高阶离散时间BAM神经网络周期解的稳定分析

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"这篇研究论文探讨了高阶离散时间双向联想记忆(BAM)神经网络周期解的存在性和全局指数稳定性。作者Ancai Zhang、Jianlong Qi和Jinhua She在2014年发表于《神经网络》期刊上，文章编号Neural Networks 50(2014)98–109。该论文是Elsevier出版的一篇学术作品，允许作者在非商业性的研究和教育用途下内部使用，包括在作者的机构内进行教学分享和与同事交流。然而，未经许可，禁止进行其他用途，如复制、分发、销售或在个人、机构或第三方网站上发布。在多数情况下，作者可以将他们的文章版本（如Word或Tex格式）上传到个人网站或机构存储库。作者如需了解更多关于Elsevier的存档和稿件政策，可访问提供的链接。论文详细分析了高阶离散时间BAM神经网络的动态行为，周期解是这类网络的重要研究对象。周期解的存在性意味着网络在特定条件下能够呈现出周期性变化的模式，这对于理解和预测神经网络的行为至关重要。全局指数稳定性则保证了当系统受到微小扰动时，系统状态会随着时间指数级地接近稳定状态，这对于实际应用中的神经网络模型至关重要，因为这确保了模型的鲁棒性和可靠性。文章可能涉及的方法包括利用Lyapunov函数、线性矩阵不等式(LMI)等工具来证明这些理论结果。Lyapunov函数是一种广泛用于稳定性分析的工具，可以帮助研究人员建立系统的稳定性条件。线性矩阵不等式则是优化问题中的一种形式，常用于处理非线性系统稳定性问题，因为它可以转化为易于处理的线性形式。此外，论文可能还讨论了如何通过控制网络参数来实现所期望的周期行为和稳定性。这通常涉及到对权重、学习速率等参数的调整，以确保网络的动态特性满足设计需求。论文的结果对于理解和设计高阶离散时间BAM神经网络具有理论指导意义，并可能为神经网络在信号处理、模式识别、控制系统等领域的应用提供理论基础。"

Author's personal copy

A. Zhang et al. / Neural Networks 50 (2014) 98–109 99

The above equations can be written in a reduced form

(t)

= −a

(t)x

(t) +

p(3−k)



j=1

(3−k)ij

(t)f

(3−k)j



(3−k)j



t − r

(3−k)ij

(t)





p(3−k)



j=1

p(3−k)



l=1

(3−k)ijl

(t)f

(3−k)j



(3−k)j



t − τ

(3−k)ij

(t)





× f

(3−k)l



(3−k)l



t − τ

(3−k)il

(t)





+ I

(t), (2)

where i = 1, 2, . . . , p(k); k = 1, 2 (k = 1 for X -layer and k = 2

for Y -layer);

p(k) =



, k = 1,

, k = 2;

(t) > 0; x

(t) is the activation of the i-th neuron in the k

layer at the time t; f

is the signal transmission function of the

i-th neuron in the k layer; b

kji

(t) and c

kjil

(t) are the first- and

second-order synoptic weights of the neural network, r

kji

(t) (≥0)

and τ

kji

(t) (≥0) stand for transmission delays; I

(t) corresponds

to the external input of the i-th neuron in the k layer; and

(t), b

kji

(t), c

kjil

(t), r

kji

(t), τ

kji

(t), and I

kij

(t) are T -periodic func-

tions in R.

Numerous attempts have been made to show the properties

of the system (2). Cao et al. examined the existence and global

exponential stability of the equilibrium with constant coefficients

(Cao, Liang, & Lam, 2004). Ho et al. studied the dynamic behaviors

with impulse terms (Ho, Liang, & Lam, 2006), and Li et al.

considered the problem of impulsive effects on the stability (Li, Lin,

Liao, & Huang, 2011). Note that a high-order Hopfield-type neural

network is a special case of (2). In other words, (2) becomes a high-

order Hopfield-type neural network when k = 1 or k = 2. Yi et al.

studied the convergence behavior of a high-order Hopfield-type

neural network in Yi, Shao, Yu, and Xiao (2008). They presented

some sufficient conditions that ensure all solutions of the network

converging to zero under a bounded condition on high-order signal

transmission functions.

Since discretization is needed in the implementation of a

continuous-time neural network, it is of both theoretical and

practical importance to study the dynamics of a discrete-time

neural network. Recently, discussions on the properties of low-

order discrete-time BAM neural networks have thrown new light

on the subject. For example, Liang et al. studied the following

discrete-time BAM neural network (Liang, Cao, & Ho, 2005)











(n + 1) = a

(n) +



j=1

(n − k(n))) + I

(n + 1) = b

(n) +



i=1

(n − l(n))) + J

(3)

where a

∈ (0, 1), b

∈ (0, 1), and k(n) and l(n) are positive in-

tegers. They used a linear matrix inequality and obtained some

sufficient conditions for the existence, uniqueness, and global ex-

ponential stability of the equilibrium point. Zhou et al. investigated

the existence and global exponential stability of periodic solution

of the following discrete-time BAM neural networks (Zhou & Liu,

2006)

(n + 1) = α

(n)x

(n) +

p(3−k)



j=1

(3−k)ji

(3−k)j



+∞



l=1

(3−k)j

(l)x

(3−k)j

(n − l)



+ I

(n). (4)

To our knowledge, there have no studies been reported on the

properties of high-order discrete-time BAM neural networks un-

til now. This motivated us to carry out a study in this paper. In this

study, we discuss the existence and exponential stability of peri-

odic solution for the following high-order discrete-time BAM neu-

ral networks

(n) = −A

(n)x

(n) +

p(3−k)



j=1

(3−k)ij

(n)f

(3−k)j



(3−k)j

(n − r

(3−k)ij

(n))



p(3−k)



j=1

p(3−k)



l=1

(3−k)ijl

(n)f

(3−k)j



(3−k)j

(n − τ

(3−k)ij

(n))



× f

(3−k)l



(3−k)l

(n − τ

(3−k)il

(n))



+ I

(n), (5)

where 1x

(n) = x

(n + 1) − x

(n), r

(3−k)ij

(n) and τ

(3−k)ij

(n) are

periodic non-negative integers, and 0 < A

(n) < 1. (5) is obtained

by applying the discretizing method in Mohamad and Gopalsamy

(2003) to the continuous-time system (2). Initial conditions of (5)

are

(n) = φ

(n), n ∈ [−τ , 0]

, i = 1, 2, . . . , p(k), k = 1, 2, (6)

where φ

(·) is a continuous function,

τ = max

k,i,j



max

n∈I

kji

(n)}, max

n∈I

{τ

kji

(n)}



{

0, 1, 2, . . . , N − 1

}

N is a positive integer,

[a, b]



a, a + 1, . . . , b − 1, b



for a, b ∈ Z, and a ≤ b.

First, we introduce some notations for convenience.

= min

n∈I

(n)}, b

kji

= max

n∈I

{|b

kji

(n)|},

kjil

= max

n∈I

{|c

kjil

(n)|},

= max

n∈I

{|I

(n)|}, M = max

k,i

sup

x∈R

{|f

(x)|},

{

0, 1, 2, . . .

}



u(n) =

N−1



n=0

u(n),

where u(n) is an N-periodic sequence. We also make the following

assumptions.

Assumption 1. A

(n), b

(3−k)ij

(n), c

(3−k)ijl

(n), r

(3−k)ij

(n),

(3−k)ij

(n), and I

(n) are N-periodic sequences on Z

Assumption 2. There exist non-negative constants L

and e

such

that

(x) ≤ L

|x| + e

, ∀x ∈ R , i = 1, 2, . . . , p(k), k = 1, 2.

Assumption 3. There exist a real number p (>2) and non-negative

constants L

and e

such that

(x)|

≤ L

|x| + e

, ∀x ∈ R, i = 1, 2, . . . , p(k), k = 1, 2.

Assumption 4. f

(x) (i = 1, 2, . . . , p(k), k = 1, 2) are continu-

ous and bounded in R.

Assumption 5. There exist a non-negative constant L

such that

(x) − f

(y)| ≤ L

|x − y|, ∀x, y ∈ R,

i = 1, 2, . . . , p(k), k = 1, 2.

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