Adaptive Projective Synchronization in An Array of Asymmetric Neural Networks

(1)

Adaptive Projective Synchronization in An Array

of Asymmetric Neural Networks

Guoliang Cai, Qin Yao , Xinghua Fan, Juan Ding

Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu 212013, China Email: [email protected]

Abstract—The main objective of this paper is further to investigate the adaptive projective synchronization problem for an array of coupled dynamical neural networks with both linear and nonlinear time-varying delays. Based on the Lyapunov stability theory, by using the technique of Young inequality, some inclusive results for the adaptive projective synchronization of the coupling networks are derived analytically. It is shown that under the controllers of this paper, the neural network will achieve adaptive projective synchronization for any coupling delays as long as the coupling strength is strong enough. Numerical simulations are shown to illustrate the effectiveness of these controllers.

Index Terms— neural networks, adaptive projective synchronization, time-varying delays, topology.

I. INTRODUCTION

The neural networks have been gaining increasing recognitions as a fundamental tool in understanding dynamical behavior and the response of real systems coming from different fields. Its application involves pattern identify, auto control, classic computation, associated memory, military application and decision support system. Combining of neural network with decision support system can realize parallel associated self-adapting in reasoning and automation of data mining. The dynamics of neural networks has been extensively investigated, with special emphasis on the interplay between the complexity in the overall topology and the local dynamical properties of the coupled nodes [1-8].

In recent years, as a typical kind of dynamics, synchronization in neural networks has become of significant interest, such as adaptive control based on BP neural networks, the adaptive current protection based on adaptive neural networks, research on neural network based adaptive user model in automatic web personalization and so on. Since synchronization in an array of linearly coupled dynamical systems was investigated in [9]. Many results on the local, global and partial synchronization in various coupled systems have also been obtained [10-14].Papers [15, 16] dealt with the global asymptotic synchronization problem for a class of chaotic neural networks with delay. The global synchronization of two chaotic neural networks with identical structure and different initial conditions was derived, which also present a procedure to construct a

synchronization controller. The anti-synchronization of a class of chaotic neural networks with time delay was investigated in paper [17]. Among various complex dynamical behaviors, projective synchronization (PS) is one of the signiﬁcant and interesting phenomenon. Historically, the synchronization of complex dynamical networks has been studied in various ﬁelds of science and engineering among [18-23]. PS is a generalized method of synchronization. Most existing studies pertaining to PS have mainly concentrated on two coupled chaotic systems. In Ref [18], the generalized projective synchronization of different chaotic systems with unknown parameters was investigated. By Lyapunov stability theory, the adaptive control method is proposed to achieve above synchronization phenomenon. Meanwhile, according to the invariance principle of differential equations, unknown parameter can be estimated accurately. The schemes are successfully applied to two groups of examples: the anti-phase synchronization between Lorenz system and Chen system; the complete synchronization between hyper-chaotic system and generalized Loren system. Then in [19], the authors proposed a linear controller and an updated law to realize the PS in drive–response dynamical networks of partially linear systems with time -varying coupling delay. Later, a number of PS studies such as adaptive projective synchronization, generalized projective synchronization, adaptive generalized function projective synchronization of some complex networks were investigated in papers [20-23].

Paper [24] investigated the global exponential synchronization for an array of asymmetric neural networks with time-varying delays and nonlinear coupling, assuming neither the differentiability for time-varying delays nor the symmetry for the inner coupling matrices. Paper [25] investigated the global synchronization in an array of linearly coupled neural networks with constant and delayed coupling. Using delayed state-feedback controller and analytic technique, the synchronization control of discrete-time neural network with delay was investigated in paper [26]. One may see more studies in papers [27-30].

At present, most of the theoretical results concerning adaptive projective synchronization of chaos are mainly focused on systems whose models are identical or similar, and parameters are exactly known in advance. But in many practical situations, the adaptive projective synchronization of chaos can be discussed for two strictly

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different systems, such as in social science and biological science. Further more, the parameters of many systems cannot be known entirely; the adaptive projective synchronization will be greatly affected by these uncertainties. Inspired by the above discussion, in this paper, we are further to investigate the adaptive projective synchronization problem for an array of coupled dynamical neural networks with delay-independent and delay-dependent dynamic behavior, where both linear and nonlinear time-varying delays are simultaneously taken into account. Based on the Lyapunov stability theory, by using the technique of Young inequality, some inclusive results for the adaptive projective synchronization of the coupling networks are derived.

The next paper is organized as follows: In section 2, system description and preliminaries are stated. An array of asymmetric cellular neural network with both linear and nonlinear time-varying delays is described. Two assumptions and a lemma are listed. In section 3, based on the Lyapunov stability theory, by using the technique of Young inequality, the adaptive projective synchronization of the array of asymmetric cellular neural network is derived. And four corollaries obtained following. The four corollaries show that some studies are just the special conditions of our work. In section 4, numerical simulations are provided to show the effectiveness of the theoretical results. We give the conclusions of this study in section 5.

II. MODEL DESCRIPTION AND PRELIMINARIES

In this paper, we first consider the following cellular neural network[24] with both linear and nonlinear time-varying delays:

1 2 3

1 1 1

4 1

( )

( ( ))

( (

( )))

( )

(

( ))

( ( ))

( (

( )))

( ).

1,2,..., .

i i i i

n n n

ij j ij j ij j

j j j

n

ij j j

x t

Cx t

Af x t

Bf x t τ t

G D x t

G D x t τ t

G D h x t

G D h x t τ t

I t

i

n

  



 



















(1)

where xi(t)=( xi1(t), xi2(t),…, xin(t))TRn is the state vector

of the ith neuron at time t, n corresponds to the number of neurons; C=diag(c1,c2,…,cn)>0, ck(k=1,2,…,n) is a

constant matrix and denotes the rate with which the kth neuron reset its potential to the resting state in isolation when disconnected from the network and external inputs;

A=(aij)n×n, B=(bij)n×n represent the connection weight

matrix and the delay connection weight matrix, respectively; (t) is time delay, f(xi(t))=(f1(xi1(t)), f2(xi2(t)),…,fn(xin(t)))T, fk (k=1,2,…,n) is the activation

function; D1, D2, D3, D4Rn×n represent the linking

matrix, the delay linking matrix, the nonlinear linking matrix and the nonlinear delay linking matrix, respectively; h(xi(t))=( h1(xi1(t)), h2(xi2(t)),…,hn(xin(t)))T, hk(k=1,2,…,n) is the coupling function which is nonlinear

function; I(t)=( I1(t), I2(t),…, In(t))T is an external input

vector; G=(Gij)n×nis the coupling configuration

representing the coupling strength and the topological

structure of the network, and satisfies the following diffusive coupling connections:

1,

0 (

),

, ( ,

1, 2,..., ).

n

ij ii ij

j j i

G

i

j

G

i j

n

 





 





For simplicity, we assume that:

ij i i

C

γ, A

α,

B

β, G

ξ,

D

θ







_









(2)

We now introduce the following assumptions and lemma:

Assumption 1. The time delay (t) satisfies

0 

τ t

( ) 1



and



satisfies

 

[ ( ),1].

τ

t

Assumption 2. Suppose there exist four positive constants Lk(k=1,2,3,4), such that:

1

2

3

4

( ( )) ( ( )) ( ) ( ) ,

( ( ( ))) ( ( ( )))

( ( )) ( ( )) ,

( ( )) ( ( )) ( ) ( ) ,

( ( ( ))) ( ( ( )))

( ( )) ( ( )) .

f x t f λy t L x t λy t

f x t τ t f λy t τ t L x t τ t λy t τ t

h x t h λy t L x t λy t

h x t τ t h λy t τ t L x t τ t λy t τ t

   



  



   

 

  



 _ _ _



   



(3)

They hold for any distributed vectors x(t), y(t), and the norm



of a vector is defined as

x



(

x x

T

) .

1/2

Lemma 1. [7] Suppose

0,

1, 1

1 1,

e



h



r



r



q



the next inequality holds:

1

1 .

r

r q r

r

eh

e

h

e

h

r

q

r











III. ADAPTIVE PROJECTIVE SYNCHRONIZATION

We refer to cellular neural network (1) as the drive system. In this section, we make the cellular neural network which has the same connection topologies as the response system:

1 2 3

1 1 1

4 1

( )

( ( ))

( (

( )))

( )

(

( ))

( ( ))

( (

( )))

( )

.

1,2,..., .

i i i i

n n n

ij j ij j ij j

j j j

n

ij j i

j

y t

Cy t

Af y t

Bf y t τ t

G D y t

G D y t τ t

G D h y t

G D h y t τ t

I t

u

i

n

  



 



















(4)

where yi(t)=( yi1(t), yi2(t),…, yin(t)) T_

Rn is the response state vector of the ith neuron at time t, ui(i=1,2,…,n) is

nonlinear controllers to be designed.

Let ei(t)=xi(t)-yi(t),  is a scaling factor, then the

(3)

1 2 1 1 3 1 4 1

( )

[ ( ( ))

( ( ))]

[ ( (

( )))

( (

( )))]

( )

(

( ))

[ ( ( ))

( ( ))]

[ ( (

( )))

( (

( )))]

.

i i i i i

n n

i ij j ij j

j j

n

ij j j

j n

ij j j i

j

e t

Ce t

A f x t

λf y t

B f x t τ t

λf y t - τ t

G D e t

G D e t τ t

G D h x t

λh y t

G D h x t τ t

λh y t - τ t

λu

   

 























(5)

By using adaptive controlling method, we get the following theorem:

Theorem 1. Suppose Assumptions 1 and 2 hold for networks (1) and (4). Choose the following adaptive controller and updated law:

3 1 4 1

1 [

( )

( (

( ))

( ( )))

( (

(

( )))

( (

( ))))

( (

( ))

( ( )))

( (

(

( )))

( (

( ))))].

i i i i i i

n

i ij j j

j n

ij j j

j

u

l e t

A f λy t

λf y t

B f λy t τ t

λ

λf y t τ t

G D h λy t

λh y t

G D h λy t τ t

λh y t τ t

 





















(6) * * *

(1

ij

) ( ) ,

r

0,

, ,

1, 2,..., .

i i i i ij

i

l

k

e t

l

L i j

n

l











(7)

where the adaptive feedback gain variable

l=(l1,l2,…,ln)TRn is to be designed. If L* satisfies: *

1 2 1 2 3 3 4 4

4

1 2 3 3 4

1

1 (

)

(( 1)

)

( -1)( +

)

1

1 ( +

).

1,2,..., .

1

n j j i

r L γ > L rα+ L β r

+ξ r

θ θ L θ L θ

δ

α

L

ξ θ

θ L θ

θ

i

n

α

δ





 













(8)

where r>1, i, j>0. Then the error system (5) is globally

stable, thus we have the networks (1) and (4) achieve adaptive projective synchronization.

Proof. Choose the following Lyapunov function:

2

1 1 ( )

2 2 4 4

1 1 ( ) 1

( )

( ) d

1

1 ( ) d

.

1

2

t

n n

r r

i i i i

i i t τ t

t

n n _r n

i ij j i i

i j t τ t i i

L B

V t

α e t

α

e s

s

δ

D

L D

r

α G

e s

s

α

l

δ

k

      













_{ }







(9)

Substitute the adaptive controller (6) to (5), we get:

1 2 1 1 3 1 4 1

( )

[ ( ( ))

(

( ))]

[ ( (

( )))

(

( )))]

( )

(

( ))

[ ( ( ))

(

( ))]

[ ( (

( )))

(

( )))]

( ).

1, 2,..., .

i i i i

i i

n n

ij j ij j

j j

n

ij j j

j n

ij j j

j i i

e t

Ce t

A f x t

f λy t

B f x t τ t

f λy t - τ t

G D e t

G D e t τ t

G D h x t

h λy t

G D h x t τ t

h λy t - τ t

l e t

i

n

   

 

























(10)

Calculating the derivative of (9) along the trajectories of (10), we get:

1 1

2 2

1 1

2 4 4 1 1 2 4 4

1 1 1

( )

sign( ( )) ( )

( )

(

( ))

1

1 ( )

1

1 (

( ))

.

1

n r

i i i i

i

n n

r r

i i i i

i i

n n _r

i ij j i j

n n _r n

i ij j i i i

i j i i

V t

α r e t

e t e t

L B

α e t

α e t τ t

δ

D

L D

α G e t

δ

D

L D

α G e t τ t

α

l l

δ

k

        

























(11)

By Assumption 1 and Lemma 1, we have:

1 ( )

1.

1 τ t

δ



_



(12)

1 1 1

1

1 ( )

(

( ))

(

( ))

( ) ,

1

1 ( )

( )

( ) ,

1

1 ( )

(

( ))

(

( ))

( ) .

r r r

i i i i

r

r r

i j j i

r

r r

i j j i

r

e t

e t τ t

e t

r

e t

r

e t

e t τ t

e t

r

  





_

_

_

_



_



_

_









_

_

_

_



(13)

Substitute (2), (7), (10), (12), (13) to (11) in proper orders, the simplified result may obtain:

*

1 2

1

1 2 3 3 4 4 4

1 2 3 3 4

1

1 ( )

[ (

)

((

1)

)

1 ( -1)( +

)

1 ( +

)] ( ) .

1

n i i n r j i j i

V t

α r γ L α L

L β r

δ

ξ r

θ θ

L θ

α

_L

ξ θ

θ

L θ

θ

e t

α

δ

 



 





 









Obviously, we have:V t( )0, if condition (8) holds, namely,

lim

i

( )

0,

t

e t



i=1, 2,… n, therefore, the error system (5) is globally stable, and then we have the networks (1) and (4) achieve adaptive projective synchronization. Thus the proof is completed.

From Theorem 1, we may get the following corollaries:

Corollary 1. If B=D2=D4=0, system (1) is reduced to

the following non-time-delay system:

1 1

3 1

( )

( ( ))

( )

( ( ))

( ).

1, 2,..., .

n

i i i ij j

j n

ij j j

x t

Cx t

Af x t

G D x t

G D h x t

I t

i

n

 

 







(14)

Choose the following adaptive controller and updated law 3 1

1 [

( )

( (

( ))

( ( )))

( (

( ))

( ( )))].

i i i i i

n

ij j j

j

u

l e t

A f λy t

λf y t

λ

G D h λy t

λh y t













*

T * *

(1

ij

) ( ) ( ),

0,

, ,

1, 2,..., .

i i i i i ij

i

l

k

e t e t

l

L i j

n

l

(4)

If the next inequality holds:

*

1 1 3 3

1

( +

).

1, 2,..., .

n j

j i

α

L

γ > L α+

ξ θ L θ

i

n

α









Then the network (14) is adaptive projective synchronization.

Proof. Consider the following Lyapunov function:

T 2

1 1

1

1 ( )

( ) ( )

.

2

n n

i i i i i

i i i

V t

α e t e t

α

l

k

 









Similarly to the proof of Theorem 1, one may easily find that the network (14) is adaptive projective synchronization.

Corollary 2. If (t) =, system (1) is reduced to the following time-delay system:

1 2 1 1 3 4 1 1

( )

( ( ))

( (

))

( )

(

)

( ( ))

( (

))

( ).

i i i i

n n

ij j ij j

j j

n n

ij j ij j

j j

x t

Cx t

Af x t

Bf x t τ

G D x t

G D x t τ

G D h x t

G D h x t τ

I t

   

 











 



(15)

Choose the following adaptive controller updated law:

3 1 4 1

1 [

( )

( (

( ))

( ( )))

( (

(

))

( (

)))

( (

( ))

( ( )))

( (

(

))

( (

)))].

i i i i i

i i

n

ij j j

j n

ij j j

j

u

l e t

A f λy t

λf y t

λ

B f λy t τ

λf y t τ

G D h λy t

λh y t

G D h λy t τ

λh y t τ

 





















* * *

(1

ij

) ( ) ,

r

0,

, ,

1, 2,..., .

i i i i ij

i

l

k

e t

l

L i j

n

l











2

1 1

2 2 4 4

1 1 1

( )

( ) d

1

1 ( ) d

.

1

2

t

n n

r r

i i i i

i i t τ

t

n n n

r

i ij j i i

i j t τ i i

L B

V t

α e t

α

e s

s

δ

D

L D

r

α G

e s

s

α l

δ

k

  _   _ 













 







Similarly to the proof of Theorem 1, one can guarantee that the network (15) is adaptive projective synchronization.

Corollary 3. If B=D3=D4=0, system (1) is reduced to

the following linear time-delays system:

1 1

2 1

( )

( ( ))

( )

(

( ))

( ).

1, 2,..., .

n

i i i ij j

j n

ij j j

x t

Cx t

Af x t

G D x t

G D x t τ t

I t

i

n

 

 











(16)

Choose the following adaptive controller updated law

1 [

( )

( (

( ))

(

( )))],

i i i i i

u

l e t

A f λy t

λf y t

λ







*

* *

(1

ij

) ( ) ,

r

0,

, ,

1, 2,..., .

i i i i ij

i

l

k

e t

l

L i j

n

l











*

1 1 2

1

1 (

)

( +

).

1, 2,..., .

1

n j j i

α

r L

γ > L rα

ξ θ

θ

i

n

α

δ











Proof. The corresponding response system is:

1 1

2 1

( )

( ( ))

( )

(

( ))

( )

.

1, 2,..., .

n

i i i ij j

j n

ij j i

j

y t

Cy t

Af y t

G D y t

G D y t τ t

I t

u

i

n

 

 











(17)

Let ei(t)=xi(t)-yi(t),  is a scaling factor, then the

following error cellular neural system of (16) and (17) can be obtained:

1 1

2 1

( )

[ ( ( ))

( ( ))]

( )

(

( ))

.

1, 2,..., .

n

i i i i ij j

j n

ij j i

j

e t

Ce t

A f x t

λf y t

G D e t

G D e t τ t

λu

i

n

 

 













(18)

Consider the following Lyapunov function:

2

1 1 1

2 1

( )

( ) d

1

1 .

2

t

n n n _r

r

i i i ij j

i i j t τ(t)

n i i i i

D

V t

α e t

α G

e s

s

δ

r

α

l

k

    

















Similarly to the proof of Theorem 1, one can verify that the error system (18) is globally stable, and then the networks (16) and (17) can achieve adaptive projective synchronization.

Corollary 4. If D1=D2=0, system (1) is reduced to the

following nonlinear time-delays system:

3 4

1 1

( )

( ( ))

( (

( )))

( ( ))

( (

( )))

( ).

i i i i

n n

ij j ij j

j j

x t

Cx t

Af x t

Bf x t τ t

G D h x t

G D h x t τ t

I t

 

 

















(19)

Choose the following adaptive controller update

3 1 4 1

1 [ ( )

( (

( ))

( ( )))

( (

(

( )))

( (

( ))))

( (

( ))

( ( )))

( (

(

( )))

( (

( ))))].

i i i i i

i i

n

ij j j

j n

ij j j

j

u

l e t

A f λy t

λf y t

λ

B f λy t τ t

λf y t τ t

G D h λy t

λh y t

G D h λy t τ t

λh y t τ t

(5)

0 5 10 15 20 0

5 10

||e

1

||

0 5 10 15 20

0 5 10

||e

2

||

0 5 10 15 20

0 5 10

t

||e

3

||

0 5 10 15 20

0 5 10

||e

4

||

0 5 10 15 20

0 5 10

||e

5

||

Figue 1. Projective synchronization errors for the drive and response networks.

0 5 10 15 20

-8 -6 -4 -2 0 2 4

t

C

11

C₁₂ C

13

C₁₅ C

15

0 5 10 15 20

-8 -6 -4 -2 0 2 4 6

t

C

23

C

32

C

44

C

45

C

55

Figue2. Estimation of the weighted matrix G with time t, where Gij=Cij.

*

* *

(1

ij

) ( ) ,

r

0,

, ,

1, 2,..., .

i i i i ij

i

l

k

e t

l

L i j

n

l











*

1 2

3 3 4 4 4 3 3 4 1

1 (

)

((

1)

)

1 ( -1)(

)

(

).

1, 2,..., .

1

n j j i

r L

γ > L rα+ L β r

δ

+ ξ r

L θ

α

L

ξ L θ

θ

i

n

α

δ





 











2

1 1 ( )

2 4 4

1 1 ( ) 1

( )

( ) d

1

1 ( ) d

.

1

2

t

n n

r r

i i i i

i i t τ t

t

n n _r n

i ij j i i

i j t τ t i i

L B

V t

α e t

α

e s

s

δ

L D

r

α G

e s

s

α

l

δ

k

  

  _ 













_{ }







Similarly to the proof of Theorem 1, one can deduce that the network (19) can achieve adaptive projective synchronization.

IV. NUMERICAL SIMULATIONS

In this section, to verify and demonstrate the effectiveness of the proposed approaches in this paper, we consider a numerical example that is Lü system:

1 2 1 1

2 2 1 3 2

3 3 1 2 3

(

)

( , ).

x

a x

x

cx

x x

C x

f t x

x

bx

x x

x



  



 

  

_

_



_{ }

 

_

  



 

  

_

_



 

  



 

where a=36, b=3, c=20,

1 3 1 2

0

0 0 ,

( , )

.

0

0 a

a

C

c

f t x

x x

b

x x





















_



_

 

_

_

















Now we consider a weighted linearly coupled neural network (16) with coupling delays consisting of 5 identical Lü systems. Taking the weight configuration coupling matrix:

5 5

6

3

2

0

1

2

5

0

2

1 (

)

1

4

1

1 .

1

3

0

1

0

1

0

2

ij

G

_









_

















_









_







The drive neural network system is defined as:

5 1 1 5

2 1

( )

( ( ))

( )

(

( ))

( ).

1, 2,...,5.

i i i ij j

j ij j

j

x t

Cx t

Af x t

G D x t

G D x t τ t

I t

i



 











(6)

For simplicity, we assume that A=D1=I(t)=I3,

( )

.

1

t t

e

τ t

e





Let the scaling factor λ=2, the initial values are given as:L1=1, r=2, γ=36, α=1, L*=I3, θ1=1, θ2=1, ξ=6, αi=1, αj=0.1, δ=0.5. xi(0)=(0.3+0.1i, 0.3+0.1i,

0.3+0.1i)T, yi(0)=(1.4+0.1i, 1.4+0.1i, 1.4+0.1i)T, i,j=1,2,…,5.

Based on Corollary 3, the response system is given as:

5 1 1 5

2 1

( )

( ( ))

( )

(

( ))

( )

.

1, 2,...,5.

i i i ij j

j

ij j i

j

y t

Cy t

Af y t

G D y t

G D y t τ t

I t

u

i



 











1 [

( )

( (

( ))

( ( )))],

1, 2,...,5.

i i i i i

u

l e t

A f λy t

λf y t

i

λ









*

* *

(1

ij

) ( ) ,

r

0,

, ,

1, 2,...,5.

i i i i ij

i

l

k

e t

l

L i j

l











The synchronization errors are shown respectively in Figs 1 and 2. The numerical results show that the error cellular neural system of Corollary 3 is globally stable in Fig.1. The estimation of the weighted matrix G shows in Fig.2. and then the network (16) is adaptive projective synchronization.

V. CONCLUSIONS

This paper has further investigated the adaptive projective synchronization problem for an array of coupled dynamical neural networks with delay-independent and delay-dependent dynamic behavior, where both linear and nonlinear time-varying delays are simultaneously taken into account, respectively. Sufficient conditions have obtained to show the adaptive projective synchronization of an array dynamical neural networks, which are different from the existing ones and have wider application fields than some results in literature. Numerical simulations are presented to show and illustrate the effectiveness of these controllers.

The technology assessment of the paper is a simple calculation and is easy to achieve. The method is a power tool to solve the time delay problem. There are large results about the synchronization of dynamical neural networks in literature, but one may find out that some studies are just the special conditions of our Theorem 1 or Corollaries 1-4. For example, the neural network of Ref. [12] is the neural networks of our Corollary 3. Experiments show that the methods presented in this paper are of high application in synchronization fields.

ACKNOWLEDGMENTS

This work was supported by the National Nature Science foundation of China (Nos. 70571030, 90610031), the Society Science Foundation from Ministry of Education of China (Nos. 08JA790057), the Advanced Talents’ Foundation of Jiangsu University (No. 07JDG054), A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education

Institutions, and the Student Research Foundation of Jiangsu University (No. 10A147).

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Guoliang Cai received the Ph.D. degree from Jiangsu University. He is currently a professor of the Nonlinear Scientific Research Center of Jiangsu University. His major research fields are chaos, nonlinear scientific research. His research activities have been supported by the the National Nature Science foundation of China (Nos. 70571030, 90610031), the Society Science Foundation from Ministry of Education of China (Nos. 08JA790057), the Advanced Talents’ Foundation of Jiangsu University (No. 07JDG054).

Qin Yao is currently a M.S candidate at the Nonlinear Scientific Research Center of Jiangsu University, Zhenjiang, Jiangsu, China. She received her BS degree in Mathematics and Applied Mathematics of Hubei Institute for Nationalities in 2009. Her major is Applied Mathematics. Her main research areas include synchronization, neural networks and sliding mode control.

Xinghua Fan is currently an associate professor of the Nonlinear Scientific Research Center of Jiangsu University. She is a Postdoc. Her main research area is nonlinear scientific research.