Synchronization of the Complex Dynamical Networks with a Gui Chaotic

Strange Attractor

Zhanji Gui

1

and Lan Kang

2

1

Department of Software Engineering, Hainan College of Software Technology, Qionghai, 571400, P.R. China

2

Department of Obstetric, Haikou Women and Children Hospital, Haikou, 570203, P.R. China

Keywords:

Gui Chaotic Strange Attractor, Neural Networks, Synchronization.

Abstract:

In this paper, impulsive neural networks with a Gui chaotic strange attractor is studied. By employing the

Lyapunov-like stability theory of impulsive functional differential equations, some criteria for synchronization

of impulsive neural networks are derived. An illustrative example is provided to show the effectiveness and

feasibility of the proposed method and results.

1 INTRODUCTION

In recent years, the additive neural networks have

been extensively studied, including both continuous

time and discrete-time settings, and applied to as-

sociative memory, model identiﬁcation, optimization

problems, etc. Many essential features of these net-

works, such as qualitative properties of stability, os-

cillation, and convergence issues have been investi-

gated (Cao, 1999; Hopﬁeld, 1982; Song and Zhang,

2008; Subashini and Sahoo, 2014; Wang and Huang,

2014).

However, as we well know, nonautonomous phe-

nomena often occur in many realistic systems. Partic-

ularly when we consider a long-term dynamical be-

haviors of a system, the parameters of the system usu-

ally will change with time. In addition, in many appli-

cations, the property of periodic oscillatory solutions

of cellular neural networks also is of great interest.

In fact, there has been considerable research on the

nonautonomous neural networks(Gopalsamy and He,

1994; Subashini and Sahoo, 2014) . The cellular neu-

ral networks with impulse effect are studied, where

the criteria on the existence, uniqueness and global

stability of periodic solution are obtained.Further, a

new chaos strange attractor was also found, known

as Gui chaos strange attractor(Zhang and Gui, 2009a;

Zhang and Gui, 2009b).

Another type of synchronization, impulsive syn-

chronization, has been developed(Amritkar and

Gupte, 1993; Anzo and Barajas-Ramłrez, 2014;

D¨orﬂer and Bullo, 2014; Wan and Cao, 2015; Xie

and Xu, 2014). It allows synchronization of chaotic

systems using only small impulses (Yang and Chua,

1999a) generated by samples of the state variables of

the driving system at discrete time instances. These

samples are called the synchronizing impulses and

they drive the response system discretely at these in-

stances. After a ﬁnite period of time, the two chaotic

systems behave in accordance with each other and

the synchronization of the two chaotic systems is

achieved. In other words, the asymptotic stability

property of the error dynamics between the driving

and response systems is reached. The impulsive syn-

chronization has been applied to a number of chaos-

based communication systems which exhibit good

performance for the synchronization purposes and for

security purposes (Yang and Chua, 1997; Yang and

Chua, 1999b) .

Motivated by the above discussions, the aim of

this paper is to study the synchronization of impul-

sive neural networks with a Gui chaotic strange attrac-

tor. By employing the Lyapunov-like stability theory

of impulsive functional differential equations, some

criteria for synchronization of impulsive neural net-

works are derived.

The remainder of the paper is organized as fol-

lows: Section 2 describes the issue of synchroniza-

tion of coupled impulsive systems with a Gui chaotic

strange attractor. In Section 3, some sufﬁcient condi-

tions for the synchronization are derivedby construct-

ing suitable Lyapunov-like function. In Section 4, an

illustrative example is given to show the effectiveness

of the proposed method. Conclusions are given in

Section 5.

257

Gui Z. and Kang L..

Synchronization of the Complex Dynamical Networks with a Gui Chaotic Strange Attractor.

DOI: 10.5220/0005504902570262

In Proceedings of the 5th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH-2015),

pages 257-262

ISBN: 978-989-758-120-5

Copyright

c

2015 SCITEPRESS (Science and Technology Publications, Lda.)

2 PRELIMINARIES AND

PROBLEM FORMULATION

In this paper, we consider the following nonau-

tonomous cellular neural networks model with im-

pulses

dx

i

dt

= −a

i

x

i

(t) +

n

∑

j=1

b

ij

f

j

(x

j

(t)) + c

i

(t),

t > 0, t 6= t

k

,

∆x

i

(t

k

) = x

i

(t

+

k

) − x

i

(t

−

k

) = d

ik

x

i

(t

−

k

),

(1)

where i = 1, 2, . . . , n;k = 1, 2, . . . ;∆x

i

(t

k

) = x

i

(t

+

k

) −

x

i

(t

−

k

) are the impulses at moments t

k

and t

1

<

t

2

< . . .is a strictly increasing sequence such that

lim

k→∞

t

k

= +∞; x

i

(t) corresponds to the state of the

ith unit at time t, f

j

(x

j

(t)) denotes the output of the

jth unit at time t, b

ij

denotes the strength of the jth

unit on the ith unit at time t, c

i

(t) is the external bias

on the ith at time t, a

i

represents the rate with which

the ith unit will reset its potential to the resting state

when disconnected from the network and external in-

puts.

As usual in the theory of impulsive differential

equations, at the points of discontinuity t

k

of the solu-

tion t 7→ x

i

(t)we assume that x

i

(t

k

) ≡ x

i

(t

−

k

). It is clear

that, in general, the derivatives x

′

i

(t

k

) do not exist. On

the other hand, according to the ﬁrst equality of (1)

there exist the limits x

′

i

(t

∓

k

). According to the above

convention, we assume x

′

i

(t

k

) ≡ x

′

i

(t

−

k

).

Throughout this paper, we assume that:

(H

1

) Functions f

j

(u) ( j = 1, 2, . . . , n) are Lipschitz

continuous and monotonically non-decreasing,

i.e., for all u

1

, u

2

∈ R = (−∞, ∞) there are con-

stants L

j

> 0 such that

0 6

f

j

(u

1

) − f

j

(u

2

)

u

1

− u

2

6 L

j

.

(H

2

) There exists a positive integer T , such that

t

k+T

= t

k

+ ω, d

i(k+T)

= d

ik

,

where k = 1, 2, . . . , i = 1, 2, . . . , n.

In (Gui and Ge, 2006) , the system (1) were found

to have a Gui chaotic strange attractor. Now we con-

sider the drive system in the form of the neural net-

works (1). For the purpose of synchronization, we

introduce the response system that is driven by (1) via

a set of signals

dy

i

dt

= −a

i

y

i

(t) +

n

∑

j=1

b

ij

f

j

(y

j

(t)) + c

i

(t),

t > 0, t 6= t

k

,

y

i

(t

+

k

) = 2d

ik

x

i

(t

−

k

) + (1− d

ik

)y

i

(t

−

k

),

(2)

where i = 1, 2, . . . , n, k = 1, 2, . . . . Letting e(t) = y

i

−

x

i

be synchronization error, where e(t) = (e

1

(t), e

2

(t),

. . . , e

n

(t))

T

, x

i

(t) and y

i

(t) are the state variables of

drive system (1) and response system (2). Thus,we

can derive the error dynamical system as follows:

(

˙e = − De(t) +WG(e(t)), t 6= t

k

,

e(t

+

k

) = (I − D

k

)e(t

−

k

), k = 1, 2, . . . ,

(3)

where

g

j

= f

j

(e

j

+ x

j

) − f

j

(x

j

),

G(e) = [g

1

, g

2

, . . . , g

n

]

T

.

I is identity matrix, and

D

k

=

d

1k

0 ·· · 0

0 d

2k

·· · 0

.

.

.

.

.

.

.

.

.

.

.

.

0 0 ·· · d

nk

.

In fact, from the analysis above, we can see that (1)

and (2) are synchronized if and only if the equilib-

rium point of (3) is asymptotically stable for any ini-

tial condition.So the global impulsivesynchronization

problem can be solved if the controller gain matrices

d

ik

are suitably designed such that the zero solution of

(3) is globally asymptotically stable.

3 MAIN RESULTS

In this section, we will derive some sufﬁcient condi-

tions for synchronization in the sense of the fact that

the error system (3)

Theorem 1. If there exist a positive constant ε >

0, α > 0, positive deﬁnite diagonal matrix P > 0, such

that

1. Linear matrix Inequality

−PD− D

T

P+ εL

2

M

λ

M

(W

T

W) − αP P

P −ε

< 0

(4)

where L

M

= max{L

j

}, λ

M

denote the largest eigen-

value of the matrix (∗).

2. τ < inf

k∈N

{t

k

− t

k−1

} < 1;

3. There exists a constant α such that

ln(ηβ

k

) + α(t

k

− t

k−1

) < 0, (5)

where β

k

= λ

2

M

(I − D

k

); then, the origin of system (3)

is globally asymptotically stable, which implies that

(1) and (2) are completely synchronized.

Proof. Construct a Lyapunov function in the form of

V(e(t)) = e

T

(t)Pe(t).

SIMULTECH2015-5thInternationalConferenceonSimulationandModelingMethodologies,Technologiesand

Applications

258

When t ∈ (t

k

− t

k−1

], the total derivative of V(e) with

respect to (3) is

˙

V(e) = ˙e

T

Pe+ e

T

P˙e,

=(−De+WG(e))

T

Pe

+ e

T

P(−De+WG(e))

= − e

T

(PD+ D

T

P)e

+ 2e

T

PWG(e).

Also, by the well-known inequality

2a

T

b 6 ε

−1

a

t

a+ εb

T

b,

for ∀ε > 0, we obtain

˙

V(e) 6 −e

T

(PD+ D

T

P)e+ ε

−1

e

T

PPe

+εL

2

M

λ

M

(W

T

W)G(e).

From the Assumption (H

1

), we can obtain

||G(e)||

2

6 L

2

M

||e||

2

This leads to

˙

V(e) 6 −e

T

(PD+ D

T

P)e+ ε

−1

e

T

PPe

+εL

2

M

λ

M

(W

T

W)e

T

e. (6)

From Linear matrix inequality (4) and (6), we have

˙

V(e) < αe

T

Pe = αV(e).

Let V(t) = V(e(t)), then

V(t) 6 V(t

+

k−1

)exp[α(t − t

k−1

)], (7)

where t ∈ (t

k−1

,t

k

], k ∈ N. From the second equation

in (3), we have

V(t

+

k

) = e

T

(t

+

k

)Pe(t

+

k

)

= [(I − D

k

)e(t

k

)]

T

P[(I − D

k

)e(t

k

)]

6 λ

2

M

(I − D

k

)V(t

k

)

= β

k

V(t

k

). (8)

For t ∈ (t

0

,t

1

], from (7) and (8), we have

V(t) 6 V(t

0

)exp[α(t − t

0

)], t ∈ (t

0

,t

1

],

which leads to

V(t

1

) 6 V(t

0

)exp[α(t

1

− t

0

)], t ∈ (t

0

,t

1

],

and

V(t

+

1

) 6 V(t

0

)β

1

exp[α(t

1

− t

0

)], t ∈ (t

0

,t

1

],

Similarly, for t ∈ (t

1

,t

2

],

V(t) 6 V(t

+

1

)exp[α(t − t

1

)],

6 V(t

0

)β

1

β

2

exp[α(t − t

0

)]

In general, for t ∈ (t

k−1

,t

k

],

V(t) 6 V(t

+

1

)exp[α(t − t

1

)],

6 V(t

0

)β

1

β

2

·· · β

k

exp[α(t − t

0

)] (9)

For t ∈ (t

k

,t

k+1

)], it follows from (5) and (9) that,

V(e(t)) 6 V(e(t

0

))β

1

β

2

·· · β

k

exp[α(t − t

0

)]

6 V(e(t

0

)){β

1

exp[ατ

1

]}{β

2

exp[ατ

2

]}

·· · {β

k

exp[ατ

k

]}exp[α(t − t

k

)]

6 V(e(t

0

))

exp[α(t − t

k

)]

η

k

(10)

From (10) , we can see that the trivial solution of sys-

tem (3) is globally asymptotically stable.This com-

pletes the proof.

4 AN ILLUSTRATIVE EXAMPLE

In order to demonstrate and verify the performance of

the proposed method, some numerical simulations are

presented in this section.

As is known to all that (1) can exhibit Gui chaotic

strange attractor (Zhang and Gui, 2009a; Zhang and

Gui, 2009b).In order to show it clearly, we give the

following example:

˙x

1

˙x

2

˙x

3

= −

x

1

x

2

x

3

+

1.2 −1.6 0

1.2 1.0 0.9

0 2.2 0.15

f

1

(x

1

)

f

2

(x

2

)

f

3

(x

3

)

+

c

1

(t)

c

2

(t)

c

3

(t)

x

i

(t

+

k

) = x

i

(t

k

) + d

ik

x

i

(t

k

), i = 1, 2, 3 k ∈ Z

+

,

(11)

where

d

1k

= 0.35, d

2k

= 0.4, d

3k

= 0.5,

f

j

(x

j

) = 0.5(|x

j

+ 1| − |x

j

− 1|), j = 1, 2.

Obviously, f

j

(x) satisfy (H

1

).

Now we investigate the inﬂuence of the period T

of impulsive effect on the system (11). Set

c

1

(t)

c

2

(t)

c

3

(t)

=

2− 2cost

2− 2sint

1+ cost

.

For T = 1, then (H

2

) isn’t satisﬁed. Periodic oscil-

lation of system (11) will be destroyed by impulses

effect. Numeric results show that system (11) still has

SynchronizationoftheComplexDynamicalNetworkswithaGuiChaoticStrangeAttractor

259

–1

0

1

2

3

4

x1

50 100 150 200

t

Figure 1: Time-series of the x

1

(t)of system (11).

0

2

4

6

8

10

x2

50 100 150 200

t

Figure 2: Time-series of the x

2

(t) of system (11).

2

4

6

8

10

x3

0 50 100 150 200

t

Figure 3: Time-series of the x

3

(t) of system (11).

a global attractor which can be a Gui chaotic strange

attractor(see Figs.1-4).

Every solutions of system (11) will ﬁnally tend to

the Gui chaotic strange attractor. As shown in Figs.

1-4, the system (11) possesses a Gui chaotic strange

attractor.

Now the response chaotic cellular neural network

1

2

3

4

x1

5

6

7

8

9

10

11

x2

6

8

10

x3

Figure 4: Phase portrait of Gui chaotic strange attractor of

system (11) with T = 1.

is designed as follows:

˙y

1

˙y

2

˙y

3

= −

y

1

y

2

y

3

+

1.2 −1.6 0

1.2 1.0 0.9

0 2.2 0.15

f

1

(y

1

)

f

2

(y

2

)

f

3

(y

3

)

+

c

1

(t)

c

2

(t)

c

3

(t)

y

i

(t

+

k

) = 2d

ik

x

i

(t

−

k

) + (1− d

ik

)y

i

(t

−

k

),

(12)

Remark 1. In (Zhang and Gui, 2009a; Zhang and

Gui, 2009b), the authors investigate the inﬂuence of

the period T of impulsive effect on the system (11).

If T =

2

5

π or T = 0.1π, then q = 5 or q = 20, respec-

tively, in (H

2

). According to Theorem 1 and Theo-

rem 2, the cellular neural networks model (11) has a

unique 2π-periodic solution which is globally asymp-

totically stable. For T =

2

5

π, γ

1

= 0.6, γ

2

= 0.85,

each positive solution tends to a unique positive 2π-

periodic solution with 5-impulses in a period. For

T = 0.1π, γ

1

= 0.4, γ

2

= 0.3, each positive solution

tends to a unique positive 2π-periodic solution with

20-impulses in a period.

Remark 2. If T = 1, then (H

2

) isn’t satisﬁed. Pe-

riodic oscillation of system (11) will be destroyed by

impulses effect. Numeric results show that system

(11) still has a Gui chaotic strange attractor(Zhang

and Gui, 2009a; Zhang and Gui, 2009b).

Let e(t) = y

i

− x

i

, then the error system (13) of

drive system (11) and respond system (12) is con-

SIMULTECH2015-5thInternationalConferenceonSimulationandModelingMethodologies,Technologiesand

Applications

260

structed as follow

˙e

1

(t)

˙e

2

(t)

˙e

3

(t)

= −

e

1

(t)

e

2

(t)

e

3

(t)

+

1.2 −1.6 0

1.2 1.0 0.9

0 2.2 0.15

·

f

1

(e

1

(t) + x

1

) − f

1

(x

1

)

f

2

(e

2

(t) + x

1

) − f

2

(x

2

)

f

3

(e

3

(t) + x

3

) − f

3

(x

3

)

e

1

(t

+

k

) = (1 − d

1k

)e

1

(t

−

k

),

e

2

(t

+

k

) = (1 − d

2k

)e

2

(t

−

k

),

e

3

(t

+

k

) = (1 − d

3k

)e

3

(t

−

k

).

(13)

e3

e2

e1

–1

–0.5

0

0.5

1

2 4 6 8

Figure 5: Synchronization errors between drive system (11)

and response system (12) with T = 1.

–0.0002

–0.00015

–0.0001

–5e–05

0

e1

–0.00016

–0.00014

–0.00012

–0.0001

–8e–05

–6e–05

–4e–05

–2e–05

0

e2

0

5e–07

1e–06

1.5e–06

2e–06

Figure 6: Phase portrait of Synchronization errors with T =

1.

If one choose L

1

= L

2

= L

3

= 1, η = 1.1, d

1k

=

0.35, d

2k

= 0.4, d

3k

= 0.5, It is easy to check the con-

ditions in Theorem 1 are satisﬁed. So, the system

(11) and (12) is synchronized. By Theorem 1, syn-

chronization can be obtained. The synchronization

performance is illustrated by Figs. 5,6. The numer-

ical simulations show that synchronization could be

quickly achieved.

Furthermore, if T = 0.4, set

c

1

(t)

c

2

(t)

c

3

(t)

=

2− 2sint

3− 3cost

1− 1sint

.

then, periodic oscillation of system (11) will be de-

stroyed by impulses effect. Numeric results show

that system (11) still has a Gui chaotic strange attrac-

tor(see Fig.7).

–2

0

2

4

6

8

x1

0

10

20

30

40

x2

0

500

1000

1500

2000

2500

x3

Figure 7: Phase portrait of Gui chaotic strange attractor of

system (11) with T = 0.4.

e3

e2

e1

–1

–0.5

0

0.5

1

0.5 1 1.5 2 2.5 3

Figure 8: Synchronization errors between drive system (11)

and response system (12) with T = 0.4.

It follows from Theorem 1 that systems (11) and

(12) are impulsively synchronized. Figs. 8 and 9 de-

pict the synchronizationerror of the state variables be-

tween the drive system and the response system.

SynchronizationoftheComplexDynamicalNetworkswithaGuiChaoticStrangeAttractor

261

–1.2e–12

0

e1

–8e–14

–6e–14

–4e–14

–2e–14

e2

–3e–16

–2e–16

–1e–16

0

1e–16

2e–16

3e–16

Figure 9: Phase portrait of Synchronization errors with T =

0.4.

5 CONCLUSIONS

In the paper,the synchronization of the impulses com-

plex dynamical network with a Gui chaotic strange at-

tractor and has been investigated based on the stabil-

ity analysis of impulsive functional differential equa-

tion. The criteria for the synchronization are derived.

An illustrative example is ﬁnally included to visual-

ize the effectiveness and feasibility of the developed

methods. Compared with the correspondingly previ-

ous works(Luo, 2008; Yang and Cao, 2007; Yang and

Cao, 2010; Zhang, 2009), our model of research is

new. As far as we know, There is no paper to deal

with such a problem.

ACKNOWLEDGEMENTS

This work was supported by the National Natural Sci-

ence Foundation of People’s Republic of China(Grant

no. 60963025), the Natural Science Foundation of

Hainan(Grant no.613166,112008).

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