adaptive hybrid chaos synchronization of lorenz-stenflo and qi 4-d chaotic systems with unknown...
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DOI:10.5121/ijfcst.2012.2102 15
ADAPTIVE HYBRID CHAOS SYNCHRONIZATION OF
LORENZ-STENFLO AND QI 4-DCHAOTIC SYSTEMS
WITH UNKNOWN PARAMETERS
Sundarapandian Vaidyanathan1
1Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
Avadi, Chennai-600 062, Tamil Nadu, [email protected]
ABSTRACT
This paper investigates the adaptive hybrid chaos synchronization of uncertain 4-D chaotic systems, viz.
identical Lorenz-Stenflo (LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and
non-identical LS and Qi systems with unknown parameters. In hybrid chaos synchronization of master
and slave systems, the odd states of the two systems are completely synchronized, while the even states ofthe two systems are anti-synchronized so that complete synchronization (CS) and anti-synchronization
(AS) co-exist in the synchronization of the two systems. In this paper, we devise adaptive control schemes
for the hybrid chaos synchronization using the estimates of parameters for both master and slave systems.
Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability
theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control
method is very effective and convenient to achieve hybrid synchronization of identical and non-identical
LS and Qi systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed
adaptive synchronization schemes for the identical and non-identical uncertain LS and Qi 4-D chaotic
systems.
KEYWORDS
Adaptive Control, Chaos, Hybrid Synchronization, Lorenz-Stenflo System, Qi System.
1.INTRODUCTION
Chaotic systems are nonlinear dynamical systems that are highly sensitive to initial conditions.
The sensitive nature of chaotic systems is commonly called as the butterfly effect[1].
Chaos is an interesting nonlinear phenomenon and has been extensively and intensively studied
in the last two decades [1-30]. Chaos theory has been applied in many scientific disciplines suchas Mathematics, Computer Science, Microbiology, Biology, Ecology, Economics, Population
Dynamics and Robotics.
In 1990, Pecora and Carroll [2] deployed control techniques to synchronize two identicalchaotic 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 [3], chemical systems [4-5], ecological systems [6], securecommunications [7-9], etc.
In most of the chaos synchronization approaches, the master-slave or drive-response formalismis used. If a particular chaotic system is called the masteror 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 systemtracks the output of the master system asymptotically.
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Since the seminal work by Pecora and Carroll [2], a variety of impressive approaches have beenproposed for the synchronization of chaotic systems such as the OGY method [10], active
control method [11-15], adaptive control method [16-22], sampled-data feedbacksynchronization method [23], time-delay feedback method [24], backstepping method [25],sliding mode control method [26-30], etc.
So far, many types of synchronization phenomenon have been presented such as completesynchronization [2], phase synchronization [31], generalized synchronization [32], anti-synchronization [33-34], projective synchronization [35], generalized projective
synchronization [36-37], etc.
Complete synchronization (CS) is characterized by the equality of state variables evolving intime, while anti-synchronization (AS) is characterized by the disappearance of the sum of
relevant variables evolving in time. Projective synchronization (PS) is characterized by the fact
that the master and slave systems could be synchronized up to a scaling factor, whereas ingeneralized projective synchronization (GPS), the responses of the synchronized dynamical
states synchronize up to a constant scaling matrix . It is easy to see that the completesynchronization (CS) and anti-synchronization (AS) are special cases of the generalized
projective synchronization (GPS) where the scaling matrix I = and ,I = respectively.
In hybrid synchronization of chaotic systems [15], one part of the system is synchronized andthe other part is anti-synchronized so that the complete synchronization (CS) and anti-
synchronization (AS) coexist in the system. The coexistence of CS and AS is highly useful insecure communication and chaotic encryption schemes.
In this paper, we investigate the hybrid chaos synchronization of 4-D chaotic systems, viz.identical Lorenz-Stenflo systems ([38], 2001), identical Qi systems ([39], 2005) and non-
identical Lorenz-Stenflo and Qi systems. We deploy adaptive control method because the
system parameters of the 4-D chaotic systems are assumed to be unknown. The adaptivenonlinear controllers are derived using Lyapunov stability theory for the hybrid chaos
synchronization of the two 4-D chaotic systems when the system parameters are unknown.
This paper is organized as follows. In Section 2, we provide a description of the 4-D chaoticsystems addressed in this paper, viz. Lorenz-Stenflo systems (2001) and Qi systems (2005). In
Section 3, we discuss the hybrid chaos synchronization of identical Lorenz-Stenflo systems. In
Section 4, we discuss the hybrid chaos synchronization of identical Qi systems. In Section 5, wederive results for the hybrid synchronization of non-identical Lorenz-Stenflo and Qi systems.
2.SYSTEMS DESCRIPTION
The Lorenz-Stenflo system ([38], 2001) is described by
1 2 1 4
2 1 3 2
3 1 2 3
4 1 4
( )
( )
x x x x
x x r x x
x x x xx x x
= +
=
=
=
&
&
&
&
(1)
where 1 2 3 4, , ,x x x x are the state variables and , , , r are positive constant parameters of the
system.
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The LS system (1) is chaotic when the parameter values are taken as
2.0, 0.7, 1.5 = = = and 26.0.r=
The strange attractor of the chaotic LS system (1) is shown in Figure 1.
The Qi system ([39], 2005) is described by
1 2 1 2 3 4
2 1 2 1 3 4
3 3 1 2 4
4 4 1 2 3
( )
( )
x a x x x x x
x b x x x x x
x cx x x x
x dx x x x
= +
= +
= +
= +
&
&
&
&
(2)
where 1 2 3 4, , ,x x x x are the state variables and , , ,a b c d are positive constant parameters of the
system.
The system (2) is chaotic when the parameter values are taken as
30, 10, 1a b c= = = and 10.d=
The strange attractor of the chaotic Qi system (2) is shown in Figure 2.
Figure 1. Strange Attractor of the Lorenz-Stenflo Chaotic System
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Figure 2. Strange Attractor of the Qi Chaotic System
3.ADAPTIVE HYBRID CHAOS SYNCHRONIZATION OF IDENTICAL LORENZ-
STENFLO SYSTEMS
3.1 Theoretical Results
In this section, we discuss the adaptive hybrid chaos synchronization of identical Lorenz-Stenflosystems ([38], 2001), where the parameters of the master and slave systems are unknown.
As the master system, we consider the LS dynamics described by
1 2 1 4
2 1 3 2
3 1 2 3
4 1 4
( )
( )
x x x x
x x r x x
x x x x
x x x
= +
=
=
=
&
&
&
&
(3)
where 1 2 3 4, , ,x x x x are the states and , , , r are unknown real constant parameters of the
system.
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As the slave system, we consider the controlled LS dynamics described by
1 2 1 4 1
2 1 3 2 2
3 1 2 3 3
4 1 4 4
( )
( )
y y y y u
y y r y y u
y y y y u
y y y u
= + +
= +
= +
= +
&
&
&
&
(4)
where 1 2 3 4, , ,y y y y are the states and 1 2 3 4, , ,u u u u are the nonlinear controllers to be designed.
The hybrid chaos synchronization error is defined by
1 1 1
2 2 2
3 3 3
4 4
e y x
e y x
e y x
e y x
=
= +
=
= +
(5)
The error dynamics is easily obtained as
1 2 1 2 4 4 1
2 2 1 1 1 3 1 3 2
3 3 1 2 1 2 3
4 1 1 4 4
( 2 ) ( 2 )
( 2 )
( 2 )
e e e x e x u
e e r e x x x y y u
e e y y x x u
e e x e u
= + +
= + + +
= + +
= + +
&
&
&
&
(6)
Let us now define the adaptive control functions
1 2 1 2 4 4 1 1
2 2 1 1 1 3 1 3 2 2
3 3 1 2 1 2 3 3
4 1 1 4 4 4
( ) ( 2 ) ( 2 )
( ) ( 2 )
( )
( ) 2
u t e e x e x k e
u t e r e x y y x x k e
u t e y y x x k e
u t e x e k e
=
= + + +
= +
= + +
(7)
where , , and rare estimates of , , and ,r respectively, and , ( 1, 2,3, 4)ik i = are
positive constants.
Substituting (7) into (6), the error dynamics simplifies to
1 2 1 2 4 4 1 1
2 1 1 2 2
3 3 3 3
4 4 4 4
( )( 2 ) ( )( 2 )
( )( 2 )( )
( )
e e e x e x k e
e r r e x k ee e k e
e e k e
= +
= +
=
=
&
&
&
&
(8)
Let us now define the parameter estimation errors as
, ,e e e = = = and .re r r= (9)
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Substituting (9) into (8), we obtain the error dynamics as
1 2 1 2 4 4 1 1
2 1 1 2 2
3 3 3 3
4 4 4 4
( 2 ) ( 2 )
( 2 )r
e e e e x e e x k e
e e e x k e
e e e k e
e e e k e
= +
= +
=
=
&
&
&
&
(10)
For the derivation of the update law for adjusting the estimates of the parameters, the Lyapunovapproach is used.
We consider the quadratic Lyapunov function defined by
( )2 2 2 2 2 2 2 21 2 3 4 1 2 3 41
( , , , , , , , )2
r rV e e e e e e e e e e e e e e e e = + + + + + + + (11)
which is a positive definite function on8.R
We also note that
, ,e e e = = = && &
& & & and r
e r= && (12)
Differentiating (11) along the trajectories of (10) and using (12), we obtain
[ ]
2 2 2 2 2 2
1 1 2 2 3 3 4 4 1 2 1 2 4 3
1 4 4 2 1 1
( 2 )
( 2 ) ( 2 )r
V k e k e k e k e e e e e x e e e
e e e x e e e x r
= + +
+ + +
&&&
&&
(13)
In view of Eq. (13), the estimated parameters are updated by the following law:
2
1 2 1 2 4 5
2
3 6
1 4 4 7
2 1 1 8
( 2 )
( 2 )
( 2 ) r
e e e x e k e
e k e
e e x k e
r e e x k e
= +
= +
= +
= + +
&
&
&
&
(14)
where 5 6 7, ,k k k and 8k are positive constants.
Substituting (14) into (12), we obtain
2 2 2 2 2 2 2 2
1 1 2 2 3 3 4 4 5 6 7 8 rV k e k e k e k e k e k e k e k e =
& (15)
which is a negative definite function on8.R
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Thus, by Lyapunov stability theory [40], it is immediate that the hybrid synchronization error
, ( 1, 2,3, 4)ie i = and the parameter estimation error , , , re e e e decay to zero exponentially
with time.
Hence, we have proved the following result.
Theorem 1. The identical hyperchaotic Lorenz-Stenflo systems (3) and (4) with unknown
parameters are globally and exponentially hybrid synchronized via the adaptive control law (7),
where the update law for the parameter estimates is given by (14) and , ( 1, 2, ,8)i
k i = K are
positive constants.
3.2 Numerical Results
For the numerical simulations, the fourth-order Runge-Kutta method with time-step610h = is
used to solve the 4-D chaotic systems (3) and (4) with the adaptive control law (14) and the
parameter update law (14) using MATLAB.
We take 4ik = for 1, 2, ,8.i = K
For the Lorenz-Stenflo systems (3) and (4), the parameter values are taken as
2.0, 0.7, 1.5 = = = and 26.0.r=
Suppose that the initial values of the parameter estimates are
(0) 6, (0) 5, (0) 2, (0) 7.a b r d = = = =
The initial values of the master system (3) are taken as
1 2 3 4(0) 4, (0) 10, (0) 15, (0) 19.x x x x= = = =
The initial values of the slave system (4) are taken as
1 2 3 4(0) 21, (0) 6, (0) 3, (0) 27.y y y y= = = =
Figure 3 depicts the hybrid-synchronization of the identical Lorenz-Stenflo systems (3) and (4).It may also be noted that the odd states of the two systems are completely synchronized, while
the even states of the two systems are anti-synchronized.
Figure 4 shows that the estimated values of the parameters, viz. , , and rconverge to the
system parameters
2.0, 0.7, 1.5 = = = and 26.0.r=
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Figure 3. Hybrid-Synchronization of Lorenz-Stenflo Chaotic Systems
Figure 4. Parameter Estimates ( ), ( ), ( ), ( )t t t r t
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4. ADAPTIVE HYBRID CHAOS SYNCHRONIZATION OF IDENTICAL QI
SYSTEMS
4.1 Theoretical Results
In this section, we discuss the adaptive hybrid chaos synchronization of identical Qi systems
([39], 2005), where the parameters of the master and slave systems are unknown.
As the master system, we consider the Qi dynamics described by
1 2 1 2 3 4
2 1 2 1 3 4
3 3 1 2 4
4 4 1 2 3
( )
( )
x a x x x x x
x b x x x x x
x cx x x x
x dx x x x
= +
= +
= +
= +
&
&
&
&
(16)
where 1 2 3 4, , ,x x x x are the states and , , ,a b c d are unknown real constant parameters of the
system.
As the slave system, we consider the controlled Qi dynamics described by
1 2 1 2 3 4 1
2 1 2 1 3 4 2
3 3 1 2 4 3
4 4 1 2 3 4
( )
( )
y a y y y y y u
y b y y y y y u
y cy y y y u
y dy y y y u
= + +
= + +
= + +
= + +
&
&
&
&
(17)
where 1 2 3 4, , ,y y y y are the states and 1 2 3 4, , ,u u u u are the nonlinear controllers to be designed.
The hybrid synchronization error is defined by
1 1 1
2 2 2
3 3 3
4 4
e y x
e y x
e y x
e y x
=
= +
=
= +
(18)
The error dynamics is easily obtained as
1 2 1 2 2 3 4 2 3 4 1
2 1 2 1 1 3 4 1 3 4 2
3 3 1 2 4 1 2 4 3
4 4 1 2 3 1 2 3 4
( 2 )
( 2 )
e a e e x y y y x x x u
e b e e x y y y x x x u
e ce y y y x x x ue de y y y x x x u
= + +
= + + +
= + +
= + + +
&
&
&
&
(19)
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Let us now define the adaptive control functions
1 2 1 2 2 3 4 2 3 4 1 1
2 1 2 1 1 3 4 1 3 4 2 2
3 3 1 2 4 1 2 4 3 3
4 4 1 2 3 1 2 3 4 4
( ) ( 2 )
( ) ( 2 )
( )( )
u t a e e x y y y x x x k e
u t b e e x y y y x x x k e
u t ce y y y x x x k eu t de y y y x x x k e
= +
= + + + +
= +
=
(20)
where , ,a b c and dare estimates of , ,a b c and ,d respectively, and , ( 1, 2, 3, 4)ik i = are
positive constants.
Substituting (20) into (19), the error dynamics simplifies to
1 2 1 2 1 1
2 1 2 1 2 2
3 3 3 3
4 4 4 4
( )( 2 )
( )( 2 )
( )
( )
e a a e e x k e
e b b e e x k e
e c c e k e
e d d e k e
=
= + +
=
=
&
&
&
&
(21)
Let us now define the parameter estimation errors as
, ,a b c
e a a e b b e c c= = = and .de d d= (22)
Substituting (22) into (21), we obtain the error dynamics as
1 2 1 2 1 1
2 1 2 1 2 2
3 3 3 3
4 4 4 4
( 2 )
( 2 )
a
b
c
d
e e e e x k e
e e e e x k e
e e e k ee e e k e
=
= + +
=
=
&
&
&
&
(23)
For the derivation of the update law for adjusting the estimates of the parameters, the Lyapunov
approach is used.
We consider the quadratic Lyapunov function defined by
( )2 2 2 2 2 2 2 21 2 3 4 1 2 3 41
( , , , , , , , )2
a b c d a b c d V e e e e e e e e e e e e e e e e= + + + + + + + (24)
which is a positive definite function on8.R
We also note that
, ,a b ce a e b e c= = = && && & & and de d=
&& (25)
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Differentiating (24) along the trajectories of (23) and using (25), we obtain
2 2 2 2
1 1 2 2 3 3 4 4 1 2 1 2
2 2
2 1 2 1 3 4
( 2 )
( 2 )
a
b c d
V k e k e k e k e e e e e x a
e e e e x b e e c e e d
= +
+ + + + +
&&
& &&(26)
In view of Eq. (26), the estimated parameters are updated by the following law:
1 2 1 2 5
2 1 2 1 6
2
3 7
2
4 8
( 2 )
( 2 )
a
b
c
d
a e e e x k e
b e e e x k e
c e k e
d e k e
= +
= + + +
= +
= +
&
&
&
&
(27)
where 5 6 7, ,k k k and 8k are positive constants.
Substituting (27) into (26), we obtain
2 2 2 2 2 2 2 2
1 1 2 2 3 3 4 4 5 6 7 8a b c d V k e k e k e k e k e k e k e k e= & (28)
which is a negative definite function on8.R
Thus, by Lyapunov stability theory [40], it is immediate that the synchronization error
, ( 1, 2,3, 4)i
e i = and the parameter estimation error , , ,a b c d e e e e decay to zero exponentially
with time.
Hence, we have proved the following result.
Theorem 2.The identical Qi systems (16) and (17) with unknown parameters are globally andexponentially hybrid synchronized by the adaptive control law (20), where the update law for
the parameter estimates is given by (27) and , ( 1, 2, ,8)i
k i = K are positive constants.
4.2 Numerical Results
For the numerical simulations, the fourth-order Runge-Kutta method with time-step610h = is
used to solve the chaotic systems (16) and (17) with the adaptive control law (14) and the
parameter update law (27) using MATLAB. We take 4ik = for 1, 2, ,8.i = K
For the Qi systems (16) and (17), the parameter values are taken as
30, 10, 1,a b c= = = 10.d=
Suppose that the initial values of the parameter estimates are
(0) 6, (0) 2, (0) 9, (0) 5a b c d = = = =
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The initial values of the master system (16) are taken as
1 2 3 4(0) 4, (0) 8, (0) 17, (0) 20.x x x x= = = =
The initial values of the slave system (17) are taken as
1 2 3 4(0) 10, (0) 21, (0) 7, (0) 15.y y y y= = = =
Figure 5 depicts the hybrid synchronization of the identical Qi systems (16) and (17).
Figure 6 shows that the estimated values of the parameters, viz. , ,a b c and dconverge to the
system parameters
30, 10, 1, 10.a b c d = = = =
Figure 5. Anti-Synchronization of the Qi Systems
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Figure 6. Parameter Estimates ( ), ( ), ( ), ( )a t b t c t d t
5. ADAPTIVE HYBRID CHAOS SYNCHRONIZATION OF NON-IDENTICAL
LORENZ-STENFLO AND QI SYSTEMS
5.1 Theoretical Results
In this section, we discuss the adaptive hybrid chaos synchronization of non-identical Lorenz-Stenflo system ([38], 2001) and Qi system ([39], 2005), where the parameters of the master and
slave systems are unknown.
As the master system, we consider the LS dynamics described by
1 2 1 4
2 1 3 2
3 1 2 3
4 1 4
( )
( )
x x x x
x x r x xx x x x
x x x
= +
=
=
=
&
&
&
&
(29)
where 1 2 3 4, , ,x x x x are the states and , , , r are unknown real constant parameters of the
system.
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As the slave system, we consider the controlled Qi dynamics described by
1 2 1 2 3 4 1
2 1 2 1 3 4 2
3 3 1 2 4 3
4 4 1 2 3 4
( )
( )
y a y y y y y u
y b y y y y y u
y cy y y y u
y dy y y y u
= + +
= + +
= + +
= + +
&
&
&
&
(30)
where 1 2 3 4, , ,y y y y are the states, , , ,a b c d are unknown real constant parameters of the system
and1 2 3 4, , ,u u u u are the nonlinear controllers to be designed.
The hybrid chaos synchronization error is defined by
1 1 1
2 2 2
3 3 3
4 4
e y x
e y x
e y x
e y x
=
= +
=
= +
(31)
The error dynamics is easily obtained as
1 2 1 2 1 4 2 3 4 1
2 1 2 2 1 1 3 1 3 4 2
3 3 3 1 2 4 1 2 3
4 4 4 1 1 2 3 4
( ) ( )
( )
e a y y x x x y y y u
e b y y x rx x x y y y u
e cy x y y y x x u
e dy x x y y y u
= + +
= + + +
= + + +
= + +
&
&
&
&
(32)
Let us now define the adaptive control functions
1 2 1 2 1 4 2 3 4 1 1
2 1 2 2 1 1 3 1 3 4 2 2
3 3 3 1 2 4 1 2 3 3
4 4 4 1 1 2 3 4 4
( ) ( ) ( )
( ) ( )
( )
( )
u t a y y x x x y y y k e
u t b y y x rx x x y y y k e
u t cy x y y y x x k e
u t dy x x y y y k e
= + +
= + + + +
= +
= + +
(33)
where , , , , , ,a b c d and rare estimates of , , , , , ,a b c d and ,r respectively, and
, ( 1, 2,3, 4)i
k i = are positive constants.
Substituting (33) into (32), the error dynamics simplifies to
1 2 1 2 1 4 1 1
2 1 2 1 2 2
3 3 3 3 3
4 4 4 4 4
( )( ) ( )( ) ( )
( )( ) ( )
( ) ( )
( ) ( )
e a a y y x x x k e
e b b y y r r x k e
e c c y x k e
e d d y x k e
=
= + +
= +
=
&
&
&
&
(34)
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Let us now define the parameter estimation errors as
, , ,
, , ,
a b c d
r
e a a e b b e c c e d d
e e e e r r
= = = =
= = = =
(35)
Substituting (35) into (32), we obtain the error dynamics as
1 2 1 2 1 4 1 1
2 1 2 1 2 2
3 3 3 3 3
4 4 4 4 4
( ) ( )
( )
a
b r
c
d
e e y y e x x e x k e
e e y y e x k e
e e y e x k e
e e y e x k e
=
= + +
= +
=
&
&
&
&
(36)
For the derivation of the update law for adjusting the estimates of the parameters, the Lyapunov
approach is used. We consider the quadratic Lyapunov function defined by
( )2 2 2 2 2 2 2 2 2 2 2 21 2 3 41
2a b c d r V e e e e e e e e e e e e = + + + + + + + + + + + (37)
which is a positive definite function on12
.R
We also note that
, , , , , , ,a b c d r e a e b e c e d e e e e r = = = = = = = = & & && && & && & & & & & & & (38)
Differentiating (37) along the trajectories of (36) and using (38), we obtain
2 2 2 2
1 1 2 2 3 3 4 4 1 2 1 2 1 2
3 3 4 4 1 2 1 4 4
3 3 1 4 2 1
( ) ( )
( )
a b
c d
r
V k e k e k e k e e e y y a e e y y b
e e y c e e y d e e x x e x
e e x e e x e e x r
= + + +
+ + +
+ + +
&&&
& &&
& & &
(39)
In view of Eq. (39), the estimated parameters are updated by the following law:
1 2 1 5 1 2 1 4 4 9
2 1 2 6 3 3 10
3 3 7 1 4 11
4 4 8 2 1 12
( ) , ( )
( ) ,
,
,
a
b
c
d r
a e y y k e e x x e x k e
b e y y k e e x k e
c e y k e e x k e
d e y k e r e x k e
= + = +
= + + = +
= + = +
= + = +
&&
& &
&&
& &
(40)
where 5 6 7 8 9 10 11, , , , , ,k k k k k k k and 12k are positive constants.
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Substituting (40) into (39), we obtain
2 2 2 2
1 1 2 2 3 3 4 4V k e k e k e k e= & 2 2
5 6a bk e k e
2 2
7 8c dk e k e
2 2
9 10k e k e 2 2
11 12 rk e k e (41)
which is a negative definite function on12
.R
Thus, by Lyapunov stability theory [30], it is immediate that the synchronization error
, ( 1, 2,3, 4)ie i = and the parameter estimation error decay to zero exponentially with time.
Hence, we have proved the following result.
Theorem 3. The non-identical Lorenz-Stenflo system (29) and Qi system (30) with unknown parameters are globally and exponentially hybrid synchronized by the adaptive control law
(33), where the update law for the parameter estimates is given by (40) and
, ( 1,2, ,12)i
k i = K are positive constants.
5.2 Numerical Results
For the numerical simulations, the fourth-order Runge-Kutta method with time-step6
10h
= is
used to solve the chaotic systems (29) and (30) with the adaptive control law (27) and the
parameter update law (40) using MATLAB. We take 4ik = for 1, 2, ,12.i = K
For the Lorenz-Stenflo system (29) and Qi system (30), the parameter values are taken as
30, 10, 1, 10, 2.0, 0.7, 1.5, 26.a b c d r = = = = = = = = (42)
Suppose that the initial values of the parameter estimates are
(0) 2, (0) 6, (0) 1, (0) 4, (0) 5, (0) 8, (0) 7, (0) 11a b c d r = = = = = = = =
The initial values of the master system (29) are taken as
1 2 3 4(0) 6, (0) 15, (0) 20, (0) 9.x x x x= = = =
The initial values of the slave system (30) are taken as
1 2 3 4(0) 20, (0) 18, (0) 14, (0) 27.y y y y= = = =
Figure 7 depicts the complete synchronization of the non-identical Lorenz-Stenflo and Qi
systems.
Figure 8 shows that the estimated values of the parameters, viz. , , , , , ,a b c d andrconverge to the original values of the parameters given in (42).
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Figure 7. Complete Synchronization of Lorenz-Stenflo and Qi Systems
Figure 8. Parameter Estimates ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( )a t b t c t d t t t t r t
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6.CONCLUSIONS
In this paper, we have derived new results for the hybrid synchronization of identical Lorenz-Stenflo systems (2001), identical Qi systems (2005) and non-identical Lorenz-Stenflo and Qi
systems with unknown parameters via adaptive control method. The hybrid synchronizationresults derived in this paper are established using Lyapunov stability theory. Since the
Lyapunov exponents are not required for these calculations, the adaptive control method is avery effective and convenient for achieving hybrid chaos synchronization for the uncertain 4-D
chaotic systems discussed in this paper. Numerical simulations are shown to demonstrate the
effectiveness of the adaptive synchronization schemes derived in this paper for the hybrid chaossynchronization of identical and non-identical uncertain Lorenz-Stenflo and Qi systems.
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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 210 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 three AIRCC control journals International
Journal of Instrumentation and Control Systems, International Journal
of Control Theory and Computer Modeling, International Journal of
Information Technology, Control and Automation. He is an Associate
Editor of the journals International Journal of Information Sciences
and Techniques, International Journal of Control Theory and
Applications, International Journal of Computer Information Systems,International Journal of Advances in Science and Technology. He has
delivered several Key Note Lectures on Control Systems, Chaos
Theory, Scientific Computing, MATLAB, SCILAB, etc.