# Periodic Solutions of a Cohen-Grossberg-Type BAM Neural Networks with Distributed Delays and Impulses

Authors: Qiming Liu; Rui Xu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

A class of Cohen-Grossberg-type BAM neural networks with distributed delays
and impulses are investigated in this paper. Sufficient conditions to guarantee the uniqueness
and global exponential stability of the periodic solutions of such networks are established
by using suitable Lyapunov function, the properties ofM-matrix, and some suitable mathematical transformation. The results in this paper improve the earlier publications.


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## Body

## 1. Introduction

The research of neural networks with delays involves not only the dynamic analysis of equilibrium point but also that of periodic oscillatory solution. The dynamic behavior of periodic oscillatory solution is very important in learning theory due to the fact that learning usually requires repetition [1, 2].Cohen and Grossberg proposed the Cohen-Grossberg neural networks (CGNNs) in 1983 [3]. Kosko proposed bi directional associative memory neural networks (BAMNNs) in 1988 [4]. Some important results for periodic solutions of delayed CGNNs have been obtained in [5–10]. Xiang and Cao proposed a class of Cohen-Grossberg BAM neural networks (CGBAMNNs) with distributed delays in 2007 [11]; in addition, many evolutionary processes are characterized by abrupt changes at certain time; these changes are called to be impulsive phenomena, which are included in many neural networks such as Hopfield neural networks, BAM neural networks, CGNNs, and CGBAMNNs and can affect dynamical behaviors of the systems just as time delays. The results for periodic solutions of CGBAMNNs with or without impulses are obtained in [11–15].The objective of this paper is to study the existence and global exponential stability of periodic solutions of CGBAMNNs with distributed delays by using suitable Lyapunov function, the properties ofM-matrix, and some suitable mathematical transformation. Comparing with the results in [13, 14], improved results are successively obtained, the conditions for the existence and globally exponential stability of the periodic solution of such system without impulses have nothing to do with inputs of the neurons and amplification functions; and we also point that CGBAMNNs model is a special case of CGNNs model, many results of CGBAMNNs can be directly obtained from the results of CGNNs.The rest of this paper is organized as follows. Preliminaries are given in Section2. Sufficient conditions which guarantee the uniqueness and global exponential stability of periodic solutions for CGBAMNNs with distributed delays and impulses are given in Section 3. Two examples are given in Section 4 to demonstrate the main results.


## 2. Preliminaries

Consider the following periodic CGNNs model with distributed delays and impulses:(2.1)ẋi(t)=-ai(xi(t))[bi(t,xi(t))-∑j=1npij(t)fj(ρjxj(t))-∑j=1nuij(t)∫0+∞kij(s)fj(ρjxj(t-s))ds-Ii(t)],t>0,t≠tk,Δxi(tk)=-γikxi(tk),t=tk,k∈Z+,
where 1≤i≤n, t>0, and Z+={1,2,…}. xi(t) denotes the state variable of the ith neuron, fj(·) denotes the signal function of the jth neuron at time t; Ii denotes input of the ith neuron at time t; ai(·) represents amplification function; bi(t,·) is appropriately behaved function; pij(t) and uij(t) are connection weights of the neural networks at time t; respectively, ρj is positive constant, which corresponds to the neuronal gain associated with the neuronal activations and kij corresponds to the delay kernel function; pij(t) and uij(t) are continuously periodic functions on [0,+∞) with common period T>0.Δxi(tk)=xi(tk+)-xi(tk-); tk is called impulsive moment and satisfies 0<t1<t2<⋯,limk→+∞tk=+∞; xi(t-) and xi(t+) denote the left-hand and right-hand limits at tk; respectively, we always assume xi(tk-)=xi(tk) and xi′(tk-)=xi′(tk),k∈Z+.For system (2.1), we assume the following.(H1)
The amplification functionai(·) is continuous, and there exist constants a̲i,a¯i such that 0<a̲i≤ai(xi(t))≤a¯i for 1≤i≤n.(H2)
The behaved functionbi(t,·) is T-periodic about the first argument; there exists continuous T-periodic function αi(t) such that(2.2)bi(t,x)-bi(t,y)x-y≥αi(t)>0,for allx≠y,1≤j≤n.(H3)
For activation functionfj(·), there exists positive constant Lj such that(2.3)Lj=sup⁡x≠y|fj(x)-fj(x)x-y|,for allx≠y, 1≤j≤n.(H4)
The kernel functionkij(s) is nonnegative continuous function on [0,+∞) and satisfies(2.4)∫0+∞seλskij(s)ds<+∞,Kij(λ)=∫0+∞eλskij(s)dsis differentiable function forλ∈[0,rij),0<rij<+∞, Kij(0)=1 and limλ→rij-Kij(λ)=+∞.(H5)
There exists positive integerk0 such that tk+k0=tk+T and γi(k+k0)=γik hold.Remark 2.1.
A typical example of kernel function is given bykij(s)=(sr/r!)rijr+1e-rijs for s∈[0,+∞), where rij∈(0,+∞),r∈{0,1,…,n}. These kernel functions are called as the gamma memory filter [16] and satisfy condition (H4).
For any continuous functionS(t) on [0,T], S̲ and S¯ denote mint∈[0,T]{|S(t)|} and maxt∈[0,T]{|S(t)|}, respectively.
For anyx(t)=(xi(t),x2(t),…,xk(t))T∈Rk, t>0, define ∥x(t)∥=∑i=1k|xi(t)|, and for any φ(s)=(φ1(s),φ2(s),…,φk(s))T∈Rk, s∈(-∞,0], define ∥φ(s)∥=sups∈(-∞,0]∑i=1k|φi(s)|.
Denote(2.5)PC((-∞,0],Rk)={ψ:[-∞,0]⟶Rk∣ψ(s)isboundedandcontinuousforallbutatmostafinitenumberofpointss∈(-∞,0],andatthesepointss,ψ(s+),ψ(s-)existandψ(s-)=ψ(s)}.
Then PC((-∞,0],Rk) is a Banach space with respect to ∥·∥.
The initial conditions of system (2.1) are given by
(2.6)xi(s)=φi(s),-∞≤s≤0,1≤i≤n,
where φ(s)=(φ1(s),φ2(s),…,φn(s))∈PC([-∞,0],Rn).
Letx(t,φ)=(x1(t,φ),x2(t,φ),…,xn(t,φ)T) denote any solution of the system (2.1) with initial value φ∈PC((-∞,0],Rn).Definition 2.2.
A solutionx(t,φ) of system (2.1) is said to be globally exponentially stable, if there exist two constants λ>0,M>0 such that
(2.7)‖x(t,ψ)-x(t,φ)‖≤M‖ψ-φ‖e-λt,t>0,
for any solutions x(t,ψ) of system (2.1).Definition 2.3.
A real matrixA=(aij)n×nis said to be a nonsingular M-matrix if aij≤0(i,j=1,2,…,n,i≠j), and all successive principle minors of A are positive.Lemma 2.4 (see [17]).
A matrix with nonpositive off-diagonal elementsA=(aij)n×n is a nonsingular M-matrix if and if only there exists a vector p=(pi)1×n>0 such that pTA>0 or Ap holds.Lemma 2.5.
Under assumptions (H1)–(H5), system (2.1) has a T-periodic solution which is globally exponentially stable, if the following conditions hold.(H6)
ℳ=A-C is a nonsingular M-matrix, where(2.8)A=diag⁡(α1̲,α2̲,…,αn̲),C=(cij)n×n,cij=(p¯ij+u¯ij)ρjLj.(H7)
ai((1-γik)s)≥|1-γik|ai(s), for all s∈R,i=1,2,…,n.Proof.
Letx(t,ψ1) and x(t,ψ2) be two solutions of system (2.1) with initial value ψ1=(φ1,φ2,…,φn) and ψ2=(ζ1,ζ2,…,ζn)∈PC((-∞,0],Rn), respectively.
Let(2.9)Fi(θ)=μi(α̲i-θa̲i)-∑j=1nμj(p¯ji+u¯jiKji(θ))ρiLi,i=1,2,…,n.
Since ℳ is a nonsingular M-matrix according to condition (H6), ℳT is also a nonsingular M-matrix; we know from Lemma 2.4 that there exists a vector p=(μ1,μ2,…,μn)T such that ℳTp>0; that is,
(2.10)μiα̲i-∑j=1nμj(p¯ji+u¯ji)ρiLi>0,
for 1≤i≤n, which indicates that Fi(0)>0. Since Fi(θ) are continuous and differential on [0,rji) and limθ→rji-Fi(θ)=-∞ according to condition (H4), Fi′(θ)<0 for θ∈[0,uji). There exist constants θi such that Fi(θi)=0 for i=1,2,…,n. So we can choose
(2.11)0<λ≤min⁡{θ1,θ2,…,θn},
such that
(2.12)Fi(λ)≥0.
Define
(2.13)Xi(t)=|xi(t,ψ2)-xi(t,ψ1)|.
Now we define a Lyapunov functionV(t) by
(2.14)V(t)=∑i=1nμi{Vi(t)+∑j=1nu¯ijLjρj∫0+∞kij(s)∫t-stXj(μ)eλ(s+μ)dμds},
in which
(2.15)Vi(t)=eλtsign⁡(xi(t,ψ2)-xi(t,ψ1))∫xi(t,ψ1)xi(t,ψ2)1ai(s)ds.
for i=1,2,…,n.
Whent≠tk,k∈Z+, calculating the upper right derivative of V(t) along solution of (2.1), similar to proof of Theorem  3.1 in [10], corresponding to case in which r→1,vijl(t)=0 in [10], we obtain from (2.12)–(2.15) that(2.16)D+V(t)∣(2.1)≤-eλt{∑i=1nμi(α̲i-λa̲i)Xi(t)-∑i=1n∑j=1nμi(p¯ij+u¯ijKij(λ))ρjLjXj(t)}=-eλt{∑i=1nμi(α̲i-λa̲i)Xi(t)-∑i=1n∑j=1nμi(p¯ji+u¯jiKji(λ))ρiLiXi(t)}=-eλt∑i=1nFi(λ)Xi(t)≤0.
When t=tk,k∈Z+, we have
(2.17)Vi(tk+)=eλt+sign⁡(xi(tk+,ψ2)-xi(tk+,ψ1))∫xi(tk+,ψ1)xi(tk+,ψ2)1ai(s)ds=eλtsign⁡(1-γik)(xi(tk,ψ2)-xi(tk,ψ1))∫(1-γik)xi(tk,ψ1)(1-γik)xi(tk,ψ2)1ai(s)ds,=eλtsign⁡(xi(tk,ψ2)-xi(tk,ψ1))∫xi(tk,ψ1)xi(tk,ψ2)|1-γik|ai((1-γik)s)ds,
which, together with (H7), leads to
(2.18)Vi(tk)-Vi(tk+)=eλtsign⁡(xi(tk,ψ2)-xi(tk,ψ1))∫xi(tk,ψ1)xi(tk,ψ2)(1ai(s)-|1-γik|ai((1-γik)s))ds,≥eλtsign⁡(xi(tk,ψ2)-xi(tk,ψ1))∫xi(tk,ψ1)xi(tk,ψ2)ai((1-γik)s)-|1-γik|ai(s)ai(s)ai((1-γik)s)ds≥0,
that is,
(2.19)Vi(tk+)≤Vi(tk).
It follows that
(2.20)V(tk+)=∑i=1nμi{Vi(t+)+∑j=1nu¯ijLjρj∫0+∞kij(s)∫t+-st+Xj(μ)eλ(s+μ)dμds}≤∑i=1nμi{Vi(t)+∑j=1nu¯ijLjρj∫0+∞kij(s)∫t-stXj(μ)eλ(s+μ)dμds}=V(tk).
Then we have
(2.21)V(t)≤V(0).
On the other hand, from (2.14), we have
(2.22)V(t)≥m0eλt∑i=1n|xi(t,ψ2)-xi(t,ψ1)|,V(0)≤M0sup⁡l∈(-∞,0]⁡∑i=1n|φi(l)-ζi(l)|,
in which
(2.23)m0=min⁡1≤i≤n⁡(μia¯i),M0=max⁡{M1,M2},M1=max⁡1≤i≤n⁡(μia̲i),M2=∑j=1nμjmax⁡1≤i≤n(u¯jiρiLi)∫0+∞seλsmax⁡1≤i≤n⁡kji(s)ds.
Hence, from (2.21) and (2.22), we know that the following inequality holds for t>0:
(2.24)‖x(t,ψ2)-x(t,ψ1)‖≤M‖ψ2-ψ1‖e-λt,
in which M=M0/m0.
We can always choose a positive integerN such that e-λ0NTM≤1/2 and define a Poincaré mapping P:C→C by P(ξ)=xT(ξ); we have
(2.25)‖PNψ2-PNψ1‖≤12‖ψ2-ψ1‖,
which implies that PN is a contraction mapping. Similar to [10], using contraction mapping principle, we know that system (2.1) has a T-periodic solution which is globally exponentially stable. This completes the proof.Remark 2.6.
The result above also holds for (2.1) without impulses, and the existence and globally exponential stability of the periodic solution for (2.1) have nothing to do with amplification functions and inputs of the neuron. The results in [5] have more restrictions than Lemma 2.5 in this paper because conditions for the ones in [5] are relevant to amplification functions.


## 3. Periodic Solutions of CGBAMNNs with Distributed Delays and Impulses

Consider the following periodic CGBAMNNs model with distributed delays:(3.1)ẋi(t)=-ai(xi(t))[bi(t,xi(t))-∑j=1mpij(t)fj(ρjyj(t))-∑j=1muij(t)∫0+∞kij(s)fj(ρjyj(t-s))ds-Ii(t)],t>0,t≠tk,Δxi(tk)=-γikxi(tk),t=tk,k∈Z+,ẏj(t)=-cj(yj(t))[dj(t,yj(t))-∑i=1nqji(t)gi(ρ̃ixi(t))-∑i=1nvji(t)∫0+∞k̃ji(s)gi(ρ̃ixi(t-s))ds-Jj(t)],t>0,t≠tk,Δyj(tk)=-δjkyj(tk),t=tk,k∈Z+
for 1≤i≤n,1≤j≤m, and Z+={1,2,…}; xi(t) and yj(t) denote the state variable of the ith neuron from the neural field FX and the jth neuron from the neural field FY at time t;  fj(·) and gi(·) denote the signal functions of the jth neuron from the neural field FY and the ith neuron from the neural field FX at time t; respectively, Ii and Jj denote inputs of the ith neuron from the neural field FX and the jth neuron from the neural field FY at time t; respectively, ai(·) and cj(·) represent amplification functions; bi(t,·) and dj(t,·) are appropriately behaved functions; pij(t),qji(t),uij(t), and vji(t) are the connection weights; ρj,ρ̃i are positive constants, which correspond to the neuronal gains associated with the neuronal activations; kij and k̃ji correspond to the delay kernel functions; uij(t),vji(t),pij(t),qji(t), Ii(t), and Jj(t) are all continuously periodic functions on [0,+∞) with common period T>0.Δxi(tk)=xi(tk+)-xi(tk-), Δyj(tk)=yj(tk+)-yj(tk-); tk is called impulsive moment and satisfies 0<t1<t2<⋯,limk→+∞tk=+∞; xi(t-),yj(t-) and xi(t+),yj(t+) denote the left-hand and right-hand limits at tk; respectively, we always assume xi(tk-)=xi(tk),yj(tk-)=yj(tk),xi′(tk-)=xi′(tk), and yj′(tk-)=yj′(tk),k∈Z+.For system (3.1), we assume the following.(H8)
Amplification functionsai(·) and cj(·) are continuous and there exist constants a̲i,a¯i and c̲j,c¯j such that 0<a̲i≤ai(xi(t))≤a¯i,0<c̲j≤cj(yj(t))≤c¯j,1≤i≤n,1≤j≤m.(H9)
bi(t,u),dj(t,u) are T-periodic about the first argument, there exist continuous, T-periodic functions αi(t) and βj(t) such that
(3.2)bi(t,x)-bi(t,y)x-y≥αi(t)>0,dj(t,x)-dj(t,y)x-y≥βj(t)>0
for all x≠y,1≤i≤n,1≤j≤m.(H10)
For activation functionsfj(·) and gi(·), there exist constant Lj and L̃i such that
(3.3)Lj=sup⁡x≠y|fj(x)-fj(y)x-y|,L̃i=sup⁡x≠y|gi(x)-gi(y)x-y|,∀x≠y∈R,1≤i≤n,1≤j≤m.(H11)
The kernel functionskij(s) and k̃ji(s) are nonnegative continuous functions on [0,+∞) and satisfy
(3.4)∫0+∞seλskij(s)ds<+∞,∫0+∞seλsk̃ji(s)ds<+∞,Kij(λ)=∫0+∞eλskij(s)ds,K̃ji(λ)=∫0+∞eλsk̃ji(s)ds,
are differentiable functions for λ∈[0,rij) and λ∈[0,r̃ji); respectively, 0<rij<+∞,0<r̃ji<+∞,  Kij(0)=1,K̃ji(0)=1, limλ→rij-Kij(λ)=+∞ and limλ→r̃ji-K̃ji(λ)=+∞.(H12)
There exists positive integerk0 such that tk+k0=tk+T and γi(k+k0)=γik,δj(k+k0)=δjk hold.We assume that system (3.1) has the following initial conditions:(3.5)xi(s)=φi(s),yj(s)=ϕj(s),i=1,2,…,n,j=1,2,…,m,-∞≤s≤0,
where ψ=(φ,ϕ)∈PC((-∞,0],Rn+m), φ(s)=(φ1(s),φ2(s),…,φn(s)),ϕ(s)=(ϕ1(s),ϕ2(s),…,ϕm(s)).LetZ(t,ψ)=(x(t,ψ),y(t,ψ)) denote any solution of the system (3.1) with initial value ψ=(φ,ϕ)∈PC, x(t,ψ)=(x1(t,ψ),x2(t,ψ),…,x(tn,ψ)), y(t,ψ)=(y1(t,ψ),y2(t,ψ),…,ym(t,ψ)).Theorem 3.1.
Under assumptions (H8)–(H12), there exists a T-periodic solution which is asymptotically stable, if the following conditions hold.(H13)
The followingℳ is a nonsingular M-matrix, and
(3.6)M=(A-C-C̃Ã),
in which
(3.7)A=diag⁡(α̲1,α̲2,…,α̲n),Ã=diag⁡(β̲1,β̲2,…,β̲m),C=(eij)m×n,C̃=(ẽij)n×m,ẽij=(q¯ji+v¯ji)ρ̃iL̃i,eij=(p¯ij+u¯ij)ρjLj.(H14)
ai((1-γik)s)≥|1-γik|ai(s),cj((1-δjk)s)≥|1-δjk|cj(s),∀s∈R,i=1,2,…,n,j=1,2,…,m.Proof.
Let(3.8)xn+j(t)=yj(t),an+j(t,xn+j(t))=cj(t,yj(t)),bn+j(t,xn+j(t))=dj(t,yj(t)),pn+j,i(t)=qji(t),pi,n+j(t)=pij(t),un+j,i(t)=vji(t),ui,n+j(t)=uij(t),Si(xi(t))=gi(xi(t)),Sn+j(xn+j(t))=fj(xj(t)),φn+j(s)=ψj(s),In+j=Jj(t),kn+j,i(s)=k̃ji(s),kij(s)=ki,n+j(s),αn+j(t)=βj(t),L̃n+j=Lj,ρ̃n+j=ρj.
It follows that system (3.1) can be rewritten as
(3.9)ẋi(t)=-ai(xi(t))[bi(t,xi(t))-∑j=1mpi,n+j(t)Sn+j(ρ̃n+jxn+j(t))-∑j=1mui,n+j(t)∫0+∞ki,n+j(s)Sn+j(ρ̃n+jxn+j(t-s))ds-Ii(t)],t≠tk,Δxi(tk)=-γikxi(tk),t=tk,k∈Z+,ẋn+j(t)=-an+j(xn+j(t))[bn+j(t,xn+j(t))-∑i=1npn+j,i(t)Si(ρ̃ixi(t))-∑i=1nun+j,i(t)∫0+∞kn+j,i(s)Si(ρ̃ixi(t-s))ds-In+j(t)],t≠tk,Δxn+j(tk)=-γn+j,kxn+j(tk),t=tk,k∈Z+,
for 1≤i≤n,1≤j≤m.
Initial conditions are given by(3.10)xi(s)=φi(s),s∈(-∞,0],i=1,2,…,(n+m).
Thus system (3.9) is a special case of system (2.1) in mathematical form, under conditions (H8)–(H14), we obtain from Lemma 2.5 that system (3.9) has a T-periodic solution which is globally exponentially stable if ai((1-γik)s)≥|1-γik|ai(s) and the following matrix ℳ′ is a M-matrix, and
(3.11)M′=A′-C′,
where
(3.12)A′=diag⁡(α̲1,α̲2,…,α̲n+m),D′=(0⋯0w1,n+1′⋯w1,n+m′⋮⋮⋮⋮0⋯0wn,n+1′⋯wn,n+m′wn+1,1′⋯wn+1,n′0⋯0⋮⋮⋮⋮wn+m,1′⋯wn+m,n′0⋯0),
in which wij′=(p¯ij+u¯ij)ρ̃jL̃j.
Then, we know from (3.8) and (3.11) that Theorem 3.1 holds.
Ifai(xi(t))=cj(yj(t))=1, bi(t,xi(t))=bi(t)xi(t) and dj(t,yj(t))=dj(t)yj(t), where bi(t) and dj(t) are positive continuous T-periodic functions for i=1,2,…,n,j=1,2,…,m. System (3.1) reduces to the following Hopfield-type BAM neural networks model:
(3.13)ẋi(t)=-bi(t)xi(t)+∑j=1mpij(t)fj(ρjyj(t))+∑j=1muij(t)∫0+∞kij(s)fj(ρjyj(t-s))ds+Ii(t),t>0,t≠tk,Δxi(tk)=-γikxi(tk),t=tk,k∈Z+,ẏj(t)=-dj(t)yj(t)+∑i=1nqji(t)gi(ρ̃ixi(t))+∑i=1nvji(t)∫0+∞k̃ji(s)gi(ρ̃ixi(t-s))ds+Jj(t),t>0,t≠tk,Δyj(tk)=-δjkyj(tk),t=tk,k∈Z+.Corollary 3.2.
Under assumptions  (H9)–(H12), there exists a  T-periodic solution which is globally asymptotically stable, if the following conditions hold. (H13′)
The followingℳ is a nonsingular M-matrix, and
(3.14)M=(A-C-C̃Ã),
in which
(3.15)A=diag⁡(b̲1,b̲2,…,b̲n),Ã=diag⁡(d̲1,d̲2,…,d̲m),C=(eij)m×n,C̃=(ẽij)n×m,eij=(p¯ij+u¯ij)ρjLj,ẽij=(q¯ji+v¯ji)ρ̃iL̃i.(H14′)
0≤γik≤2,0≤δjk≤2 for i=1,2,…,n,j=1,2,…,m,k∈Z+.Proof.
Asbi(t,xi(t))=bi(t)xi(t) and dj(t,yj(t))=dj(t)yj(t), we obtain αi(t)=bi(t) and βj(t)=dj(t) in (H2), (H13′) implies (H13) holds. Since ai(xi(t))=cj(yj(t))≡1, then condition (H14) reduces to (H14′). Corollary 3.2 Holds from Theorem 3.1.Remark 3.3.
The conditions for the existence and globally exponential stability of the periodic solution of (3.1) without impulses have nothing to do with inputs of the neuron and amplification functions. The results in [13, 14] have more restrictions than Theorem 3.1 in this paper because conditions for the ones in [13, 14] are relevant to amplification functions and inputs of neurons our results should be better. In addition, Corollary 3.2 is similar to Theorem  2.1 in [15]; our results generalize the results in [15].Remark 3.4.
In view of proof of Theorem3.1, since CGBAMNNs model is a special case of CGNNs model in form as BAM neural networks model is a special case of Hopfield neural networks model, many results of CGBAMNNs can be directly obtained from the ones of CGNNs, needing no repetitive discussions. Since system (3.1) reduces to autonomous system, Theorem 3.1 still holds, which means that system (3.1) has a equilibrium which is globally asymptotically stable; we know that many results in [18] can be directly obtained from the results in [19].


## 4. Two Simple Examples

Example 4.1.
Consider the following CGNNs model with distributed delays:(4.1)ẋ1(t)=-2[x1(t)-0.2tanh(x1(t))-sin⁡⁡t∫0+∞e-stanh(x2(t-s))ds],ẋ2(t)=-(2+cos⁡(x2(t)))[x2(t)-0.3∫0+∞e-stanh(x1(t-s))ds-5].
Obviously, system (4.1) satisfies (H1)–(H5).
Note that(4.2)M=(1-1.2-0.31),
it is a nonsingular M-matrix and system (4.1) also satisfies condition (H6). According to Lemma 2.5, system (4.1) has a 2π-periodic solution which is globally exponentially stable. Figure 1 shows the dynamic behaviors of system (4.1) with initial condition (0.1,0.2).
However, It is easy to check that system (4.1) does not satisfy Theorem 4.3 or  4.4 in [5], so theorems in [5] cannot are used to ascertain the existence and stability of periodic solutions of system (4.1).Time response of state variablesx1,x2 and phase plot in space (t,x1,x2) for system (4.1).
(a)(b)Example 4.2.
Consider the following CGBAMNNs model with distributed delays and impulses:(4.3)ẋ1(t)=-(2+sin⁡(x1(t)))[2x1(t)-sin⁡t∫0+∞e-s|y1(t-s)|ds-1],t>0,t≠tk,Δx1(tk)=-γ1kxi(tk),t=tk,k∈Z+,ẏ1(t)=-(3+cos⁡(y1(t)))[(3+cos⁡t)y1(t)-sin⁡t∫0+∞e-s|x1(t-s)|ds-1],t>0,t≠tk,Δy1(tk)=-δ1ky1(tk),t=tk,k∈Z+,
where tk=πk,k∈Z+.
Obviously, system (4.3) satisfies (H8)–(H12).Case 1.
γ1k=0,δ1k=0. Note that
(4.4)M=(2-1-12),
it is a nonsingular M-matrix and system (4.3) also satisfies condition (H13). According to Theorem 3.1, system (4.3) without impulses has a 2π-periodic solution which is globally exponentially stable. Figure 2 shows the dynamic behaviors of system (4.3) with initial condition (0.1,0.2).
However, it is easy to check that system (4.3) without impulses does not satisfy Theorem  1 in [13] and theorems in [14]; so theorems in [13, 14] cannot be used to ascertain the existence and stability of periodic solutions of system (4.3).Time response of state variablesx1,y1 and phase plot in space (t,x1,y1) for system (4.3) without impulsive effects.
(a)(b)Case 2.
γ1k=0.7,δ1k=(1-0.5sin(tk+1)). Note that a1(s)=2+sins,c1(s)=3+coss, and a̲1/a¯1=0.5>|1-γ1k|=0.3 and |1-δ1k|<0.5<c̲1/c¯1=2/3, which means condition (H14) also holds for system (4.3). Hence, system (4.3) with impulses still has that there exists a 2π-periodic solution which is globally asymptotically stable. Figure 3 shows the dynamic behaviors of system (4.3) with initial condition (0.1,0.2).
This example illustrates the feasibility and effectiveness of the main results obtained in this paper, and it also shows that the conditions for the existence and globally exponential stability of the periodic solutions of CGBAMNNs without impulses have nothing to do with inputs of the neurons and amplification functions. If impulsive perturbations exist, the periodic solutions still exist and they are globally exponentially stable when we give some restrictions on impulsive perturbations.Time response of state variablesx1,y1 and phase plot in space (t,x1,y1) for system (4.3) with impulsive effects.
(a)(b)


## 5. Conclusions

A class of CGBAMNNs with distributed delays and impulses are investigated by using suitable Lyapunov functional, the properties ofM-matrix, and some suitable mathematical transformation in this paper. Sufficient conditions to guarantee the uniqueness and global exponential stability of the periodic solutions of such networks are established without assuming the boundedness of the activation functions. Lemma 2.5 improves the results in [5], and Theorem 3.1 improves the results in [13, 14] and generalize the results in [15]. In addition, we point that CGBAMNNs model is a special case of CGNNs model; many results of CGBAMNNs can be directly obtained from the ones of CGNNs, needing no repetitive discussions. Our results are new, and two examples have been provided to demonstrate the effectiveness of our results.



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*Source: 643418-2012-04-04.xml*