The International Journal of Information Technology, Control and Automation (IJITCA)

International Journal of Information Technology, Control and Automation (IJITCA) Vol.2, No.1, January 2012
DOI:10.5121/ijitca.2012.2102 15
HYBRID CHAOS SYNCHRONIZATION OF
UNCERTAIN LORENZ-STENFLO AND QI 4-D
CHAOTIC SYSTEMS BY ADAPTIVE 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
This paper investigates the 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. In hybrid chaos synchronization of master and slave systems, the odd states
of the two systems are completely synchronized, while the even states of the 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 shall assume that the parameters of both master
and slave systems are unknown and 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 such
as Mathematics, Computer Science, Microbiology, Biology, Ecology, Economics, Population
Dynamics and Robotics.
In 1990, Pecora and Carroll [2] deployed control techniques 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 [3], chemical systems [4-5], ecological systems [6], secure
communications [7-9], etc.
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

16
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 [2], a variety of impressive approaches have been
proposed for the synchronization of chaotic systems such as the OGY method [10], active
control method [11-15], adaptive control method [16-22], sampled-data feedback
synchronization 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 complete
synchronization [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 in
time, 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 in
generalized projective synchronization (GPS), the responses of the synchronized dynamical
states synchronize up to a constant scaling matrix .
α It is easy to see that the complete
synchronization (CS) and anti-synchronization (AS) are special cases of the generalized
projective synchronization (GPS) where the scaling matrix I
α = and ,
I
α = − respectively.
In hybrid synchronization of chaotic systems [15], one part of the system is synchronized and
the 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 in
secure 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 adaptive
nonlinear 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 chaotic
systems 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, we
derive 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 x
x x x
α γ
β
α
= − +
= − −
= −
= − −
&
&
&
&
(1)
where 1 2 3 4
, , ,
x x x x are the state variables and , , ,r
α β γ are positive constant parameters of the

17
system.
The LS system (1) is chaotic when the parameter values are taken as
2.0, 0.7, 1.5
α β γ
= = = and 26.0.
r =
The state orbits of the chaotic LS system (1) are shown in Figure 1.
The Qi system ([29], 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 hyperchaotic when the parameter values are taken as
30, 10, 1
a b c
= = = and 10.
d =
The state orbits of the chaotic Qi system (2) are shown in Figure 2.
Figure 1. State Orbits of the Lorenz-Stenflo Chaotic System

18
Figure 2. State Orbits of the Qi Chaotic System
3. HYBRID SYNCHRONIZATION OF IDENTICAL LORENZ-STENFLO SYSTEMS
3.1 Theoretical Results
In this section, we discuss the hybrid synchronization of identical Lorenz-Stenflo systems ([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
α γ
β
α
= − +
= − −
= −
= − −
&
&
&
&
(3)
where 1 2 3 4
, , ,
x x x x are the states and , , ,r
α β γ are unknown real constant parameters of the
system.

19
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
α γ
β
α
= − + +
= − − +
= − +
= − − +
&
&
&
&
(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 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
α γ
β
α
= − − + − +
= − + + − − +
= − + − +
= − + − +
&
&
&
&
(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
ˆ ˆ
( ) ( 2 ) ( 2 )
ˆ
( ) ( 2 )
ˆ
( )
ˆ
( ) 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
α γ
β
α
= − − − − − −
= − + + + −
= − + −
= + + −
(7)
where ˆ
ˆ ˆ
, ,
α β γ and r̂ are estimates of , ,
α β γ and ,
r respectively, and ,( 1,2,3,4)
i
k 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
ˆ ˆ
( )( 2 ) ( )( 2 )
ˆ
( )( 2 )
ˆ
( )
ˆ
( )
e e e x e x k e
e r r e x k e
e e k e
e e k e
α α γ γ
β β
α α
= − − − + − − −
= − + −
= − − −
= − − −
&
&
&
&
(8)
Let us now define the parameter estimation errors as
ˆ
ˆ ˆ
, ,
e e e
α β γ
α α β β γ γ
= − = − = − and ˆ.
r
e r r
= − (9)

20
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
α γ
β
α
= − − + − −
= + −
= − −
= − −
&
&
&
&
(10)
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
1 2 3 4 1 2 3 4
1
( , , , , , , , )
2
r r
V e e e e e e e e e e e e e e e e
α β γ α β γ
= + + + + + + + (11)
which is a positive definite function on 8
.
R
We also note that
ˆ
ˆ ˆ
, ,
e e e
α β γ
α β γ
= − = − = −
&
& &
& & & and ˆ
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
ˆ
ˆ
( 2 )
ˆ
( 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
α β
γ
α β
γ
 
 
= − − − − + − − − − + − −
   
 
 
+ − − + + −
 
&
&
&
&
&
(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
ˆ ( 2 )
ˆ
ˆ ( 2 )
ˆ ( 2 ) r
e e e x e k e
e k e
e e x k e
r e e x k e
α
β
γ
α
β
γ
= − − − +
= − +
= − +
= + +
&
&
&
&
(14)
where 5 6 7
, ,
k k k and 8
k 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 r
V k e k e k e k e k e k e k e k e
α β γ
= − − − − − − − −
& (15)
which is a negative definite function on 8
.
R

21
Thus, by Lyapunov stability theory [40], it is immediate that the hybrid synchronization error
,( 1,2,3,4)
i
e i = and the parameter estimation error , , , r
e e e e
α β γ 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-step 6
10
h −
= 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 4
i
k = 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
α β γ
= = = and 26.0.
r =
Suppose that the initial values of the parameter estimates are
ˆ ˆ
ˆ ˆ
(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. ˆ
ˆ ˆ
, ,
α β γ and r̂ converge to the
system parameters
2.0, 0.7, 1.5
α β γ
= = = and 26.0.
r =

22
Figure 3. Hybrid-Synchronization of Lorenz-Stenflo Chaotic Systems
Figure 4. Parameter Estimates ˆ
ˆ ˆ ˆ
( ), ( ), ( ), ( )
t t t r t
α β γ

23
4. HYBRID SYNCHRONIZATION OF IDENTICAL QI SYSTEMS
In this section, we discuss the hybrid 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
, , ,
The hybrid synchronization error is defined by
1 1 1
2 2 2
3 3 3
4 4 4
e y x
e y x
e y x
e y x
= −
= +
= −
= +
(18)
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 u
e de y y y x x x u
= − − + − +
= + + − − +
= − + − +
= − + + +

(19)

24
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
ˆ
( ) ( 2 )
ˆ
( ) ( 2 )
ˆ
( )
ˆ
( )
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 e
u t de y y y x x x k e
= − − − − + −
= − + + + + −
= − + −
= − − −
(20)
where ˆ
ˆ ˆ
, ,
a b c and d̂ are estimates of , ,
a b c and ,
d respectively, and ,( 1,2,3,4)
i
k i = are
positive constants.
1 2 1 2 1 1
2 1 2 1 2 2
3 3 3 3
4 4 4 4
ˆ
( )( 2 )
ˆ
( )( 2 )
ˆ
( )
ˆ
( )
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)
ˆ
ˆ ˆ
, ,
a b c
e a a e b b e c c
= − = − = − and ˆ.
d
e d d
= − (22)
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 e
e e e k e
= − − −
= + + −
= − −
= − −

(23)
approach is used.
We consider the quadratic Lyapunov function defined by
( )
2 2 2 2 2 2 2 2
1 2 3 4 1 2 3 4
1
( , , , , , , , )
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)
.
R
We also note that
ˆ
ˆ ˆ
, ,
a b c
e a e b e c
= − = − = −

and ˆ
d
e d
= −

(25)

25
2 2 2 2
1 1 2 2 3 3 4 4 1 2 1 2
2 2
2 1 2 1 3 4
ˆ
( 2 )
ˆ ˆ
ˆ
( 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)
1 2 1 2 5
2 1 2 1 6
2
3 7
2
4 8
ˆ ( 2 )
ˆ ( 2 )
ˆ
ˆ
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 8
2 2 2 2 2 2 2 2
1 1 2 2 3 3 4 4 5 6 7 8
a b c d
V k e k e k e k e k e k e k e k e
= − − − − − − − −
(28)
.
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.
Theorem 2. The identical Qi systems (16) and (17) with unknown parameters are globally and
exponentially 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.
10
h −
= 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 4
i
k = 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 =
ˆ ˆ
ˆ ˆ
(0) 6, (0) 2, (0) 9, (0) 5
a b c d
= = = =

26
1 2 3 4
(0) 4, (0) 8, (0) 17, (0) 20.
x x x x
= = = =
1 2 3 4
(0) 10, (0) 21, (0) 7, (0) 15.
y y y y
= = = =
Figure 5 depicts the complete synchronization of the identical Qi systems (16) and (17).
Figure 6 shows that the estimated values of the parameters, viz. ˆ
ˆ ˆ
, ,
a b c and d̂ converge to the
system parameters
30, 10, 1, 10.
a b c d
= = = =
Figure 5. Anti-Synchronization of the Qi Systems

27
Figure 6. Parameter Estimates ˆ ˆ
ˆ ˆ
( ), ( ), ( ), ( )
a t b t c t d t
5. HYBRID SYNCHRONIZATION OF NON-IDENTICAL LORENZ-STENFLO
AND QI SYSTEMS
In this section, we discuss the adaptive 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 x
x x x x
x x x
α γ
β
α
= − +
= − −
= −
= − −

(29)
where 1 2 3 4
, , ,
x x x x are the states and , , ,r
α β γ are unknown real constant parameters of the
system.

28
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
and 1 2 3 4
, , ,
The hybrid chaos synchronization error is defined by
1 1 1
2 2 2
3 3 3
4 4 4
e y x
e y x
e y x
e y x
= −
= +
= −
= +
(31)
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
α γ
β
α
= − − − − + +
= + − + − − +
= − + + − +
= − − − + +

(32)
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
ˆ ˆ
ˆ
( ) ( ) ( )
ˆ ˆ
( ) ( )
ˆ
ˆ
( )
ˆ ˆ
( )
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
α γ
β
α
= − − + − + − −
= − + + − + + −
= − − + −
= + + − −
(33)
where ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ
, , , , , ,
a b c d α β γ and r̂ are estimates of , , , , , ,
a b c d α β γ and ,
r respectively, and
,( 1,2,3,4)
i
k i = are positive constants.
1 2 1 2 1 4 1 1
2 1 2 1 2 2
3 3 3 3 3
4 4 4 4 4
ˆ ˆ
ˆ
( )( ) ( )( ) ( )
ˆ ˆ
( )( ) ( )
ˆ
ˆ
( ) ( )
ˆ ˆ
( ) ( )
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
α α γ γ
β β
α α
= − − − − − − − −
= − + + − −
= − − + − −
= − − − − −

(34)

29
ˆ ˆ
ˆ ˆ
, , ,
ˆ
ˆ ˆ ˆ
, , ,
a b c d
r
e a a e b b e c c e d d
e e e e r r
α β γ
α α β β γ γ
= − = − = − = −
= − = − = − = −
(35)
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
α γ
β
α
= − − − − −
= + + −
= − + −
= − − −

(36)
approach is used. We consider the quadratic Lyapunov function defined by
( )
2 2 2 2 2 2 2 2 2 2 2 2
1 2 3 4
1
2
a b c d r
V e e e e e e e e e e e e
α β γ
= + + + + + + + + + + + (37)
.
R
We also note that
ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ ˆ
, , , , , , ,
a b c d r
e a e b e c e d e e e e r
α β γ
α β γ
= − = − = − = − = − = − = − = −

(38)
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
ˆ
ˆ
( ) ( )
ˆ ˆ
ˆ ( )
ˆ ˆ ˆ
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
α
β γ
α
β γ
 
 
= − − − − + − − + + −
   
 
   
 
+ − − + − − + − − − −
   
 
 
     
+ − + − − + −
 
 
 
 

(39)
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
ˆ
ˆ ( ) , ( )
ˆ ˆ
( ) ,
ˆ
ˆ ,
ˆ ˆ
,
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
α
β
γ
α
β
γ
= − + = − − − +
= + + = +
= − + = − +
= − + = +

(40)
where 5 6 7 8 9 10 11
, , , , , ,
k k k k k k k and 12

30
2 2 2 2
1 1 2 2 3 3 4 4
V k e k e k e k e
= − − − −
2 2
5 6
a b
k e k e
− − 2 2
7 8
c d
k e k e
− − 2 2
9 10
k e k e
α β
− − 2 2
11 12 r
k e k e
γ
− − (41)
.
R
Thus, by Lyapunov stability theory [30], it is immediate that the synchronization error
,( 1,2,3,4)
i
e i = and the parameter estimation error decay to zero exponentially with time.
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.
10
h −
= 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 4
i
k = 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
α β γ
= = = = = = = = (42)
ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ ˆ
(0) 2, (0) 6, (0) 1, (0) 4, (0) 5, (0) 8, (0) 7, (0) 11
a b c d r
α β γ
= = = = = = = =
1 2 3 4
(0) 6, (0) 15, (0) 20, (0) 9.
x x x x
= = = =
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. ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ
, , , , , ,
a b c d α β γ and
r̂ converge to the original values of the parameters given in (42).

31
Figure 7. Complete Synchronization of Lorenz-Stenflo and Qi Systems
Figure 8. Parameter Estimates ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ ˆ
( ), ( ), ( ), ( ), ( ), ( ), ( ), ( )
a t b t c t d t t t t r t
α β γ

32
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 synchronization
results 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 a
very 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 chaos
synchronization 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 175 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 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.

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