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Neural Computing and Applications ISSN 0941-0643Volume 23Combined 3-4 Neural Comput & Applic (2013)23:815-821DOI 10.1007/s00521-012-0998-y

Dynamic behaviors of memristor-baseddelayed recurrent networks

Shiping Wen, Zhigang Zeng & TingwenHuang

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ORIGINAL ARTICLE

Dynamic behaviors of memristor-based delayed recurrentnetworks

Shiping Wen • Zhigang Zeng • Tingwen Huang

Received: 1 November 2011 / Accepted: 23 May 2012 / Published online: 10 June 2012

� Springer-Verlag London Limited 2012

Abstract This paper investigates the problem of the

existence and global exponential stability of the periodic

solution of memristor-based delayed network. Based on the

knowledge of memristor and recurrent neural network, the

model of the memristor-based recurrent networks is estab-

lished. Several sufficient conditions are obtained, which

ensure the existence of periodic solutions and global expo-

nential stability of the memristor-based delayed recurrent

networks. These results ensure global exponential stability

of memristor-based network in the sense of Filippov solu-

tions. And, it is convenient to estimate the exponential

convergence rates of this network by the results. An illus-

trative example is given to show the effectiveness of the

theoretical results.

Keywords Memristor � Recurrent networks � Time delays

1 Introduction

As the missing 4-th passive circuit element (see [1]), sci-

entists took almost 40 years to invent the memristor, until a

team at Hewlett-Packard Labs announced the development

of a memristor in Nature on May 1, 2008 (see [2]). This

new circuit element shares many properties of resistors and

shares the same unit of measurement (ohm) and offers a

nonvolatile memory storage within a simple device struc-

ture attractive for potential applications. In electronics

field, it will be useful for low-power computation and

storage, if the circuit element is able to store information/

data without the need of power (see [3]).

In addition, we notice that recurrent neural networks

have been widely studied in recent years (see [4–20]),

because of their immense application prospective. Many

applications have been developed in different areas such as

combinatorial optimization, knowledge acquisition, and

pattern recognition. Such applications depend on the sta-

bility of networks. Therefore, stability is one main property

of networks. In electronic implementation of analog net-

works, time delay is usually time-varying due to the finite

switching speed of amplifiers. It also knows that time delay

is the main cause of instability and poor performance of

networks. Therefore, it is very important to study the global

stability of neural networks with time-varying delays.

In this paper, based on the works (see [21–26]), we will

study the dynamic behaviors for a general class of memr-

istor-based recurrent networks with time-varying delays in

view of its many potential applications, for example, super-

dense nonvolatile computer memory and neural synapses.

2 Memristor-based recurrent network

The memrsitor-based recurrent network can be imple-

mented by very large-scale integration (VLSI) circuits as

shown in Fig. 1, and the connection weights are imple-

mented by the memristors. fj and gj are activation func-

tions, si(t) is the time-varying delay, for the i-th subsystem,

S. Wen � Z. Zeng (&)

Department of Control Science and Engineering,

Huazhong University of Science and Technology,

Wuhan 430074, China

e-mail: zgzeng527@126.com

S. Wen � Z. Zeng

Key Laboratory of Image Processing and Intelligent Control

of Education Ministry of China, Wuhan 430074, China

T. Huang

Texas A&M University at Qatar, Doha 5825, Qatar

123

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DOI 10.1007/s00521-012-0998-y

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xi(t) is the voltage of the capacitor Ci, fj(xj(t)), gj(xj(t -

si(t))) are the functions about xi(t) with and without vary-

ing-time delays, respectively, Mfij is the memristor between

the feedback function fj(xj(t)) and xi(t), Mgij is the memr-

istor between the feedback function gj(xj(t - si(t))) and

xi(t), Ri is the resistor parallel to the capacitor Ci, Ii is an

external input or bias, where i; j ¼ 1; 2; . . .; n:

From this circuit, we know the currents Ifij; Igij; ði; j ¼1; 2; � � � ; nÞ are

Ifij ¼ uijðfjðxjðtÞÞ � xiðtÞÞWfijðxjðtÞÞ;Igij ¼ uijðgjðxjðt � siðtÞÞÞ � xiðtÞÞWgijðxjðtÞÞ;

where uij ¼1; i 6¼ j;�1; i ¼ j;

�and Wfij(xj(t)), Wgij(xj(t)) are the

memductances of the memristors Mfij, Mgij.

The memductance of the memristors can be depicted as

in the Fig. 2 (see [22]), which are bounded.

Thus, by Kirchoff’s current law, the equation of the i-th

subsystem is written as the following:

CdxiðtÞ

dtþ xiðtÞ

Ri

¼Xn

j¼1

ðIfij þ IgijÞ þ Ii

¼Xn

j¼1

uijðfjðxjðtÞÞ � xiðtÞÞWfijðxjðtÞÞ

þXn

j¼1

uijðgjðxjðt � siðtÞÞÞ

� xiðtÞÞWgijðxjðtÞÞ þ Ii; ð1Þ

then, we can get

Ci

dxiðtÞdt¼ �

Xn

j¼1

ðWfijðxjðtÞÞ þWgijðxjðtÞÞÞ þ1

Ri

" #xiðtÞ

þXn

j¼1

uijfjðxjðtÞÞWfijðxjðtÞÞ

þXn

j¼1

uijgjðxjðt � siðtÞÞÞWgijðxjðtÞÞ þ Ii; ð2Þ

Fig. 1 Circuit of memristor-based recurrent network, where xi(.) is

the state of the i-th subsystem, fj(.), gj(.) are the amplifiers, Mfij is the

connection memritor between the amplifier fj(.) and state xi(.) and Mgij

is the connection memritor between the amplifier gj(.) and state

xi(.), Ri and Ci are the resistor and capacitor, Ii is the external input,

ai, bi are the outputs, i; j ¼ 1; 2; . . .; n

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therefore

dxiðtÞdt¼ �riðxðtÞÞxiðtÞ þ

Xn

j¼1

aijðxjðtÞÞfjðxjðtÞÞ

þXn

j¼1

bijðxjðtÞÞgjðxjðt � siðtÞÞÞ þ ui; ð3Þ

where

aijðxjðtÞÞ ¼uij

Ci

Wfij; bijðxjðtÞÞ ¼uij

Ci

WgijðxjðtÞÞ;

riðxðtÞÞ ¼1

Ci

Xn

j¼1

ðWfijðxjðtÞÞ þWgijðxjðtÞÞÞ þ1

Ri

� �;

ui ¼Ii

Ci

:

then we can get

dxðtÞdt¼ �rðxðtÞÞxðtÞ þ AðxðtÞÞf ðxðtÞÞ

þ BðxðtÞÞgðxðt � sðtÞÞÞ þ u; ð4Þ

where

rðxðtÞÞ ¼ diagðr1ðxðtÞÞ;r2ðxðtÞÞ; . . .rnðxðtÞÞÞ;AðxðtÞÞ ¼ ½aijðxjðtÞÞ�n�n;BðxðtÞÞ ¼ ½bijðxjðtÞÞ�n�n;

u ¼ ðu1; u2; . . .unÞT :

Remark 1 According to the analysis above, r(x(t)),

A(x(t)), B(x(t)) in this system are changed according to

the state of the system, so this network based on

memristors is a state-dependent switching system. System

(4) represents a class of memristor-based recurrent

networks with varying time delays, which demonstrates

plentiful characteristics and quite different from general

switched systems.

Remark 2 System (4) represents a general class of memr-

istor-based recurrent networks with or without time delays. In

particular, when fj is a sigmoid function and gj : 0, (j =

1, 2, …, n), system (4) is the memristor-based Hopfield

neural network. Similarly, when fj(x) = (|x ? 1| - |x - 1|)/2

and gj : 0, (j = 1, 2, …, n) or gj : fj, (j = 1, 2, …, n),

system (4) represents the memristor-based cellular neural

networks without or with time delays, respectively.

Global stability of neural networks has been widely

investigated (see [4–20, 27–36]); however, few works have

been done on the memristor-based networks, so it is nec-

essary to investigate such class of networks.

3 Preliminaries

Denote |u| as the absolute-value vector; that is,

|u| = (|u1|, |u2|, …, |un|)T. Denote u ¼ ðu1; . . .; unÞT ; jjxjjpas the p-norm of the vector x with p, 1� p\1: jjxjj1 ¼maxi2f1;2;...;ng jxij is the vector infinity norm. Denote ||D||p

as the p-norm of the matrix D with p. Denote C0 as the set

of continuous functions. Let

f ðxÞ ¼ ðf1ðx1Þ; f2ðx2Þ; . . .; fnðxnÞÞT ;gðxÞ ¼ ðg1ðx1Þ; g2ðx2Þ; . . .; gnðxnÞÞT :

From (3)

dxiðtÞdt2 �½H�i ;Hþi �xiðtÞ þ

Xn

j¼1

½C�ij ;Cþij �fjðxjðtÞÞ

þXn

j¼1

½!�ij ;!þij �gjðxjðt � sijðtÞÞÞ þ ui; ð5Þ

where

H�i ¼ inft0 � t\þ1

ðriðxðtÞÞÞ;Hþi ¼ supt0 � t\þ1

ðriðxðtÞÞÞ;

K�ij ¼ inft0 � t\þ1

ðaijðxjðtÞÞÞ;Kþij ¼ supt0 � t\þ1

ðaijðxjðtÞÞÞ;

!�ij ¼ inft0 � t\þ1

ðbijðxjðtÞÞÞ;!þij ¼ supt0 � t\þ1

ðbijðxjðtÞÞÞ;

or equivalently, for i; j 2 f1; 2; . . .; ng; there exist Hi 2½H�i ;Hþi �;Cij 2 ½C�ij ;Cþij �;!ij 2 ½!�ij ;!

þij �; such that

dxiðtÞdt¼ �HixiðtÞ þ

Xn

j¼1

CijfjðxjðtÞÞ

þXn

j¼1

!ijgjðxjðt � sijðtÞÞÞ þ ui: ð6Þ

Consequently, from (4), we can get

dxðtÞdt¼ �HxðtÞ þ Kf ðxðtÞÞ þ !gðxðt � sðtÞÞÞ þ u; ð7Þ

−1 −0.5 0 0.5 1−1.5

−1

−0.5

0

0.5

1

1.5 x 10−4

V

I

Fig. 2 Typical i-v characteristic of memristor [1, 22]. The pinched

hysteresis loop is due to the nonlinear relationship between the

memristance current and voltage. The memristor exhibits the feature

of pinched hysteresis, which means that a lag occurs between the

application and the removal of a field and its subsequent effect, just as

the neurons in the human brain have

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where

H ¼ diagðH1; . . .;HnÞ;K ¼ ðKijÞn�n;! ¼ ð!ijÞn�n:

In the following discussions, it is assumed that

H1 The activation functions fj and gj are Lipschitz con-

tinuous; that is, for j 2 f1; 2; . . .; ng; 8r1; r2; r3; r4 2 <;there exist real number ij and kj such that

jfjðr1Þ � fjðr2Þj � ijjr1 � r2j;jgjðr3Þ � gjðr4Þj � kjjr3 � r4j:

H2 For i; j 2 f1; 2; . . .; ng;½H�i ;Hþi �xi � ½H�i ;Hþi �yi � ½H�i ;Hþi �ðxi � yiÞ;½K�ij ;K

þij �fjðxjÞ � ½K�ij ;K

þij �fjðyjÞ � ½K�ij ;K

þij ��fjðxjÞ � fjðyjÞ

�;

½!�ij ;!þij �gjðxjÞ � ½!�ij ;!

þij �gjðyjÞ � ½!�ij ;!

þij ��fjðxjÞ � gjðyjÞ

�:

Several definitions and lemmas, which will be used later,

are listed there.

Definition 1 If an arbitrary solution x(t) with uðsÞ 2

C�½t0 � s; t0�;<n

�of system (4) satisfies

jxiðtÞj �Xn

j¼1

supt0�s� s� t0

juðsÞjðsÞj !

bi expf�aiðt � t0Þg;

ð8Þ

where t C t0 C 0, bi and ai are positive constants, then, the

equilibrium point of system (4) is said to be globally

exponentially stable, and �a ¼ �limt!1ð�ðln jjxðtÞjj1Þ=tÞ is

called the rate of exponential convergence of (4).

Definition 2 If P = [pij]n 9 n satisfies: (i) pij� 0; i 6¼ j;

i; j ¼ 1; 2; . . .; n; (ii) there exists a vector u [ 0 such that

Pu [ 0, then P is called a nonsingular M matrix. When

t C t0, assume v1ðtÞ; v2ðtÞ; . . .; vnðtÞ satisfy

DþviðtÞ ¼Xn

j¼1

cijvjðtÞ þXn

j¼1

dijvjðt � sijðtÞÞ;

i ¼ 1; 2; . . .; n; ð9Þ

where cii is a real number, cij (i = j) and dij are nonneg-

ative real numbers, DþviðtÞ ¼ lim suph!0þðviðt þ hÞ �viðtÞÞ=h for i; j ¼ 1; 2; . . .; n:

Definition 3 The periodic solution x�ðtÞ with uðsÞ 2Cð½t0 � s; t0�;<nÞ of system (4) is said to be global expo-

nential stability, if for any solution x(t) of system (4), there

exists

jxiðtÞ � x�ðtÞj �Xn

i¼1

supt0�s� s� t0

jujðsÞj !

bi expf�aiðt � t0Þg; ð10Þ

where t C t0 C 0, bi and ai are positive constants, then, the

equilibrium point of the network (4) is said to be globally

exponentially stable, and �a ¼ �limt!1ð�ðln jjxðtÞjj1Þ=tÞ is

called the rate of exponential convergence of (4).

Lemma 1 [19] For all i 2 f1; 2; . . .; ng; t t0; if there

exists gi 2 C0 ½t0 � s; t0�;<nð Þ; <� ¼ ð�1; 0�; such that

ðcii�giðtÞÞ exp

Z t

t0

giðsÞds

8<:

9=;þ

Xn

j¼1;j6¼i

cij exp

Z t

t0

gjðsÞds

8<:

9=;

þXn

j¼1

dij exp

Zt�sijðtÞ

t0

gjðsÞds

8><>:

9>=>;�0; ð11Þ

then, the solution vi(t) of (9) satisfies jviðtÞj ��vðt0Þexp

R t

t0giðsÞds where �vðt0Þ ¼

Pni¼1 supt0�s� h� t0

jviðhÞj

Lemma 2 Let P be a given n 9 n matrix and

�P :¼ 0 P

PT 0

;

If k is an eigenvalue of �P; then -k is also an eigenvalue of�P and k2 is an eigenvalue of PPT and PTP.

4 Main results

First, we will illustrate the existence of the equilibrium

point for memristor-based recurrent neural network (4).

The local existence of a solution x(t) to (4) on

[0, t0], t0 [ 0, with x(0) = x0 is a straightforward conse-

quence of Theorem 1 in [36]. Moreover, since Hi 2½H�i ;Hþi �;Cij 2 ½C�ij ;Cþij �;!ij 2 ½!�ij ;!

þij �; are bounded on

<n, it is seen that if 8~R [ 0 is sufficiently large, then it

results

ti ¼ sgnð�HixiðtÞÞ; i ¼ 1; 2; . . .; n;

for jjxðtÞjj2 ~< and for any

t 2 /ðxðtÞÞ ¼ �HxðtÞ þ Kf ðxðtÞÞ þ !gðxðt � sðtÞÞÞ þ u:

In another way, we can say /(x(t)) points toward the

interior on the boundary of a sufficiently large sphere.

Therefore, if we let �R ¼ maxfjjx0jj2; ~Rg; it follows that

jjxðtÞjj2� �R on ½0;þ1Þ: This means that x(t) is bounded

and hence defined on ½0;þ1Þ: And this method to prove

the stability of systems has been used in [18–20].

Let the periodic solution x�ðtÞ ¼ ðx�1ðtÞ; x�2ðtÞ; . . .;

x�nðtÞÞT ; t 2 ½t0;þ1Þ: Therefore

0 ¼ �Hx� þ Kf ðx�Þ þ !gðx�Þ þ u: ð12Þ

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According to Assumption H2, the memristor-based

network (7) can be rewritten as:

dzðtÞdt¼ �HzðtÞ þ KFðzðtÞÞ þ !Gðzðt � sðtÞÞÞ; ð13Þ

where

zðtÞ ¼ xðtÞ � x�;FðzðtÞÞ ¼ f ðzðtÞ þ x�Þ � f ðx�Þ;Gðzðt � sÞÞ ¼ gðzðt � sðtÞÞ þ x�Þ � gðx�Þ:

Next, we present the main theorem of this paper. Denote

jPj and jWj as the matrices jPj ¼ ½jpijj�n�n;W ¼ ½jwijj�n�n

respectively,

jpijj ¼ maxt0 � t�þ1

fjaijðxðtÞÞjg; jwijj ð14Þ

Theorem 1 If there exists positive number h; a1; a2; . . .;

an such that for i 2 f1; 2; . . .; ng;

aið�ðHiÞ� þ hÞ þXn

j¼1

ajjpijjij

þXn

j¼1

ajjwijjkj expfhsiðtÞg� 0; ð15Þ

then the memristor-based network (13) has a unique x-

periodic solution, which is globally exponentially stable.

Proof Let viðtÞ ¼ jziðtÞj=ai; i ¼ 1; 2; . . .; n: It follows

from (13) that

DþviðtÞ� �HiviðtÞ

þ a�1i

Xn

j¼1

jpijjajijvjðtÞ þXn

j¼1

jwijjajkjvjðt � siðtÞÞ( )

:

Take gi(t): - h, then, Lemma 1 and (15) imply that

jviðtÞj ��vðt0Þ expf�hðt � t0Þg; where �vðt0Þ ¼Pn

i¼1 supt0�s� e� t0jxiðeÞj=ai; which implies the conclusion holds.

Corollary 1 If H� � jPjL� jWjK is a nonsingular

M-matrix, then (4) has a unique x-periodic solution, which

is globally exponentially stable, where

L ¼ ½iij�n�n;K ¼ ½kij�n�n;H� ¼ diagðH�1 ;H�2 ; . . .;H�n Þ:

Proof According to the work done above, it is easy to

know the memristor-based network (4) has a periodic

solution x�ðtÞ: Taking zðtÞ ¼ xðtÞ � x�ðtÞ; (4) is modified

as (13).

Next, we will prove the periodic solution x�ðtÞ is glob-

ally exponentially stable. Since H� � jPjL� jWjK is a

nonsingular M matrix, there exist positive numbers

a1; a2; . . .; an such that

H�i ai [Xn

j¼1

ajjpijjij þXn

j¼1

ajjwijjkj: ð16Þ

Let

piðkiÞ ¼ aiki �H�i ai þXn

j¼1

ajjpijjij

þXn

j¼1

ajjwijjkj expfkjsjðtÞg: ð17Þ

It is easy to see pi(0) \ 0, while piðH�i Þ 0; dpiðkiÞ=dki [ 0; thus there exists si 2 ð0;H�i Þ such that

pi(si) = 0, and when s 2 ð0; siÞ; piðsÞ\0: Taking smin :=

minisi,pk(smin) B 0 for all k 2 f1; 2; . . .; ng: Taking

h = smin, condition (15) of Theorem 1 holds, then Corol-

lary 1 holds.

Remark 2 Corollary 1 provides an easily testable criterion

to ascertain the global exponential stability of memristor-

based networks with time-varying delays. In addition,

Corollary 1 provides a method to estimate the rate of

exponential convergence of memristor-based networks

by solving transcendental equation piðsiÞ ¼ 0; ði ¼1; 2; . . .; nÞ; and taking the smallest si as the estimated rate of

exponential convergence of system (13). Moreover, since pi

is monotone, pi(0) \ 0 and piðH�i Þ 0; it is easy to estimate

the rate of convergence by using a bi-section method.

Theorem 2 If there exists h[ 0, such that �2H� þ hþkPð1þ i2

maxÞ þ kW þ k2maxkW expfhsðtÞg� 0; then, net-

work (4) has a unique periodic solution, which is globally

exponentially stable, where imax :¼ max ii; kmax :¼max ki; i ¼ 1; 2; . . .; n:

Proof According to the work done above, it is easy to

know the memristor-based network (4) has at least one

equilibrium point x�ðtÞ: Taking zðtÞ ¼ xðtÞ � x�ðtÞ; (4) is

modified as (13). Let V(x(t)) = xT(t)x(t), then

dVðzðtÞÞdt

¼�2zTðtÞHzðtÞþðzTðtÞ;FTðzðtÞÞÞPðzTðtÞ;

FTðzðtÞÞÞTþðzTðtÞ;GTðzðt�sðtÞÞÞÞWðzTðtÞ;Tðzðt�sðtÞÞÞÞT �

��2H�þkPð1þi2

maxÞþkW

þk2maxkW expfhsðtÞg

�

jjzðtÞjj22��hjjzðtÞjj22¼�hVðzðtÞÞ:ð18Þ

So, system (4) is globally stability.

Corollary 2 If �2H�kPð1þ i2maxÞ þ kWð1þ k2

maxÞ exp

fhsðtÞg\0; then the memristor-based network (4) has a

unique x-periodic solution, which is globally exponentially

stable.

Proof The process to prove the existence of the equilib-

rium set in this part is the same as in Corollary 1.

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Remark 3 This paper is quite different from [19] as (i) it

investigates the existence and global exponential stability

of the periodic solution of memristor-based delayed net-

work; (ii) all the results are in the sense of Filippov.

5 An illustrative example

In this section, based on the works [1, 4, 8], we consider the

following numerical example to validate the derived

results. Consider a memristor-based recurrent network with

time delays,

_xðtÞ ¼ �rðxðtÞÞxðtÞ þ AðxðtÞÞf ðxðtÞÞþ BðxðtÞÞgðxðt � sðtÞÞÞ þ IðtÞ; ð19Þ

rðxðtÞÞ ¼r1ðx1ðtÞÞ 0

0 r2ðx2ðtÞÞ

;

AðxðtÞÞ ¼a11ðx1ðtÞÞ a12ðx2ðtÞÞa21ðx1ðtÞÞ a22ðx2ðtÞÞ

;

BðxðtÞÞ ¼b11ðx1ðtÞÞ b12ðx2ðtÞÞb21ðx1ðtÞÞ b22ðx2ðtÞÞ

;

r1ðx1ðtÞÞ 2 ½12þ sin2ðffiffiffiffiffiffi3ppÞ; 15þ cos2ð

ffiffiffiffiffiffi2pp�;

r2ðx2ðtÞÞ 2 ½7:5þ cos2ðffiffiffiffiffiffi2ppÞ; 18þ sin2ð

ffiffiffiffiffiffi3pp�;

a11ðx1ðtÞÞ 2 ½�1; 0:5�; a12ðx2ðtÞÞ 2 ½�2; 3�;a21ðx1ðtÞÞ 2 ½�4; 3:5�; a22ðx2ðtÞÞ 2 ½�0:5;�0:25�;b11ðx1ðtÞÞ 2 ½0:5; 1�; b12ðx2ðtÞÞ 2 ½�0:5;�0:25�;b21ðx1ðtÞÞ 2 ½�1; 2�; a22ðx2ðtÞÞ 2 ½�2; 1:5�;

IðtÞ ¼ ð� sinðtÞ; cosðtÞÞT ;f ðxðtÞÞ ¼ ð1� e�xðtÞÞ=ð1þ e�xðtÞÞ; gðxðtÞÞ ¼ cosðtÞ;

the time delays si(t) C 0, (i = 1, 2) are any bounded

continuous functions. And i1 ¼ i2 ¼ k1 ¼ k2 ¼ 1; H�1 ¼12þ sin2ð

ffiffiffiffiffiffi3ppÞ; H�2 ¼ 7:5þ cos2ð

ffiffiffiffiffiffi2ppÞ; jp11j ¼ 1; jp12j

¼ 3; jp21j ¼ 4; jp22j ¼ 0:5; jw11j ¼ 1; j w12j ¼ 0:5; jw21j ¼2; jw22j ¼ 2; let a1 ¼ 1; a2 ¼ 2:5;

a1H�1 [ 12 [ a1jp11j þ a2jp12j þ a1jw11j

þ a2jw12j ¼ 10:75;

a2H�2 [ 18:75 [ a1jp21j þ a2jp22j

þ a1jw21j þ a2jw22j ¼ 14:5:

According to Corollary 1, this memristor-based network is

globally exponentially stable. According to (16), there exist

h1, h2 [ 0 such that for s(t) B s, we can obtain the equations:

a1h1 � a1H�1 þ a1jp11j þ a2jp12j þ a1jw11j expfh1sg

þ a2jw12j expfh2sg� 0;

a2h2 � a2H�2 þ a1jp21j þ a2jp22j þ a1jw21j expfh1sg

þ a2jw22j expfh2sg� 0:

According to Remark 2, the exponential convergence rate

of the network is at least equal to H�:For numerical simulation, with the random initial state,

and siðtÞ ¼ 0:1 sinðtÞ; ði ¼ 1; 2Þ; Figs. 3 and 4 depict the

responses of variables x1(t) and x2(t). That confirm the

results in Theorem 1.

6 Conclusions

This paper investigates the problem of the existence and

global exponential stability of the periodic solution of

memristor-based delayed network. Several sufficient con-

ditions are obtained, which ensure the existence and global

exponential stability of memristor-based recurrent net-

works with time-varying delays. All the results are in the

sense of Filippov. And these stability conditions are mild

and some are easy to verify.

Acknowledgments The work is supported by the Natural Science

Foundation of China under Grants 60974021 and 61125303, the

973 Program of China under Grant 2011CB710606, the Fund for

Distinguished Young Scholars of Hubei Province under Grant

2010CDA081.

0 2 4 6 8 10−2

−1

0

1

2

time

x 1

0 1 2 3 4 5 6−2

−1

0

1

2

time

x 2

Fig. 3 Transient behavior of the memristor-based network (19)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x1

x 2

Fig. 4 Transient behavior of the memristor-based network (19)

820 Neural Comput & Applic (2013) 23:815–821

123

Author's personal copy

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