Global attractive periodic solutions of BAM neural networks with continuously distributed delays in the leakage terms

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Abstract

Bidirectional associative memory (BAM) model is considered with the introduction of continuously distributed delays in the leakage (or forgetting) terms. By using continuation theorem in coincidence degree theory and the Lyapunov functional, some very verifiable and practical algebraic mean delay dependent criteria on the existence and global attractive periodic solutions are derived.

Introduction

In this paper, we are concerned with the existence and global attractivity of periodic solutions for bidirectional associative memory neural networks with continuously distributed delays in the leakage terms: xi(t)=ai0hi(1)(s)xi(ts)ds+j=1paijfj(0hij(s)yj(ts)ds)+Ii(t)yj(t)=bj0hj(2)(s)xj(ts)ds+i=1mbjigi(0lji(s)xi(ts)ds)+Jj(t)}, where

(H1) ai,bj,aij and bji are real constants, ai>0,bj>0, Ii,JjC(R,R), Ii and Jj are ω-periodic functions, where ω>0, i=1,2,,m,j=1,2,,p;

(H2) hi(1),hj(2)C(R+,R+), 0hi(1)(s)ds=0hj(2)(s)ds=1, 0shi(1)(s)ds< and 0shj(2)(s)ds<, hij,ljiC(R+,R+), 0hij(s)ds=0lji(s)ds=1, i=1,2,,m, j=1,2,,p, R+=[0,);

(H3) There exist constants Ljf>0,Lig>0, such that |fj(u)fj(v)|Ljf|uv|,|gi(u)gi(v)|Lig|uv|for anyu,vR, i=1,2,,m, j=1,2,,p.

A class of networks known as bidirectional associative memory (BAM) neural networks has been introduced and studied by Kosko [1], [2], [3]. Subsequently, BAM neural networks with axonal transmission delays such as xi(t)=aixi(t)+j=1paijfj(yj(tσj(2)))+Ii,i=1,2,,myj(t)=bjyj(t)+i=1mbjigi(xi(tσi(1)))+Jj,j=1,2,,p}, and xi(t)=aixi(t)+j=1paijfj(0hij(s)yj(ts)ds)+Iiyj(t)=bjyj(t)+i=1mbjigi(0lji(s)xi(ts)ds)+Jj}, have been studied by several authors (see [4], [5], [6], [7], [8], [9], [10], [11] and the references therein). Neural networks have been designed to solve a variety of problems; when neural networks are designed to solve optimization problems, it is expected that such networks have a unique equilibrium which is globally asymptotically or exponentially stable. For BAM neural networks with periodic coefficients and discrete delays (or continuously distributed delays), the existence and global asymptotical or exponential stability of periodic solutions also have been discussed in [12], [13], [14], [15].

Almost all the models of the BAM are variations of the coupled systems of differential equations in which the positive constants ai,bj denote the timescales of the respective layers of the network; the first terms in each of the right side of (1.2) (or (1.3)) correspond to a stabilizing negative feedback of the systems which act instantaneously without time delay; these terms are variously known as forgetting or leakage terms (see for instance Kosko [1], Haykin [16]). It is known from the literature on population dynamics (see Gopalsamy [17]) that time delays in the stabilizing negative feedback terms will have a tendency to destabilize a system. Since time delays in the leakage terms are usually not easy to handle, such delays have only been considered in [18] for the following BAM neural networks xi(t)=aixi(tτi(1))+j=1naijfj(yj(tσj(2)))+Ii,i=1,2,,nyj(t)=bjyj(tτj(2))+i=1nbjigi(xi(tσi(1)))+Jj,j=1,2,,n}. By constructing the degenerate Lyapunov–Kravsovskii functional, Gopalsamy [18] has obtained some delay dependent sufficient conditions for (1.4) to have a stable equilibrium.

Motivated by the idea of [18], we consider (1.1) with the incorporation of continuously distributed delays in the leakage terms and obtain delay dependent sufficient conditions for the system to have a global periodic attractor under periodic inputs by using the coincidence degree theory and Lyapunov functionals.

Section snippets

Preliminaries

Let X and Z be real normed vector spaces let L:DomLXZ be a linear mapping, and N:XZ is continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimkerL=codimImL<+ and ImL is closed in Z. If L is a Fredholm mapping of index zero there exist continuous projections P:XX and Q:ZZ such that ImP=kerL, ImL=kerQ=Im(IQ). It follows that L|DomLkerP:(IP)XImL is invertible. We denote its inverse by KP. If Ω is an open bounded subset of X, the mapping N will be called L

Existence of periodic solutions

Definition 3.1

[20]

A real matrix A=(aij)n×n is said to be an M-matrix, if

  • (1)

    aii>0(i=1,2,,n),aij0(ij,i,j=1,2,,n);

  • (2)

    det(a11a1iai1aii)>0,i=1,2,,n.

Theorem 3.1

Under assumptions (H1)–(H3) , if(H4)W=(ABCD)n×n is an M-matrix .

Then(1.1)has at least one ω-periodic solution, where n=m+p , A=diag(a1(1a1τ̄1(1)),,am(1amτ̄m(1))) , D=diag(b1(1b1τ̄1(2)),,bp(1bpτ̄p(2))) , B=(|aij|Ljf(1+aiτ̄i(1)))m×p , C=(|bji|Lig(1+bjτ̄j(2)))p×m , τ̄i(1)=0shi(1)(s)ds and τ̄j(2)=0shj(2)(s)ds denote the mean time delays on [0,) respectively, i

Global attractivity

Definition 4.1

If fC(R,R), then the upper right derivative D+f(t) of f(t) is defined as D+f(t)=lim suph0+f(t+h)f(t)h.

Theorem 4.1

Under assumptions (H1)–(H4) , suppose that

(H5) 0s2hi(1)(s)ds< and 0s2hj(2)(s)ds< , 0shij(s)ds< , 0slji(s)ds< , i=1,2,,m , j=1,2,,p;

and(H6){pi=2ai(1aiτ̄i(1))j=1p[|aij|Ljf(1+aiτ̄i(1))+|bji|Lig(1+bjτ̄j(2))]>0,qj=2bj(1bjτ̄j(2))i=1m[|bji|Lig(1+bjτ̄j(2))+|aij|Ljf(1+aiτ̄i(1))]>0,hold. Then(1.1)has an ω-periodic solution which is globally attractive.

Proof

From Theorem 3.1, (3.1), (3.2)

References (20)

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    BAM are able to develop attractors allowing the network to perform various types of recall under noisy conditions and successfully carry out pattern completion. Many advances were made since Kosko introduced the BAM in 1988 [2–21]. As is well known, time delays are often encountered unavoidably in many practical systems such as biological, artificial neural networks, automatic control systems, population models, and so on.

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This work was supported by the Natural Science Foundation of Guangdong Province under Grant 8351009001000002 and the National Natural Science Foundation of China under Grant 60572073.

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