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Exponential Synchronization of Linearly Coupled Neural
Networks with Impulsive Disturbances
Jianquan Lu, Daniel W. C. Ho, Senior Member, IEEE,
Jinde Cao, Senior Member, IEEE, and Jürgen Kurths
Abstract—This brief investigates globally exponential synchro-
nization for linearly coupled neural networks (NNs) with time-
varying delay and impulsive disturbances. Since the impulsive
effects discussed in this brief are regarded as disturbances,
the impulses should not happen too frequently. The concept of
average impulsive interval is used to formalize this phenomenon.
By referring to an impulsive delay differential inequality, we
investigate the globally exponential synchronization of linearly
coupled NNs with impulsive disturbances. The derived sufficient
Manuscript received May 20, 2010; revised August 1, 2010; accepted
December 11, 2010. Date of publication January 13, 2011; date of current
version February 9, 2011. The work of J. Lu was supported in part by
the National Natural Science Foundation of China (NSFC), under Grant
11026182, the Natural Science Foundation of Jiangsu Province of China,
under Grant BK2010408, the Innovation Fund of Basic Scientific Research
Operating Expenses 3207010501, and the Alexander von Humboldt Founda-
tion of Germany. The work of D. W. C. Ho was partially supported by a grant
CityU 7002561 and GRF of HKSAR (CityU 1011109). The work of J. Cao
was partially supported by the National Natural Science Foundation of China
under Grant 11072059 and Grant 60874088, the Specialized Research Fund
for the Doctoral Program of Higher Education under Grant 20070286003,
and the Natural Science Foundation of Jiangsu Province of China under Grant
BK2009271. The work of J. Kurths was partially supported by Supermodeling
by Combining Imperfect Models, European Union, and Evolving Complex
Networks, Wissenschaftsgemeinschaft Gottfried Wilhelm Leibniz.
J. Lu is with the Department of Mathematics, Southeast University,
Nanjing 210096, China. He is also with the Potsdam Institute for Climate
Impact Research, Potsdam 14415, Germany (e-mail: jqluma@seu.edu.cn;
jqluma@gmail.com).
D. W. C. Ho is with the Department of Mathematics, City University of
Hong Kong, Hong Kong (e-mail: madaniel@cityu.edu.hk).
J. Cao is with the Department of Mathematics, Southeast University,
Nanjing 210096, China (e-mail: jdcao@seu.edu.cn).
J. Kurths is with the Potsdam Institute for Climate Impact Research,
Telegraphenberg, Potsdam 14415, Germany. He is also with the Department
of Physics, Humboldt University Berlin, Berlin 12489, Germany, and also
with the Institute for Complex Systems and Mathematical Biology, University
of Aberdeen, Aberdeen AB24 3UE, U.K. (e-mail: Juergen.Kurths@pik-
potsdam.de).
Color versions of one or more of the figures in this brief are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNN.2010.2101081
condition is closely related with the time delay, impulse strengths,
average impulsive interval, and coupling structure of the systems.
The obtained criterion is given in terms of an algebraic inequality
which is easy to be verified, and hence our result is valid for large-
scale systems. The results extend and improve upon earlier work.
As a numerical example, a small-world network composing of
impulsive coupled chaotic delayed NN nodes is given to illustrate
our theoretical result.
Index Terms—Desynchronizing impulses, globally exponential
synchronization, linearly coupled neural networks.
I. INTRODUCTION
Recently, synchronization of complex systems and dynam-
ical networks has attracted a great deal of attention [1]–[12].
A complex dynamical network is a large set of interconnected
nodes, and each node acts a nonlinear dynamical system.
From the literature, there are two common phenomena in
many evolving networks: delay effects and impulsive effects
[13]–[17]. Time delay is ubiquitous in the implementation
of electronic networks due to the finite switching speed of
amplifiers and finite signal propagation time [13], [15], [17].
Delayed chaotic neural networks (NNs) coupled with a linear
coupling configuration are easy to physically implement and
hence have potential applications especially in secure com-
munications based on synchronization [18], [19]. On the other
hand, the states of electronic networks and biological networks
are often subject to instantaneous disturbances and experience
abrupt changes at certain instants, which may be caused by
switching phenomenon, frequency change, or other sudden
noise, i.e., they exhibit impulsive effects [14], [16], [20], [21].
Impulsive dynamical networks, which are characterized by
abrupt changes in the state differences of the systems at certain
instants, have sparked the interest of many researchers for their
various applications in information science, economic systems,
automated control systems, etc [22]. Since delays and impulses
can heavily affect the dynamical behaviors of the networks, it
is necessary to investigate both the delay and impulsive effects
on the synchronization of dynamical networks.
Synchronization of linearly coupled systems with impulses
has also been studied in many papers [12], [14], [17],
[22]–[28]. Many NNs rely on a synchronous behavior for
a proper functioning, e.g., information transmission, pattern
recognition, and learning [29], [30]. Hence, this brief will be
devoted to studying the synchronization of coupled NNs. In
most works [14], [22], [25]–[28], impulsive controllers are
designed for the stabilization of complex dynamical networks.
However, sometimes impulses can play a negative role for the
synchronization of dynamical networks [17], [27]. In [17], it
was shown that synchronization of impulsive coupled delayed
dynamical networks is heavily dependent on the impulsive
effects of the network connections. In [27], robust stability
of complex impulsive dynamical systems was studied when
the impulsive effects are destabilizing or stabilizing.
When the impulsive effects are destabilizing, the impulses
should not happen too frequently [12]. In [17], [20], and [27],
the lower bound of the impulsive intervals are used to represent
this phenomenon, and such a formalization would lead the
obtained results to be rather conservative. Hence, by referring
1045–9227/$26.00 © 2011 IEEE
330 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 22, NO. 2, FEBRUARY 2011
to the concept of average dwell time (describing switched
system) [31]–[33], a similar concept named “average impul-
sive interval” will be used to represent the impulse sequences
in this brief [12]. The synchronization problem of dynamical
networks with destabilizing impulses will be studied by using
the concept of average impulsive interval. The Lyapunov
method and the Kronecker product are used to derive the main
results of this brief. A numerical example is also given to
illustrate the effectiveness of the obtained result.
The main merits of this brief are as follows: 1) impulsive
effects are destabilizing; 2) impulses occur in the process of
the connected nodes’ state coupling; 3) time-varying delay is
only required to be bounded without any derivative constraint;
4) the concept of average-impulsive interval is used to release
the constraint on the lower bound of the destabilizing impul-
sive interval, and hence makes the result available for a wider
range of impulsive signals; and 5) our criterion is applicable
to large-scale dynamical networks.
Notations: Standard notations will be used in this brief.
Throughout this brief, for real symmetric matrices X and
Y , the notation X ≤Y (respectively, X < Y) means that the
matrix X − Y is negative semidefinite (respectively, nega-
tive definite). I
n
is the identity matrix of order n.Weuse
λ
max
(·) (respectively λ
2
(·)) to denote the maximum (respec-
tively second largest) eigenvalue of a real symmetric matrix.
R
n
denotes the n-dimensional Euclidean space. R
n×n
are n×n
real matrices. The superscript “T ” represents the transpose.
diag{···}stands for a block-diagonal matrix. Matrices, if not
explicitly stated, are assumed to have compatible dimensions.
Let PC(m) denote the class of piecewise right continuous
function ϕ :[t
0
− τ,t]→R
m
(m ∈ N) with the norm
defined by ϕ(t)
τ
= sup
−τ ≤s≤0
ϕ(t + s).Forϕ : R → R,
denote ϕ(t
+
) = lim
s→0
+
ϕ(t + s), ϕ(t
−
) = lim
s→0
−
ϕ(t + s),
¯ϕ(t) = sup
−τ ≤s≤0
{ϕ(t+s)}. The Dini derivative of ϕ(t) is defined
as D
+
ϕ(t) = lim sup
s→0
+
(ϕ(t + s) − ϕ(t))/(s).
II. M
ODEL AND PRELIMINARIES
We consider a complex system consisting of N linearly
coupled identical NNs. Each node is an n-dimensional NN
composed of a linear term, a nonlinear term, a time-varying
delay term, and an external input vector. The ith NN can be
described by following differential equation:
˙x
i
(t) = Cx
i
(t) + B
1
f (x
i
(t)) + B
2
f (x
i
(t − τ(t))) + I (t) (1)
where x
i
(t) =[x
i1
(t), x
i2
(t),...,x
in
(t)]
T
is the state vector
of the ith NN at time t; C ∈ R
n×n
, B
1
∈ R
n×n
and B
2
∈
R
n×n
are matrices; f (x
i
(t)) =[f
1
(x
i1
(t)), f
2
(x
i2
(t)),...,
f
n
(x
in
(t))]
T
; I (t) =[I
1
(t), I
2
(t),...,I
n
(t)]∈R
n
is an ex-
ternal input vector; and τ(t) is a time-varying delay satisfying
0 <τ(t) ≤ τ
∗
.
For the nonlinear functions f
k
(·), we have the following
assumption.
Assumption 1: Assume that f
k
(·)(k = 1, 2,...,n) are
globally Lipschitz continuous functions, i.e., there exist con-
stants l
k
> 0 (k = 1, 2,...,n) such that | f
k
(x
1
) − f
k
(x
2
)|≤
l
k
|x
1
− x
2
| (k = 1, 2,...,n) hold for any x
1
, x
2
∈ R. Denote
L = diag{l
1
, l
2
,...,l
n
} for convenience.
The dynamical behavior of linearly coupled NNs can be
described by the following delayed differential equations
[19], [34]:
˙x
i
(t) = Cx
i
(t) + B
1
f (x
i
(t)) + B
2
f (x
i
(t − τ(t))) + I(t)
+ c
N
j=1, j=i
a
ij
(x
j
(t) − x
i
(t)), i = 1,...,N (2)
where = diag{γ
1
,γ
2
,...,γ
n
} (satisfying γ
i
> 0fori =
1, 2,...,n) is the diagonal inner coupling matrix between two
connected nodes i and j (i = j) at time t for all 1 ≤ i, j ≤ N;
c is the coupling strength; and a
ij
is defined as follows: if there
is a connection from node j to node i ( j = i), then a
ij
> 0;
otherwise, a
ij
= 0. It means that the network is directed and
the coupling matrix A = (a
ij
)
N×N
is asymmetric.
In the process of signal transmission, the coupled states
x
j
(t) − x
i
(t) between connected nodes j and i are suddenly
changed in the form of impulses at discrete times t
k
.Thatis,
x
i
(t
+
k
) −x
j
(t
+
k
) = S
k
·(x
i
(t
−
k
) −x
j
(t
−
k
)).Forthesakeofana-
lytical simplification, we shall choose a constant impulse gain
in the form of S
k
= μI,whereI is an n × n identity matrix.
Remark 9: This simplification for the impulsive matrix is
only for the simplification of the expression, and it does not
cause any loss of generality in the sense of synchronization
analysis. The matrix product can be used to describe the
synchronization criterion if the constant matrix S
k
is used to
describe the impulsive amplification.
Let a
ii
=−
N
j=1, j=i
a
ij
. Then, the impulsive dynamical
system can be obtained in the following form:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙x
i
(t) = Cx
i
(t) + B
1
f (x
i
(t)) + B
2
f (x
i
(t − τ(t)))
+ I (t) + c
N
j=1
a
ij
x
j
(t), t ≥ 0, t = t
k
, k ∈ N
x
j
(t
+
k
) − x
i
(t
+
k
) = μ · (x
j
(t
−
k
) − x
i
(t
−
k
)),
for k ∈ N,(i, j) satisfying a
ij
> 0,
x
i
(t) = φ
i
(t), −τ
∗
≤ t ≤ 0
(3)
where {t
1
, t
2
, t
3
, ...} is a sequence of strictly increasing im-
pulsive moments, and |μ| > 1 represents the strength of
impulsive disturbances between the coupling of the connected
nodes. Here, A = (a
ij
)
N×N
is a Laplacian matrix of the
corresponding network [35]. We always assume that x
i
(t) is
right continuous at t = t
k
,i.e.,x(t
k
) = x(t
+
k
). Therefore, the
solutions of (3) are piecewise right-hand continuous functions
with discontinuities at t = t
k
for k ∈ N.
By referring to the concept of average dwell time [31], [32],
we propose a new concept named average impulsive interval
to describe the impulse signal in [12]. This concept has been
utilized for the derivation of a unified synchronization criterion
of dynamical networks in [12]. Since |μ| > 1, which means
that the impulses can potentially destroy the synchronization,
we need to require that they do not happen too frequently.
Definition 2 is then given to enforce an upper bound on the
number of impulsive times.
Definition 2 ([12] average impulsive interval): The aver-
age impulsive interval of the impulsive sequence ζ =
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 22, NO. 2, FEBRUARY 2011 331
{t
1
, t
2
,...} is said to be not less than T
a
if there exist a positive
integer N
0
and a positive number T
a
such that
N
ζ
(T, t) ≤
T − t
T
a
+ N
0
∀T ≥ t ≥ 0(4)
where N
ζ
(T, t) denotes the number of impulsive times of the
impulsive sequence ζ on the interval (t, T ).
Remark 10: When N
0
= 1, the consecutive impulses must
be separated by at least T
a
units of time. When N
0
≥ 2, for
arbitrarily small and any T
a
> 0, many impulsive sequences
{t
1
, t
2
,...} can be constructed such that the lower bound of
the impulsive intervals are less than and simultaneously the
average impulsive intervals are not less than T
a
.LetN
0
≥ 2,
one simple example being ζ
∗
={t
0
+ 2T
a
, t
0
+ 2T
a
+ , t
0
+
4T
a
, t
0
+ 4T
a
+ ,...}. For the impulsive sequence ζ
∗
,the
lower bound of the impulsive interval is . Since the lower
bound is used to represent the frequency of the impulsive
sequence in [17], [20] and [27], the results obtained in [17],
[20] and [27] are not available for the impulsive sequence ζ
∗
with sufficiently small .
Remark 11: The idea behind this concept is as follows.
High-density impulses (such as “t
0
+ 2T
a
, t
0
+ 2T
a
+ ”)
are allowed to happen in a certain interval, and low-density
impulses (such as “t
0
+2T
a
+, t
0
+4T
a
”) should follow for
compensation.
Definition 3 ([36]): The impulsive dynamical network (3)
is said to be globally exponentially synchronized if there exist
an η>0, T > 0, and M
0
> 0 such that for any initial values
φ
i
(s)(i = 1, 2,...,N)
x
i
(t) − x
j
(t)≤M
0
e
−ηt
hold for all t > T ,andforanyi, j = 1, 2,...,N.
Definition 4 ([37]): For an irreducible square matrix A
with nonnegative off-diagonal elements, the functions α(A)
and β(A) are defined as follows: decompose A uniquely as
A = L
A
+ D
A
,whereL
A
is a zero row sum matrix and
D
A
≤ 0 is a diagonal matrix. Let ξ = (ξ
1
,ξ
2
,...,ξ
N
)
T
be the unique positive vector such that ξ
T
L
A
= 0and
N
i=1
ξ
i
= 1. The existence of such a positive vector ξ
can be guaranteed by the following Lemma 2. Let =
diag(ξ).Thenα(A) = λ
2
(A + A
T
) and β(A) =
λ
max
(A + A
T
).
Lemma 1 ([37]): Let A be an irreducible matrix with non-
negative off-diagonal elements and nonnegative row sums with
the decomposition A = L
A
+ D
A
as shown in Definition 4.
Then α(A)<0, β(A) ≤ 0. Furthermore, β(A)<0ifand
only if D
A
= 0.
Lemma 2: For any vectors x, y ∈ R
n
,wehave2x
T
y ≤
x
T
x + y
T
y.
Lemma 3 ([19], [38] (Perron–Frobenius Theorem)): For an
irreducible matrix A with nonnegative off-diagonal ele-
ments, which satisfies the diffusive coupling condition a
ii
=
−
N
j=1, j=i
a
ij
, we have the following propositions.
1) If λ is an eigenvalue of A and λ = 0, then Re(λ) < 0.
2) A has an eigenvalue 0 with multiplicity 1 and the right
eigenvector [1, 1,...,1]
T
.
3) Suppose that ξ =[ξ
1
,ξ
2
,...,ξ
N
]
T
∈ R
N
satisfying
N
i=1
ξ
i
= 1 is the normalized left eigenvector of A
corresponding to eigenvalue 0. Then, ξ
i
> 0 hold for all
i = 1, 2,...,N.Furthermore,ifA is symmetric, then
we have ξ
i
= 1/N for i = 1, 2,...,N.
Lemma 4 ([20]): Suppose p > q ≥ 0andu(t) satisfies the
scalar impulsive differential inequality
⎧
⎨
⎩
D
+
u(t) ≤−pu(t) + q ¯u(t), t = t
k
, t ≥ t
0
u(t
+
k
) ≤ ρ
k
u(t
−
k
), k ∈ N
u(t) = φ(t), t ∈[t
0
− τ,t
0
]
(5)
where u(t) is continuous at t = t
k
, t ≥ t
0
, u(t
k
) = u(t
+
k
),
and u(t
−
k
) exists, φ ∈ PC(1),and ¯u(t) = sup
−τ ≤s≤0
{u(t + s)}.
Then
u(t) ≤φ(t
0
)
τ
·
'
(
t
0
<t
k
≤t
ρ
k
)
· e
−λ(t−t
0
)
, t ≥ t
0
(6)
where λ>0 is the unique solution of the equation λ − p +
qe
λτ
= 0.
Remark 12: The result of Lemma 4 extends the famous
Halanay differential inequality [39] to impulsive delay differ-
ential systems, and will be used for the proof of the following
Proposition 1.
Proposition 1: Suppose p > q ≥ 0andu(t) satisfies the
scalar impulsive differential inequality
⎧
⎨
⎩
D
+
u(t) ≤−pu(t) + q ¯u(t), t = t
k
, t ≥ t
0
u(t
+
k
) ≤ ρu(t
−
k
), k ∈ N
u(t) = φ(t), t ∈[t
0
− τ,t
0
]
(7)
where u(t) is continuous at t = t
k
, t ≥ t
0
, u(t
k
) = u(t
+
k
),
u(t
−
k
) exists, φ ∈ PC(1), |ρ| > 1, and the average impulsive
interval of the impulsive sequence ζ ={t
1
, t
2
,...} is not less
than T
a
. Then there exists a constant M
0
such that
u(t) ≤ M
0
e
−(λ−
ln |ρ|
T
a
)(t−t
0
)
, t ≥ t
0
(8)
where λ>0 is the unique solution of the equation λ − p +
qe
λτ
= 0.
Proof: See Appendix.
Remark 13: Proposition 1 plays an important role in the
synchronization analysis of dynamical networks with impul-
sive disturbances in this brief, because it shows the utilization
of the concept of average impulsive interval.
III. S
YNCHRONIZATION ANALYSIS
In this section, globally exponential synchronization of
linearly coupled NNs with impulsive disturbances will be
studied. Let x(t) = (x
T
1
(t), x
T
2
(t),...,x
T
N
(t))
T
, F(x(t)) =
( f
T
(x
1
(t)), f
T
(x
2
(t)),..., f
T
(x
N
(t)))
T
,andI(t) =
(I
T
(t), I
T
(t),...,I
T
(t))
T
, then the impulsive dynamical
system (3) can be rewritten in the following Kronecker-
product form:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙x(t) = (I
N
⊗ C)x(t) + (I
N
⊗ B
1
)F(x(t))
+ (I
N
⊗ B
2
)F(x(t − τ(t))) + I(t)
+ c(A ⊗ )x(t), t ≥ 0, t = t
k
, k ∈ N,
x
j
(t
+
k
) − x
i
(t
+
k
) = μ · (x
j
(t
−
k
) − x
i
(t
−
k
)),
for k ∈ N,(i, j) satisfying a
ij
> 0,
x
i
(t) = φ
i
(t), −τ
∗
≤ t ≤ 0.
(9)
332 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 22, NO. 2, FEBRUARY 2011
Suppose that ξ = (ξ
1
,ξ
2
,...,ξ
N
)
T
is the normalized
left eigenvector of the configuration coupling matrix A with
respect to the eigenvalue 0 satisfying
N
i=1
ξ
i
= 1. Since the
coupling configuration matrix A = (a
ij
)
N×N
is irreducible,
according to the Lemma 3, we can conclude that ξ
i
> 0
for i = 1, 2,...,N.Let = diag{ξ
1
,ξ
2
,...,ξ
N
} > 0, and
W
= (w
ij
)
N×N
= − ξξ
T
.
Theorem 2: Consider the linearly coupled NNs (9) with
impulsive disturbances and the asymmetric irreducible cou-
pling matrix A.Letγ =−α(A)/λ
max
(W ), p =−λ
max
(2C −
cγ + B
1
B
T
1
+ L
T
L + B
2
B
T
2
),andq = λ
max
(L
T
L),where
L is the Lipschitz matrix given in Assumption 1, and α(A)
is defined in Definition 4. Suppose that Assumption 1 holds,
and then the average impulsive interval of the impulsive
sequence ζ ={t
1
, t
2
,...} is not less than T
a
. Then, the
linearly coupled NNs (9) with impulsive disturbances will be
globally exponentially synchronized with the convergence rate
λ − (2ln|μ|)/(T
a
) if
p > q and λ −
2ln|μ|
T
a
> 0 (10)
where λ>0 is the unique solution of the equation λ − p +
qe
λτ
∗
= 0.
Proof: Let V (t) = x
T
(t)(W ⊗ I
n
)x(t). By a detailed cal-
culation, we get that V (t) = (1/2)
N
i=1
N
j=1, j=i
−w
ij
(x
i
−
x
j
)
T
(x
i
− x
j
). It can be observed that V (t) is a smooth and
positive semidefinite function since −w
ij
= ξ
i
ξ
j
> 0. In fact,
the function V (t) vanishes at the synchronization manifold
M ={x
1
(t) = x
2
(t) = ... = x
N
(t)}. The derivative of V (t)
along the trajectories of the systems (9) can be obtained as
follows:
D
+
V (x)
= 2x
T
(t)(W ⊗ C)x(t) + 2x
T
(t)(W ⊗ B
1
)F(x(t))
+ 2x
T
(t)(W ⊗ B
2
)F(x(t − τ(t)))
+ 2x
T
(t)(W ⊗ I
n
)I(t) +2cx
T
(t)(WA⊗ )x(t),
t ∈ (t
k−1
, t
k
], k ∈ N. (11)
From Assumption 1, we have
( f (x
i
) − f (x
j
))
T
( f (x
i
) − f (x
j
))
≤ (x
i
− x
j
)
T
L
T
L(x
i
− x
j
). (12)
Noting that WA = ( − ξξ
T
)A = A − ξ(ξ
T
A) = A
and (W ⊗ I
n
)I(t) = 0, it follows that
D
+
V (x)
≤−
N
i=1
N
j=1, j=i
w
ij
*
(x
i
− x
j
)
T
(C −
1
2
cγ+
1
2
B
1
B
T
1
+
1
2
L
T
L +
1
2
B
2
B
T
2
)(x
i
− x
j
) +
1
2
x
i
(t − τ(t))
− x
j
(t − τ(t))
T
L
T
L
x
i
(t − τ(t)) − x
j
(t − τ(t))
+
+ cx
T
(t)
(A + A
T
) ⊗ + W ⊗ γ
x(t),
t ∈ (t
k−1
, t
k
], k ∈ N (13)
where w
ij
=−ξ
i
ξ
j
< 0fori = j, inequality (12), and
Lemma 2 are used.
Since p =−λ
max
(2C −cγ+ B
1
B
T
1
+ L
T
L + B
2
B
T
2
) and
q = λ
max
(L
T
L), it follows from (13) that
D
+
V (x)
≤−p
N
i=1
N
j=1, j=i
−
1
2
w
ij
(x
i
− x
j
)
T
(x
i
− x
j
)
+ q
N
i=1
N
j=1, j=i
−
1
2
w
ij
[(x
i
(t − τ(t)) − x
j
(t − τ(t)))
T
× (x
i
(t − τ(t)) − x
j
(t − τ(t)))]
+ cx
T
(t)
(A + A
T
) ⊗ + W ⊗ γ
x(t)
=−pV(t) + qV(t − τ(t))
+ cx
T
(t)
(A + A
T
) ⊗ + W ⊗ γ
x(t)
≤−pV(t) + q
¯
V (t) + cx
T
(t)
(A + A
T
) ⊗
+ W ⊗ γ
x(t), t ∈ (t
k−1
, t
k
], k ∈ N (14)
where
¯
V (t) = sup
−τ
∗
≤s≤0
{V (t + s)}.
Let
˜
A = A + A
T
. From the Perron–Frobenius theorem
[38], the eigenvalues of the matrix
˜
A can be arranged as
follows: 0 = λ
1
(
˜
A)>λ
2
(
˜
A) ≥ λ
3
(
˜
A) ≥ ...≥ λ
N
(
˜
A),where
λ
2
(
˜
A) = α(A) by Definition 4.
From matrix decomposition theory [38], there exists a
unitary matrix U such that
˜
A = UU
T
,where =
diag{0,λ
2
(
˜
A),...,λ
N
(
˜
A)},andU =[u
1
, u
2
,...,u
N
] with
u
1
= ((1/
√
N), (1/
√
N),...,(1/
√
N))
T
.
Let y(t) = (U
T
⊗ I
n
)x(t), then one has x(t) = (U ⊗
I
n
)y(t).Lety
i
(t) ∈ R
n
(i = 1, 2,...,N) be such that y(t) =
[y
T
1
(t), y
T
2
(t),...,y
T
N
(t)]
T
, and it follows that
x
T
(t)
(A + A
T
) ⊗
x(t)
= y
T
(t)(U
T
⊗ I
n
)(
˜
A ⊗ )(U ⊗ I
n
)y(t)
=
N
i=2
λ
i
(
˜
A)y
T
i
(t)y
i
(t)
≤ λ
2
(
˜
A)
N
i=2
y
T
i
(t)y
i
(t)
= α(A)
N
i=2
y
T
i
(t)y
i
(t). (15)
From the construction of matrix W, it can be observed that
W is a zero row sum irreducible symmetric matrix with nega-
tive off-diagonal elements. Hence it follows that λ
max
(W )>0.
Moreover, W ·u
1
= (0, 0,...,0)
T
:= O
n
∈ R
N
.Thenweget
U
T
WU =
0 O
T
n
O
n
˜
U
T
W
˜
U
,where
˜
U =[u
2
, u
3
,...,u
N
]
satisfying
˜
U
T
˜
U = I
N−1
. Hence, one has
x
T
(t)(W ⊗ γ)x(t)
= γ y
T
(t)(U
T
WU ⊗ )y(t)
= γ ˜y
T
(t)(
˜
U
T
W
˜
U ⊗ ) ˜y(t)
≤ γλ
max
(W ) ˜y
T
(t)(
˜
U
T
˜
U ⊗ ) ˜y(t)
= γλ
max
(W )
N
i=2
y
T
i
(t)y
i
(t) (16)
where ˜y(t) =[y
T
2
(t),...,y
T
N
(t)]
T
.
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 22, NO. 2, FEBRUARY 2011 333
It follows from (15), (16), and the equality γ =
−α(A)/2λ
max
(W ) that
cx
T
(t)
(A + A
T
) ⊗ + W ⊗ γ
= c(α(A) + γλ
max
(W ))
N
i=2
y
T
i
(t)y
i
(t)
= 0. (17)
Recalling (14), one gets
D
+
V (x) ≤−pV(t) + q
¯
V (t), t ∈ (t
k−1
, t
k
], k ∈ N. (18)
Since A is irreducible, for any pair of suffixes i and j
(i = j), there exist suffixes s
1
, s
2
,...,s
m
such that a
js
1
> 0,
a
s
1
s
2
> 0,..., and a
s
m
i
> 0. From (9), we have x
j
(t
+
k
) −
x
i
(t
+
k
) = μ · (x
j
(t
−
k
) − x
i
(t
−
k
)), k ∈ N, for each pair of (i, j)
satisfying a
ij
> 0. Then, for any pair of suffixes i and j,
we get
x
j
(t
+
k
) − x
i
(t
+
k
)
= (x
j
(t
+
k
) − x
s
1
(t
+
k
)) + (x
s
1
(t
+
k
) − x
s
2
(t
+
k
))
+···+ (x
s
m
(t
+
k
) − x
i
(t
+
k
))
= μ · (x
j
(t
−
k
) − x
s
1
(t
−
k
)) + μ · (x
s
1
(t
−
k
) − x
s
2
(t
−
k
))
+···+μ · (x
s
m
(t
−
k
) − x
i
(t
−
k
))
= μ · (x
j
(t
−
k
) − x
i
(t
−
k
)). (19)
Hence, for t = t
k
, k ∈ N,weget
V (t
+
k
)
=
1
2
N
i=1
N
j=1
j=i
−w
ij
(x
i
(t
+
k
) − x
j
(t
+
k
))
T
(x
i
(t
+
k
) − x
j
(t
+
k
))
=
μ
2
2
N
i=1
N
j=1
j=i
−w
ij
(x
i
(t
−
k
) − x
j
(t
−
k
))
T
(x
i
(t
−
k
) − x
j
(t
−
k
))
= μ
2
V (t
−
k
). (20)
Employing Proposition 1 and from (18) and (20), we can
conclude that there exists constant M
0
such that
V (t) ≤ M
0
e
−(λ−
2ln|μ|
T
a
)(t)
, t ≥ 0 (21)
where λ>0 is the unique solution of the equation
λ−p+qe
λτ
∗
= 0. Hence, we have (1/2)ξ
i
ξ
j
x
i
(t)−x
j
(t)
2
≤
V (t) ≤ M
0
e
−(λ−(2ln|μ|)/(T
a
))(t)
. Since λ −(2ln|μ|)/(T
a
)>0,
globally exponential synchronization of linearly coupled NNs
(9) with impulsive disturbances is achieved according to
Definition 3. The proof is completed.
Remark 14: In this brief, we require this network to be
strongly connected, which means that there is a path from
each node in the network to every other node. By referring to
the concept of β(A) in Definition 4, the results in Theorem 2
and techniques in [11], the results can be extended to the case
of reducible coupling (containing rooted spanning tree [37]),
which means that the network is not required to be strongly
connected. Details will not be discussed here due limitations
of space.
−
15
−
10
−
50 5101520
−
1
−
0.5
0
0.5
1
1.5
x
i2
(t)
(a)
x
i1
(t)
Fig. 1. Chaotic attractor of system (22).
IV. NUMERICAL EXAMPLE
As an application of the theoretical result, the chaos syn-
chronization problem of a typical small-world network with
100 NNs is discussed in this section. A chaotic NN is chosen
as the isolated node of the network, which can be described
by the following [40]:
˙x(t) =−Cx(t) + B
1
f (x(t)) + B
2
f (x(t − τ(t))) + I(t) (22)
with C =
'
10
01
)
, A =
'
2.0 −0.11
−5.03.2
)
, B =
'
−1.6 −0.1
−0.18 −2.4
)
, I =
'
0
0
)
,andτ(t) = (e
t
/1 + e
t
),
where x(t) = (x
1
(t), x
2
(t))
T
is the state vector of the network,
and f (x(t)) = g(x(t)) = (tanh(x
1
), tanh(x
2
))
T
. Then the
Lipschitz constants can be obtained as l
1
= l
2
= 1. The single
NN model (22) has a chaotic attractor as shown in Fig. 1 with
the initial values x
1
(s) = 0.2, x
2
(s) = 0.5, ∀s ∈[−1, 0].
We consider an NW small-world network with 100 dy-
namical nodes [41]. The NW small-world model algorithm is
presented as follows. 1) Begin with a nearest neighbor coupled
network consisting of N nodes arranged in a ring, where each
node i is adjacent to its neighbor nodes, i = 1, 2,...,(k/2),
with k being even. 2) Add a connection between each pair
of nodes with probability p. In this example, the parameters
are set as N = 100, k = 2, and p = 0.04. Then small-world
network can be generated with the coupling matrix A.Inthis
example, we generate 100 small-world networks randomly. By
calculation, the quantities γ =−α(A)/λ
max
(W ) of these 100
networks belong to [2.7146, 4.7901]. From Theorem 2, we
can observe that the remaining 99 small-world networks can
be synchronized if the small-world network with minimum
γ =−α(A)/λ
max
(W ) = 2.7146, in which α(A) =−0.0785
and λ
max
(W ) = 0.0289 can be synchronized. Hence, the
small-world network with γ = 2.7146 is selected for
simulation in this example.
Let c = 8and = diag{5, 5}. By a simple computation,
we obtain p = 65.9471, q = 1andτ
∗
= 1. The unique
solution of the equation λ − p + qe
λτ
∗
= 0isλ = 4.1243.
Let the impulsive strength μ be μ = 1.2. According to
