# Existence and Global Stability of a Periodic Solution for Discrete-Time Cellular Neural Networks

Authors: Haijian Shao; Haikun Wei; Haoxiang Wang
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

A novel sufficient condition is developed to obtain the discrete-time analogues of cellular neural network (CNN) with periodic coefficients in the three-dimensional space. Existence and global stability of a periodic solution for the discrete-time cellular neural network (DT-CNN) are analysed by utilizing continuation theorem of coincidence degree theory and Lyapunov stability theory, respectively. In addition, an illustrative numerical example is presented to verify the effectiveness of the proposed results.


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## Body

## 1. Introduction

Cellular neural networks (CNNs) are the basis of both discrete-time cellular neural networks (DT-CNNs) [1] and the cellular neural networks universal machine (CNNs-UM). The dynamical behaviour of Chua and Yang cellular neural network (CY-CNN) is given by the state equation
(1.1)Cdxijdt=-1Rxij+∑C(k,l)∈Nr(ij)Aklykl+∑C(k,l)∈Nr(ij)Bklukl+Iij,yij=f(xij)=12(|xij+1|-|xij-1|),i=1,…,m,j=1,…,n,
where I, u, y, and x denotes input bias, input, output, and state variable of each cell, respectively. Nr(ij) is the t-neighbourhood of cell C(k,l) as Nr(ij)={C(k,l)∣max{|k-i|,|l-j|}≤r},   i and j denote the position of the cell in the network, k and l denote the position of the neighbour cell relative to the cell in consideration. B is the nonlinear weights template matrices for input feedback and A is the corresponding template matrices for the outputs of neighbour cells. Non-linearity means that templates can change over time.A large number of cellular neural networks (CNNs) models have appeared in the literature [2–4], and these models differ in cell complexity, parameterization, cell dynamics, and network topology. Various generalizations of cellular neural networks have attracted attention of scientific community due to their promising potential for tasks of classification, associative memory, parallel computation [5–9], pattern recognition, computer vision, and solving any optimization problem [10–13]. Such applications rely on the existence of equilibrium points and the qualitative properties of cellular neural networks.Discrete-time cellular neural networks (DT-CNNs) have been studied both in theory and applications. Previous results introduced many properties of DT-CNN in the two dimensional plane. For instance, [14] has been successfully applied to investigate the discrete-time analogues of cellular neural network (CNN) with variable coefficients in the two-dimensional plane. However, three-dimensional structure is more accurate, specific, and closer to real structures of CNN. Based on the above discussion, this paper proposes some effective results of DT-CNN in the three-dimensional space.Motivated by the constructing of continuous system (1.1), the discrete analogue of the system (1.1) is considered as follows:
(1.2)Xij[n+1]=e-hXij[n]+(1-e-h)(∑C(k,l)∈Nr(ij)AklYkl[n]+∑C(k,l)∈Nr(ij)BklUkl+Iij),∀n∈Z0+.For anyh>0, the discrete-time analogues (1.2) converge to the continuous-time system (1.1) will be provided. Without loss of generality, (1.2) can be substituted in the DT-CNNs model:
(1.3)xij[n+1]=e-hxij[n]+(1-e-h)∑C(kijhlijh)∈Nr(kijhlijh)(Akijhlijhykijhlijh[n]+Bkijhlijhukijhlijh)+Iij[n],∀n∈Z0+,xij[n]=φij[n],∀n∈Z0-={0,-1,-2,…}.Then, the spatial structure with respect to (1.3) is shown in Figure 1, where r=maxC(kijhlijh)∈Nr(ijh)(|xkijhlijh-∂Ω|), Ω≜{x∈𝒳⊂Nr(ijh),∥x∥<Θ}, Nr(ijh) is ther-neighbourhood of a cell C(k,l)=Ckijhlijh, and Θ will be denoted by the proof of Theorem 3.1 in Section 3.Figure 1
Spatial structure with respect to (1.3).The rest of the paper is organized as follows: in Section2, system description and preliminaries are developed in detail and some definitions, assumptions, and lemmas are stated. Section 3 gives sufficient conditions for a periodic solution for DT-CNN in three-dimensional space by utilizing continuation theorem of coincidence degree theory. Section 4 proposes global stability of a periodic solution for the DT-CNN. A numerical simulation is given to show correctness of our analysis in Section 5 and concluded in Section 6.


## 2. System Preliminaries and Description

Consider the following model which is equivalent to the (1.3):
(2.1)xij[n+1]=α(h)xij[n]+β(h)∑h=1mh∑j=1mj(Akijhlijhykijhlijh[n]+Bkijhlijhukijhlijh)+Iij[n],∀n∈Z0+,xij[n]=φij[n],∀n∈Z0-={0,-1,-2,…},kijh=k(i,j,h)∈N+,lijh=l(i,j,h)∈N+,i=1,2,…,mi,j=1,2,…,mj,k=1,2,…,mk,
where α(h)=e-h,β(h)=1-α(h), for all h>0, φij(n) are N(h)-periodic sequences, that is, φij(n)=φij(n+N(h)).Throughout the paper, the following definitions and lemmas will be introduced.Definition 2.1 (Fredholm operator).
Let𝒳 and 𝒴 be a Banach space, an operator L is called Fredholm operator if L is a bounded linear operator between 𝒳 and 𝒴 whose kernel and cokernel are finite-dimensional and whose range is closed. Equivalently, an operator L:𝒳→𝒴 is Fredholm if it is invertible modulo compact operator, that is, if there exists a bounded linear operator S:𝒴→𝒳 such that Id𝒳-SL, Id𝒴-LS are compact operators on 𝒳 and 𝒴, respectively, where  Id𝒳 and Id𝒴 are the identity operator.Definition 2.2 (L-compact).
An operatorN will be called L-compact on Ω- if the open bounded set QN(Ω-) is bounded and Kp(I-Q)N:Ω-→𝒳 is compact, where Kp is the inverse operator of N. Since ImQ is isomorphic to KerL, there exists an isomorphism J:ImQ→KerL.The index of a Fredholm operator isindL=dimKerL–codimImL, then operator L will be called a Fredholm operator of index zero if dimKerL=codimImL<+∞ and ImL is closed in 𝒴. Then a following abstract equation in Banach space 𝒳 is defined by
(2.2)Lx=λNx.LetL:DomL⊂𝒳→𝒴 be linear operator, and N:𝒳→𝒴 be a continuous operator. If L is a Fredholm operator of index zero, there must exist continuous projectors P:𝒳→𝒳 and Q:𝒴→𝒴, such that:
(2.3)P:𝒳∩DomL→KerL,KerL=ImP,Q:𝒴→𝒴/ImL,ImL=KerQ.In other words,L|DomL∩KerP:DomL∩KerP→ImL is invertible, and the inverse of the operator L is denoted by Kp.Lemma 2.3 (Gaines and Mawhin [15]).
Let𝒳 be a Banach space, L be a Fredholm operator of index zero, and let N:Ω-→𝒳 be L-compact on Ω-,Ω⊂𝒳, where Ω is an open bounded set, suppose:
(2.4)(i)Lx≠λNx,for  any(x,λ)∈(∂Ω∩DomL)×(0,1);(ii)QNx≠0,for  anyx∈∂Ω∩KerL;(iii)deg(JQN,Ω∩KerL,0)≠0.
Then Lx=Nx has at least one solution in DomL∩Ω-.Lemma 2.4.
If a and b are some certain nonnegative vectors, then there exists a positive constantβ, such that ab≤(β/2)a2+(1/2β)b2.Proof.
Assuminga and b are some certain non-negative vectors, β is a positive constant, then
(2.5)2ab=2aβ(β)-1b≤βa2+1βb2⇒ab≤β2a2+12βb2.
Thus, the proof of Lemma 2.4 is completed.Assumption 2.5.
A
k
i
j
h
l
i
j
h
,
B
k
i
j
h
l
i
j
h
,
I
i
j
(
i
=
1
,
…
,
m
i
,
j
=
1
,
…
,
m
j
,
h
=
1
,
…
,
m
h) are N-periodic sequence of Z0+. For the sake of convenience, we use the following notations: ∥x∥ℝ22=∑j=1mj∑i=1mimaxn∈IN|xij[n]|2. For each operator P:Nr(ijh)→Nr(ijh) and any s=s(u,v,w), t=t(i,j,h)∈Nr(ijh), such that:
(2.6)|P(s)-P(t)|≤(|mi||u-i|2+|mj||v-j|2+|mh||w-h|2)1/2≤2mijh×max{dist(s,oijh),dist(t,oijh)}≤2mijhr,
where oijh is the spherical centre of Nr(ijh) with a radius length r, mijh=max{|mi|,|mj|,|mh|}, then it is easy to obtain: |P(s)-P(t)|≤∥P∥|(s-t)|≤2mijhr<+∞.Assumption 2.6.
There is a positive constantCykl, such that |ykijhlijh[x1]-ykijhlijh[x2]|≤Cykl|x1-x2|, for all x1≠x2∈R.


## 3. Existence of a Periodic Solution with respect to (2.1)

In many cases, many proposed results are not ideal and therefore it is necessary to formulate a novel and effective result for DT-CNN in the three-dimensional space. Can we obtain the result about the existence and stability of a periodic solution for DT-CNN in three-dimensional space? This is the topic we wish to address in this paper. The aim of the present work is to develop a strategy to determine the existence and global stability of a periodic solution with respect to (2.1) in the three-dimensional space. Consequently, we processed with the following result.Theorem 3.1.
Suppose that Assumptions2.5 and 2.6 hold, and the following condition holds:
(3.1)∑h=1mh|B~kijhlijhukijhlijh|2-γij>0,
where γij=∑h=1mh|A~kijhlijhykijhlijh|2-I~2ij, i=1,…,mi, j=1,…,mj, n∈IN={0,1,…,N-1}, then (2.1) has at least one N-periodic solution.Proof.
In this section, by means of using Mawhin’s continuation theorem of coincidence degree theory, we will study the existence of at least one periodic solution with respect to (2.1), for convenience, some following notations will be used:
(3.2)IN={0,1,…,N-1},f_=minn∈IN{|f(n)|},f-=maxn∈IN{|f(n)|},
where f(n) is any function. Let 𝒳=𝒴=x[n]={(x11[n],…,x1mj[n],…,xmi1[n],…,xmimj[n])T: xij[n]=xij[n+N]∈R+mimj,N∈N+,i=1,…,mi,j=1,…,mj}, and yN⊂𝒳=𝒴 be the subspace of all N-periodic sequence; equip it with the norm ∥x∥ℝ22=∑j=1mj∑i=1mimaxn∈IN|xij[n]|2. For any ε>0, {xim}im=1mi⊂Nr(ijh), there exists N(ε)>0 and ε>0, such that im>N(ε)⇒d(xim,xim+1)≤d(xim,oijh)+d(xim+1,oijh)<ε. Thus, {xim}im=1mi is a Cauchy sequence in Nr(ijh) and oijh is the spherical centre of Nr(ijh), d(x,y)=max{|x-y|:x∈X}. By utilizing the meaning of Nr(ijh) and Bolzano-Weierstrass theorem (Each bounded sequence in Rn has a convergent subsequence, here Rmimj⊂Rn, dim(Rmimj)<+∞), it is easy to know that (𝒳,∥·∥) is a Banach space.
Set(3.3)Nx=(x^11[n],…,x^1mj[n],x^21[n],…,x^2mj[n],x^mi1[n],…,x^mimj[n])T,Lx=(Δx11[n],…,Δx1mj[n],Δx21[n],…,Δx2mj[n],Δxmi1[n],…,Δxmimj[n])T,Px=Qx=(x~11[n],…,x~1mj[n],x~21[n],…,x~2mj[n],x~mi1[n],…,x~mimj[n])T,
that is
(3.4)x[n]=(x11[n],…,x1mj[n],…,xmi1[n],…,xmimj[n])T∈𝒳,KerL={x={x[n]}∈yN⊂𝒳:x[n]=c∈Rmimj,n∈IN},ImL={x={x[n]}∈yN⊂𝒳:∑n=0N-1xij[n]=0,i=1,…,mi,j=1,…,mj,n∈IN},
where Δxij[n]=xij[n+1]-xij[n]=-β(h)[xij[n]-∑h=1mh∑j=1mj(Akijhlijhykijhlijh[n]+Bkijhlijhukijhlijh)] + Iij[n],x^ij[n]=-β(h)[xij[n]-∑h=1mh∑j=1mj(Akijhlijhykijhlijh[n]+Bkijhlijhukijhlijh)]+Iij[n],x~=(1/N)∑n=0N-1x[n],  for all n∈IN. Then we will learn that dimKerL=codimImL<+∞, it is easy to prove that L is a bounded linear operator, P and Q are two continuous operators such that KerL=ImP, ImL=KerQ=Im(I-Q), and Kp|ImL:ImL→DomL∩KerP, that is
(3.5)Kp(x[n])=∑s=0N-1x(s)-1N∑s=1N∑t=1s-1x(t),n∈IN,
that is
(3.6)QNx=β(h)1+β(h)×(∑h=1mh(A~k11hl11hyk11hl11h[n]+B~k11hl11huk11hl11h)+(1+(β(h))-1)I~11⋮∑h=1mh(A~k1mjhl1mjhyk1mjhl1mjh[n]+B~k1mjhl1mjhuk1mjhl1mjh)+(1+(β(h))-1)I~1mj⋮∑h=1mh(A~kmi1hlmi1hykmi1hlmi1h[n]+B~kmi1hlmi1hukmi1hlmi1h)+(1+(β(h))-1)I~mi1⋮∑h=1mh(A~kmimjhlmimjhykmimjhlmimjh[n]+B~kmimjhlmimjhukmimjhlmimjh)+(1+(β(h))-1)I~mimj)n∈IN,Kp(I-Q)Nx=-β(h)×(δ∑n=0N-1([x11[n]-∑h=1mh(Ak11hl11hyk11hl11h[n]+Bk11hl11huk11hl11h)]+(β(h))-1I11[n])-1N∑s=1N∑t=1s-1([x11[t]-∑h=1mh(Ak11hl11hyk11hl11h[t]+Bk11hl11huk11hl11h)]+(β(h))-1I11[n])⋮δ∑n=0N-1([xmimj[n]-∑h=1mh(Akmimjhlmimjhykmimjhlmimjh[n]+Bkmimjhlmimjhukmimjhlmimjh)]+(β(h))-1Ikmimjhlmimjh[n]([xmimj[n]-∑h=1mh(Akmimjhlmimjhykmimjhlmimjh[n]+Bkmimjhlmimjhukmimjhlmimjh)])miAkmimjhlmimjhBkmimjhlmimjhukmimjhlmimjh-1N∑s=1N∑t=1s-1([xmimj[t]-∑h=1mh(Akmimjhlmimjhykmimjhlmimjh[t]+Bkmimjhlmimjhukmimjhlmimjh)]+(β(h))-1Ikmimjhlmimjh[n][xmimj[t]-∑h=1mh(Akmimjhlmimjhykmimjhlmimjh[t]+Bkmimjhlmimjhukmimjhlmimjh)][xmimj[t]-∑h=1mh(Akmimjhlmimjhykmimjhlmimjh[t]+Bkmimjhlmimjhukmimjhlmimjh)]∑h=1mh)mimjAkmimjhlmimjhBkmimjhlmimjhukmimjhlmimjh)n∈IN,
where δ(β(h),n,N) is a constant, which is only depended on variables h, n, and N.
Obviously, employing the Lebesgue’s convergence theorem, we can easily learn thatQN(Ω-) is bounded, Kp(I-Q)N(Ω-) is compact for any open bounded set Ω⊂𝒳⊂Nr(ijh) by using Ascoli-Arzela’s theorem (A subset ℱ of 𝒞(𝒳) is compact if and only if it is closed, bounded and equi-continuous). Thus, N is L-compact on a closed set Ω- with any open bounded set Ω⊂𝒳⊂Nr(ijh).
Suppose thatx[n]=(x11[n],…,x1mj[n],…,xmi1[n],…,xmimj[n])T∈𝒳 is a solution with respect to (2.1), for certain λ∈(0,1). Then the following equation can be derived by (2.2):
(3.7)Δxij[n]=xij[n+1]-xij[n]=-λ{β(h)[xij[n]-∑h=1mh∑j=1mj(Akijhlijhykijhlijh[n]+Bkijhlijhukijhlijh)]+Iij[n]},∀n∈IN.
Then, the following results can be derived by utilizing (3.7):
(3.8)maxn∈IN|xij[n]|=maxn∈IN|xij[n+1]|≤maxn∈IN[∑h=1mh∑j=1mj(|Akijhlijhykijhlijh[n]|+|Bkijhlijhukijhlijh|)(1-λβ(h))|xij[n]|+λβ(h)×∑h=1mh∑j=1mj(|Akijhlijhykijhlijh[n]|+|Bkijhlijhukijhlijh|)+|λIij[n]|]≤maxn∈IN(1-λβ(h))n|φij[0]|+(λ(1-α(h)))n×∑h=1mh∑j=1mj(|A-kijhlijhykijhlijh[0]|)+λnI-ij+σij≤maxn∈IN{|φij[0]|+(1-α(h))n∑h=1mh∑j=1mjA-kijhlijh}+σij<+∞,
where σij=maxn∈IN(1-α(h))∑h=1mh∑j=1mj|B-kijhlijhu-kijhlijh|+I-ij,foralln∈IN. Therefore, the solution with respect to (2.1) is bounded for certain λ∈(0,1). In other words,
(3.9)maxn∈IN|xij[n]|≤maxn∈IN{|φij[0]|+(1-α(h))n∑h=1mh∑j=1mj(|A-kijhlijhykijhlijh[0]|)+(1-α(h))∑h=1mh∑j=1mj|B-kijhlijhu-kijhlijh|+I-ij}≜Θij.
Then the open bounded setΩ is presented as follows:
(3.10)Ω≜{x∈𝒳⊂Nr(ijh),∥x∥<∑i=1mi∑j=1mjΘij}.
ThusLx≠λNx for any (x,λ)∈(∂Ω∩DomL)×(0,1), the Ω satisfies condition (i) in Lemma 2.3.
In Figure2, the nonlinear weights template matrices B and the boundary of Ω are shown, respectively. Then for any two dimensional plane of any spherical neighbourhood is denoted. Thus, for any x∈∂Ω∩KerL=∂Ω∩Rmimj, KerL={x={x[n]}∈yN⊂𝒳:x[n]=c∈Rmimj,n∈IN}, it is easy to learn that x is a constant vector in Rmimj with ∥x∥=Θ; Thus, we have
(3.11)QNx=(QNx11,…,QNx1mj,…,QNxmi1,…,QNxmimj)T,
where QNxij[n]=(β(h)/(1+β(h)))∑h=1mh∑j=1mj(A~kijhlijhykijhlijh[n]+B~kijhlijhukijhlijh)+I~ij[n],foralln∈IN. Furthermore, we can calculate the bound of QNx as follows:
(3.12)∥QNx∥2=∑j=1mj∑i=1mimaxn∈IN|xij[n]|2=∑j=1mj∑i=1mimaxn∈IN|β(h)1+β(h)∑h=1mh∑j=1mj(A~kijhlijhykijhlijh[n]+B~kijhlijhukijhlijh)+I~ij[n]|2≥∑j=1mj∑i=1mi|β(h)1+β(h)|2×maxn∈IN{∑h=1mh|B~kijhlijhukijhlijh|2+I~2ij[n]-∑h=1mh|A~kijhlijhykijhlijh|2}≥∑j=1mj∑i=1mi(maxn∈IN(∑h=1mh|B~kijhlijhukijhlijh|2-γij))>0,
where γij=∑h=1mh|A~kijhlijhykijhlijh|2-I~2ij, i=1,…,mi, j=1,…,mj, n∈IN. Thus for any x=∂Ω∩KerL, QNx≠0, this proves the condition (ii) in Lemma 2.3.
In order to prove the condition (iii) is satisfied with respect to (2.1), we only need to prove that deg(JQN,Ω∩KerL,0)≠0. Define Φ:DomL→𝒳 by
(3.13)Φ(x11,…,x1mj,…,xmi1,…,xmimj)T=(x~11,…,x~1mj,…,x~mi1,…,x~mimj)T,
where x~ij[n]=(β(h)/(1+β(h)))∑h=1mh∑j=1mj(A~kijhlijhykijhlijh[n]+B~kijhlijhukijhlijh)+I~ij[n],  for  all n∈IN.
Now we will prove thatx∈∂Ω∩KerL=∂Ω∩Rmimj, Φ(x11,…,x1mj,…,xmi1,…,xmimj)T≠(0,0,…,0)T. If this is not true, then x∈∂Ω∩KerL=∂Ω∩Rmimj, Φ(x11,…,x1mj,…,xmi1,…,xmimj)T=(0,0,…,0)T, thus, for constant vector x∈∂Ω, we have:
(3.14)x~ij[n]=β(h)1+β(h)∑h=1mh∑j=1mj(A~kijhlijhykijhlijh[n]+B~kijhlijhukijhlijh)+I~ij[n]=0.
Equivalently, (3.14) can be written as the following form:
(3.15)∑h=1mh∑j=1mj(A~kijhlijhykijhlijh[n]+B~kijhlijhukijhlijh)=-1+β(h)β(h)I~ij[n]⇒∑h=1mh∑j=1mjA~kijhlijhykijhlijh[n]=-(1+β(h)β(h)I~ij[n]+∑h=1mh∑j=1mjB~kijhlijhukijhlijh).
Combining (3.12) and (3.15), the following results are obtained:
(3.16)maxn∈IN∑h=1mh∑j=1mj|A~kijhlijhykijhlijh|=maxn∈IN|1+β(h)β(h)I~ij+∑h=1mh∑j=1mjB~kijhlijhukijhlijh|.
Thus, the following result is derived by calculating the (3.16):
(3.17)|1+β(h)β(h)I~ij+∑h=1mh∑j=1mjB~kijhlijhukijhlijh|-∑h=1mh∑j=1mj|B~kijhlijhukijhlijh|≤|1+β(h)β(h)I~ij|<|I~ij|<maxn∈IN{∑h=1mh∑j=1mj|B~kijhlijhukijhlijh|+I~ij}.
Obviously, (3.17) is a contradiction since ((1+β(h))/β(h))>1, then for any x=∂Ω∩KerL, KerL=ImP, Px=(x~11[n],…,x~1mj[n],x~mi1[n],…,x~mimj[n])T≠(0,…,0)T, Φ(x11,…,x1mj,…,xmi1,…,xmimj)T≠(0,…,0)T. Thus, deg(JQN,Ω∩KerL,0)≠0. Therefore, (2.1) has at least one N-periodic solution, thus the proof of Theorem 3.1 is completed.Figure 2
InputB-template and the boundary of Ω.Corollary 3.2.
Suppose that Assumptions2.5 and 2.6 hold, and the following condition holds:
(3.18)∑h=1mh|A~kijhlijhykijhlijh|2-ηij>0,
where ηij=∑h=1mh|B~kijhlijhukijhlijh|2-I~2ij,  i=1,…,mi, j=1,…,mj, n∈IN, then (2.1) has at least one N-periodic solution.Proof.
Similar to the proof of Theorem3.1, so it is omitted.


## 4. Globally Stability of a Periodic Solution with respect to (2.1)

The existence of a periodic solution for the system (2.1) is derived in the Theorem 3.1. Then global stability of a periodic solution with respect to (2.1) in the three-dimensional space is presented in the following.Theorem 4.1.
Suppose that Assumptions2.5 and 2.6 hold, and the following condition holds:
(4.1)∑h=1mh∑j=1mjA-kijhlijh≤1Cyklmhmj,
where i=1,2,…,mi, Cykl,mj and mh are positive constants, then the periodic solution with respect to (2.1) is global stability.Proof.
It follows from the Theorem3.1 that (2.1) has at least a periodic solution, without loss of generality, the periodic solution can be described by:
(4.2)x*[n]=(x11*[n],…,x1mj*[n],…,xmi1*[n],…,xmimj*[n])T∈𝒳.
Then we can define the following formula:(4.3)uij[n]=|xij[n]-xij*[n]|,i=1,…,mi,j=1,…,mj.
Now, we show that the a periodic solutionx*[n] is globally stable, and the following inequality is obtained by utilizing (2.1) and (4.3):
(4.4)uij[n+1]=|xij[n+1]-xij*[n+1]|≤{α(h)|xij[n]-xij*[n]|+Cyklβ(h)∑h=1mh∑j=1mjAkijhlijh|ykijhlijh[n]-ykijhlijh*[n]|}≤{α(h)[|xij[n]-xoijh[n]|+|xoijh[n]-xij*[n]|]+Cyklβ(h)∑h=1mh∑j=1mjAkijhlijh×[|xkijhlijh[n]-xoijh[n]|+|xoijh[n]-xkijhlijh*[n]|]∑h=1mh∑j=1mjA-kijhlijh}≤2r(α(h)+Cyklβ(h)mhmj∑h=1mh∑j=1mjA-kijhlijh)<+∞.
We design the following Lyapunov-type sequenceV[n] by
(4.5)V[n]=∑i=1mi∑j=1mj|xij[n]-xij*[n]|.
Then, we can calculate theΔV[n] by combining (2.1) and (4.5):
(4.6)ΔV[n]=V[n+1]-V[n]=∑i=1mi∑j=1mj(|xij[n+1]-xij*[n+1]|-|xij[n]-xij*[n]|)≤∑i=1mi∑j=1mj{-β(h)|xij[n]-xij*[n]|+Cyklβ(h)∑h=1mh∑j=1mjAkijhlijh|ykijhlijh[n]-ykijhlijh*[n]|}≤∑h=1mh∑j=1mj2rβ(h)(Cyklmhmj∑h=1mh∑j=1mjA-kijhlijh-1)≤0.
Thus, it is easy to obtainV[n]≤V[0] by the meaning of the (4.6), and furthermore,
(4.7)∑i=1mi∑j=1mj|xij[n]-xij*[n]|≤β(h){Cyklmhmj∑h=1mh∑j=1mjA-kijhlijh-1}sups∈Z0-d[s]⇒∑i=1mi∑j=1mj|xij[n]-xij*[n]|≤β(h){Cyklmhmj∑h=1mh∑j=1mjA-kijhlijh-1}sups∈Z0-d[s]≤0⇒xij[n]=xij*[n],n∈IN={0,1,….,N-1},
where d[s]=maxs∈Z0-{|xkijhlijh[s]-xkijhlijh*[s]|,|xij[s]-xij*[s]|}. Obviously, from the proof of Theorem 4.1, the globally stable of a periodic solution with respect to (2.1) is derived. Then, existence and global stability of a periodic solution for DT-CNNs are obtained by utilizing the conditions of the proposed theorems in an arbitrary diameter plane of a convex space. Thus the proof of Theorem 4.1 is completed.


## 5. Numerical Simulation

In this section, we give an example to show the effectiveness and improvement of the derived results. Consider the following continuous cellular neural networks:(5.1)Cdxijdt=-1Rxij+∑C(k,l)∈Nr(ij)Aklykl+∑C(k,l)∈Nr(ij)Bklukl+Iij,yij=f(xij)=12(|xij+1|-|xij-1|),i=1,2,3,j=1,2,
for t>0, where C=1,R=1/2,ykl=(1/2)(|xij+1|-|xij-1|),Ak111l111=Ak311l311=Ak321l321 = 0.1sin(8πt),Ak121l121=Ak211l211=Ak221l221=0.1cos(8πt),Bki11li11uki11li11=-0.06+0.08sin(8πt),Bki21li21uki21li21=-0.02+0.06cos(8πt),Ii1=sin(8πt),Ii2=cos(8πt),i=1,2,3,x[0]=(5.386,-4.836,8.863,-3.683,6.386,-4.836)T. Then, state trajectories of xij(i=1,2,3,j=1,2) are denoted in Figure 3.Figure 3
State trajectories of the neuronsx11, x12, x21, x22, x31, x32.From Figure3, it is easy to know that a (1/4)-periodic solution of the continuous cellular neural networks is globally stable. Compared to the system (5.1), we design the discrete-time analogue of the continuous cellular neural network as follows:
(5.2)xij[n+1]=α(h)xij[n]+β(h)∑h=1mh∑j=1mj(Akijhlijhykijhlijh[n]+Bkijhlijhukijhlijh)+Iij[n],n∈Z0+,i=1,2,3,j=1,2,x[0]=(x11[0],x12[0],x21[0],x22[0],x31[0],x32[0])T=(5.386,-4.836,8.863,-3.683,6.386,-4.836)T,
for h>0, by using Assumptions 2.5 and 2.6 in Section 2, each variable is denoted as:
(5.3)α(h)=e-h,β(h)=1-e-h,Ak111l111=Ak311l311=Ak321l321=120(1-e-2h)sin(8πnh),ykijhlijh[n]=12(|xkijhlijh[n]+1|-|xkijhlijh[n]-1|),Ak121l121=Ak211l211=Ak221l221=120(1-e-2h)sin(8πnh),Ii1=120(1-e-2h)sin(8πnh),Ii2=120(1-e-2h)cos(8πnh),Bki11li11uki11li11=-0.02+0.08sin(8πnh),i=1,2,3,Bki21li21uki21li21=-0.08+0.06cos(8πnh),i=1,2,3.
The derived results of this paper are verified by the following steps.(1) According to the illustrations of the neighbourhood distancer for cell C(k,l)=Ckijhlijh which is given by Nr(ijh) function, and by (3.9) and (3.10), the exact values of distance r and Ω are illustrated as:
(5.4)Ω≜{x∈𝒳⊂Nr(ijh),∥x∥<∑i=13∑j=12Θij},maxn∈IN|xij[n]|≤maxn∈IN{|φij[0]|+(1-α(h))n∑h=11∑j=12(|A-kijhlijhykijhlijh[0]|)+(1-α(h))∑h=11∑j=12|B-kijhlijhu-kijhlijh|+I-ij}≜Θij,i=1,2,3.Then,Θij(i=1,2,3, j=1,2) is calculated below,
(5.5)maxn∈IN|x1j[n]|≤maxn∈IN{|φ1j[0]|+(1-α(h))n∑h=11∑j=12(|A-k1jhl1jhyk1jhl1jh[0]|)+(1-α(h))∑h=11∑j=12|B-k1jhl1jhu-k1jhl1jh|+I-1j}≤5.386+4.836+0.05×4+0.24=10.662≜Θ1j,j=1,2,maxn∈IN|x2j[n]|≤maxn∈IN{|φ2j[0]|+(1-α(h))n∑h=11∑j=12(|A-k2jhl2jhyk2jhl2jh[0]|)+(1-α(h))∑h=11∑j=12|B-k2jhl2jhu-k2jhl2jh|+I-2j}≤8.863+3.683+0.05×4+0.24=12.986≜Θ2j,j=1,2,maxn∈IN|x3j[n]|≤maxn∈IN{|φ3j[0]|+(1-α(h))n∑h=11∑j=12(|A-k3jhl3jhyk3jhl3jh[0]|)+(1-α(h))∑h=11∑j=12|B-k3jhl3jhu-k3jhl3jh|+I-3j}≤6.386+4.836+0.05×4+0.24=11.662≜Θ3j,j=1,2.Thus, the subsetΩ of function Nr(ijh) is derived by the following:
(5.6)Ω≜{x∈𝒳⊂N35.31(ijh),∥x∥<∑i=13∑j=12Θij=35.31}.(2) We will verify the condition of Theorem3.1 if we want to utilize Theorem 4.1. After strictly calculating the condition of Theorem 3.1, it is easy to obtain that the function [n]=|B~kij1lij1ukij1lij1|2-|A~kij1lij1ykij1lij1|2+I~2ij∈R+, i=1,2,3, j=1,2, n∈IN; therefore, the condition of the Theorem 3.1 is critically satisfied as well.(3) According to (4.1), the condition of the Theorem 4.1 will be derived as follows:
(5.7)∑h=1mh∑j=1mjA-kijhlijh<310<11×1×2=1Cyklmhmj.
Then state trajectories of neurons x11, x12, x21, x22, x31, x32 are shown in Figures 4 and 5.Figure 4
State trajectories of neuronsx11, x12, x21, x22, x31, x32(h=1/4).Figure 5
State trajectories of neuronsx11, x12, x21, x22, x31, x32(h=1/2).From Figures4 and 5, we can learn that all the periodic solution converges to a unique a 1/4h-periodic solution, then the DT-CNN (2.1) has a globally stable 1/4h-periodic solution. Thus, all conditions of Theorems 3.1 and 4.1 are strictly satisfied; therefore all conditions of proposed theorems are critically verified.


## 6. Conclusions

Existence and global stability are important dynamical properties in CNN. In this paper, we consider the discrete-time analogues of CNN with periodic coefficients and obtain some new results for the DT-CNN in the three-dimensional space. Comparisons between our results and the previous results have also been made. And it has been demonstrated that our criteria are more general and effective than those reported in the literature.



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*Source: 645214-2012-12-10.xml*