adaptive hybrid chaos synchronization of lorenz-stenflo and qi 4-d chaotic systems with unknown...

8/3/2019 Adaptive Hybrid Chaos Synchronization of Lorenz-Stenflo and Qi 4-D Chaotic Systems with Unknown Parameters

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International Journal in Foundations of Computer Science & Technology,Vol. 2, No.1, January 2012

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


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)


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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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.


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)


, ,a b c

e a a e b b e c c= = = and .de 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 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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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)


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.


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.


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.


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=


(0) 6, (0) 2, (0) 9, (0) 5a b c d = = = =


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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 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


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)


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)


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.


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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, , ,

, , ,

a b c d

r

e a a e b b e c c e d d

e e e e r r

= = = =

= = = =

(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

=

= + +

= +

=

&

&

&

&

(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)


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)


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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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.


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.


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)


(0) 2, (0) 6, (0) 1, (0) 4, (0) 5, (0) 8, (0) 7, (0) 11a b c d r = = = = = = = =


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. , , , , , ,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.

adaptive hybrid chaos synchronization of lorenz-stenflo and qi 4-d chaotic systems with unknown...

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