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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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