization between two different chaotic systems,” Adaptive
Commun. Nonlinear Sci. Num. Simulation, Vol. 12, pp. 976-985.
[18] Liao, T.L. Tsai. S.H. (2000) “Adaptive synchronization of chaotic systems and its applications to
secure communications,” Chaos, Solitons and Fractals, Vol. 11, pp. 1387-1396.
[19] Sundarapandian, V. (2011) “Adaptive control and synchronization of hyperchaotic Newton-Leipnik
system,” International Journal of Advanced Information Technology, Vol. 1, No. 3, pp 22-33.
[20] Sundarapandian, V. (2011) “Adaptive control and synchronization of hyperchaotic Cai system,”
International Journal of Control Theory and Computer Modeling, Vol. 1, No. 1, pp 1-13.
[21] Sundarapandian, V. (2011) “Adaptive synchronization of uncertain Sprott H and I chaotic systems,”
International Journal of Computer Information Systems, Vol. 1, No. 5, pp 1-7.
[22] Yang, T. Chua, L.O. (1999) “Control of chaos using sampled-data feedback control”, Internat. J.
Bifurcat. Chaos, Vol. 9, pp 215-219.
[23] Park, J.H. Kwon, O.M. (2003) “A novel criterion for delayed feedback control of time-delay
chaotic systems”, Chaos, Solitons and Fractals, Vol. 17, pp 709-716.
[24] Yu, Y.G. Zhang, S.C. (2006) “Adaptive backstepping synchronization of uncertain chaotic
systems”, Chaos, Solitons and Fractals, Vol. 27, pp 1369-1375.
[25] Konishi, K., Hirai, M. Kokame, H. (1998) “Sliding mode control for a class of chaotic systems,”
Phys. Lett. A, Vol. 245, pp 511-517.
[26] Sundarapandian, V. (2011) “Global chaos synchronization of Pehlivan systems by sliding mode
control,” International Journal on Computer Science and Engineering, Vol. 3, No. 5, pp 2163-2169.
[27] Sundarapandian, V. Sivaperumal, S. (2011) “Global chaos synchronization of hyperchaotic Chen
systems by sliding mode control,” International Journal of Engineering Science and Technology,
Vol. 3, No. 5, pp 4265-4271.

11
[28] Sundarapandian, V. (2011) “Global chaos synchronization of four-wing chaotic systems by sliding
mode control,” International Journal of Control Theory and Computer Modeling, Vol. 1, No. 1, pp
15-31.
[29] Srisuchinwong, B. Munmuangsaen, B. (2010) “A highly chaotic attractor for a dual-channel single-
attractor, private communication system”, Proceedings of the Third Chaotic Modeling and Simulation
International Conference, Chania, Crete, Greece, June 2010, pp 177-184.
[30] 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, etc.
He has published over 170 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 90 papers in National
Conferences. He is a Senior Member of AIRCC
and is the Editor-in-Chief of the AIRCC
journals – International Journal of
Instrumentation and Control Systems,
International Journal of Control Theory and
Computer Modeling, etc. He is an Associate
Editor of the journals – International Journal of
Control Theory and Applications, International
Journal of Advances in Science and
Technology, International Journal of Computer
Information Systems, etc. He has delivered
several Key Note Lectures on Control Systems,
Chaos Theory, Scientific Computing,
MATLAB, SCILAB, etc.

ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTOR

More Related Content

What's hot (19)

Similar to ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTOR (20)

More from ijistjournal (20)

Recently uploaded (20)

ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTOR