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Convergence behavior of delayed discrete cellular neural network without periodic coefficients
Neural Networks 53 (2014) 61–68
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Neural Networks journal homepage: www.elsevier.com/locate/neunet
Convergence behavior of delayed discrete cellular neural network without periodic coefficients✩ Jinling Wang, Haijun Jiang ∗ , Cheng Hu, Tianlong Ma College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, Xinjiang, PR China
article
info
Article history: Received 5 May 2013 Received in revised form 28 October 2013 Accepted 16 January 2014 Keywords: Discrete cellular neural network Periodic solution Convergence Without periodic coefficient
abstract In this paper, we study convergence behaviors of delayed discrete cellular neural networks without periodic coefficients. Some sufficient conditions are derived to ensure all solutions of delayed discrete cellular neural network without periodic coefficients converge to a periodic function, by applying mathematical analysis techniques and the properties of inequalities. Finally, some examples showing the effectiveness of the provided criterion are given. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Since cellular neural networks (CNNs) were introduced by Chua and Roska (1990) in 1990, they have been successfully applied in signal and image processing, pattern recognition and optimization, especially in image processing and solving nonlinear algebraic equations. Hence, CNNs have been the aim of intensive analysis by numerous authors in recent years (Cao, 2003; Liu & Huang, 2006; Liu & Liao, 2004; Mohamad & Gopalsamy, 2000, 2003; Orman, 2012; Wang, Li, & Xu, 2011; Yang, 2012; Zhang & Chen, 2008; Zhang & Sun, 2005). In implementing the continuous-time network for computer simulation, experimental and computational purposes, it is usual to discretize the continuous-time network. Certainly, the discretetime analog inherits the dynamical characteristics of continuoustime networks under mild or no restriction on the discretization step-size, and also remains functional similarity to the continuoustime system. We refer to Mohamad (2008), Wu, Li, and Huang (2012) and Wang, Liu, and Liu (2005) for related discussions on the important and the need for discrete-time analogs to reflect the dynamics of their continuous-time counterparts. Recently, many authors have studied the dynamical behaviors of discrete-time analogs of CNNs with delays and variable coefficients. Li (2004) has investigated the stability and existence of periodic solutions of CNNs without periodic coefficients by using
✩ This was supported by the National Natural Science Foundation of People’s Republic of China (Grant NO. 61164004). ∗ Corresponding author. Tel.: +86 13079900716. E-mail address: [email protected] (H. Jiang).
http://dx.doi.org/10.1016/j.neunet.2014.01.007 0893-6080/© 2014 Elsevier Ltd. All rights reserved.
Mawhin’s continuation theorem of coincidence degree theory and by constructing Lyapunov functions. Zhou, Li, and Zhang (2008) have studied the convergence behaviors of delayed continuoustime cellular neural networks without periodic coefficients, and some sufficient conditions are established to ensure that all solutions of networks converge to a periodic function. Chen and Zhao (2009) have obtained discrete analog of highorder Cohen–Grossberg neural networks with time-varying delay by analysis and approximation techniques, and also sufficient conditions are given to guarantee global exponential stability of the periodic solution. Yang, Cui, and Long (2009) have obtained global exponential stability of periodic solution of a cellular neural network difference equation with delays and impulses by using contraction mapping theorem and inequality techniques. Hence, most authors of the bibliographies listed above obtained that all solutions of continue-time delayed CNNs without periodic coefficients converge to a periodic function and existence of periodic solutions of delayed discrete CNNs with periodic coefficients. However, we see, for delayed discrete CNNs without periodic coefficients, up until now, study works are very few. In this paper, our main purpose is to investigate the convergence behavior of delayed discrete CNNs without periodic coefficients. First, we give several conditions to ensure delayed discrete CNNs having a unique periodic solution. Then, we derive some sufficient conditions ensuring that all solutions of delayed discrete CNNs without periodic coefficients converge to a periodic function, by applying mathematical analysis techniques. The results are new and complement previously known results. The remainder of this paper is organized as follows. In Section 2, we introduce the discrete-time analogs discussed in
62
J. Wang et al. / Neural Networks 53 (2014) 61–68
this paper. In Section 3, some sufficient conditions are given to ensure the delayed discrete CNNs having a unique periodic solution. In Section 4, sufficient conditions are given to ensure that all solutions of the delayed discrete CNNs without periodic coefficients converge to a periodic function. Moreover, in Section 5, some examples are given to show the usefulness of our results. At last, Section 6 concludes the work of this paper. 2. Model description We consider a continuous-time cellular neural network consisting of m interconnected cells described by a system of delay differential equations of the form dxi (t ) dt
= −bi (t )xi (t ) +
m
aij (t )fj (xj (t ))
j =1
+
m
(1)
for i ∈ L = {1, 2, . . . , m}, where bi (t ) > 0, aij (t ), bij (t ), Ii (t ) ∈ R, vij > 0. Respectively, m corresponds to the number of units in a neural network; xi (t ) corresponds to the state of the ith unit at time t; fj (xj (t )) denotes the output of the jth unit on the ith unit at time t; aij (t ), bij (t ) denote the strength of the jth unit on the ith unit at time t; Ii (t ) denotes the external bias on the ith unit at time t; vij denotes the transmission delay of the ith unit along the axon of the jth unit at the time t; bi (t ) represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs. Then, we shall use a semi-discretization technique to obtain a discrete-time system of continuous-time cellular neural network system (1). For convenience in our study, we use the following notations: Let Z denote the set of all integers, i.e. Z = {. . . , −1, 0, 1, . . .}. Let Z + = {1, 2, . . .} and Z0+ = {0, 1, 2, . . .}. For c , d ∈ Z and c ≤ d we denote the discrete interval [c , d]Z = {c , c + 1, . . . , d − 1, d}. If d = ∞, then [c , ∞)Z = {c , c + 1, . . .}. While there is no unique way in obtaining a discrete-time analog of continuous-time network (1), such as Euler scheme, Runge–Kutta scheme, we begin by reformulating network (1) with the following approximation:
dt
t
= −bi +
h xi (t ) +
h
m
m
t
aij
h
j =1
h fj
t
bij
h fj
h
j =1
xj
t
h
h−
xj
t
h
h
v ij
h
h
+ Ii
t
h
h ,
(2)
for i ∈ L, t ∈ [nh, (n + 1)h), n ∈ Z + , where h is a fixed positive real number denoting a uniform discretization step-size and [r ] denotes the integer part of the real number r. Clearly, for t ∈ v [nh, (n + 1)h), n ∈ Z + , we have [ ht ] = n. We let [ hij ] = τij and we have τij ∈ Z + because of vij ≥ 0. For convenience in the following we use the notation xi (nh) = xi (n). Then, we can integrate (3) over [nh, t ) and by allowing t → (n + 1)h in the above, we obtain after some simplification that xi (n + 1) = e−bi (n)h xi (n) + θi (h)
m
j =1
1≤i,j≤m
(H1 ) h ∈ (0, ∞), bi (n) : Z → (0, ∞), aij (n), bij (n), Ii (n) : Z → R, τij ∈ Z + , i, j = 1, 2, . . . , m. (H2 ) fi (·) ∈ C (R, R) is bounded on R, and ∃Li > 0, s.t .|fi (x) − fi (y)| ≤ Li |x − y|. (H3 ) bi (n), aij (n), bij (n), Ii (n) are all ω-periodic functions, where ω In the following, for convenience, we will use the following notations. aM ij = max |aij (n)|, 1≤i,j≤m
bi = min |bi (n)|, 1≤i≤m
bij (n)fj (xj (n − τij )) + θi (h)Ii (n),
bM ij = max |bij (n)|, 1≤i,j≤m
bi = max |bi (n)|. 1≤i≤m
3. Existence of periodic solution In this section, we give several conditions to ensure system (3) having a unique w -periodic solution. Lemma 1. Assume that (H1 )–(H3 ) hold. Then system (3) has at least one w -periodic solution. The result of Lemma 1 was given by Theorem 3.1 in Li (2004). Theorem 1. Assume that (H1 )–(H3 ) hold. Furthermore, assume that (H4 ) there exists a constant λ > 1 such that bi −
m
aM ij Lj −
j =1
m
τij bM > 0. ij Lj λ
j=1
Then system (3) has a unique w -periodic solution. Proof. It follows from Lemma 1 that system (3) has an ω-periodic solution, x(n) = (x1 (n), x2 (n), . . . , xm (n))T . Obviously, if this periodic solution is global attractive, then it is unique. Let x(n) = (x1 (n), x2 (n), . . . , xm (n))T be an arbitrary solution of system (3). From system (3), we can get xi (n + 1) − xi (n + 1) = (xi (n) − xi (n))e−bi (n)h
+ θi (h)
m
aij (n)[fj (xj (n)) − fj (xj (n))]
j =1
+ θi (h)
m
bij (n)[fj (xj (n − τij )) − fj (xj (n − τij ))].
j =1
Thus, by equality above we obtain
|xi (n + 1) − xi (n + 1)| ≤ |xi (n) − xi (n)|e−bi (n)h m + θi (h) aM ij Lj |(xj (n)) − xj (n)| j =1
j =1
τ = max {τij },
where ϕi (·) is a real-valued continuous function defined on [τ , 0]Z . In this paper, for system (3) we introduce the following assumptions.
aij (n)fj (xj (n))
+ θi (h)
m
+ θi (h)
xi (s) = ϕi (s), s ∈ [−τ , 0]Z ,
is a positive integer.
bij (t )fj (xj (t − vij )) + Ii (t ),
j =1
dxi (t )
−bi (n)h for i ∈ L, n ∈ Z0+ , where θi (h) = 1−eb (n) , i ∈ L. It is not difficult i to verify that θi (h) > 0 if bi > 0, h > 0 and θi (h) ≈ h + O(h2 ) for small h > 0. System (3) is discrete-time analog of continuous-time system (1). One can show that discrete-time system (3) converges towards continuous-time system (1) when h → 0+ . System (3) is supplemented with initial conditions given by
(3)
m
bM ij Lj |xj (n − τij ) − xj (n − τij )|.
j =1
Define function
J. Wang et al. / Neural Networks 53 (2014) 61–68
m
Fi (λ) = 1 − λ + λθi (h) bi −
aM ij Lj −
m
j =1
τij bM ij Lj λ
,
j =1
where i = 1, 2, . . . , m, then from condition (H4 ) and θi (h) > 0, we have Fi (1) > 0. So there exists a constant λ0 > 1, such that Fi (λ0 ) > 0, i = 1, 2, . . . , m. Let ui (n) = λn0 |xi (n) − xi (n)|, n ∈ [−τ , +∞)Z , then ui (n + 1) ≤ λ
n +1 0
|xi (n) − xi (n)|e−bi (n)h
+ θi (h)
m
aM ij Lj |(xj (n)) − xj (n)|
λe−bi h − 1 + Li λ
bM ij Lj
|xj (n − τij ) − xj (n − τij )|
j=1 m
= λ0 e−bi (n)h ui (n) + λ0 θi (h)
aM ij Lj uj (n)
j =1 m
τij bM ij Lj λ uj (n − τij ).
Let i=1,...,m s∈[−τ ,0] Z
aM ji θj (h) + Li
m
|xi (s) − xi (s)|}.
ui (n) ≤ K , n ∈ [−τ , n0 ]Z ,
Proof. It follows from Lemma 1 that system (3) has an ω-periodic solution, x(n) = (x1 (n), x2 (n), . . . , xm (n))T . Obviously, if this periodic solution is global attractive, then it is unique. Let x(n) = (x1 (n), x2 (n), . . . , xm (n))T be an arbitrary solution of system (3). From system (3), we can get
−bi0 (n0 −1)h
ui0 (n0 ) > K .
aij (n)[fj (xj (n)) − fj (xj (n))]
+ θi (h)
m
bij (n)[fj (xj (n − τij )) − fj (xj (n − τij ))].
j=1
|xi (n + 1) − xi (n + 1)| ≤ |xi (n) − xi (n)|e−bi (n)h + θi (h)
ui (n0 − 1)
m
m
By directly calculating,
By further calculating, we have
m
aM ij Lj |(xj (n)) − xj (n)|
j =1
aM ij Lj uj (n0 − 1)
+ θi (h)
j=1
+ λ0 θi0 (h)
j =1
j=1
i ̸= i0 ;
ui0 (n) ≤ K , n ∈ [−τ , n0 − 1]Z ;
+ λ0 θi0 (h)
τji +1 bM < −η < 0. ji θj (h)λ
Then system (3) has a unique w -periodic solution.
+ θi (h)
Then we claim that ui (n) ≤ K , n ∈ Z0+ , i = 1, . . . , m. Otherwise, we can choose constants i0 ∈ {i = 1, . . . , m}, n0 ∈ Z0+ such that
K < ui0 (n0 ) ≤ λ0 e
m
xi (n + 1) − xi (n + 1) = (xi (n) − xi (n))e−bi (n)h
j =1
K = max { sup
Corollary 1. Assume that (H1 )–(H3 ) and (H4 )′ hold. Then system (3) has a unique w -periodic solution.
j =1
m
+ λ0 θi (h)
However, in Theorem 1, we get the same result by using the properties of inequalities. Besides, the condition (H4 ) in Theorem 1 is different from the condition in Ref. Li (2004), and it is the key to obtain the convergence of the solution.
Theorem 2. Assume that (H1 )–(H3 ) hold. Furthermore, assume that (H5 ) there exist constants η > 0, λ > 1 such that
j=1
+ θi (h)
63
m
bM ij Lj |xj (n − τij ) − xj (n − τij )|.
j=1
m
τij bM ij Lj λ uj (n0 − 1 − τi0 j )
j =1
m = λ0 − λ0 θi0 (h)bi + λ0 θi0 (h) aM ij Lj
Let ui (n) = λn |xi (n) − xi (n)|, n ∈ [−τ , +∞)Z , then we can easily obtain
ui ( n + 1 ) ≤ λ
n +1
|xi (n) − xi (n)|e−bi (n)h
j =1
+ θi0 (h)
m
τi 0 j bM ij Lj
λ
+ θi (h)
K
j =1
which is a contradiction. So ui (n) ≤ K , n ∈ Z0+ , i = 1, . . . , m. Hence,
|xi (n) − xi (n)| ≤
1
λ0
n max { sup
i=1,...,m s∈[−τ ,0] Z
|xi (s) − xi (s)|}.
Therefore, the ω-periodic solution of (3) is unique. The proof is completed. |x (n)−x (n)|
Remark 1. If we choose ui (n) = λn0 i θ (h)i i changed into (H4 )′ there exists a constant λ > 1 such that bi θi (h) −
m j=1
aM ij Lj θj (h) −
m
aM ij Lj |(xj (n)) − xj (n)|
j=1
< K,
m
, then (H4 ) can be
τij bM ij Lj λ θj (h) > 0.
j =1
Moreover, in Ref. Li (2004), the author obtained system (3) had a unique w -periodic solution by applying Lyapunov function.
+ θi (h)
m
bM ij Lj |xj (n − τij ) − xj (n − τij )|
j =1
= λe−bi (n)h ui (n) + λθi (h)
m
aM ij Lj uj (n)
j=1
+ λθi (h)
m
τij bM ij Lj λ uj (n − τij ).
j =1
Now, consider the following Lyapunov function V (n) =
m i=1
ui (n) +
m
τij+1 bM i ij Lj
j=1
λ
θ (h)
n−1
uj (s) .
s=n−τij
By applying conditions (H3 ) and (H5 ), we can calculate the difference ∆V (n) = V (n + 1) − V (n) as follows:
∆V (n) = V (n + 1) − V (n)
64
J. Wang et al. / Neural Networks 53 (2014) 61–68 m
=
m
ui (n + 1) +
i =1
−
ui (n) −
m m
n
τij+1 θi (h) bM ij Lj λ
4. Convergence
uj ( s )
s=n+1−τij
i =1 j =1
m i =1
≤
m
n−1
τij+1 θi (h) bM ij Lj λ
In this section, we obtain sufficient conditions to ensure that all solutions of system without periodic coefficients converge to a periodic function. Now, we consider the following delayed discrete cellular neural networks without periodic coefficients:
uj (s)
s=n−τij
i =1 j =1
m m m (λe−bi h − 1)ui (n) + λ aM ij Lj θi (h)uj (n) i =1
+
∗ xi (n + 1) = e−bi (n)h xi (n) + θi∗ (h)
i=1 j=1
m m
θ (h)uj (n − τij )
m
+ θi∗ (h)
i =1 j =1 m
+
m
τij+1 θi (h)(uj (n) − uj (n − τij )) bM ij Lj λ
λe−bi h − 1 + Li λ
i =1
+ Li
m
x∗i (s) = ϕi∗ (s), s ∈ [−τ , 0]Z ,
aM ji θj (h)
j =1 m
θ (h)λ
τji +1
ui (n)
< 0.
n→+∞
m
n→+∞
Further, ui (n) ≤
i =1
m
ui (0) +
i =1
1 + Li
i=1
m
τij+1 bM i ij Lj
λ
θ (h)
a∗ij
uj ( s )
M
= max |a∗ij (n)|,
b∗ij
1≤i,j≤m
bi = min |bi (n)|,
lim (Ii (n) − Ii∗ (n)) = 0.
n→+∞
M
= max |b∗ij (n)|, 1≤i,j≤m
bi = max |b∗i (n)|.
∗
∗
s=−τij
j =1 m
−1
∗
1≤i≤m
1≤i≤m
τji+1 bM τji ji θj (h)λ
Theorem 3. Assume that (H1 )–(H4 ) and (H7 ) hold, and x(n) = (x1 (n), x2 (n), . . . , xm (n))T is the ω-periodic solution of system (3). Then every solution x∗ (n) = (x∗1 (n), x∗2 (n), . . . , x∗m (n))T of system (4) with any initial conditions (5) converges to the ω-periodic solution x(n) of system (3). In simple terms,
j =1
sup {|xi (s) − xi (s)|}.
s∈[−τ ,0]Z
That is m
lim (aij (n) − a∗ij (n)) = 0,
n→+∞
In the following, for convenience, we will use the following notations.
i =1
×
1≤i,j≤m
lim (bij (n) − b∗ij (n)) = 0,
ui (n) ≤ V (n) ≤ V (0).
≤
τ = max sup{τij (n), n ∈ Z }, (5)
lim (bi (n) − b∗i (n)) = 0,
So we have
m
(4)
where ϕi (·) is a real-valued continuous function defined on [τ , 0]Z . We assume that coefficients bi (n), aij (n), bij (n), Ii (n), b∗i (n), ∗ aij (n), b∗ij (n), Ii∗ (n) satisfy the following conditions. (H7 )
bM ji j
j=1
m
b∗ij (n)fj (xj (n − τij )) + θi∗ (h)Ii∗ (n),
where b∗i (n), a∗ij (n), b∗ij (n), Ii∗ (n) are not ω-periodic functions. System (4) is supplemented with initial conditions given by
i =1 j =1
=
j =1
m
a∗ij (n)fj (xj (n))
j =1
τij+1 bM i ij Lj
λ
m
|xi (n) − xi (n)| ≤
i =1
1
λ
max
×
sup
n i=1,...,m
s∈[−τ ,0]Z
1 + Li
m
θ (h)λ
bM ji j
τji+1
lim |x∗i (n) − xi (n)| = 0.
τji
n→+∞
j =1
|xi (s) − xi (s)|.
Proof. Let ∗
Therefore, the ω-periodic solution of (3) is unique. The proof is complete. Remark 2. If we choose ui (n) = λn0
|xi (n)−xi (n)| . θi (h)
j =1
Then (H5 ) can be
changed into (H5 ) there exist constants η > 0, λ > 1 such that ′
λe−bi h − 1 + Li θi (h)λ
m
δi (n) = (e−bi (n)h − e−bi (n)h )xi (n) m m ∗ ∗ + θi (h) aij − θi (h) aij fj (xj (n))
aM ji + Li θi (h)
m
θi (h) ∗
+
j =1
m
bij − θi (h) ∗
j =1
m
bij
fj (xj (n − τij ))
j =1
+ θi∗ (h)Ii∗ (n) − θi (h)Ii (n),
τji +1 bM < −η < 0. ji λ
Moreover, we can get the same result under the condition of (H5 ) .
where i = 1, 2, . . . , m. According to conditions (H4 ) and (H7 ), we can choose constants N > 0, λ0 > 1, n0 ∈ Z such that
Corollary 2. Assume that (H1 )–(H3 ) and (H5 )′ hold. Then system (3) has a unique w -periodic solution.
|δi (n)| ≤
j =1
j =1
′
Remark 3. In fact, there are some differences between Theorems 1 and 2. In Theorem 1, by (H4 ), the result is obtained through the inequalities approach. In Theorem 2, by (H5 ), the result is obtained by applying Lyapunov function. Hence, we can find that different methods need different conditions to obtain the result.
1 n
λ00
ε,
for any n > N , ε > 0. From system (4), x∗i (n + 1) − xi (n + 1)
= e−bi (n)h x∗i (n) + θi∗ (h) ∗
m j =1
a∗ij fj (x∗j (n)) + θi∗ (h)
m j =1
b∗ij fj (x∗j (n
J. Wang et al. / Neural Networks 53 (2014) 61–68
− τij )) + θi∗ (h)Ii∗ (n) − e−bi (n)h xi (n) − θi (h)
+ θi∗0 (h)
aij fj (xj (n))
− θi (h)
bij fj (xj (n − τij )) − θi (h)Ii (n) − e
−b∗ (n)h i
+e
xi (n) − θi (h) ∗
m
aij fj (xj (n)) + θi (h) ∗
∗
m
Fi (λ) = 1 − λ + λθi (h) bi − aij fj (xj (n))
m
b∗ij fj (xj (n − τij )) + θi∗ (h)
j =1
b∗ij fj (xj (n − τij ))
j =1
= (x∗i (n) − xi (n))e−bi (n)h + θi∗ (h) ∗
m
∗
u∗i0 (n0 ) < K ∗ + ε,
i
+ θi∗ (h)
m
|a∗ij (n)|Lj |(x∗j (n)) − xj (n)|
j =1
|b∗ij (n)|Lj |x∗j (n − τij ) − xj (n − τij )| + |δi (n)|.
j =1
Let u∗i (n) = λn0 |x∗i (n) − xi (n)|, n ∈ [−τ , +∞)Z , then we can calculate the inequality as follows: n +1 0
m
|xi (n) − x∗i (n)|e−bi (n)h ∗
aij Lj |(xj (n)) − xj (n)| ∗M
∗
j =1 m
b∗ij Lj |x∗j (n − τij ) − xj (n − τij )| + |δi (n)| M
= λ0 e−bi (n)h u∗i (n) + λ0 θi∗ (h) ∗
ui ( n) ≤ K ∗ ,
n ∈ Z0+ , i = 1, . . . , m.
Hence,
a∗ij Lj u∗j (n) + λ0 θi∗ (h) M
1
n max { sup
λ0
i=1,...,m s∈[−τ ,0] Z
|x∗i (s) − xi (s)|}.
Furthermore, we get lim |x∗i (n) − xi (n)| = 0.
The proof is complete.
Remark 4. In Theorem 3, we consider the convergent behavior for discrete-time delayed CNNs. However, in Zhou et al. (2008), the author study the convergent behavior for continue-time delayed CNNs. Hence, as we know, the discrete-time system has the same convergent behavior with the continuous-time system. Theorem 4. Assume that (H1 )–(H3 ), (H5 )′ and (H7 ) hold, and x(n) = (x1 (n), x2 (n), . . . , xm (n))T is the ω-periodic solution of system (3). Then every solution x∗ (n) = (x∗1 (n), x∗2 (n), . . . , x∗m (n))T of system (4) with any initial conditions (5) converges to the ωperiodic solution x(n) of system (3). In simple terms,
Proof. Let ∗
M
j =1
j=1
Next, we can set ∗
K = max { sup
i=1,...,m s∈[−τ ,0] Z
|xi (s) − xi (s)|}. ∗
j =1
m
θi (h) ∗
+
bij − θi (h) ∗
j=1
Then we claim that u∗i (n) ≤ K ∗ , n ∈ Z0+ , i = 1, . . . , m. Otherwise, there exist some constants i0 ∈ {i = 1, . . . , m}, n0 ∈ Z0+ such that u∗i (n) ≤ K ∗ ,
n ∈ [−τ , n0 ]Z , i ̸= i0 ;
u∗i0 (n) ≤ K ∗ ,
n ∈ [−τ , n0 − 1]Z ;
u∗i0 (n0 ) > K ∗ .
−b∗i (n0 −1)h ∗ 0
ui0 (n0 ) ≤ λ0 e
ui (n0 − 1) + λ0 θi0 (h) ∗
m
aij Lj uj (n0 − 1) ∗M
∗
j =1
b∗ij Lj λτij u∗j (n0 − 1 − τi0 j ) + λ00 |δi (n0 − 1)| n
M
m
bij
fj (xj (n − τij ))
j =1
+ θi∗ (h)Ii∗ (n) − θi (h)Ii (n). According to conditions (H5 )′ and (H7 ), we can choose constants N > 0, λ0 > 1, n0 ∈ Z such that
|δi (n)| ≤
Then, we obtain
m
δi (n) = (e−bi (n)h − e−bi (n)h )xi (n) m m ∗ ∗ + θi (h) aij − θi (h) aij fj (xj (n))
b∗ij Lj λτij u∗j (n − τij ) + λn0+1 |δi (n)|.
+ λ0 θi∗0 (h)
n→+∞
m
j =1
∗
∀ε > 0.
lim |x∗i (n) − xi (n)| = 0.
j =1
×
,
j =1
n→+∞
m
m
bij Lj λ
τij
i = 1, 2, . . . , m.
−b∗ (n)h
+ θi∗ (h)
∗M
which is a contradiction. So
|xi (n) − xi (n)| ≤
|x∗i (n + 1) − xi (n + 1)|
+ θi (h)
m
where i = 1, 2, . . . , m, then we get Fi∗ (1) > 0. So there exists a constant λ0 > 1, such that Fi∗ (λ0 ) > 0, i = 1, 2, . . . , m. Then,
∗
Hence, by inequality above
∗
aij Lj −
j =1
j =1
ui (n + 1) ≤ λ
∗M
u∗i0 (n0 ) ≤ K ∗ ,
× [fj (x∗j (n)) − fj (xj (n))] m + θi∗ (h) b∗ij (n)[fj (x∗j (n − τij )) − fj (xj (n − τij ))] + δi (n).
+ θi∗ (h)
m
That is to say a∗ij (n)
j =1
≤ |x∗i (n) − xi (n)|e
∗
∗
j =1
m
n
K ∗ + λ00 |δi (n0 − 1)|.
∗
j =1
−θi∗ (h)
M
According to Theorem 1,
x i ( n)
j =1
−b∗i (n)h
τi0 j
b∗ij Lj λ
j =1
j =1 m
65
m
m
1 n
λ00
ε,
for any n > N , ε > 0. From Theorem 1, we have
|x∗i (n + 1) − xi (n + 1)| ≤ |x∗i (n) − xi (n)|e−bi (n)h + θi∗ (h) ∗
j =1
m
|a∗ij (n)|Lj |(x∗j (n)) − xj (n)|
j =1
m M ≤ λ0 − λ0 θi∗0 (h)b∗i + λ0 θi∗0 (h) a∗ij Lj j =1
+ θi∗ (h)
m j=1
|b∗ij (n)|Lj |x∗j (n − τij ) − xj (n − τij )| + |δi (n)|.
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J. Wang et al. / Neural Networks 53 (2014) 61–68
λn+1
Let u∗i (n + 1) = θ ∗ (h) |x∗i (n + 1) − xi (n + 1)|, n ∈ [−τ , +∞)Z , i then we can calculate the inequality as follows:
λn+1 ∗ |x∗i (n) − xi (n)|e−bi (n)h ui (n + 1) ≤ ∗ θi (h) m + θi∗ (h) |a∗ij (n)|Lj |(x∗j (n)) − xj (n)|
Case (ii). If there exists a constant n0 ≥ N such that M (n0 ) =
∥u∗ (n0 )∥. So we get 0 ≤ ∥∆u∗ (n0 )∥ =
∗
i=1
i =1
m
|b∗ij (n)|Lj |x∗j (n − τij ) − xj (n − τij )| + |δi (n)|
i=1 j=1 m m
+λ
b∗ji M θi∗ (h)Lj λτji u∗i (n0 − τji ) +
−b∗i (n)h ∗
ui (n) + λ
i =1
m
aij θj (h)Lj uj (n) ∗M ∗
≤ (λe
∗
−b∗i h
− 1) + λ max
j =1
+λ
m
b∗ij θj∗ (h)Lj λτij u∗j (n − τij ) + M
λn+1 |δi (n)|. θi∗ (h)
So, we get
△u∗i (n) =
m
i =1
u∗i (n) − u∗i (n)
i=1
i =1
+λ
i=1 j=1
b∗ij θj∗ (h)Lj λτij u∗j (n − τij ) + M
i=1 j=1
m λn+1 |δi (n)|. θ ∗ (h) i =1 i
Let M (n) = max ∥u∗ (s)∥ = max −τ ≤s≤n
−τ ≤s≤n
m
|u∗i (s)| .
i=1
It is obvious that ∥u (n)∥ ≤ M (n), and M (n) is non-decreasing. Now, we consider two cases: Case (i). Suppose M (n) > ∥u∗ (n)∥, for all n > N. Then, we claim that M (n) ≡ M (N ) is a constant for all n > N. By way of contradiction, we assume it does not hold. Consequently, there is a constant n1 > N such that M (N ) ̸= M (n1 ), that is to say, M (n1 ) > M (N ). In addition, we have ∗
M (n1 ) = max{ max ∥u∗ (s)∥, −τ ≤s≤N
max ∥u∗ (s)∥}
N
= max{M (N ), max ∥u∗ (s)∥} N
N
So there must exist β ∈ (N , n1 ]Z such that ∥u∗ (β)∥ = M (n1 ) ≥ M (β), which is a contradiction. Hence, M (n) ≡ M (N ), for all n ≥ N. That is to say, ∥u∗ (n)∥ ≤ M (N ). So, we have
+ λ max
λ |x∗ (n) − xi (n)| < M (N ). θi∗ (h) i Then,
τji
bji θi (h)Lj λ ∗M ∗
1
1
1
∥u∗ (n0 )∥ + ηε 2
2
2
If M (n0 ) > ε , then ∥∆u∗ (n0 )∥ < 0 which is a contradiction. So we have M (n0 ) < ε , that is to say, ∥∆u∗ (n0 )∥ < ε. Furthermore, if M (n) = ∥u∗ (n)∥ for any n > n0 , then ∗ ∥u (n0 )∥ < ε. If M (n) > ∥u∗ (n)∥, n > n0 , we can choose n0 ≤ n2 ≤ n such that M (n2 ) = ∥u∗ (n2 )∥, ∥u∗ (n2 )∥ < ε , and M (s) > ∥u∗ (s)∥, ∀s ∈ (n2 , n]Z . Using the same method in the proof of case (i), we can get M (s) = ∥u∗ (n2 )∥,
∀s ∈ [n3 , n]Z ,
∥u∗ (n)∥ < M (n) = M (n2 ) = ∥u∗ (n2 )∥ < ε. In summary, there must exist a constant N ∥u∗ (n)∥ < ε, for all n > N. Then,
|x∗i (k) − xi (k)| < M (N )
∞ |x∗ (k) − xi (k)| i
θi (h) ∗
> 0 such that
lim |x∗i (n) − xi (n)| = 0.
n→+∞
The proof is complete.
Remark 5. Obviously, system (4) is a delayed CNNs without periodic coefficients, therefore all the results in the Refs. Cao (2003), Dong, Ma, and Huang (2002), Huang, Cao, and Wang (2002), Lu and Chen (2004) and Liu and Liao (2004) are not applicable for proving that all solutions of system (4) converge to a periodic function. The results of this paper are essentially new.
xi (n + 1) = e−bi (n)h xi (n) + θi (h)
m
aij (n)fj (xj (n))
j =1
+ θi (h)
m
bij (n)fj (xj (n)) + θi (h)Ii (n),
( 3′ )
j =1 ∗ xi (n + 1) = e−bi (n)h xi (n) + θi∗ (h)
m
a∗ij (n)fj (xj (n))
j =1
1
. λk
m
+ θi∗ (h)
b∗ij (n)fj (xj (n)) + θi∗ (h)Ii∗ (n).
( 4′ )
j =1
Furthermore,
< M (N )
Thus, lim |x∗i (n) − xi (n)| = 0.
n→+∞
m j =1
n
k =1
aij θj (h)Lj
At last, we choose τij = 0, for any i, j = 1, . . . , m. System (3) and system (4) can be rewritten as follows:
= max ∥u∗ (s)∥.
θi∗ (h)
∗M ∗
which implies that
1
|δi (n0 )|
< − ηM (n0 ) + ηε.
m m m ∗ M (λe−bi (n)h − 1)u∗i (n) + λ a∗ij θj∗ (h)Lj u∗j (n) ≤ m m
m
θi∗ (h)
j=1
j =1
m
m λn0 +1
i=1 j=1
j =1
= λe
△u∗i (n0 )
m m m ∗ ≤ (λe−bi (n0 )h − 1)u∗i (n0 ) + λ a∗ji M θi∗ (h)Li u∗i (n0 )
j =1
+ θi∗ (h)
m
∞ 1 < ∞. λk k =1
Corollary 3. Assume that (H1 )–(H3 ), (H7 ) hold, and bi −
m
j=1
aM ij Lj −
> 0. Suppose that x(n) = (x1 (n), x2 (n), . . . , xm (n))T is the ω-periodic solution of system (3)′ . Then every solution x∗ (n) = (x∗1 (n), x∗2 (n), . . . , x∗m (n))T of system (4)′ with any initial conditions (5) converges to the ω-periodic solution x(n) of system (3).
m
M j=1 bij Lj
J. Wang et al. / Neural Networks 53 (2014) 61–68
67
Corollary 4. Assume H3 ), (H7 ) hold, and λe−bi h − m Mthat (H1 )–( m 2 1 + Li θi (h)λ j=1 aji + Li θi (h) j=1 bM < 0. Suppose that ji λ
x(n) = (x1 (n), x2 (n), . . . , xm (n))T is the ω-periodic solution of system (3)′ . Then every solution x∗ (n) = (x∗1 (n), x∗2 (n), . . . , x∗m (n))T of system (4)′ with any initial conditions (5) converges to the ωperiodic solution x(n) of system (3).
Remark 6. Note that (3)′ and (4)′ are discrete-time CNNs without delays. In Liu and Huang (2006), the convergent behavior of continue-time CNNs has been studied. However, as we know, from Corollaries 3 and 4, the discrete-time system has the same convergent behavior. Fig. 1. Time responses in Example 1.
5. Example In the following, we give two examples to demonstrate the results obtained in Theorems 3 and 4.
∗ x1 ( n + 1) = e−b1 (n)h x1 (n) + θ1∗ (h) 2 2 ∗ ∗ ∗ a1j (n)fj (xj (n)) + b1j (n)fj (xj (n − τ1j )) + I1 (n) × j=1
Example 1. Consider the following 2-dimension discrete delayed CNNs without periodic coefficients:
∗ x1 (n + 1) = e−b1 (n)h x1 (n) + θ1∗ (h) 2 2 ∗ ∗ ∗ a1j (n)fj (xj (n))+ b1j (n)fj (xj (n − τ1j )) + I1 (n) × j =1
j =1
, (6)
1 + 1+2nn2 , where b1 (n) = 8 + n+1 , a11 (n) = 41 + 1+nn2 , a∗12 (n) = 36 I1∗ (n) = 1 + 1n + sin( 51 n), b∗11 (n) = 1 + 1+nn2 , b∗12 (n) = 31 + 1+nn2 , 1
∗
5n , a∗21 (n) = 1 + 1+nn2 , a∗22 (n) = 41 + 1+ n2 1 1 1 1 1 1 ∗ ∗ ∗ I2 (n) = 1 + n + sin( 5 n), b21 (n) = 5 + n , b22 (n) = 2 + n2 , f1 (x) = 1+xx2 , f2 (x) = 2+xx2 , h = 0.2.
b∗2 (n) = 10 +
1 , 2n+1
Note the following CNNs:
x (n + 1) = e−b1 (n)h x (n) + θ (h) 1 1 1 2 2 a1j (n)fj (xj (n)) + b1j (n)fj (xj (n − τ1j )) + I1 (n) × j =1
j =1
(7)
j =1
(n) = sin( 51 n) + 1, ∗ b11 (n) = 1, b2 (n) = 10, a21 (n) = 1, b12 (n) = 13 , a22 (n) = 41 , I2 (n) = sin( 51 n) + 1, b21 (n) = 15 , b22 (n) = 21 , f1 (x) = 1+xx2 , f2 (x) = 2+xx2 , h = 0.2. 1 , 4
a12 (n) =
1 ,I 36 1
By further calculating, we have bi −
m j =1
aM ij Lj −
1 , n
m
∗
τij bM > 0, ij Lj λ
j =1
where i, j = 1, 2. Since system (6) and (7) satisfies (H1 )–(H4 ), (H7 ), hence, system (7) has the 10π -periodic solution. Moreover, all solutions of system (6) converge to the periodic solution of system (7). A numeric simulation for example is shown in Fig. 1, and the initial values are random. Example 2. Consider the following 2-dimension discrete delayed CNNs without periodic coefficients:
cos 0.2n, a∗12 (n) = ∗
1 36 n , 1+n2 1 , n2 x , 1 +x 2
+
Note the following CNNs:
x (n + 1) = e−b1 (n)h x (n) + θ (h) 1 1 1 2 2 a1j (n)fj (xj (n))+ b1j (n)fj (xj (n − τ1j )) + I1 (n) × j =1 j =1 , (9) x2 (n + 1) = e−b2 (n)h x2 (n) + θ2 (h) 2 2 a2j (n)fj (xj (n))+ b2j (n)fj (xj (n − τ2j )) + I2 (n) × j =1
1 , where b1 (n) = 4(sin 0.2n + 5), a11 (n) = 41 cos 0.2n, a12 (n) = 36 I1 (n) = 1, b11 (n) = 1 + cos 0.2n − sin n, b12 (n) = 1, b∗2 (n) = 5(cos 0.2n + 4), a21 (n) = 13 sin 0.2, a22 (n) = 17 , I2∗ (n) = 1 + 1n ,
b21 (n) = cos 0.2, b22 (n) = 13 , f1 (x) = 1+xx2 , f2 (x) = 2+xx2 , h = 0.2. By further calculating, we have
j =1
where b1 (n) = 8, a11 (n) =
1 4
(n) = 1 + b11 (n) = 1 + cos 0.2n − sin n, b12 (n) = 1 + b∗2 (n) = 5(cos 0.2n + 4), a∗21 (n) = 13 sin 0.2, a∗22 (n) = 17 + I2∗ (n) = 1 + 1n , b∗21 (n) = cos 0.2, b∗22 (n) = 13 + 1n , f1 (x) = f2 (x) = 2+xx2 , h = 0.2. 1 ∗ ,I n 1
j =1
x2 (n + 1) = e−b2 (n)h x2 (n) + θ2 (h) 2 2 a2j (n)fj (xj (n)) + b2j (n)fj (xj (n − τ2j )) + I2 (n) , ×
(8)
j =1
where b∗1 (n) = 4(sin 0.2n + 5), a∗11 (n) =
j =1
j =1
∗
∗
j =1
x2 (n + 1) = e−b2 (n)h x2 (n) + θ2∗ (h) 2 2 ∗ b∗2j (n)fj (xj (n − τ2j )) + I2∗ (n) a ( n ) f ( x ( n ))+ × j j 2j ∗
j =1
x2 (n + 1) = e−b2 (n)h x2 (n) + θ2∗ (h) 2 2 ∗ ∗ ∗ a2j (n)fj (xj (n)) + b2j (n)fj (xj (n − τ2j )) + I2 (n) , ×
λe−bi h − 1 + Li λ
m j =1
aM ji θj (h) + Li
m
τji +1 bM < 0, ji θj (h)λ
j =1
where i, j = 1, 2. Since system (8) and (9) satisfies (H1 )–(H3 ), (H5 )′ , (H7 ), hence, system (9) has the 10π -periodic solution. Moreover, all solutions of system (8) converge to the periodic solution of system (9). A numeric simulation for example is shown in Fig. 2, and the initial values are random. 6. Conclusions In this paper, several conditions have been obtained to ensure delayed discrete CNNs having a unique w -periodic solution by using the properties of inequalities which are different from the previous results. Consequently, we derive some sufficient conditions ensuring that all solutions of discrete delayed CNNs without periodic coefficients converge to a periodic function, by applying mathematical analysis techniques. The results are new and complement previously known results.
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J. Wang et al. / Neural Networks 53 (2014) 61–68
Fig. 2. Time responses in Example 2.
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Liu, B., & Huang, L. (2006). Existence and exponential stability of periodic solutions for cellular neural networks with time-varying delays. Physics Letters A, 349, 474–483. Liu, Z., & Liao, L. (2004). Existence and global exponential stability of periodic solutions of cellular neural networks with time-vary delays. Journal of Mathematical Analysis and Applications, 290(2), 247–262. Lu, W., & Chen, T. (2004). On periodic Dynamical systems. Chinese Annals of Mathematics B, 25, 455–462. Mohamad, S. (2008). Computer simulations of exponentially convergent networks with large impulses. Mathematics and Computers in Simulation, 77, 331–344. Mohamad, S., & Gopalsamy, K. (2000). Dynamics of a class of discrete-time neural networks and their continuous-time counterparts. Mathematics and Computers in Simulation, 53, 1–39. Mohamad, S., & Gopalsamy, K. (2003). Exponential stability of continuous-time and discrete-time cellular neural networks with delays. Applied Mathematics and Computation, 135(1), 17–38. Orman, Z. (2012). New sufficient conditions for global stability of neutral-type neural networks with time delays. Neurocomputing, 97, 141–148. Wang, Z., Liu, Y., & Liu, X. (2005). On global asymptotic stability of neural networks with discrete and distributed delays. Physics Letters A, 345, 299–308. Wang, X., Li, S., & Xu, D. (2011). Globally exponential stability of periodic solutions for impulsive neutral-type neural networks with delays. Nonlinear Dynamics, 64, 65–75. Wu, S., Li, K., & Huang, T. (2012). Global exponential stability of static neural networks with delay and impulses: Discrete-time case. Communications in Nonlinear Science and Numerical Simulation, 17, 3947–3960. Yang, W. (2012). Existence of an exponential periodic attractor of periodic solutions for general BAM neural networks with time-varying delays and impulses. Applied Mathematics and Computation, 219, 569–582. Yang, X., Cui, X., & Long, Y. (2009). Existence and global exponential stability of periodic solution of a cellular neural networks difference equation with delays and impulses. Neural Networks, 22, 970–976. Zhang, H., & Chen, L. (2008). Asymptotic behavior of discrete solutions to delayed neural networks with impulses. Neurocomputing, 71, 1032–1038. Zhang, Y., & Sun, J. (2005). Stability of impulsive neural networks with time delays. Physics Letters A, 348, 44–50. Zhou, J., Li, Q., & Zhang, F. (2008). Convergence behavior of delayed cellular neural networks without periodic coefficients. Applied Mathematics Letters, 21, 1012–1017.
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Neural Networks journal homepage: www.elsevier.com/locate/neunet
Convergence behavior of delayed discrete cellular neural network without periodic coefficients✩ Jinling Wang, Haijun Jiang ∗ , Cheng Hu, Tianlong Ma College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, Xinjiang, PR China
article
info
Article history: Received 5 May 2013 Received in revised form 28 October 2013 Accepted 16 January 2014 Keywords: Discrete cellular neural network Periodic solution Convergence Without periodic coefficient
abstract In this paper, we study convergence behaviors of delayed discrete cellular neural networks without periodic coefficients. Some sufficient conditions are derived to ensure all solutions of delayed discrete cellular neural network without periodic coefficients converge to a periodic function, by applying mathematical analysis techniques and the properties of inequalities. Finally, some examples showing the effectiveness of the provided criterion are given. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Since cellular neural networks (CNNs) were introduced by Chua and Roska (1990) in 1990, they have been successfully applied in signal and image processing, pattern recognition and optimization, especially in image processing and solving nonlinear algebraic equations. Hence, CNNs have been the aim of intensive analysis by numerous authors in recent years (Cao, 2003; Liu & Huang, 2006; Liu & Liao, 2004; Mohamad & Gopalsamy, 2000, 2003; Orman, 2012; Wang, Li, & Xu, 2011; Yang, 2012; Zhang & Chen, 2008; Zhang & Sun, 2005). In implementing the continuous-time network for computer simulation, experimental and computational purposes, it is usual to discretize the continuous-time network. Certainly, the discretetime analog inherits the dynamical characteristics of continuoustime networks under mild or no restriction on the discretization step-size, and also remains functional similarity to the continuoustime system. We refer to Mohamad (2008), Wu, Li, and Huang (2012) and Wang, Liu, and Liu (2005) for related discussions on the important and the need for discrete-time analogs to reflect the dynamics of their continuous-time counterparts. Recently, many authors have studied the dynamical behaviors of discrete-time analogs of CNNs with delays and variable coefficients. Li (2004) has investigated the stability and existence of periodic solutions of CNNs without periodic coefficients by using
✩ This was supported by the National Natural Science Foundation of People’s Republic of China (Grant NO. 61164004). ∗ Corresponding author. Tel.: +86 13079900716. E-mail address: [email protected] (H. Jiang).
http://dx.doi.org/10.1016/j.neunet.2014.01.007 0893-6080/© 2014 Elsevier Ltd. All rights reserved.
Mawhin’s continuation theorem of coincidence degree theory and by constructing Lyapunov functions. Zhou, Li, and Zhang (2008) have studied the convergence behaviors of delayed continuoustime cellular neural networks without periodic coefficients, and some sufficient conditions are established to ensure that all solutions of networks converge to a periodic function. Chen and Zhao (2009) have obtained discrete analog of highorder Cohen–Grossberg neural networks with time-varying delay by analysis and approximation techniques, and also sufficient conditions are given to guarantee global exponential stability of the periodic solution. Yang, Cui, and Long (2009) have obtained global exponential stability of periodic solution of a cellular neural network difference equation with delays and impulses by using contraction mapping theorem and inequality techniques. Hence, most authors of the bibliographies listed above obtained that all solutions of continue-time delayed CNNs without periodic coefficients converge to a periodic function and existence of periodic solutions of delayed discrete CNNs with periodic coefficients. However, we see, for delayed discrete CNNs without periodic coefficients, up until now, study works are very few. In this paper, our main purpose is to investigate the convergence behavior of delayed discrete CNNs without periodic coefficients. First, we give several conditions to ensure delayed discrete CNNs having a unique periodic solution. Then, we derive some sufficient conditions ensuring that all solutions of delayed discrete CNNs without periodic coefficients converge to a periodic function, by applying mathematical analysis techniques. The results are new and complement previously known results. The remainder of this paper is organized as follows. In Section 2, we introduce the discrete-time analogs discussed in
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J. Wang et al. / Neural Networks 53 (2014) 61–68
this paper. In Section 3, some sufficient conditions are given to ensure the delayed discrete CNNs having a unique periodic solution. In Section 4, sufficient conditions are given to ensure that all solutions of the delayed discrete CNNs without periodic coefficients converge to a periodic function. Moreover, in Section 5, some examples are given to show the usefulness of our results. At last, Section 6 concludes the work of this paper. 2. Model description We consider a continuous-time cellular neural network consisting of m interconnected cells described by a system of delay differential equations of the form dxi (t ) dt
= −bi (t )xi (t ) +
m
aij (t )fj (xj (t ))
j =1
+
m
(1)
for i ∈ L = {1, 2, . . . , m}, where bi (t ) > 0, aij (t ), bij (t ), Ii (t ) ∈ R, vij > 0. Respectively, m corresponds to the number of units in a neural network; xi (t ) corresponds to the state of the ith unit at time t; fj (xj (t )) denotes the output of the jth unit on the ith unit at time t; aij (t ), bij (t ) denote the strength of the jth unit on the ith unit at time t; Ii (t ) denotes the external bias on the ith unit at time t; vij denotes the transmission delay of the ith unit along the axon of the jth unit at the time t; bi (t ) represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs. Then, we shall use a semi-discretization technique to obtain a discrete-time system of continuous-time cellular neural network system (1). For convenience in our study, we use the following notations: Let Z denote the set of all integers, i.e. Z = {. . . , −1, 0, 1, . . .}. Let Z + = {1, 2, . . .} and Z0+ = {0, 1, 2, . . .}. For c , d ∈ Z and c ≤ d we denote the discrete interval [c , d]Z = {c , c + 1, . . . , d − 1, d}. If d = ∞, then [c , ∞)Z = {c , c + 1, . . .}. While there is no unique way in obtaining a discrete-time analog of continuous-time network (1), such as Euler scheme, Runge–Kutta scheme, we begin by reformulating network (1) with the following approximation:
dt
t
= −bi +
h xi (t ) +
h
m
m
t
aij
h
j =1
h fj
t
bij
h fj
h
j =1
xj
t
h
h−
xj
t
h
h
v ij
h
h
+ Ii
t
h
h ,
(2)
for i ∈ L, t ∈ [nh, (n + 1)h), n ∈ Z + , where h is a fixed positive real number denoting a uniform discretization step-size and [r ] denotes the integer part of the real number r. Clearly, for t ∈ v [nh, (n + 1)h), n ∈ Z + , we have [ ht ] = n. We let [ hij ] = τij and we have τij ∈ Z + because of vij ≥ 0. For convenience in the following we use the notation xi (nh) = xi (n). Then, we can integrate (3) over [nh, t ) and by allowing t → (n + 1)h in the above, we obtain after some simplification that xi (n + 1) = e−bi (n)h xi (n) + θi (h)
m
j =1
1≤i,j≤m
(H1 ) h ∈ (0, ∞), bi (n) : Z → (0, ∞), aij (n), bij (n), Ii (n) : Z → R, τij ∈ Z + , i, j = 1, 2, . . . , m. (H2 ) fi (·) ∈ C (R, R) is bounded on R, and ∃Li > 0, s.t .|fi (x) − fi (y)| ≤ Li |x − y|. (H3 ) bi (n), aij (n), bij (n), Ii (n) are all ω-periodic functions, where ω In the following, for convenience, we will use the following notations. aM ij = max |aij (n)|, 1≤i,j≤m
bi = min |bi (n)|, 1≤i≤m
bij (n)fj (xj (n − τij )) + θi (h)Ii (n),
bM ij = max |bij (n)|, 1≤i,j≤m
bi = max |bi (n)|. 1≤i≤m
3. Existence of periodic solution In this section, we give several conditions to ensure system (3) having a unique w -periodic solution. Lemma 1. Assume that (H1 )–(H3 ) hold. Then system (3) has at least one w -periodic solution. The result of Lemma 1 was given by Theorem 3.1 in Li (2004). Theorem 1. Assume that (H1 )–(H3 ) hold. Furthermore, assume that (H4 ) there exists a constant λ > 1 such that bi −
m
aM ij Lj −
j =1
m
τij bM > 0. ij Lj λ
j=1
Then system (3) has a unique w -periodic solution. Proof. It follows from Lemma 1 that system (3) has an ω-periodic solution, x(n) = (x1 (n), x2 (n), . . . , xm (n))T . Obviously, if this periodic solution is global attractive, then it is unique. Let x(n) = (x1 (n), x2 (n), . . . , xm (n))T be an arbitrary solution of system (3). From system (3), we can get xi (n + 1) − xi (n + 1) = (xi (n) − xi (n))e−bi (n)h
+ θi (h)
m
aij (n)[fj (xj (n)) − fj (xj (n))]
j =1
+ θi (h)
m
bij (n)[fj (xj (n − τij )) − fj (xj (n − τij ))].
j =1
Thus, by equality above we obtain
|xi (n + 1) − xi (n + 1)| ≤ |xi (n) − xi (n)|e−bi (n)h m + θi (h) aM ij Lj |(xj (n)) − xj (n)| j =1
j =1
τ = max {τij },
where ϕi (·) is a real-valued continuous function defined on [τ , 0]Z . In this paper, for system (3) we introduce the following assumptions.
aij (n)fj (xj (n))
+ θi (h)
m
+ θi (h)
xi (s) = ϕi (s), s ∈ [−τ , 0]Z ,
is a positive integer.
bij (t )fj (xj (t − vij )) + Ii (t ),
j =1
dxi (t )
−bi (n)h for i ∈ L, n ∈ Z0+ , where θi (h) = 1−eb (n) , i ∈ L. It is not difficult i to verify that θi (h) > 0 if bi > 0, h > 0 and θi (h) ≈ h + O(h2 ) for small h > 0. System (3) is discrete-time analog of continuous-time system (1). One can show that discrete-time system (3) converges towards continuous-time system (1) when h → 0+ . System (3) is supplemented with initial conditions given by
(3)
m
bM ij Lj |xj (n − τij ) − xj (n − τij )|.
j =1
Define function
J. Wang et al. / Neural Networks 53 (2014) 61–68
m
Fi (λ) = 1 − λ + λθi (h) bi −
aM ij Lj −
m
j =1
τij bM ij Lj λ
,
j =1
where i = 1, 2, . . . , m, then from condition (H4 ) and θi (h) > 0, we have Fi (1) > 0. So there exists a constant λ0 > 1, such that Fi (λ0 ) > 0, i = 1, 2, . . . , m. Let ui (n) = λn0 |xi (n) − xi (n)|, n ∈ [−τ , +∞)Z , then ui (n + 1) ≤ λ
n +1 0
|xi (n) − xi (n)|e−bi (n)h
+ θi (h)
m
aM ij Lj |(xj (n)) − xj (n)|
λe−bi h − 1 + Li λ
bM ij Lj
|xj (n − τij ) − xj (n − τij )|
j=1 m
= λ0 e−bi (n)h ui (n) + λ0 θi (h)
aM ij Lj uj (n)
j =1 m
τij bM ij Lj λ uj (n − τij ).
Let i=1,...,m s∈[−τ ,0] Z
aM ji θj (h) + Li
m
|xi (s) − xi (s)|}.
ui (n) ≤ K , n ∈ [−τ , n0 ]Z ,
Proof. It follows from Lemma 1 that system (3) has an ω-periodic solution, x(n) = (x1 (n), x2 (n), . . . , xm (n))T . Obviously, if this periodic solution is global attractive, then it is unique. Let x(n) = (x1 (n), x2 (n), . . . , xm (n))T be an arbitrary solution of system (3). From system (3), we can get
−bi0 (n0 −1)h
ui0 (n0 ) > K .
aij (n)[fj (xj (n)) − fj (xj (n))]
+ θi (h)
m
bij (n)[fj (xj (n − τij )) − fj (xj (n − τij ))].
j=1
|xi (n + 1) − xi (n + 1)| ≤ |xi (n) − xi (n)|e−bi (n)h + θi (h)
ui (n0 − 1)
m
m
By directly calculating,
By further calculating, we have
m
aM ij Lj |(xj (n)) − xj (n)|
j =1
aM ij Lj uj (n0 − 1)
+ θi (h)
j=1
+ λ0 θi0 (h)
j =1
j=1
i ̸= i0 ;
ui0 (n) ≤ K , n ∈ [−τ , n0 − 1]Z ;
+ λ0 θi0 (h)
τji +1 bM < −η < 0. ji θj (h)λ
Then system (3) has a unique w -periodic solution.
+ θi (h)
Then we claim that ui (n) ≤ K , n ∈ Z0+ , i = 1, . . . , m. Otherwise, we can choose constants i0 ∈ {i = 1, . . . , m}, n0 ∈ Z0+ such that
K < ui0 (n0 ) ≤ λ0 e
m
xi (n + 1) − xi (n + 1) = (xi (n) − xi (n))e−bi (n)h
j =1
K = max { sup
Corollary 1. Assume that (H1 )–(H3 ) and (H4 )′ hold. Then system (3) has a unique w -periodic solution.
j =1
m
+ λ0 θi (h)
However, in Theorem 1, we get the same result by using the properties of inequalities. Besides, the condition (H4 ) in Theorem 1 is different from the condition in Ref. Li (2004), and it is the key to obtain the convergence of the solution.
Theorem 2. Assume that (H1 )–(H3 ) hold. Furthermore, assume that (H5 ) there exist constants η > 0, λ > 1 such that
j=1
+ θi (h)
63
m
bM ij Lj |xj (n − τij ) − xj (n − τij )|.
j=1
m
τij bM ij Lj λ uj (n0 − 1 − τi0 j )
j =1
m = λ0 − λ0 θi0 (h)bi + λ0 θi0 (h) aM ij Lj
Let ui (n) = λn |xi (n) − xi (n)|, n ∈ [−τ , +∞)Z , then we can easily obtain
ui ( n + 1 ) ≤ λ
n +1
|xi (n) − xi (n)|e−bi (n)h
j =1
+ θi0 (h)
m
τi 0 j bM ij Lj
λ
+ θi (h)
K
j =1
which is a contradiction. So ui (n) ≤ K , n ∈ Z0+ , i = 1, . . . , m. Hence,
|xi (n) − xi (n)| ≤
1
λ0
n max { sup
i=1,...,m s∈[−τ ,0] Z
|xi (s) − xi (s)|}.
Therefore, the ω-periodic solution of (3) is unique. The proof is completed. |x (n)−x (n)|
Remark 1. If we choose ui (n) = λn0 i θ (h)i i changed into (H4 )′ there exists a constant λ > 1 such that bi θi (h) −
m j=1
aM ij Lj θj (h) −
m
aM ij Lj |(xj (n)) − xj (n)|
j=1
< K,
m
, then (H4 ) can be
τij bM ij Lj λ θj (h) > 0.
j =1
Moreover, in Ref. Li (2004), the author obtained system (3) had a unique w -periodic solution by applying Lyapunov function.
+ θi (h)
m
bM ij Lj |xj (n − τij ) − xj (n − τij )|
j =1
= λe−bi (n)h ui (n) + λθi (h)
m
aM ij Lj uj (n)
j=1
+ λθi (h)
m
τij bM ij Lj λ uj (n − τij ).
j =1
Now, consider the following Lyapunov function V (n) =
m i=1
ui (n) +
m
τij+1 bM i ij Lj
j=1
λ
θ (h)
n−1
uj (s) .
s=n−τij
By applying conditions (H3 ) and (H5 ), we can calculate the difference ∆V (n) = V (n + 1) − V (n) as follows:
∆V (n) = V (n + 1) − V (n)
64
J. Wang et al. / Neural Networks 53 (2014) 61–68 m
=
m
ui (n + 1) +
i =1
−
ui (n) −
m m
n
τij+1 θi (h) bM ij Lj λ
4. Convergence
uj ( s )
s=n+1−τij
i =1 j =1
m i =1
≤
m
n−1
τij+1 θi (h) bM ij Lj λ
In this section, we obtain sufficient conditions to ensure that all solutions of system without periodic coefficients converge to a periodic function. Now, we consider the following delayed discrete cellular neural networks without periodic coefficients:
uj (s)
s=n−τij
i =1 j =1
m m m (λe−bi h − 1)ui (n) + λ aM ij Lj θi (h)uj (n) i =1
+
∗ xi (n + 1) = e−bi (n)h xi (n) + θi∗ (h)
i=1 j=1
m m
θ (h)uj (n − τij )
m
+ θi∗ (h)
i =1 j =1 m
+
m
τij+1 θi (h)(uj (n) − uj (n − τij )) bM ij Lj λ
λe−bi h − 1 + Li λ
i =1
+ Li
m
x∗i (s) = ϕi∗ (s), s ∈ [−τ , 0]Z ,
aM ji θj (h)
j =1 m
θ (h)λ
τji +1
ui (n)
< 0.
n→+∞
m
n→+∞
Further, ui (n) ≤
i =1
m
ui (0) +
i =1
1 + Li
i=1
m
τij+1 bM i ij Lj
λ
θ (h)
a∗ij
uj ( s )
M
= max |a∗ij (n)|,
b∗ij
1≤i,j≤m
bi = min |bi (n)|,
lim (Ii (n) − Ii∗ (n)) = 0.
n→+∞
M
= max |b∗ij (n)|, 1≤i,j≤m
bi = max |b∗i (n)|.
∗
∗
s=−τij
j =1 m
−1
∗
1≤i≤m
1≤i≤m
τji+1 bM τji ji θj (h)λ
Theorem 3. Assume that (H1 )–(H4 ) and (H7 ) hold, and x(n) = (x1 (n), x2 (n), . . . , xm (n))T is the ω-periodic solution of system (3). Then every solution x∗ (n) = (x∗1 (n), x∗2 (n), . . . , x∗m (n))T of system (4) with any initial conditions (5) converges to the ω-periodic solution x(n) of system (3). In simple terms,
j =1
sup {|xi (s) − xi (s)|}.
s∈[−τ ,0]Z
That is m
lim (aij (n) − a∗ij (n)) = 0,
n→+∞
In the following, for convenience, we will use the following notations.
i =1
×
1≤i,j≤m
lim (bij (n) − b∗ij (n)) = 0,
ui (n) ≤ V (n) ≤ V (0).
≤
τ = max sup{τij (n), n ∈ Z }, (5)
lim (bi (n) − b∗i (n)) = 0,
So we have
m
(4)
where ϕi (·) is a real-valued continuous function defined on [τ , 0]Z . We assume that coefficients bi (n), aij (n), bij (n), Ii (n), b∗i (n), ∗ aij (n), b∗ij (n), Ii∗ (n) satisfy the following conditions. (H7 )
bM ji j
j=1
m
b∗ij (n)fj (xj (n − τij )) + θi∗ (h)Ii∗ (n),
where b∗i (n), a∗ij (n), b∗ij (n), Ii∗ (n) are not ω-periodic functions. System (4) is supplemented with initial conditions given by
i =1 j =1
=
j =1
m
a∗ij (n)fj (xj (n))
j =1
τij+1 bM i ij Lj
λ
m
|xi (n) − xi (n)| ≤
i =1
1
λ
max
×
sup
n i=1,...,m
s∈[−τ ,0]Z
1 + Li
m
θ (h)λ
bM ji j
τji+1
lim |x∗i (n) − xi (n)| = 0.
τji
n→+∞
j =1
|xi (s) − xi (s)|.
Proof. Let ∗
Therefore, the ω-periodic solution of (3) is unique. The proof is complete. Remark 2. If we choose ui (n) = λn0
|xi (n)−xi (n)| . θi (h)
j =1
Then (H5 ) can be
changed into (H5 ) there exist constants η > 0, λ > 1 such that ′
λe−bi h − 1 + Li θi (h)λ
m
δi (n) = (e−bi (n)h − e−bi (n)h )xi (n) m m ∗ ∗ + θi (h) aij − θi (h) aij fj (xj (n))
aM ji + Li θi (h)
m
θi (h) ∗
+
j =1
m
bij − θi (h) ∗
j =1
m
bij
fj (xj (n − τij ))
j =1
+ θi∗ (h)Ii∗ (n) − θi (h)Ii (n),
τji +1 bM < −η < 0. ji λ
Moreover, we can get the same result under the condition of (H5 ) .
where i = 1, 2, . . . , m. According to conditions (H4 ) and (H7 ), we can choose constants N > 0, λ0 > 1, n0 ∈ Z such that
Corollary 2. Assume that (H1 )–(H3 ) and (H5 )′ hold. Then system (3) has a unique w -periodic solution.
|δi (n)| ≤
j =1
j =1
′
Remark 3. In fact, there are some differences between Theorems 1 and 2. In Theorem 1, by (H4 ), the result is obtained through the inequalities approach. In Theorem 2, by (H5 ), the result is obtained by applying Lyapunov function. Hence, we can find that different methods need different conditions to obtain the result.
1 n
λ00
ε,
for any n > N , ε > 0. From system (4), x∗i (n + 1) − xi (n + 1)
= e−bi (n)h x∗i (n) + θi∗ (h) ∗
m j =1
a∗ij fj (x∗j (n)) + θi∗ (h)
m j =1
b∗ij fj (x∗j (n
J. Wang et al. / Neural Networks 53 (2014) 61–68
− τij )) + θi∗ (h)Ii∗ (n) − e−bi (n)h xi (n) − θi (h)
+ θi∗0 (h)
aij fj (xj (n))
− θi (h)
bij fj (xj (n − τij )) − θi (h)Ii (n) − e
−b∗ (n)h i
+e
xi (n) − θi (h) ∗
m
aij fj (xj (n)) + θi (h) ∗
∗
m
Fi (λ) = 1 − λ + λθi (h) bi − aij fj (xj (n))
m
b∗ij fj (xj (n − τij )) + θi∗ (h)
j =1
b∗ij fj (xj (n − τij ))
j =1
= (x∗i (n) − xi (n))e−bi (n)h + θi∗ (h) ∗
m
∗
u∗i0 (n0 ) < K ∗ + ε,
i
+ θi∗ (h)
m
|a∗ij (n)|Lj |(x∗j (n)) − xj (n)|
j =1
|b∗ij (n)|Lj |x∗j (n − τij ) − xj (n − τij )| + |δi (n)|.
j =1
Let u∗i (n) = λn0 |x∗i (n) − xi (n)|, n ∈ [−τ , +∞)Z , then we can calculate the inequality as follows: n +1 0
m
|xi (n) − x∗i (n)|e−bi (n)h ∗
aij Lj |(xj (n)) − xj (n)| ∗M
∗
j =1 m
b∗ij Lj |x∗j (n − τij ) − xj (n − τij )| + |δi (n)| M
= λ0 e−bi (n)h u∗i (n) + λ0 θi∗ (h) ∗
ui ( n) ≤ K ∗ ,
n ∈ Z0+ , i = 1, . . . , m.
Hence,
a∗ij Lj u∗j (n) + λ0 θi∗ (h) M
1
n max { sup
λ0
i=1,...,m s∈[−τ ,0] Z
|x∗i (s) − xi (s)|}.
Furthermore, we get lim |x∗i (n) − xi (n)| = 0.
The proof is complete.
Remark 4. In Theorem 3, we consider the convergent behavior for discrete-time delayed CNNs. However, in Zhou et al. (2008), the author study the convergent behavior for continue-time delayed CNNs. Hence, as we know, the discrete-time system has the same convergent behavior with the continuous-time system. Theorem 4. Assume that (H1 )–(H3 ), (H5 )′ and (H7 ) hold, and x(n) = (x1 (n), x2 (n), . . . , xm (n))T is the ω-periodic solution of system (3). Then every solution x∗ (n) = (x∗1 (n), x∗2 (n), . . . , x∗m (n))T of system (4) with any initial conditions (5) converges to the ωperiodic solution x(n) of system (3). In simple terms,
Proof. Let ∗
M
j =1
j=1
Next, we can set ∗
K = max { sup
i=1,...,m s∈[−τ ,0] Z
|xi (s) − xi (s)|}. ∗
j =1
m
θi (h) ∗
+
bij − θi (h) ∗
j=1
Then we claim that u∗i (n) ≤ K ∗ , n ∈ Z0+ , i = 1, . . . , m. Otherwise, there exist some constants i0 ∈ {i = 1, . . . , m}, n0 ∈ Z0+ such that u∗i (n) ≤ K ∗ ,
n ∈ [−τ , n0 ]Z , i ̸= i0 ;
u∗i0 (n) ≤ K ∗ ,
n ∈ [−τ , n0 − 1]Z ;
u∗i0 (n0 ) > K ∗ .
−b∗i (n0 −1)h ∗ 0
ui0 (n0 ) ≤ λ0 e
ui (n0 − 1) + λ0 θi0 (h) ∗
m
aij Lj uj (n0 − 1) ∗M
∗
j =1
b∗ij Lj λτij u∗j (n0 − 1 − τi0 j ) + λ00 |δi (n0 − 1)| n
M
m
bij
fj (xj (n − τij ))
j =1
+ θi∗ (h)Ii∗ (n) − θi (h)Ii (n). According to conditions (H5 )′ and (H7 ), we can choose constants N > 0, λ0 > 1, n0 ∈ Z such that
|δi (n)| ≤
Then, we obtain
m
δi (n) = (e−bi (n)h − e−bi (n)h )xi (n) m m ∗ ∗ + θi (h) aij − θi (h) aij fj (xj (n))
b∗ij Lj λτij u∗j (n − τij ) + λn0+1 |δi (n)|.
+ λ0 θi∗0 (h)
n→+∞
m
j =1
∗
∀ε > 0.
lim |x∗i (n) − xi (n)| = 0.
j =1
×
,
j =1
n→+∞
m
m
bij Lj λ
τij
i = 1, 2, . . . , m.
−b∗ (n)h
+ θi∗ (h)
∗M
which is a contradiction. So
|xi (n) − xi (n)| ≤
|x∗i (n + 1) − xi (n + 1)|
+ θi (h)
m
where i = 1, 2, . . . , m, then we get Fi∗ (1) > 0. So there exists a constant λ0 > 1, such that Fi∗ (λ0 ) > 0, i = 1, 2, . . . , m. Then,
∗
Hence, by inequality above
∗
aij Lj −
j =1
j =1
ui (n + 1) ≤ λ
∗M
u∗i0 (n0 ) ≤ K ∗ ,
× [fj (x∗j (n)) − fj (xj (n))] m + θi∗ (h) b∗ij (n)[fj (x∗j (n − τij )) − fj (xj (n − τij ))] + δi (n).
+ θi∗ (h)
m
That is to say a∗ij (n)
j =1
≤ |x∗i (n) − xi (n)|e
∗
∗
j =1
m
n
K ∗ + λ00 |δi (n0 − 1)|.
∗
j =1
−θi∗ (h)
M
According to Theorem 1,
x i ( n)
j =1
−b∗i (n)h
τi0 j
b∗ij Lj λ
j =1
j =1 m
65
m
m
1 n
λ00
ε,
for any n > N , ε > 0. From Theorem 1, we have
|x∗i (n + 1) − xi (n + 1)| ≤ |x∗i (n) − xi (n)|e−bi (n)h + θi∗ (h) ∗
j =1
m
|a∗ij (n)|Lj |(x∗j (n)) − xj (n)|
j =1
m M ≤ λ0 − λ0 θi∗0 (h)b∗i + λ0 θi∗0 (h) a∗ij Lj j =1
+ θi∗ (h)
m j=1
|b∗ij (n)|Lj |x∗j (n − τij ) − xj (n − τij )| + |δi (n)|.
66
J. Wang et al. / Neural Networks 53 (2014) 61–68
λn+1
Let u∗i (n + 1) = θ ∗ (h) |x∗i (n + 1) − xi (n + 1)|, n ∈ [−τ , +∞)Z , i then we can calculate the inequality as follows:
λn+1 ∗ |x∗i (n) − xi (n)|e−bi (n)h ui (n + 1) ≤ ∗ θi (h) m + θi∗ (h) |a∗ij (n)|Lj |(x∗j (n)) − xj (n)|
Case (ii). If there exists a constant n0 ≥ N such that M (n0 ) =
∥u∗ (n0 )∥. So we get 0 ≤ ∥∆u∗ (n0 )∥ =
∗
i=1
i =1
m
|b∗ij (n)|Lj |x∗j (n − τij ) − xj (n − τij )| + |δi (n)|
i=1 j=1 m m
+λ
b∗ji M θi∗ (h)Lj λτji u∗i (n0 − τji ) +
−b∗i (n)h ∗
ui (n) + λ
i =1
m
aij θj (h)Lj uj (n) ∗M ∗
≤ (λe
∗
−b∗i h
− 1) + λ max
j =1
+λ
m
b∗ij θj∗ (h)Lj λτij u∗j (n − τij ) + M
λn+1 |δi (n)|. θi∗ (h)
So, we get
△u∗i (n) =
m
i =1
u∗i (n) − u∗i (n)
i=1
i =1
+λ
i=1 j=1
b∗ij θj∗ (h)Lj λτij u∗j (n − τij ) + M
i=1 j=1
m λn+1 |δi (n)|. θ ∗ (h) i =1 i
Let M (n) = max ∥u∗ (s)∥ = max −τ ≤s≤n
−τ ≤s≤n
m
|u∗i (s)| .
i=1
It is obvious that ∥u (n)∥ ≤ M (n), and M (n) is non-decreasing. Now, we consider two cases: Case (i). Suppose M (n) > ∥u∗ (n)∥, for all n > N. Then, we claim that M (n) ≡ M (N ) is a constant for all n > N. By way of contradiction, we assume it does not hold. Consequently, there is a constant n1 > N such that M (N ) ̸= M (n1 ), that is to say, M (n1 ) > M (N ). In addition, we have ∗
M (n1 ) = max{ max ∥u∗ (s)∥, −τ ≤s≤N
max ∥u∗ (s)∥}
N
= max{M (N ), max ∥u∗ (s)∥} N
N
So there must exist β ∈ (N , n1 ]Z such that ∥u∗ (β)∥ = M (n1 ) ≥ M (β), which is a contradiction. Hence, M (n) ≡ M (N ), for all n ≥ N. That is to say, ∥u∗ (n)∥ ≤ M (N ). So, we have
+ λ max
λ |x∗ (n) − xi (n)| < M (N ). θi∗ (h) i Then,
τji
bji θi (h)Lj λ ∗M ∗
1
1
1
∥u∗ (n0 )∥ + ηε 2
2
2
If M (n0 ) > ε , then ∥∆u∗ (n0 )∥ < 0 which is a contradiction. So we have M (n0 ) < ε , that is to say, ∥∆u∗ (n0 )∥ < ε. Furthermore, if M (n) = ∥u∗ (n)∥ for any n > n0 , then ∗ ∥u (n0 )∥ < ε. If M (n) > ∥u∗ (n)∥, n > n0 , we can choose n0 ≤ n2 ≤ n such that M (n2 ) = ∥u∗ (n2 )∥, ∥u∗ (n2 )∥ < ε , and M (s) > ∥u∗ (s)∥, ∀s ∈ (n2 , n]Z . Using the same method in the proof of case (i), we can get M (s) = ∥u∗ (n2 )∥,
∀s ∈ [n3 , n]Z ,
∥u∗ (n)∥ < M (n) = M (n2 ) = ∥u∗ (n2 )∥ < ε. In summary, there must exist a constant N ∥u∗ (n)∥ < ε, for all n > N. Then,
|x∗i (k) − xi (k)| < M (N )
∞ |x∗ (k) − xi (k)| i
θi (h) ∗
> 0 such that
lim |x∗i (n) − xi (n)| = 0.
n→+∞
The proof is complete.
Remark 5. Obviously, system (4) is a delayed CNNs without periodic coefficients, therefore all the results in the Refs. Cao (2003), Dong, Ma, and Huang (2002), Huang, Cao, and Wang (2002), Lu and Chen (2004) and Liu and Liao (2004) are not applicable for proving that all solutions of system (4) converge to a periodic function. The results of this paper are essentially new.
xi (n + 1) = e−bi (n)h xi (n) + θi (h)
m
aij (n)fj (xj (n))
j =1
+ θi (h)
m
bij (n)fj (xj (n)) + θi (h)Ii (n),
( 3′ )
j =1 ∗ xi (n + 1) = e−bi (n)h xi (n) + θi∗ (h)
m
a∗ij (n)fj (xj (n))
j =1
1
. λk
m
+ θi∗ (h)
b∗ij (n)fj (xj (n)) + θi∗ (h)Ii∗ (n).
( 4′ )
j =1
Furthermore,
< M (N )
Thus, lim |x∗i (n) − xi (n)| = 0.
n→+∞
m j =1
n
k =1
aij θj (h)Lj
At last, we choose τij = 0, for any i, j = 1, . . . , m. System (3) and system (4) can be rewritten as follows:
= max ∥u∗ (s)∥.
θi∗ (h)
∗M ∗
which implies that
1
|δi (n0 )|
< − ηM (n0 ) + ηε.
m m m ∗ M (λe−bi (n)h − 1)u∗i (n) + λ a∗ij θj∗ (h)Lj u∗j (n) ≤ m m
m
θi∗ (h)
j=1
j =1
m
m λn0 +1
i=1 j=1
j =1
= λe
△u∗i (n0 )
m m m ∗ ≤ (λe−bi (n0 )h − 1)u∗i (n0 ) + λ a∗ji M θi∗ (h)Li u∗i (n0 )
j =1
+ θi∗ (h)
m
∞ 1 < ∞. λk k =1
Corollary 3. Assume that (H1 )–(H3 ), (H7 ) hold, and bi −
m
j=1
aM ij Lj −
> 0. Suppose that x(n) = (x1 (n), x2 (n), . . . , xm (n))T is the ω-periodic solution of system (3)′ . Then every solution x∗ (n) = (x∗1 (n), x∗2 (n), . . . , x∗m (n))T of system (4)′ with any initial conditions (5) converges to the ω-periodic solution x(n) of system (3).
m
M j=1 bij Lj
J. Wang et al. / Neural Networks 53 (2014) 61–68
67
Corollary 4. Assume H3 ), (H7 ) hold, and λe−bi h − m Mthat (H1 )–( m 2 1 + Li θi (h)λ j=1 aji + Li θi (h) j=1 bM < 0. Suppose that ji λ
x(n) = (x1 (n), x2 (n), . . . , xm (n))T is the ω-periodic solution of system (3)′ . Then every solution x∗ (n) = (x∗1 (n), x∗2 (n), . . . , x∗m (n))T of system (4)′ with any initial conditions (5) converges to the ωperiodic solution x(n) of system (3).
Remark 6. Note that (3)′ and (4)′ are discrete-time CNNs without delays. In Liu and Huang (2006), the convergent behavior of continue-time CNNs has been studied. However, as we know, from Corollaries 3 and 4, the discrete-time system has the same convergent behavior. Fig. 1. Time responses in Example 1.
5. Example In the following, we give two examples to demonstrate the results obtained in Theorems 3 and 4.
∗ x1 ( n + 1) = e−b1 (n)h x1 (n) + θ1∗ (h) 2 2 ∗ ∗ ∗ a1j (n)fj (xj (n)) + b1j (n)fj (xj (n − τ1j )) + I1 (n) × j=1
Example 1. Consider the following 2-dimension discrete delayed CNNs without periodic coefficients:
∗ x1 (n + 1) = e−b1 (n)h x1 (n) + θ1∗ (h) 2 2 ∗ ∗ ∗ a1j (n)fj (xj (n))+ b1j (n)fj (xj (n − τ1j )) + I1 (n) × j =1
j =1
, (6)
1 + 1+2nn2 , where b1 (n) = 8 + n+1 , a11 (n) = 41 + 1+nn2 , a∗12 (n) = 36 I1∗ (n) = 1 + 1n + sin( 51 n), b∗11 (n) = 1 + 1+nn2 , b∗12 (n) = 31 + 1+nn2 , 1
∗
5n , a∗21 (n) = 1 + 1+nn2 , a∗22 (n) = 41 + 1+ n2 1 1 1 1 1 1 ∗ ∗ ∗ I2 (n) = 1 + n + sin( 5 n), b21 (n) = 5 + n , b22 (n) = 2 + n2 , f1 (x) = 1+xx2 , f2 (x) = 2+xx2 , h = 0.2.
b∗2 (n) = 10 +
1 , 2n+1
Note the following CNNs:
x (n + 1) = e−b1 (n)h x (n) + θ (h) 1 1 1 2 2 a1j (n)fj (xj (n)) + b1j (n)fj (xj (n − τ1j )) + I1 (n) × j =1
j =1
(7)
j =1
(n) = sin( 51 n) + 1, ∗ b11 (n) = 1, b2 (n) = 10, a21 (n) = 1, b12 (n) = 13 , a22 (n) = 41 , I2 (n) = sin( 51 n) + 1, b21 (n) = 15 , b22 (n) = 21 , f1 (x) = 1+xx2 , f2 (x) = 2+xx2 , h = 0.2. 1 , 4
a12 (n) =
1 ,I 36 1
By further calculating, we have bi −
m j =1
aM ij Lj −
1 , n
m
∗
τij bM > 0, ij Lj λ
j =1
where i, j = 1, 2. Since system (6) and (7) satisfies (H1 )–(H4 ), (H7 ), hence, system (7) has the 10π -periodic solution. Moreover, all solutions of system (6) converge to the periodic solution of system (7). A numeric simulation for example is shown in Fig. 1, and the initial values are random. Example 2. Consider the following 2-dimension discrete delayed CNNs without periodic coefficients:
cos 0.2n, a∗12 (n) = ∗
1 36 n , 1+n2 1 , n2 x , 1 +x 2
+
Note the following CNNs:
x (n + 1) = e−b1 (n)h x (n) + θ (h) 1 1 1 2 2 a1j (n)fj (xj (n))+ b1j (n)fj (xj (n − τ1j )) + I1 (n) × j =1 j =1 , (9) x2 (n + 1) = e−b2 (n)h x2 (n) + θ2 (h) 2 2 a2j (n)fj (xj (n))+ b2j (n)fj (xj (n − τ2j )) + I2 (n) × j =1
1 , where b1 (n) = 4(sin 0.2n + 5), a11 (n) = 41 cos 0.2n, a12 (n) = 36 I1 (n) = 1, b11 (n) = 1 + cos 0.2n − sin n, b12 (n) = 1, b∗2 (n) = 5(cos 0.2n + 4), a21 (n) = 13 sin 0.2, a22 (n) = 17 , I2∗ (n) = 1 + 1n ,
b21 (n) = cos 0.2, b22 (n) = 13 , f1 (x) = 1+xx2 , f2 (x) = 2+xx2 , h = 0.2. By further calculating, we have
j =1
where b1 (n) = 8, a11 (n) =
1 4
(n) = 1 + b11 (n) = 1 + cos 0.2n − sin n, b12 (n) = 1 + b∗2 (n) = 5(cos 0.2n + 4), a∗21 (n) = 13 sin 0.2, a∗22 (n) = 17 + I2∗ (n) = 1 + 1n , b∗21 (n) = cos 0.2, b∗22 (n) = 13 + 1n , f1 (x) = f2 (x) = 2+xx2 , h = 0.2. 1 ∗ ,I n 1
j =1
x2 (n + 1) = e−b2 (n)h x2 (n) + θ2 (h) 2 2 a2j (n)fj (xj (n)) + b2j (n)fj (xj (n − τ2j )) + I2 (n) , ×
(8)
j =1
where b∗1 (n) = 4(sin 0.2n + 5), a∗11 (n) =
j =1
j =1
∗
∗
j =1
x2 (n + 1) = e−b2 (n)h x2 (n) + θ2∗ (h) 2 2 ∗ b∗2j (n)fj (xj (n − τ2j )) + I2∗ (n) a ( n ) f ( x ( n ))+ × j j 2j ∗
j =1
x2 (n + 1) = e−b2 (n)h x2 (n) + θ2∗ (h) 2 2 ∗ ∗ ∗ a2j (n)fj (xj (n)) + b2j (n)fj (xj (n − τ2j )) + I2 (n) , ×
λe−bi h − 1 + Li λ
m j =1
aM ji θj (h) + Li
m
τji +1 bM < 0, ji θj (h)λ
j =1
where i, j = 1, 2. Since system (8) and (9) satisfies (H1 )–(H3 ), (H5 )′ , (H7 ), hence, system (9) has the 10π -periodic solution. Moreover, all solutions of system (8) converge to the periodic solution of system (9). A numeric simulation for example is shown in Fig. 2, and the initial values are random. 6. Conclusions In this paper, several conditions have been obtained to ensure delayed discrete CNNs having a unique w -periodic solution by using the properties of inequalities which are different from the previous results. Consequently, we derive some sufficient conditions ensuring that all solutions of discrete delayed CNNs without periodic coefficients converge to a periodic function, by applying mathematical analysis techniques. The results are new and complement previously known results.
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J. Wang et al. / Neural Networks 53 (2014) 61–68
Fig. 2. Time responses in Example 2.
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