具有时变时滞的复值BAM神经网络的概周期解.pdf

第 卷第期V o l ,N o 滨州学院学报J o u r n a l o fB i n z h o uU n i v e r s i t y 年月A p r,【微分方程与动力系统研究】具有时变时滞的复值B AM神经网络的概周期解收稿日期:基金项目:云南省教育厅自然科学基金项目(J );云南大学教育教学改革基金项目(Y )作者简介:赵莉莉(),女,白族,云南大理人,讲师,博士,主要从事非线性微分方程与动力系统研究.E m a i l:l l z h a o y n u e d u c n赵莉莉(云南大学 数学与统计学院,云南 昆明 )摘要:研究了一类具有时变时滞的复值B AM神经网络,利用B a n a c h空间中的不动点定理、实数集上的指数二分性,以及若干微分不等式技巧,获得了该类复值神经网络的概周期解存在、唯一,以及一致稳定的充分条件.最后,通过实例验证了所得结果的有效性与可行性.关键词:复值神经网络;B AM神经网络;概周期解;不动点定理;一致稳定性中图分类号:O 文献标识码:AD O I:/j c n k i 引言神经网络是在许多学科的基础上发展起来的综合性、交叉性很强的学科,它能够从人脑的神经系统结构出发,研究大脑的工作机制,进而揭示人工智能的本质.自神经网络的数学模型建立后,研究人员对实值神经网络的研究取得了大量的成果 .虽然实值神经网络在自动控制、模式识别、图像处理、医疗卫生等领域得到广泛应用,但其无法直接处理复数数据.因此,作为实值神经网络的推广,复值神经网络应运而生,它解决了一些实值神经网络不能解决的问题.近年来,对复值神经网络的研究主要集中在探讨其平衡点、周期解,以及反周期解的存在性、各类稳定性、无源性,以及耗散性,也取得了大量的研究成果 .相较于周期现象,概周期现象是频繁的、经常发生的,而且概周期解的存在性与稳定性对描述动力系统的动力学行为是非常重要的,然而少有文献研究复值神经网络概周期解的相关问题.文献 研究了一类具有时变时滞的复值单层神经网络概周期解的存在性与稳定性.但是,与传统的单层神经网络模型相比,采用异联想原理,实现网络状态在两层神经元之间来回传递的B AM神经网络不仅有理论意义,还在联想记忆、复杂优化问题、模式识别、信号处理以及自动控制等领域有强大的应用功能.基于此,本文利用B a n a c h空间中的不动点定理和指数型二分性,探讨具有时变时滞的复值B AM神经网络u i(t)ci(t)ui(t)mjai j(t)fj(vj(t)mji j(t)hj(vj(ti j(t)Ii(t),v j(t)dj(t)vj(t)nibj i(t)gi(ui(t)nilj i(t)i(ui(tj i(t)Jj(t)()的概周期解的存在性与稳定性问题.其中:i,n,j,m;ui(t),vj(t)C,分别表示第I层第i个神经元、第J层第j个神经元在t时刻的状态;ci(t),dj(t)表示自反馈连接权重;ai j(t),i j(t),第期赵莉莉具有时变时滞的复值B AM神经网络的概周期解bj i(t),lj i(t)C表示神经元之间的连接权重;Ii(t),Jj(t)C分别表示第I层第i个神经元、第J层第j个神经元在t时刻的外部输入;fj,hj,gi,iC表示神经元的激活函数;i j(t),j i(t)表示神经元的传输时滞,且满足i j(t),i j(t),为常数.概周期实值系统与复值系统定义对于复值函数u(t)uR(t)iuI(t),其中uRR e(u(t),uI(t)I m(u(t),若uR(t)与uI(t)均为概周期函数,则称u(t)为概周期复值函数.定义对于矩阵函数A(t)(ai j(t),若每一个ai j(t)都是概周期函数,则称A(t)为概周期矩阵函数;对于向量函数u(t)(u(t),u(t),un(t)T,若每一个ui(t)都是概周期函数,则称u(t)为概周期向量函数.考虑概周期实值系统x(t)A(t)x(t)()以及x(t)A(t)x(t)f(t),()其中A(t)是关于t的概周期矩阵函数,f(t)是关于t的概周期向量函数.定义 如果存在投影算子P以及正常数M,使得系统()的基解矩阵X(t)满足X(t)P X(s)Me(ts)(ts),X(t)(IP)X(s)Me(st)(st),则称系统()在实数集上满足指数型二分性.引理 若系统()在实数集上满足指数型二分性,则系统()存在唯一的概周期解x(t)tX(t)P X(s)f(s)dstX(t)(IP)X(s)f(s)ds.引理 若概周期函数a(t)的平均值Ma(t)l i mtststsa()d,则存在正常数,使得e x p(tsa()d)e x p(ts)(ts).为了探讨神经网络()概周期解的存在性与稳定性,做如下假设:(C)uiuRiiuIi,vjvRjivIj,uRi,uIi,vRj,vIj R,那么fj(vj),hj(vj),gi(ui),i(ui)可以表示为fj(vj)fRj(vRj,vIj)ifIj(vRj,vIj),hj(vj)hRj(vRj,vIj)ihIj(vRj,vIj),gi(ui)gRi(uRi,uIi)igIi(uRi,uIi),i(ui)Ri(uRi,uIi)iIi(uRi,uIi),其中fRj,fIj,hRj,hIj,gRi,gIi,Ri,Ii:R R.(C)ai j(t)aRi j(t)iaIi j(t),i j(t)Ri j(t)iIi j(t),bj i(t)bRj i(t)ibIj i(t),lj i(t)lRj i(t)ilIj i(t),Ii(t)IRi(t)iIIi(t),Jj(t)JRi(t)iJIj(t),其中aRi j,aIi j,Ri j,Ii j,bRj i,bIj i,lRj i,lIj i,IRi,IIi,JRj,JIj:R R.由假设(C)(C),系统()可以化为实值系统(uRi)(t)ci(t)uRi(t)mj(aRi j(t)fRj(vRj(t),vIj(t)aIi j(t)fIj(vRj(t),vIj(t)mj(Ri j(t)hRj(vRj(ti j(t),vIj(ti j(t)Ii j(t)hIj(vRj(ti j(t),vIj(ti j(t)IRi(t),()(uIi)(t)ci(t)uIi(t)mj(aRi j(t)fIj(vRj(t),vIj(t)aIi j(t)fRj(vRj(t),vIj(t)mj(Ri j(t)hIj(vRj(ti j(t),vIj(ti j(t)Ii j(t)hRj(vRj(ti j(t),vIj(ti j(t)IIi(t),()(vRj)(t)dj(t)vRj(t)ni(bRj i(t)gRi(uRi(t),uIi(t)bIj i(t)gIi(uRi(t),uIj(t)滨州学院学报第 卷 mj(lRj i(t)Ri(uRi(tj i(t),uIi(tj i(t)lIj i(t)Ii(uRi(tj i(t),uIi(tj i(t)JRj(t),()(vIj)(t)dj(t)vIj(t)ni(bRj i(t)gIi(uRi(t),uIi(t)bIj i(t)gRi(uRi(t),uIj(t)ni(lRj i(t)Ii(uRi(tj i(t),uIi(tj i(t)lIj i(t)Ri(uRi(tj i(t),uIi(tj i(t)JIj(t).()其中:i,n;j,m.如无特殊说明,下文i,j取值与此相同.因此,对复值神经网络()的概周期解的研究,可以转化为对实值神经网络()(包含式()()四个式子)的概周期解的研究.系统()的初值条件为uRi(s)Ri(s),uIi(s)Ii(s),vRj(s)Rnj(s),vIj(s)nj(s),st,t,t,其中(s)(R(s),I(s),Rn(s),In(s),Rn(s),In(s),Rnm(s),Inm(s)T:t,t Rnm为连续的向量函数.定义具有初值的系统()的解U(t)(uR(t),uI(t),uRn(t),uIn(t),vR(t),vRm(t),vIm(t)T(tt)称为一致稳定的,如果对于任意的和t,存在与t无关的正常数(),使得任意的具有初值的系统()的解X(t)(xR(t),xI(t),xRn(t),xIn(t),yR(t),yI(t),yRm(t),yIm(t)T只要,就有U(t)X(t),(tt),其中 m a xknmm a xs u pst,t|R(s)R(s)|,s u pst,t|I(s)I(s)|,U(t)X(t)m a xm a xin|uRi(t)xRi(t)|,m a xin|uIi(t)xIi(t)|,m a xjm|vRj(t)yRj(t)|,m a xjm|vIj(t)yIj(t)|.概周期解的存在性与稳定性对于实值神经网络(),还需要做如下假设(C)ci(t),dj(t),aRi j(t),aIi j(t),Ri j(t),Ii j,bRj i(t),bIj i(t),lRj i(t),IIj i(t),IRi(t),IIi(t),JRj(t),JIj(t)都是概周期函数,且Mci(t),Mdj(t),i,n,j,m.(C)存在函数fj(xRj,xIj),fj(xRj,xIj),hj(xRj,xIj),hj(xRj,xIj),gi(yRi,yIi),gi(yRi,yIi),i(yRi,yIi),i(yRi,yIi),以及常数Mj,Nj,i,i,pj,qj,pj,qj,pj,qj,pj,qj,pi,qi,pi,qi,pi,qi,pi,qi,使得对任意的(xRj,xIj),(zRj,zIj),(yRi,yIi),(uRi,uIi)R,有fj(xRj,xIj)fj(xRj,xIj)fj(xRj,xIj),fj(,),|fj(xRj,xIj)|Mj,|fj(xRj,xIj)fj(zRj,zIj)|pj|xRjzRj|qj|xIjzIj|,|fj(xRj,xIj)fj(zRj,zIj)|pj|xRjzRj|qj|xIjzIj|,hj(xRj,xIj)hj(xRj,xIj)hj(xRj,xIj),hj(,),|hj(xRj,xIj)|Nj,|hj(xRj,xIj)hj(zRj,zIj)|pj|xRjzRj|qj|xIjzIj|,|hj(xRj,xIj)hj(zRj,zIj)|pj|xRjzRj|qj|xIjzIj|,gi(yRi,yIi)gj(yRi,yIi)gi(yRi,yIi),gi(,),|gi(yRi,yIi)|i,|gi(yRi,yIi)gi(uRi,uIi)|pi|yRiuRi|qi|yIiuIi|,|gi(yRi,yIi)gi(uRi,uIi)|pi|yRiuRi|qi|yIiuIi|,第期赵莉莉具有时变时滞的复值B AM神经网络的概周期解i(yRi,yIi)i(yRi,yIi)i(yRi,yIi),i(,),|i(yRi,yIi)|i,|i(yRi,yIi)i(uRi,uIi)|pi|yRiuRi|qi|yIiuIi|,|i(yRi,yIi)i(uRi,uIi)|pi|yRiuRi|qi|yIiuIi|,其中,R,I.(C)m a xm a xinRici,Iici,m a xjmRjdj,Ijdj,其中cii n ftRci(t),dji n ftRdj(t),RimjaRi j(pRjqRj)(MRj(pRjqRj)aIi j(pIjqIj)(MIj(pIjqIj)Ri j(pRjqRj)(NRj(pRjqRj)Ii j(pIjqIj)(NIj(pIjqIj),IimjaRi j(pIjqIj)(MIj(pIjqIj)aIi j(pRjqRj)(MRj(pRjqRj)Ri j(pIjqIj)(NIj(pIjqIj)Ii j(pRjqRj)(NRj(pRjqRj),RjnibRj i(pRiqRi)(Ri(pRiqRi)bIj i(pIiqIi)(Ii(pIiqIi)lRj i(pRiqRi)(Ri(pRiqRi)lIj i(pIiqIi)(Ii(pIiqIi),IjmibRj i(pIiqIi)(Ii(pIiqIi)bIj i(pRiqRi)(Ri(pRiqRi)lRj i(pIiqIi)(Ii(pIiqIi)lIj i(pRiqRi)(Ri(pRiqRi),aRi js u ptR|aRi j(t)|,aIi js u ptR|aIi j(t)|,Ri js u ptR|Ri j(t)|,Ii js u ptR|Ii j(t)|,bRj is u ptR|bRj i(t)|,bIj is u ptR|bIj i(t)|,lRj is u ptR|lRj i(t)|,lIj is u ptR|lIj i(t)|,IRis u ptR|IRi(t)|,IIis u ptR|IIi(t)|,m a xm a xinRici,Iici,m a xjmRjdj,Ijdj,m a xm a xinIRici,IIici,m a xjmJRidj,JIidj ,Rimj(aRi jMRj(pRjqRj)aIi jMIj(pIjqIj)Ri jNRj(pRjqRj)Ii jNIj(pIjqIj),Iimj(aRi jMIj(pIjqIj)aIi jMRj(pRjqRj)Ri jNIj(pIjqIj)Ii jNRj(pRjqRj),Rjni(bRj iRi(pRiqRi)bIj iIi(pIiqIi)lRj iRi(pRiqRi)lIj iIi(pIiqIi),Ijni(bRj iIi(pIiqIi)bIj iRi(pRiqRi)lRj iIj(pIiqIi)lIj iRi(pRiqRi),JRjs u ptR|JRj(t)|,JIjs u ptR|JIj(t)|,j,m,i,n.用A P(R,R(n m)表示从R到R(nm)的全体概周期函数构成的集合,设U(t)|U(t)(uR(t),uI(t),uRn(t),uIn(t),vR(t),vI(t),vRm(t),vIm(t)TA P(R,R(nm).对任意的U(t)定义范数U m a xm a xins u ptRuRi(t),uIi(t),m a xjms u ptRvRj(t),uIj(t),则在该范数下成为一个B a n a c h空间.定理若条件(C)(C)成立,则系统()在U|U,U 中存在唯一的一致稳定的概周期解.证明对于任意的滨州学院学报第 卷(t)(R(t),I(t),Rn(t),In(t),Rn(t),In(t),Rnm(t),Inm(t)T,考虑线性概周期微分系统(uRi)(t)ci(t)uRi(t)mj(aRi j(t)fRj(Rnj(t),Inj(t)aIi j(t)fIj(Rnj(t),Inj(t)mj(Ri j(t)hRj(Rnj(ti j(t),Inj(ti j(t)Ii j(t)hIj(Rnj(ti j(t),Inj(ti j(t)IRi(t),(uIi)(t)ci(t)uIi(t)mj(aRi j(t)fIj(Rnj(t),Inj(t)aIi j(t)fRj(Rnj(t),Inj(t)mj(Ri j(t)hIj(Rnj(ti j(t),Inj(ti j(t)Ii j(t)hRj(Rnj(ti j(t),Inj(ti j(t)IIi(t),(vRj)(t)dj(t)vRj(t)ni(bRj i(t)gRi(Ri(t),Ii(t)bIj i(t)gIi(Ri(t),Ii(t)mj(lRj i(t)Ri(Ri(tj i(t),Ii(tj i(t)lIj i(t)Ii(Ri(tj i(t),Ii(tj i(t)JRj(t),(vIj)(t)dj(t)vIj(t)ni(bRj i(t)gIi(Ri(t),Ii(t)bIj i(t)gRi(Ri(t),Ii(t)ni(lRj i(t)Ii(Ri(tj i(t),Ii(tj i(t)lIj i(t)Ri(Ri(tj i(t),Ii(tj i(t)JIj(t).()由条件(C)、引理、引理可得系统()存在唯一的概周期解U(t)(u)R(t),(u)I(t),(u)Rn(t),(u)In,(v)R(t),(v)I(t),(v)Rm(t),(v)Im(t)T,其中(u)Ri(t)tetsci(u)du(mj(aRi j(s)fRj(Rnj(s),Inj(s)aIi j(s)fIj(Rnj(s),Inj(s)mj(Ri j(s)hRj(Rnj(si j(s),Inj(si j(s)Ii j(s)hIj(Rnj(si j(s),Inj(si j(s)IRi(s)ds,(u)Ii(t)tetsci(u)du(mj(aRi j(s)fIj(Rnj(s),Inj(s)aIi j(s)fRj(Rnj(s),Inj(s)mj(Ri j(s)hIj(Rnj(si j(s),Inj(si j(s)Ii j(s)hRj(Rnj(si j(s),Inj(si j(s)IIi(s)ds,(v)Rj(t)tetsdj(u)du(ni(bRj i(s)gRi(Ri(s),Ii(s)bIj i(s)gIi(s)(Ri(s),Ii(s)ni(lRj i(s)Ri(Ri(sj i(s),Ii(sj i(s)lIj i(s)Ii(Ri(sj i(s),Ii(sj i(s)JRj(s)ds,(v)Ij(t)tetsdj(u)du(nibRj i(s)gIi(Ri(s),Ii(s)bIj i(s)gRi(Ri(s),Ii(s)ni(lRj i(s)Ri(Ri(sj i(s),Ii(sj i(s)lIj i(s)Ri(Ri(sj i(s),Ii(sj i(s)JIj(s)ds.()因为U(t),所以可以在上定义映射:为U.接下来证明U(t).由()式及条件(C)(C)可得|(u)Ri(t)|tetsci(u)du(mj(aRi j(s)fRj(Rnj(s),Inj(s)aIi j(s)fIj(Rnj(s),Inj(s)第期赵莉莉具有时变时滞的复值B AM神经网络的概周期解mj(Ri j(s)hRj(Rnj(si j(s),Inj(si j(s)Ii j(s)hIj(Rnj(si j(s),Inj(si j(s)IRi(s)ds|tetsci(u)du(mj(aRi j|fRj(Rnj(s),Rnj(s)fRj(Rnj(s),Rnj(s)|aIi j|fIj(Rnj(s),Rnj(s)fIj(Rnj(s),Rnj(s)|)mj(Ri j|hRj(Rnj(si j(s),Inj(si j(s)hRj(Rnj(si j(s),Inj(si j(s)|Ii j|hIj(Rnj(si j(s),Inj(si j(s)hIj(Rnj(si j(s),Inj(si j(s)|)IRi)dstetsci(u)du(mj(aRi jMRj(pRjqRj)aIi jMIj(pIjqIj)mj(Ri jNRj(pRjqRj)Ii j(pIjqIj)IRi)dsc(mj(aRi jMRj(pRjqRj)aIi jMIj(pIjqIj)mj(Ri jNRj(pRjqRj)Ii jNIj(pIjqIj)IRi)RiciIRici,类似于上面的计算,可得|(u)Ii(t)|ci(mjaRi jMIj(pIjqIj)aIi jMRj(pRjqRj)Ri jNIj(pIjqIj)Ii jNRj(pRjqRj)IIi)IiciIIici,|(v)Rj(t)|dj(nibRj iRi(pRiqRi)bIj iIi(pIiqIi)lRj iRi(pRiqRi)lIj iIi(pIiqIi)JRj)RjdjJRjdj,|(v)Ij(t)|dj(nibRj iIi(pIiqIi)bIj iRi(pRiqRi)lRj iIi(pIiqIi)lIj iRi(pRiqRi)JIj)IjdjJIjdj.综上所述U m a xm a xins u ptR|(u)Ri(t)|,m a xins u ptR|(u)Ii(t)|,m a xjms u ptR|(v)Rj(t),m a xjms u ptR|(v)Ij(t)|r,即U(t)(t),从而对任意的都有,映射是从到的自反映射.下面证明:是一个压缩映射.对任意的,设(t)(R(t),I(t),Rn(t),In(t),Rn(t),In(t),Rnm(t),Inm(t)T,(t)(R(t),I(t),Rn(t),In(t),Rn(t),In(t),Rnm(t),Inm(t)T,则|()Ri(t)()Ri(t)|tetsci(u)du(mj(aRi j(s)(fRj(Rnj(s),Inj(s)fRj(Rnj(s),Inj(s)aIi j(s)(fIj(Rnj(s),Inj(s)fIj(Rnj(s),Inj(s)滨州学院学报第 卷mj(Ri j(s)(hRj(Rnj(si j(s),Inj(si j(s)hRj(Rnj(si j(s),Inj(si j(s)Ii j(s)(hIj(Rnj(si j(s),Inj(si j(s)hIj(Rnj(si j(s),Inj(si j(s)ds|tetsci(u)du(mj(|aRi j(s)fRj(Rnj(s),Inj(s)fRj(Rnj(s),Inj(s)fRj(Rnj(s),Inj(s)fRj(Rnj(s),Inj(s)|aIi j(s)fIj(Rnj(s),Inj(s)fIj(Rnj(s),Inj(s)fIj(Rnj(s),Inj(s)fIj(Rnj(s),Inj(s)|)mj(|Ri j(s)hRj(Rnj(si j(s),Inj(si j(s)hRj(Rnj(si j(s),Inj(si j(s)hRj(Rnj(si j(s),Inj(si j(s)hRj(Rnj(si j(s),nj(si j(s)|Ii j(s)hIj(Rnj(si j(s),Inj(si j(s)hIj(Rnj(si j(s),Inj(si j(s)hIj(Rnj(si j(s),Inj(si j(s)hIj(Rnj(si j(s),Inj(si j(s)|)dstetsci(u)dumjaRi j(MRj(pRjqRj)(pRjqRj)(pRjqRj)aIi j(MIj(pIjqIj)(pIjqIj)(pIjqIj)mj(Ri j(NRj(pRjqRj)(pRjqRj)(pRjqRj)ds.类似于上面的计算,还可以得到|()Ii(t)()Ii(t)|cimjaRi j(pIjqIj)(MIj(pIjqIj)aIi j(pRjqRj)(MRj(pRjqRj)Ri j(pIjqIj)(NIj(pIjqIj)Ii j(pRjqRj)(NRj(pRjqRj)Iici,|()Rnj(t)()Rnj(t)|djnibRj i(pRiqRi)(Ri(pRiqRi)bIj i(pIiqIi)(Ii(pIiqIi)lRj i(pRiqRi)(Ri(pRiqRi)lIj i(pIiqIi)(Ii(pIiqIi)Rjdj,|()Inj(t)()Inj(t)|djmibRj i(pIiqIi)(Ii(pIiqIi)bIj i(pRiqRi)(Ri(pRiqRi)lRj i(pIiqIi)(Ii(pIiqIi)lIj i(pRiqRi)(Ri(pRiqRi)Ijdj.所以 m a xm a xins u ptR()Ri(t)()Ri(t),()Ii(t)()Ii(t),m a xjms u piR()Rnj(t)()Rnj(t),()Inj(t)()Inj(t).因为,所以映射:是一个压缩映射.由B a n a c h空间不动点定理可得,在中存在唯一第期赵莉莉具有时变时滞的复值B AM神经网络的概周期解的不动点,因此系统()在中存在唯一的概周期解U(t)(uR(t),uI(t),uRn(t),uIn(t),vR(t),vI(t),vRm(t),vIm(t)T,且U.最后证明系统()的概周期解U(t)是一致稳定的.设U(t)是系统()具有初值的解,X(t)(xR(t),xI(t),xRn(t),xIn(t),yR(t),yRm(t),yIm(t)T是系统()具有初值的任一解,则uRi(t)Ri(t)ettci(u)duttetsci(u)dumj(aRi j(s)fRj(vRj(s),vIj(s)aIi j(s)fIj(vRj(s),vIj(s)mj(Ri j(s)hRj(vRj(si j(s),vIj(si j(s)Ii j(s)hIj(vRj(si j(s),vIj(si j(s)IRi(s)ds,uIi(t)Ii(t)ettci(u)duttetsci(u)dumj(aRi j(s)fIj(vRj(s),vIj(s)aIi j(s)fRj(vRj(s),vIj(s)mj(Ri j(s)hIj(vRj(si j(s),vIj(si j(s)Ii j(s)hRj(vRj(si j(s),vIj(si j(s)IIi(s)ds,vRj(t)Rnj(t)ettdj(u)duttetsdj(u)duni(bRj i(s)gRi(uRi(s),uIi(s)bIj i(s)gIi(uRi(s),uIi(s)ni(lRj i(s)Ri(uRi(sj i(s),uIi(si j(s)lIj i(s)Ii(uRi(sj i(s),uIi(sj i(s)JRj(s)ds,vIj(t)Inj(t)ettdj(u)duttetsdj(u)duni(bRj i(s)gIi(uRi(s),uIi(s)bIj i(s)gRi(uRi(s),uIi(s)ni(lRj i(s)Ii(uRi(sj i(s),uIi(sj i(s)lIj i(s)Ri(uRi(sj i(s),uIi(sj i(s)JIj(s)ds.()xRi(t)Ri(t)ettci(u)duttetsci(u)dumj(aRi j(s)fRj(yRj(s),yIj(s),aIi j(s)fIj(yRj(s),yIj(s)mj(Ri j(s)hRj(yRj(si j(s),yIj(si j(s)Ii j(s)hIj(yRj(si j(s),yIj(si j(s)IRi(s)ds,xIi(t)Ii(t)ettci(u)duttetsci(u)dumj(aRi j(s)fIj(yRj(s),yIj(s)aIi j(s)fRj(yRj(s),yIj(s)mj(Ri j(s)hIj(yRj(si j(s),yIj(si j(s)Ii j(s)hRj(yRj(si j(s),yIj(si j(s)IIi(s)ds,yRj(t)Rnj(t)ettdj(u)duttetsdj(u)duni(bRj i(s)gRi(xRi(s),xIi(s)bIj i(s)gIi(xRi(s),xIi(s)ni(lRj i(s)Ri(xRi(sj i(s),xIi(si j(s)lIj i(s)Ii(xRi(sj i(s),xIi(sj i(s)JRj(s)ds,yIj(t)Inj(t)ettdj(u)duttetsdj(u)duni(bRj i(s)gIi(xRi(s),xIi(s)bIj i(s)gRi(xRi(s),xIi(s)ni(lRj i(s)Ii(xRi(sj i(s),xIi(sj i(s)lIj i(s)Ri(xRi(sj i(s),xIi(sj i(s)JIj(s)ds.()对任意的,取(),则当 时,必有U(t)X(t),tt.()若不然,则一定存在tt使得U(t)X(t),ttt,()且 U(t)X(t).()滨州学院学报第 卷此时,由式()()()()可得|uRi(t)xRi(t)|Ri(t)Ri(t)|ettci(u)duttetsci(u)dumj(aRi j(s)(fRj(vRj(s),vIj(s)fRj(yRj(s),yIj(s)aIi j(s)(fIj(vRj(s),vIj(s)fIj(yRj(s),yIj(s)mj(Ri j(hRj(vRj(si j(s),vIj(si j(s)hRj(yRj(si j(s),yIj(si j(s)Ii j(s)(hIj(vRj(si j(s),vIj(si j(s)hRj(yIj(si j(s),yIj(si j(s)dsettci(u)duttetsci(u)dumjaRi j(pRjqRj)(MRj(pRjqRj)aIi j(pIjqIj)(MIj(pIjqIj)Ri j(pRjqRj)(NRj(pRjqRj)Ii j(pIjqIj)(NIj(pIjqIj)Rici,同理可得|uIi(t)xIi(t)|Iici,|vRj(t)yRj(t)|Rjdj,|vIj(t)yIj(t)|Ijdj,所以 U(t)X(t)m a xm a xin|uRi(t)xRi(t)|,|uIi(t)xIi(t)|,m a xjm|vRj(t)yRj(t)|,|vIj(t)yIj(t)|()(),这与式()矛盾,故式()成立,系统()的概周期解是一致稳定的.数值模拟考虑如下具有时变时滞的复值B AM神经网络u i(t)ci(t)ui(t)mjai j(t)fj(vj(t)mji j(t)hj(vj(ti j(t)Ii(t),i,v j(t)dj(t)vj(t)nibj i(t)gi(ui(t)nilj i(t)i(ui(tj i(t)Jj(t),j,.()其中:c(t)|s i n(t)|,c(t)|c o s(t)|,d(t)s i nt,d(t)c o st,a(t)s i n(t)i c o s(t),a(t)c o s(t)i s i n(t),a(t)s i nt i|c o st|,a(t)c o st i|s i n(t)|,(t)|s i nt|i c o st,(t)|s i n(t)|i s i nt,(t)|s i n(t)|i s i nt,(t)|c o s(t)|i c o st,b(t)s i nt i c o st,b(t)c o st i s i nt,b(t)|s i n(t)|i|c o st|,b(t)|c o s(t)|i s i nt,l(t)s i nt i|c o s(t)|,l(t)|s i n(t)|i c o st,l(t)|s i nt|i c o st,l(t)c o st i s i nt,I(t)s i nt i c o s(t),I(t)s i n(t)i c o st,J(t)s i n(t)i c o s(t),J(t)s i n(t)i c o s(t),(t)|s i n(t)|,(t)c o st,(t)s i nt,|c o s(t)|,(t)s i nt,(t)|s i n(t)|,(t)(t)c o st,第期赵莉莉具有时变时滞的复值B AM神经网络的概周期解fj(vj)(vRjvIj)s i n(vRjvIj)ivIj,fRj(vRj,vIj)vRjvIj,fRj(vRj,vIj)s i n(vRjvIj),fIjvIj,fIj,gi(ui)(uRiuIi)s i n(uRiuIi)iuIi,gRi(uRi,uIi)uRiuIi,gRi(uRi,uIi)s i n(uRiuIi),gIiuIi,gIi,hj(vj)vRjs i nvRj i(vRjvIj)c o s(vRjvIj),hRj(vRj,vIj)vRj,hRj(vRj,vIj)s i n(vIj),hIj(vRj,vIj)vRjvIj,hIj(vRj,vIj)c o s(vRjvIj),i(ui)uRic o suRi i(uRiuIi)s i n(uRiuIi),Ri(uRi,uIi)uRi,Ri(uRi,uIi)c o suRi,Ii(uRi,uIi)uRiuIi,Ii(uRi,uIi)s i n(uRiuIi).直接计算后,可得cc,dd,MRjMIjNRjNIjRiIiRi,pRjqRjpIjqIj,pRjqRjpIjqIj,pRjqRjpIjqIj,pRjqRjpIjqIj,pRiqRipIiqIi,pRiqRipIiqIi,pRiqRipIiqIi,pRiqRipIiqIi,aR ,aI ,aR ,aI ,aR ,aI ,aR ,aI ,R ,I ,R ,I ,R ,I ,R ,I ,bR ,bI ,bR ,bI ,bR ,bI ,bR ,bI ,lR ,lI ,lR ,lI ,lR ,lI ,lR ,lI ,IR,II,IR,II,JR,JI,JR,JI,RI,RI,RI,RI ,RI ,RI ,RI ,RI ,.因为,所以由定理可得,系统()在|,中存在唯一的、一致稳定的概周期解.参考文献:周立群,宋协慧一类具比例时滞脉冲递归神经网络的全局多项式稳定性J电子科技大学学报,():刘新,陈丽丽,黄帅一类随机细胞神经网络的均方指数稳定性J哈尔滨理工大学学报,():宋协慧,周立群一类具多比例时滞脉冲递归神经网络的稳定性分析J天津师范大学学报(自然科学版),():刘锦,赵维锐具有时变时滞的中立型神经网络的稳定性分析J华东师范大学学报(自然科学版),():徐西睿改进神经网络的舰船电力系统稳定性容错控制J船舰科学技术,():N I T T AT S o l v i n gt h eX O Rp r o b l e ma n dt h ed e t e c t i o no f s y mm e t r yu s i n gas i n g l ec o m p l e x v a l u e dn e u r o nJ N e u r a ln e t w o r k s,():冯靓,胡成,于娟脉冲耦合复值神经网络的全局渐进同步J西南师范大学学报(自然科学版),():滨州学院学报第 卷陈宇,周博,宋乾坤具有不确定性的分数阶时滞复值神经网络无源性J应用数学和力学,():刘苑醒,张玮玮,张红梅分数阶复值神经网络的准一致同步J安庆师范大学学报(自然科学版),():舒含奇,宋乾坤带有时滞的C l i f f o r d值神经网络的全局指数稳定性J应用数学和力学,():WANP,J I ANJG I m p u l s i v es t a b i l i z a t i o na n ds y n c h r o n i z a t i o no f f r a c t i o n a l o r d e r c o m p l e x v a l u e dn e u r a ln e t w o r k sJ N e u r a l p r o c e s s i n g l e t t e r s,:方聪娜,宾红华具有时变时滞的复值神经网络的概周期解J厦门大学学报(自然科学版),():F I NK A M A l m o s tp e r i o d i cd i f f e r e n t i a le q u a t i o n sM N e w Y o r k:S p r i n g e r V e r l a g,:何崇佑概周期微分方程M北京:高等教育出版社,:WANG Q Y T h ee x i s t e n c ea n du n i q u e n e s sa n ds t a b i l i t yo fa l m o s tp e r i o d i cs o l u t i o n sJ A c t am a t h e m a t i c ss i n i c a,():T h eA l m o s tP e r i o d i cS o l u t i o no fC o m p l e x v a l u e dB AM N e u r a lN e t w o r k sw i t hT i m e v a r y i n gD e l a y sZ HAOL i l i(S c h o o l o fM a t h e m a t i c sa n dS t a t i s t i c s,Y u n n a nU n i v e r s i t y,K u n m i