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International Journal of Modern Physics BVol. 27, No. 20 (2013) 1350109 (9 pages)c© World Scientific Publishing Company

DOI: 10.1142/S0217979213501099

ADAPTIVE FUNCTION Q-S SYNCHRONIZATION OF

DIFFERENT CHAOTIC (HYPER-CHAOTIC) SYSTEMS

JIAKUN ZHAO∗,†,‡, YING WU† and YUYING WANG∗

∗School of Mathematics and Statistics, Xi’an Jiaotong University,

Xi’an 710049, P. R. China†School of Aerospace, Xi’an Jiaotong University,

Xi’an 710049, P. R. China‡zhaojk@mail.xjtu.edu.cn

Received 4 April 2009Accepted 20 April 2009Published 4 July 2013

This paper presents the general method for the adaptive function Q-S synchronizationof different chaotic (hyper-chaotic) systems. Based upon the Lyapunov stability theory,the dynamical evolution can be achieved by the Q-S synchronization with a desiredscaling function between the different chaotic (hyper-chaotic) systems. This approachis successfully applied to two examples: Chen hyper-chaotic system drives the Lorenzhyper-chaotic system; Lorenz system drives Lu hyper-chaotic system. Numerical simu-lations are used to validate and demonstrate the effectiveness of the proposed scheme.

Keywords: Function Q-S synchronization; chaotic system; hyper-chaotic system; scalingfunction; Lyapunov function.

PACS numbers: 05.45.Xt, 05.45.Vx

1. Introduction

For identical or complete synchronization of two systems, the pioneering work of

FerritePecora and Carroll1 presented a criterion of the sub-Lyapunov exponents

to determine the synchronization of two systems connected with common signals

in 1990. Chaos synchronization has become a topic of great interest.2–6 Synchro-

nization phenomena have been reported in the recent literature. There exist many

types of synchronization such as complete synchronization, lag synchronization,

anticipated synchronization, phase synchronization and generalized synchroniza-

tion.2–7

Amongst all kinds of chaos synchronization, the Q-S synchronization is a more

general synchronization definition. It is presented by Yang,15 between two dynam-

ical systems, i.e., for two dynamical systems

x = f(x) , (1.1)

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J. Zhao, Y. Wu & Y. Wang

y = g(x, y) , (1.2)

where x = (x1, x2, . . . , xm)T ∈ Rm and y = (y1, y2, . . . , ym)T ∈ Rn are the state

vectors, f : Rm → Rm and g: Rm+n → Rm are two continuous vector functions.

Let Q1(x), Q2(x), . . . , Qh(x) and S1(y), S2(y), . . . , Sh(y) be observable variables of

systems (1.1) and (1.2), respectively. Systems (1.1) and (1.2) are said to syn-

chronize with respect to (Q1(x), Q2(x), . . . , Qh(x)) and (S1(y), S2(y), . . . , Sh(y)) if

limt→+∞[Qi(x)−Si(y)] = 0, i = 1, 2, . . . , h. Yan first called it Q-S synchronization8

and adopted the back-stepping design method9 to investigate the Q-S synchroniza-

tion between the Rossler system10 and the unified Lorenz–Chen–Lu system.11 Hu

and Xu12 studied the Q-S synchronization between two chaotic (hyper-chaotic)

systems with strictly different structures or different dimensions and proposed a

general scheme for Q-S synchronization based on the Lyapunov stability theory.

The properties of (right) inverse matrix play a critical role throughout the study

process. To the author’s best knowledge, the Q-S synchronization of the chaotic

systems with the scaling function and with the updating laws of the parameters

has received less attention. This work investigates the Q-S synchronization with the

preceding two points and obtains the general approach for the adaptive function

Q-S synchronization.

The rest of this paper is organized as follows. The problem formulation for

the function Q-S synchronization is given in Sec. 2. Two numerical examples are

provided to illustrate the effectiveness of the obtained scheme in Secs. 3 and 4.

Conclusions and further works are finally drawn in Sec. 5.

2. Problem Formulation for the Function Q-S Synchronization

Consider an m-dimensional chaotic (hyper-chaotic) system described by

x = f(x) , (2.1)

where x = (x1, x2, . . . , xm)T ∈ Rm is the state vector of the system, f : C(Rm →

Rm) is a vector function including nonlinear terms. We assume Eq. (2.1) as the

drive system. The controlled response system is given by

y = g(y) + U , (2.2)

where y = (y1, y2, . . . , yn)T ∈ Rn is the state vector, g: C(Rn → Rn) is a vector

function. The purpose of chaos synchronization is to design a controller U(U ∈

Rn), which is able to Q-S synchronize two identical or different chaotic (hyper-

chaotic) systems with a scaling function. Let Q(x) = (Q1(x), Q2(x), . . . , Qh(x))T

and S(y) = (S1(y), S2(y), . . . , Sh(y))T be observable variables of the systems (2.1)

and (2.2), respectively. Qi(x) =∑m

j=1 aijxj , i = 1, 2, . . . , h, where each of aij is

constant and Si(y) =∑m

j=1 bijyj , i = 1, 2, . . . , h, where each of bij is also constant.

In other words, Qix) is the linear. Let the function Q-S synchronization error of

the two chaotic (hyper-chaotic) systems be

e(t) = A(t)Q(x) − S(y) ,

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Adaptive Function Q-S Synchronization of Different Chaotic Systems

= (α1(t)Q1(x)− S1(y), α2(t)Q2(x)

−S2(y), . . . , αh(t)Qh(x) − Sh(y))T , (2.3)

where A(t) = diag(α1(t), α2(t), . . . , αh(t)), αi(t) ∈ C1(R → R), i = 1, 2, . . . , h.

Then the dynamical system between the drive system (2.1) and the response

system (2.2) can be written as

e(t) = A(t) ·Q(x) +A(t) ·DQ · f(x) −DS · (g(y) + U)

= h(x, t)−DS · (g(y) + U) , (2.4)

where A(t) is the differential matrix about the independent variable t,

A(t) =

α1(t) 0 · · · 0

0 α2(t) · · · 0

......

. . ....

0 0 · · · αh(t)

, (2.5)

h(x, t) = A(t)Q(x) +A(t)DQ · f(x) , (2.6)

and DQ(x), DS(y) are the Jacobian matrices of the vector function Q(x) and S(y),

respectively, i.e.,

DQ(x) =

∂Q1(x)

∂x1

∂Q1(x)

∂x2

· · ·∂Q1(x)

∂xm

∂Q2(x)

∂x1

∂Q2(x)

∂x2

· · ·∂Q2(x)

∂xm...

.... . .

...

∂Qh(x)

∂x1

∂Qh(x)

∂x2

· · ·∂Qh(x)

∂xm

, (2.7)

DS(y) =

∂S1(y)

∂y1

∂S1(y)

∂y2· · ·

∂S1(y)

∂yn

∂S2(y)

∂y1

∂S2(y)

∂y2· · ·

∂S2(y)

∂yn...

.... . .

...

∂Sh(y)

∂y1

∂Sh(y)

∂y2· · ·

∂Sh(y)

∂yn

. (2.8)

Therefore, by using the adaptive control techniques, the controller can be decided

as

U = −g(y) +DS−1 ·H(x, t) , (2.9)

where H(x, t) ∈ C(Rm ×R → Rn).

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J. Zhao, Y. Wu & Y. Wang

Definition 1. For the above systems (2.1) and (2.2), it is said that they are function

Q-S synchronization about the scaling function matrix A(t). If there exists the

updating laws of the estimated parameters such that the error vector (2.3) will

approach to zero as time goes on, i.e.,

limt→+∞

‖e(t)‖ = limt→+∞

‖A(t)Q(x) − S(y)‖ = 0 , (2.10)

then the states of the response system and the drive system are synchronized asymp-

totically.

The main objective of this paper is to give a general scheme to investigate

the function Q-S synchronization between the drive system (2.1) and the response

system (2.2) based on the Lyapunov stability theory and the adaptive methods. In

order to establish our main results, we need the following hypothesis.

Hypothesis 1.12 Real matrix DS is row full-rank, which means that the inverse

matrix DS−1 exists when the matrix is a square matrix. When the matrix is not a

square matrix, we further assume the number of rows less than the one of columns,

then according to the generalized inverse matrix theory,13,14 the right inverse matrix

DS−1R exists and DS−1

R = DST (DSDST )−1. For simplicity, we use DS−1 to denote

the inverse or right inverse matrix of DS(y) throughout this paper.

Theorem 1. Let e(t) be Eq. (2.3) for the drive system (2.1) and the response

system (2.2). If the active control U(x, y, t) under the Hypothesis 1 is given by

Eq. (2.9), then

limt→+∞

‖e(t)‖ = limt→+∞

‖A(t)Q(x)− S(y)‖ = 0

that is to say, the function Q-S synchronization occurs between the drive

system (2.1) and the response system (2.2) with respect to (α1(t)Q1(x),

α2(t)Q2(x), . . . , αh(t)Qh(x)) and (S1(y), S2(y), . . . , Sh(y)).

Proof. Construct a positive Lyapunov function candidate in the form of

V =1

2eT e . (2.11)

The time rate of change of V along the solution in Eq. (2.4) will be smaller than zero

with the reasonable control function H(x, t), Let H(e, x, t) = h(x, t) + Pe, where

P is an h-order positive definite matrix in order to complete this proof. Noting the

active controller (2.9), the time derivative of V along the solutions of Eq. (2.4) (at

e 6= 0) is

V = eT e = −eTΩe < 0 . (2.12)

According to the Lyapunov stability theory, the error variables become zero as time

t tends to positive infinity, i.e., limt→+∞ ‖e(t)‖ = limt→+∞ ‖A(t)Q(x)−S(y)‖ = 0.

This means that the drive system (2.1) and the response system (2.2) achieve the

function Q-S synchronization with respect to functions αi(t)Qi(x(t)) and Si(y(t))

under the controller U(x, y, t) given in the form (2.9) for any initial conditions. This

completes the proof.

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Adaptive Function Q-S Synchronization of Different Chaotic Systems

3. Function Q-S Synchronization of Chen Hyper-Chaotic System

and Lorenz Hyper-Chaotic System

We consider Chen hyper-chaotic system15 as the drive system and Lorenz hyper-

chaotic system is described as follows16 as the response system. The drive

system is

x1 = a1(x2 − x1) + x4 ,

x2 = d1x1 − x1x3 + c1x2 ,

x3 = x1x2 − b1x3 ,

x4 = x2x3 + r1x4 ,

(3.1)

and the response system is

y1 = a2(y2 − y1) + u1 ,

y2 = −y1y3 + b2y1 + y2 − y4 + u2 ,

y3 = y1y2 − c2y3 + u3 ,

y4 = d2y2y3 + u4 .

(3.2)

Let

A(t) =

−x1 0 0 0

0 1 0 0

0 0 x3 + 1 0

0 0 0 2x4 + 2

and P =

p1 0 0 0

0 p2 0 0

0 0 p3 0

0 0 0 p4

,

where Pi > 0, i = 1, 2, 3, 4.

We get

A(t) =

−a1(x2 − x1)− x4 0 0 0

0 0 0 0

0 0 x1x2 − b1x3 0

0 0 0 2x2x3 + 2r1x4

.

Assume the observable variables of systems (3.1) and (3.2) are

Q(x) =

Q1(x)

Q2(x)

Q3(x)

Q4(x)

=

x1

x2

x3

x4

and S(y) =

S1(y)

S2(y)

S3(y)

S4(y)

=

y1

y2

y3

y4

, (3.3)

respectively. We have

A(t)Q(x) =

−x21

x2

x3 + x23

2x4 + x24

, DQ(x) =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

and

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J. Zhao, Y. Wu & Y. Wang

DS(y) =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

. (3.4)

Then

DS−1 =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

.

We can obtain the controller

U = (u1, u2, u3, u4)T = −g(y) +DS−1 · (A(t) ·Q(x) +A(t) ·DQ · f(x) + Pe) ,

i.e.,

u1 = −a2(y2 − y1)− 2x1ω1 + p1e1 ,

u2 = y1y3 − b2y1 − y2 + y4ω2 + p2e2 ,

u3 = −y1y2 + c2y3 + 2x3ω3 + ω3 + p3e3 ,

u4 = −d2y2y3 + 2ω4 + 4x4ω4 + p4e4 ,

(3.5)

where ω1 = a1(x2 − x1) + x4, ω2 = d1x1 − x1x3 + c1x2, ω3 = x1x2 + b1x3, ω4 =

x2x3 + r1x4.

To simplify the analysis, we set k1 = k2 = k3 = k4 = 1.0 (in the whole paper).

We select the parameters of the drive system (3.1) as a1 = 35.0, b1 = 3.0, c1 = 12.0,

d1 = 7.0, r1 = 0.5 to ensure the hyper-chaotic behavior and the parameters of the

Fig. 1. Dynamics of synchronization error states for systems (3.1) and (3.2) with time t.

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Adaptive Function Q-S Synchronization of Different Chaotic Systems

response system (3.2) as a2 = 10.0, b2 = 28.0, c2 = 8/3, d2 = 0.1. The initial

values of two systems are x1(0) = 5.0, x2(0) = 8.0, x3(0) = 0.1, x4(0) = 0.3

and y1(0) = 3.0, y2(0) = 4.0, y3(0) = 5.0, y4(0) = 5.0. The four function Q-S

synchronization error states for the systems (3.1) and (3.2) are shown in Fig. 1.

4. The Lorenz System Drives the Hyper-Chaotic Lu System

Let the Lorenz system drive the Lu hyper-chaotic system.17 The drive system is

x1 = a(x2 − x1) ,

x2 = −x2 − x1x3 + bx1 ,

x3 = x1x2 − c1x3 ,

(4.1)

and the response system is

y1 = a2(y2 − y1) + y4 + u1 ,

y2 = −y1y3 + b2y2 + u2 ,

y3 = y1y2 − c2y3 + u3 ,

y4 = y1y3 + r2y4 + u4 .

(4.2)

Let A(t) =

(

x1 0 0

0 x2 0

0 0 x3

)

and P =

(

p1 0 0

0 p2 0

0 0 p3

)

, where pi > 0, i = 1, 2, 3. we

get

A(t) =

a1(x2 − x1) 0 0

0 −x2 − x1x3 + b1x1 0

0 0 x1x2 − c1x3

.

The observable variables of systems (3.6) and (3.7) are

Q(x) =

Q2(x)

Q2(x)

Q2(x)

=

x2

x2 − x3

−x1

and S(y) =

y2

y1 + y4

y3

,

respectively.

We have

DQ(x) =

0 1 0

0 1 −1

−1 0 0

, A(t)Q(x) =

x1x2

x22 − x2x3

−x1x3

and

DS(y) =

0 1 0 0

1 0 0 1

0 0 1 0

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J. Zhao, Y. Wu & Y. Wang

!"#$% &

Fig. 2. Dynamics of synchronization error states for systems (4.1) and (4.2) with time t.

then

DS−1 =

01

20

1 0 0

0 0 1

01

20

, (4.3)

u1 = −a2(y2 − y1)− y4 + x2λ2 −1

2x2λ3 −

1

2x3λ2 +

1

2p2e2 ,

u2 = y1y3 − b2y2 + x1λ2 + x2λ1 + p1e1 ,

u3 = −y1y2 + c2y3 − x1λ3 − x3λ1 + p3e3 ,

u4 = −y1y3 − r2y4 + x2λ2 −1

2x2λ3 −

1

2x3λ2 +

1

2p2e2 ,

where λ1 = a1(x2 − x1), λ2 = −x2 − x1x3 + b1x1, λ3 = x1x2 − c1x3.

Set k1 = k2 = k3 = k4 = 1.0 and select the parameters of the drive system (4.1)

as a1 = 10C, b1 = 28, c1 = 8/3 to ensure the hyper-chaotic behavior and the

parameters of the response system (4.2) as a1 = 36.0, b1 = 3.0, c1 = 20.0, r1 = 1.0.

The initial values of two systems are x1(0) = 2.0, x2(0) = −13.0, x3(0) = 14.0

and y1(0) = 3.0, y2(0) = 4.0, y3(0) = 5.0, y4(0) = 5.0. The three function Q-S

synchronization error states for the systems (4.1) and (4.2) are shown in Fig. 2.

5. Conclusion

This study proposed the problem formulation for the adaptive function Q-S syn-

chronization based on the Lyapunov stability theory. The acquired approach shows

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Adaptive Function Q-S Synchronization of Different Chaotic Systems

that the function Q-S synchronization of two chaotic (or hyper-chaotic) systems

can be achieved. It expands the work of the general Q-S synchronization of two dif-

ferent dynamical systems. Furthermore, two numerical examples have been given

to illustrate the effectiveness of the proposed scheme. The technique of designing

the scaling function A(t) will make the problem of the general Q-S synchronization

more interesting and attractive. Our future work will study the more general Q-S

synchronization with the scaling function and the updating laws of the unknown

parameters.

Acknowledgments

This work was supported by the China Postdoctoral Science Foundation Funded

Project (No: 2013M532030) and the Natural Science Foundation of China (NSFC)

under Grant 11272242

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