# Stochastic Dynamics of Discrete-Time Fuzzy Random BAM Neural Networks with Time Delays

Authors: Sufang Han; Tianwei Zhang; Guoxin Liu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

By using the semidiscrete method of differential equations, a new version of discrete analogue of stochastic fuzzy BAM neural networks was formulated, which gives a more accurate characterization for continuous-time stochastic neural networks than that by the Euler scheme. Firstly, the existence of the2p-th mean almost periodic sequence solution of the discrete-time stochastic fuzzy BAM neural networks is investigated with the help of Minkowski inequality, Hölder inequality, and Krasnoselskii’s fixed point theorem. Secondly, the 2p-th moment global exponential stability of the discrete-time stochastic fuzzy BAM neural networks is also studied by using some analytical skills in stochastic theory. Finally, two examples with computer simulations are given to demonstrate that our results are feasible. The main results obtained in this paper are completely new, and the methods used in this paper provide a possible technique to study 2p-th mean almost periodic sequence solution and 2p-th moment global exponential stability of semidiscrete stochastic fuzzy models.


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## Body

## 1. Introduction

In [1, 2], Kosko introduced the bidirectional associative memory (BAM) neural networks, which have been widely applied in psychophysics, parallel computing, perception, robotics, adaptive pattern recognition, associative memory, image processing pattern recognition, combinatorial optimization, and so on. All of these applications heavily depend on the (almost) periodicity and global exponential stability [3–14]. In the past twenty years, a large, fast growing body of investigations focused on the existence and global exponential stability of the equilibrium point, periodic, and almost periodic solutions of BAM neural networks with time delays in literatures [15–27].Fuzzy theory was conceived in the 1960s by L. A. Zadeh, and it took about 20 years until the broader use of this theory in practice. Fuzzy technology joined forces with artificial neural networks and genetic algorithms under the title of computational intelligence or soft computing. In recent years, the research on the dynamical behaviours of fuzzy neural networks has attracted much attention (see [28–33]). On the other hand, uncertain models described by the stochastic differential equations have caused great concern since uncertain models have widely applications in practice such as engineering, physics, chemistry, and biology [34–38]. In the actual situations, random factors have consequences on the performance of the neural networks. In neural networks, the connection weights of the neurons depend on certain resistance and capacitance values that include modeling errors or uncertainties during the parameter identification process. Therefore, it is worth studying the dynamical behaviors of fuzzy stochastic neural networks.The discrete-time neural networks become more important than the continuous-time counterparts when implementing the neural networks in a digital way. In order to investigate the dynamical characteristics with respect to digital signal transmission, it is essential to formulate the discrete analogue of neural networks. Mohamad and Gopalsamy [39, 40] proposed a novel method (i.e., semidiscretization technique) in formulating a discrete-time analogue of the continuous-time neural networks, which faithfully preserved the characteristics of their continuous-time counterparts. With the help of the semidiscretization technique [39], many scholars obtained the semidiscrete analogue of the continuous-time neural networks and some meaningful results were gained for the dynamical behaviors of the semidiscrete neural networks, such as periodic solutions, almost periodic solutions and global exponential stability (see [41–47]).For instance, Huang et al. [44] discussed the almost periodic dynamics of the following semidiscrete cellular neural networks:(1)xik+1=e−aikxik+1−e−aikaik∑j=1nbijkfjxjk+Iik,where k∈ℤ, in which ℤ denotes the set of integers, and i=1,2,…,n.In [45], Huang et al. considered the following semidiscrete models for a class of general neural networks:(2)xik+1=e−aikxik+1−e−aikaik∑l=1m∑j=1nbijlkfjxjk−τijl+Iik,where k∈ℤ and i=1,2,…,n. The authors [45] derived the existence of locally exponentially convergent 2N almost periodic sequence solutions of system (2).By careful observation, it easily discovers that the disquisitive models in literatures [41–47] are deterministic, such as (1) and (2). Stimulated by this point, it is necessary to consider random factors in the determinant models. Therefore, this paper considers the semidiscrete models for the following stochastic fuzzy BAM neural networks:(3)dxit=−aitxit+∑j=1mbijtfjyjt+∑j=1mcijtfjyjt−τijt+∧j=1mαijfjyjt−τijt+∧j=1mTijμj+∨j=1mβijfjyjt−τijt+∨j=1mSijμj+Iitdt+∑j=1mdijtσjyjt−ηijtdwjt,dyjt=−a˜jtyjt+∑i=1nb˜jitf˜ixit+∑i=1nc˜jitf˜ixit−τ˜jit+∧i=1nα˜jif˜ixit−τ˜jit+∧i=1nT˜jiμ˜i+∨i=1nβ˜jif˜ixit−τ˜jit+∨i=1mS˜jiμ˜i+Jjtdt+∑i=1nd˜jitσ˜ixit−η˜jitdw˜it.System (3) is composed of two layers, that is, X-layer and Y-layer. xit denotes the membrane potentials of the set of n neurons in X-layer, and yjt denotes the membrane potentials of the set of m neurons in Y-layer at time t; fj and f˜i represent the measures of activation to its incoming potentials of the unit j from Y-layer and the unit i from X-layer, respectively; bij corresponds to the synaptic connection weight of the unit j on the unit i, and b˜ji corresponds to the synaptic connection weight of the unit i on the unit j; Ii and Jj signify the bounded external bias or input; ai and a˜j denote rate with which the ith unit and jth unit will reset their potentials to the resting state in isolation when it is disconnected from the network and external inputs, respectively; αij, βij, Tij, Sij α˜ji, β˜ji, T˜ji, and S˜ji are elements of fuzzy feedback MIN, MAX template, fuzzy feed forward MIN, and MAX template, respectively; ∧ and ∨ denote the fuzzy AND and fuzzy OR operation, respectively; wj and w˜i are the standard Brownian motions, where i=1,2,…,n and j=1,2,…,m. The other coefficients in system (3) are similarly specified.The main aim of this paper is to investigate the dynamics of the semidiscrete analogue of system (3) by using semidiscretization technique [39] and stochastic theory. The main contributions of this paper are summed up as follows: (1) the semidiscrete analogue is established for stochastic fuzzy BAM neural networks (3); (2) a Volterra additive equation is derived for the solution of the semidiscrete model; (3) the existence of 2p-th mean almost periodic sequence solutions is obtained; (4) a decision theorem is acquired for the 2p-th moment global exponential stability; and (5) the methods used in this article can be applied to study the dynamics of other discrete stochastic fuzzy models.The work of this paper is a continuation of literatures [44–47], and the results in this paper complement the corresponding results in [44–47]. The paper is organized as follows. In Section 2, the discrete analogue of system (3) is established and some useful lemmas are given. In Section 3, we employ Krasnoselskii’s fixed point theorem to obtain sufficient conditions for the existence of at least one 2p-th mean almost periodic sequence solution. In Section 4, we consider the 2p-th moment global exponential stability. Two illustrative examples and computer simulations are given in Section 5. In Section 6, the conclusions and future works of this paper are presented.Throughout this paper, we use the following notations. Letℝ denote the set of real numbers. ℝn denotes the n-dimensional real vector space. Let Ω,ℱ,P be a complete probability space. Denote by BCℤ,LpΩ,ℝn the vector space of all bounded continuous functions from ℤ to LpΩ,ℝn, where LpΩ,ℝn denotes the collection of all p-th integrable ℝn-valued random variables. Then, BCℤ,LpΩ,ℝn is a Banach space with the norm Xp=supk∈ℤXp, Xp=max1≤i≤nExikp1/p, ∀X=xi≔x1,x2,…,xn∈BCℤ,LpΩ,ℝn, where p>1 and E⋅ stands for the expectation operator with respect to the given probability measure P. a,bℤ=a,b∩ℤ, ∀a,b∈ℝ.


## 2. Model Formulation and Preliminaries

### 2.1. Discrete Analogue of System (3)

Consider the following stochastic functional differential equations:(4)dxt=−atxtdt+Ft,xt,xt−τtdt+Gt,xt,xt−ηtdwt,which yields the following stochastic functional differential equations with piecewise constant arguments:(5)dxt=−atxtdt+Ft,xt,xt−τtdt+Gt,xt,xt−ηtΔwtdt,where t denotes the integer part of t. Here, the discrete analogue of the stochastic parts of system (4) is obtained by the Euler scheme, i.e., dwt=Δwtdt=wt+1−wtdt. For t∈ℝ, there exists an integer k∈ℤ such that k≤t<k+1. Then, the above equation becomes(6)dxt=−akxtdt+Fk,xk,xk−τkdt+Gk,xk,xk−ηkΔwkdt.Integrating the above equation fromk to t and letting t⟶k+1, we achieve the discrete analogue of system (4) as follows:(7)xk+1=e−akxk+1−e−akakFk,xk,xk−τk+Gk,xk,xk−ηkΔwk,where k∈ℤ. System (7) is the discrete analogue of system (4) by the semidiscrete method. By a similar discussion as that in system (7), we obtain the semidiscrete analogue of system (3) as follows:(8)xik+1=e−aikxik+1−e−aikaik∑j=1mbijkfjyjk+∑j=1mcijkfjyjk−τijk+∧j=1mαijfjyjk−τijk+∧j=1mTijμj+∨j=1mβijfjyjk−τijk+∨j=1mSijμj+Iik+∑j=1mdijkσjyjk−ηijkΔwjk,yjk+1=e−a˜jkyjk+1−e−a˜jka˜jk∑i=1nb˜jikf˜ixik+∑i=1nc˜jikf˜ixik−τ˜jik+∧i=1nα˜jif˜ixik−τ˜jik+∧i=1nT˜jiμ˜i+∨i=1nβ˜jif˜ixik−τ˜jik+∨i=1nS˜jiμ˜i+Jjk+∑i=1nd˜jikσ˜ixik−η˜jikΔw˜ik,k∈ℤ,where i=1,2,…,n and j=1,2,…,m.


### 2.2. Volterra Additive Equation for the Solution of System (8)

Lemma 1.
Assume thatX=xi is a solution of system (8), and then X can be expressed as(9)xik=∏s=k0k−1e−aisxik0+∑v=k0k−1∏s=v+1k−1e−ais1−e−aivaiv∑j=1mbijvfjyjv+∑j=1mcijvfjyjv−τijv+∧j=1mαijfjyjv−τijv+∧j=1mTijμj+∨j=1mβijfjyjv−τijv+∨j=1mSijμj+Iiv+∑j=1mdijvσjyjv−ηijvΔwjv,yjk=∏s=k0k−1e−a˜jsyjk0+∑v=k0k−1∏s=v+1k−1e−a˜js1−e−a˜jva˜jv∑i=1nb˜jivf˜ixiv+∑i=1nc˜jivf˜ixiv−τ˜jiv+∧i=1nα˜jif˜ixiv−τ˜jiv+∧i=1nT˜jiμ˜i+∨i=1nβ˜jif˜ixiv−τ˜jiv+∨i=1nS˜jiμ˜i+Jjv+∑i=1nd˜jivσ˜ixiv−η˜jivΔw˜iv,where k0∈ℤ, k∈k0,+∞ℤ, i=1,2,…,n, and j=1,2,…,m.Proof.
Let(10)Fik,x≔∑j=1mbijkfjyjk+∑j=1mcijkfjyjk−τijk+∧j=1mαijfjyjk−τijk+∧j=1mTijμj+∨j=1mβijfjyjk−τijk+∨j=1mSijμj+Iik+∑j=1mdijkσjyjk−ηijkΔwjk,k∈ℤ,i=1,2,…,n.
ByΔukvk=Δukvk+uk+1Δvk and the first equation of system (8), it gets(11)Δ∏s=0k−1eaisxik=∏s=0keais1−e−aikaikFik,x,where i=1,2,…,n and k∈ℤ. So,(12)∑v=k0k−1Δ∏s=0v−1eaisxiv=∑v=k0k−1∏s=0veais1−e−aivaivFiv,xis equivalent to(13)∏s=0k−1eaisxik=∏s=0k0−1eaisxik0+∑v=k0k−1∏s=0veais1−e−aivaivFiv,x,where i=1,2,…,n and k∈ℤ. By the above equations, we can easily derive the first equation of system (9). Similarly, we can obtain the second equation of system (9). This completes the proof.


### 2.3. Some Lemmas

Lemma 2 (Minkowski inequality) [48].
Assume thatp≥1, Eξp<∞, Eηp<∞, and then(14)Eξ+ηp1/p≤Eξp1/p+Eηp1/p.Lemma 3 (Hölder inequality) [48].
Assume thatp>1, and then(15)∑kakbk≤∑kak1−1/p∑kakbkp1/p.
Ifp=1, then ∑kakbk≤∑kaksupkbk.Lemma 4 [35].
Suppose thatg∈L2a,b,ℝ, and then(16)Esupt∈a,b∫atgsdωsp≤CpE∫abgt2dtp/2,where(17)Cp=32pp/2,0<p<2,4,p=2,pp+12p−1p−1p/2,p>2.Lemma 5.
Assume thatxk:k∈ℤ is real-valued stochastic process and wk is the standard Brownian motion, and then(18)ExkΔwkp≤CpExkp,k∈ℤ,where Cp is defined as that in Lemma 4, in which p>0.Proof.
By Lemma 4, it follows that(19)ExkΔwkp=E∫kk+1xkdwsp≤CpE∫kk+1x2kdsp/2≤CpExkp,k∈ℤ.
This completes the proof.Lemma 6 [49].
SupposeX=xi and Y=yi are two states of system (8), and then we have(20)∧j=1nαijfjxj−∧j=1nαijfjyj≤∑j=1nαijfjxj−fjyj,∨j=1nβijfjxj−∨j=1nβijfjyj≤∑j=1nβijfjxj−fjyj,i=1,2,…,n.Lemma 7 [50].
Assume thatΛ is a closed convex nonempty subset of a Banach space X. Suppose further that B and C map Λ into X such that(1)x,y∈Λ implies that Bx+Cy∈Λ(2)B is continuous and BΛ is contained in a compact set(3)C is a contraction mapping
Then, there exists az∈Λ with z=Bz+Cz.Setfu=supk∈ℤfk and fl=infk∈ℤfk for bounded function f defined on ℤ. Throughout this paper, suppose that the following conditions are satisfied:H1 ai and a˜j are bounded sequences defined on ℤ with ail>0 and a˜jl>0, i=1,2,…,n, and j=1,2,…,m.H2 There exists several constants Ljf, Ljσ, Ljf˜, and Ljσ˜ such that(21)fju−fjv≤Ljfu−v,σju−σjv≤Ljσu−v,f˜iu−f˜iv≤Ljf˜u−v,σ˜ju−σ˜jv≤Ljσ˜u−v,for allu,v∈ℝ, i=1,2,…,n, and j=1,2,…,m.


## 2.1. Discrete Analogue of System (3)

Consider the following stochastic functional differential equations:(4)dxt=−atxtdt+Ft,xt,xt−τtdt+Gt,xt,xt−ηtdwt,which yields the following stochastic functional differential equations with piecewise constant arguments:(5)dxt=−atxtdt+Ft,xt,xt−τtdt+Gt,xt,xt−ηtΔwtdt,where t denotes the integer part of t. Here, the discrete analogue of the stochastic parts of system (4) is obtained by the Euler scheme, i.e., dwt=Δwtdt=wt+1−wtdt. For t∈ℝ, there exists an integer k∈ℤ such that k≤t<k+1. Then, the above equation becomes(6)dxt=−akxtdt+Fk,xk,xk−τkdt+Gk,xk,xk−ηkΔwkdt.Integrating the above equation fromk to t and letting t⟶k+1, we achieve the discrete analogue of system (4) as follows:(7)xk+1=e−akxk+1−e−akakFk,xk,xk−τk+Gk,xk,xk−ηkΔwk,where k∈ℤ. System (7) is the discrete analogue of system (4) by the semidiscrete method. By a similar discussion as that in system (7), we obtain the semidiscrete analogue of system (3) as follows:(8)xik+1=e−aikxik+1−e−aikaik∑j=1mbijkfjyjk+∑j=1mcijkfjyjk−τijk+∧j=1mαijfjyjk−τijk+∧j=1mTijμj+∨j=1mβijfjyjk−τijk+∨j=1mSijμj+Iik+∑j=1mdijkσjyjk−ηijkΔwjk,yjk+1=e−a˜jkyjk+1−e−a˜jka˜jk∑i=1nb˜jikf˜ixik+∑i=1nc˜jikf˜ixik−τ˜jik+∧i=1nα˜jif˜ixik−τ˜jik+∧i=1nT˜jiμ˜i+∨i=1nβ˜jif˜ixik−τ˜jik+∨i=1nS˜jiμ˜i+Jjk+∑i=1nd˜jikσ˜ixik−η˜jikΔw˜ik,k∈ℤ,where i=1,2,…,n and j=1,2,…,m.


## 2.2. Volterra Additive Equation for the Solution of System (8)

Lemma 1.
Assume thatX=xi is a solution of system (8), and then X can be expressed as(9)xik=∏s=k0k−1e−aisxik0+∑v=k0k−1∏s=v+1k−1e−ais1−e−aivaiv∑j=1mbijvfjyjv+∑j=1mcijvfjyjv−τijv+∧j=1mαijfjyjv−τijv+∧j=1mTijμj+∨j=1mβijfjyjv−τijv+∨j=1mSijμj+Iiv+∑j=1mdijvσjyjv−ηijvΔwjv,yjk=∏s=k0k−1e−a˜jsyjk0+∑v=k0k−1∏s=v+1k−1e−a˜js1−e−a˜jva˜jv∑i=1nb˜jivf˜ixiv+∑i=1nc˜jivf˜ixiv−τ˜jiv+∧i=1nα˜jif˜ixiv−τ˜jiv+∧i=1nT˜jiμ˜i+∨i=1nβ˜jif˜ixiv−τ˜jiv+∨i=1nS˜jiμ˜i+Jjv+∑i=1nd˜jivσ˜ixiv−η˜jivΔw˜iv,where k0∈ℤ, k∈k0,+∞ℤ, i=1,2,…,n, and j=1,2,…,m.Proof.
Let(10)Fik,x≔∑j=1mbijkfjyjk+∑j=1mcijkfjyjk−τijk+∧j=1mαijfjyjk−τijk+∧j=1mTijμj+∨j=1mβijfjyjk−τijk+∨j=1mSijμj+Iik+∑j=1mdijkσjyjk−ηijkΔwjk,k∈ℤ,i=1,2,…,n.
ByΔukvk=Δukvk+uk+1Δvk and the first equation of system (8), it gets(11)Δ∏s=0k−1eaisxik=∏s=0keais1−e−aikaikFik,x,where i=1,2,…,n and k∈ℤ. So,(12)∑v=k0k−1Δ∏s=0v−1eaisxiv=∑v=k0k−1∏s=0veais1−e−aivaivFiv,xis equivalent to(13)∏s=0k−1eaisxik=∏s=0k0−1eaisxik0+∑v=k0k−1∏s=0veais1−e−aivaivFiv,x,where i=1,2,…,n and k∈ℤ. By the above equations, we can easily derive the first equation of system (9). Similarly, we can obtain the second equation of system (9). This completes the proof.


## 2.3. Some Lemmas

Lemma 2 (Minkowski inequality) [48].
Assume thatp≥1, Eξp<∞, Eηp<∞, and then(14)Eξ+ηp1/p≤Eξp1/p+Eηp1/p.Lemma 3 (Hölder inequality) [48].
Assume thatp>1, and then(15)∑kakbk≤∑kak1−1/p∑kakbkp1/p.
Ifp=1, then ∑kakbk≤∑kaksupkbk.Lemma 4 [35].
Suppose thatg∈L2a,b,ℝ, and then(16)Esupt∈a,b∫atgsdωsp≤CpE∫abgt2dtp/2,where(17)Cp=32pp/2,0<p<2,4,p=2,pp+12p−1p−1p/2,p>2.Lemma 5.
Assume thatxk:k∈ℤ is real-valued stochastic process and wk is the standard Brownian motion, and then(18)ExkΔwkp≤CpExkp,k∈ℤ,where Cp is defined as that in Lemma 4, in which p>0.Proof.
By Lemma 4, it follows that(19)ExkΔwkp=E∫kk+1xkdwsp≤CpE∫kk+1x2kdsp/2≤CpExkp,k∈ℤ.
This completes the proof.Lemma 6 [49].
SupposeX=xi and Y=yi are two states of system (8), and then we have(20)∧j=1nαijfjxj−∧j=1nαijfjyj≤∑j=1nαijfjxj−fjyj,∨j=1nβijfjxj−∨j=1nβijfjyj≤∑j=1nβijfjxj−fjyj,i=1,2,…,n.Lemma 7 [50].
Assume thatΛ is a closed convex nonempty subset of a Banach space X. Suppose further that B and C map Λ into X such that(1)x,y∈Λ implies that Bx+Cy∈Λ(2)B is continuous and BΛ is contained in a compact set(3)C is a contraction mapping
Then, there exists az∈Λ with z=Bz+Cz.Setfu=supk∈ℤfk and fl=infk∈ℤfk for bounded function f defined on ℤ. Throughout this paper, suppose that the following conditions are satisfied:H1 ai and a˜j are bounded sequences defined on ℤ with ail>0 and a˜jl>0, i=1,2,…,n, and j=1,2,…,m.H2 There exists several constants Ljf, Ljσ, Ljf˜, and Ljσ˜ such that(21)fju−fjv≤Ljfu−v,σju−σjv≤Ljσu−v,f˜iu−f˜iv≤Ljf˜u−v,σ˜ju−σ˜jv≤Ljσ˜u−v,for allu,v∈ℝ, i=1,2,…,n, and j=1,2,…,m.


## 3. 2p-th Mean Almost Periodic Sequence Solution

Define(22)au≔max1≤i≤naiu,al≔min1≤i≤nail,a˜u≔max1≤j≤ma˜ju,a˜l≔max1≤j≤ma˜jl,D∗≔max1≤i≤n∑j=1mbiju+ciju+αij+βijLjf,D˜∗≔max1≤j≤m∑i=1nb˜jiu+c˜jiu+α˜ji+β˜jiLif˜,K∗≔max1≤i≤n∑j=1mdijuLjσ,K˜∗≔max1≤j≤m∑i=1nd˜jiuLiσ˜,r2p≔max1−e−aual1−e−alD∗+K∗C2p1/2p,1−e−a˜ua˜l1−e−a˜lD˜∗+K˜∗C2p1/2p,β2p≔α2p1−r2p,where(23)α2p≔maxe2p,e˜2p,e2p≔1−e−aual1−e−almax1≤i≤n∑j=1mbiju+ciju+αij+βijfj0+∑j=1mTij+Sijμj+Iiu+∑j=1mdijuσj0C2p1/2p,e˜2p≔1−e−a˜ua˜l1−e−a˜lmax1≤j≤m∑i=1nb˜jiu+c˜jiu+α˜ji+β˜jif˜i0+∑i=1nT˜ji+S˜jiμ˜i+Jju+∑i=1nd˜jiuσ˜i0C2p1/2p.Definition 3.1 [34]. A stochastic process X∈BCℤ;LpΩ;ℝn is said to be p-th mean almost periodic sequence if for each ϵ>0, there exists an integer lϵ>0 such that each interval of length lϵ contains an integer ω for which(24)Xk+ω−Xkp=max1≤i≤nExik+ω−xikp1/p<ϵ,∀k∈ℤ.A stochastic processX, which is 2-nd mean almost periodic sequence, is called square-mean almost periodic sequence. Like for classical almost periodic functions, the number ω is called an ϵ-translation of X.Theorem 1.
Assume that all of the coefficients in system (8) are almost periodic sequences, and then H1–H2 hold and the following condition is satisfied:H3 r2p<1, where 2p>1
Then, there exists a2p-th mean almost periodic sequence solution X of system (8) with X2p≤β2p.Proof.
LetΛ⊆BCℤ;L2pΩ;ℝm+n be the collection of all 2p-th mean almost periodic sequences X satisfying the inequality X2p≤β2p.
DefineΦXk=BXk+CXk, where(25)ΦXk=ΦXik;ΦXn+jk=ΦX1k,…,ΦXnk,ΦXn+1k,…,ΦXn+mkT,ΦXik=BXik+CXik,ΦXn+jk=BXn+jk+CXn+jk,(26)BXik=∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaiv∑j=1mbijvfjyjv+∑j=1mcijvfjyjv−τijv+∧j=1mαijfjyjv−τijv+∧j=1mTijμj+∨j=1mβijfjyjv−τijv+∨j=1mSijμj+Iiv,BXn+jk=∑v=−∞k−1∏s=v+1k−1e−a˜js1−e−a˜jva˜jv∑i=1nb˜jivf˜ixiv+∑i=1nc˜jivf˜ixiv−τ˜jiv+∧i=1nα˜jif˜ixiv−τ˜jiv+∧i=1nT˜jiμ˜i+∨i=1nβ˜jif˜ixiv−τ˜jiv+∨i=1nS˜jiμ˜i+Jjv,(27)CXik=∑v=∞k−1∏s=v+1k−1e−ais1−e−aivaiv∑j=1mdijvσjyjv−ηijvΔwjv,CXn+jk=∑v=−∞k−1∏s=v+1k−1e−a˜js1−e−a˜jva˜jv∑i=1nd˜jivσ˜ixiv−η˜jivΔw˜iv,where i=1,2,…,n, j=1,2,…,m, and k∈ℤ.
LetX0=xi0;yj0 be defined as(28)xi0k=∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaiv∑j=1mbijvfj0+∑j=1mcijvfj0+∧j=1mαijfj0+∧j=1mTijμj+∨j=1mβijfj0+∨j=1mSijμj+∑j=1mdijvσj0Δwjv+Iiv,yj0k=∑v=−∞k−1∏s=v+1k−1e−a˜js1−e−a˜jva˜jv∑i=1nb˜jivf˜i0+∑i=1nc˜jivf˜i0+∧i=1nα˜jif˜i0+∧i=1nT˜jiμ˜i+∨i=1nβ˜jif˜i0+∨i=1nS˜jiμ˜i+∑i=1nd˜jivσ˜i0Δw˜iv+Jjv,where i=1,2,…,n, j=1,2,…,m, and k∈ℤ. By Minkoswki inequality in Lemma 2 and Hölder inequality in Lemma 3, we have(29)xi02p≤max1≤i≤nsupk∈ℤE∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaiv∑j=1mbijvfj0+cijvfj02p1/2p+E∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaiv∧j=1mαij+∨j=1mβijfj02p1/2p+E∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaiv∧j=1mTij+∨j=1mSijμj2p1/2p+E∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaiv∑j=1mdijvσj0Δwjv2p1/2p+E∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaivIiv2p1/2p.
From Lemma 6, it gets from the above inequality that(30)xi02p≤max1≤i≤nsupk∈ℤ1−e−aual1−e−al∑j=1mbijufj0+cijufj0+∑j=1mαij+βijfj0+∑j=1mTij+Sijμj+Iiu+∑j=1mdijuσj0∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaiv1−1/2p×∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaivEΔwjv2p1/2p≤1−e−aual1−e−almax1≤i≤n∑j=1mbiju+ciju+αij+βijfj0+∑j=1mTij+Sijμj+Iiu+∑j=1mdijuσj0C2p1/2p≔e2p.
Similarly, we have(31)yj02p≤1−e−a˜ua˜l1−e−a˜lmax1≤j≤m∑i=1nb˜jiu+c˜jiu+α˜ji+β˜jif˜i0+∑i=1nT˜ji+S˜jiμ˜i+Jju+∑i=1nd˜jiuσ˜i0C2p1/2p≔e˜2p.
By the above inequalities, it concludes(32)X02p≤maxe2p,e˜2p≔α2p.
It follows (25)–(27) that(33)ΦXik−xi0ki=1n2p≤max1≤i≤nsupk∈ℤ∑j=1mbijuLjfE∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaivyjv2p1/2p+max1≤i≤n,1≤j≤msupk∈ℤDi∗∗E∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaivyjv−τijv2p1/2p+max1≤i≤n,1≤j≤msupk∈ℤK∗E∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaivyjv−ηijvΔwjv2p1/2p,which yields from Lemmas 2 and 3 that(34)ΦXik−xi0ki=1n2p≤max1≤i≤nsupk∈ℤ∑j=1mbijuLjf∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaivp−1×∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaivEyjv2p1/2p+max1≤i≤n,1≤j≤msupk∈ℤDi∗∗∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaivp−1×∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaivEyjv−τijv2p1/2p+max1≤i≤n,1≤j≤msupk∈ℤK∗∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaivp−1×∑v=−∞k−1∏s=v+1k−1e−ais1−e−aivaivEyjv−ηijvΔwjv2p1/2p,where Di∗∗=D∗−∑​j=1mbijuLjf, i=1,2,…,n. Applying Lemma 5 to the above inequality, it derives(35)ΦXik−xi0ki=1n2p≤1−e−aual1−e−alD∗+K∗C2p1/2pX2p≤r2pX2p≤r2pα2p1−r2p.
Similarly,(36)ΦXn+jk−yj0kj=1m2p≤1−e−a˜ua˜l1−e−a˜lD˜∗+K˜∗C2p1/2pX2p≤r2pX2p≤r2pα2p1−r2p.
Together with the above inequalities, we obtain(37)ΦX−X02p≤r2pα2p1−r2p.
Hence, forX=xi;yj∈Λ, it leads from (32) and (37) to(38)ΦX2p≤X02p+ΦX−X02p≤α2p+r2pα2p1−r2p=α2p1−r2p≔β2p.
From (38), BΛ is uniformly bounded. Together with the continuity of B, for any bounded sequence φn in Λ, we know that there exists a subsequence φnk in Λ such that Bφnk is convergent in BΛ. Therefore, B is compact on Λ. Then, condition (2) of Lemma 7 is satisfied.
The next step is proving condition (1) of Lemma 7. Now, we consist in proving the2p-th mean almost periodicity of BX⋅ and CX⋅. Since X⋅ is a 2p-th mean almost periodic sequence and all the coefficients in system (8) are almost periodic sequences, for all ϵ>0 there exists lϵ>0 such that every interval of length lϵ>0 contains a ω with the property that(39)Exik+ω−xik2p1/2p<ϵ,aik+ω−aik<ϵ,a˜jk+ω−a˜jk<ϵ,bijk+ω−bijk<ϵ,b˜jik+ω−b˜jik<ϵ,cijk+ω−cijk<ϵ,c˜jik+ω−c˜jik<ϵ,dijk+ω−dijk<ϵ,d˜jik+ω−d˜jik<ϵ,τijk+ω−τijk<ϵ,τ˜jik+ω−τ˜jik<ϵ,ηijk+ω−ηijk<ϵ,η˜jik+ω−η˜jik<ϵ,Iik+ω−Iik<ϵ,Jjk+ω−Jjk<ϵ,where i=1,2,…,n, j=1,2,…,m, and k∈ℤ. By (26), (27), and H2, we could easily find a positive constant M such that(40)EBXik+ω−BXik2p1/2p≤Mmax1≤i≤nsupk∈ℤExik+ω−xik2p1/2p<Mϵ,EBXn+jk+ω−BXn+jk2p1/2p≤Mmax1≤j≤msupk∈ℤEyjk+ω−yjk2p1/2p<Mϵ,(41)ECXik+ω−CXik2p1/2p≤Mmax1≤i≤nsupk∈ℤExik+ω−xik2p1/2p<Mϵ,ECXn+jk+ω−CXn+jk2p1/2p≤Mmax1≤j≤msupk∈ℤEyjk+ω−yjk2p1/2p<Mϵ,where i=1,2,…,n, j=1,2,…,m, and k∈ℤ. From (40) and (41), BX⋅ and CX⋅ are 2p-th mean almost periodic processes. Further, by (38), it is easy to obtain that BX+CY∈Λ, ∀X,Y∈Λ. Then, condition (1) of Lemma 7 holds.
Finally,∀X=xi;yj,X∗=xi∗;yj∗∈Λ, from (27), it yields(42)CXik−CX∗iki=1n2p≤1−e−aualmax1≤i≤nsupk∈ℤE∑v=−∞k−1∏s=v+1k−1e−al×∑j=1mdijvσjxjv−ηijv−σjyjv−ηijvΔwjv2p≤1−e−aualmax1≤i≤n,1≤j≤msupk∈ℤK∗∑v=−∞k−1∏s=v+1k−1e−al2p−1×∑v=−∞k−1∏s=v+1k−1e−alExjv−ηijv−yjv−ηijvΔwjv2p≤K∗C2p1/2p1−e−aual1−e−alX−Y2p≤r2pX−Y2p,similarly,(43)CXn+jk−CX∗n+jkj=1m2p≤K˜∗C2p1/2p1−e−a˜ua˜l1−e−a˜lX−Y2p≤r2pX−Y2p.
From the above inequalities, it leads(44)CX−CY2p≤r2pX−Y2p.
In view ofH3, C is a contraction mapping. Hence, condition (3) of Lemma 7 is satisfied. Therefore, all the conditions in Lemma 7 hold. By Lemma 7, system (8) has a 2p-th mean almost periodic sequence solution. This completes the proof.According to Theorem 1, we can easily obtain the following theorem.Theorem 2.
Assume that all conditions in Theorem 1 hold and all coefficients in system (8) are periodic sequences, and then system (8) admits at least one 2p-th mean periodic sequence solution.Remark 1.
In view of the definition ofr2p in Theorem 1, ai and a˜j in system (8) are bigger, then the probability of r2p<1 is bigger, where i=1,2,…,n, and j=1,2,…,m. The parameters Tij, Sij, μij, T˜ji, S˜ji, μ˜ji, Ii, and Jj have no effect on the value of r2p, but they have effects on the boundedness of almost periodic sequence solutions of system (8), where i=1,2,…,n and j=1,2,…,m. In order to derive the existence of almost periodic sequence solutions of system (8), we would better choose bigger ai and a˜j, and the other parameters should be smaller except Tij, Sij, μij, T˜ji, S˜ji, μ˜ji, Ii, and Jj, where i=1,2,…,n and j=1,2,…,m.


## 4. 2p-th Moment Global Exponential Stability

Suppose thatX=xi;yj with initial value φ=φi;ψj and X∗=xi∗;yj∗ with initial value φ∗=φi∗;ψj∗ are arbitrary two solutions of system (8). For convenience, let(45)γ2p=max1≤i≤n,1≤j≤msups∈−μ0,0ℤEφis−φi∗s2p1/2p,Eψjs−ψj∗s2p1/2p,μ0=maxi,jτiju,ηiju,τ˜jiu,η˜jiu.Definition 4.1 [35]. System (8) is said to be 2p-th moment global exponential stability if there are positive constants k0, M, and λ such that(46)Xk−X∗k2p=max1≤i≤n,1≤j≤mExik−xi∗k2p1/2p,Eyjk−yj∗k2p1/2p<Mγ2pe−λk,where ∀k>k0,k∈ℤ. The 2-nd moment global exponential stability is called square-mean global exponential stability.Theorem 3.
Assume thatH1–H3 hold, and then system (8) is 2p-th moment globally exponentially stable.Proof.
From Lemmas 1 and 6, it follows that(47)xik−xi∗k≤∏s=0k−1e−aisφi0−φi∗0+1−e−aual∑v=0k−1∏s=v+1k−1e−ais∑j=1mbijuLjfyjv−yj∗v+ciju+αij+βijLjfyjv−τijv−yj∗v−τijv+dijuLjσyjv−ηijv−yj∗v−ηijvΔwjv≤e−alkφi0−φi∗0+1−e−aual∑v=0k−1e−alk−v−1∑j=1mbijuLjfyjv−yj∗v+ciju+αij+βijLjfyjv−τijv−yj∗v−τijv+dijuLjσyjv−ηijv−yj∗v−ηijvΔwjv,(48)yjk−yj∗k≤∏s=0k−1e−a˜jsψj0−ψj∗0+1−e−a˜ua˜l∑v=0k−1∏s=v+1k−1e−a˜js∑i=1nb˜jiuLif˜xiv−xi∗v+c˜jiu+α˜ji+β˜jiLif˜xiv−τ˜jiv−xi∗v−τ˜jiv+d˜jiuLiσ˜xiv−η˜jiv−xi∗v−η˜jivΔw˜iv≤e−a˜lkψj0−ψj∗0+1−e−a˜ua˜l∑v=0k−1e−a˜lk−v−1∑i=1nb˜jiuLif˜xiv−xi∗v+c˜jiu+α˜ji+β˜jiLif˜xiv−τ˜jiv−xi∗v−τ˜jiv+d˜jiuLiσ˜xiv−η˜jiv−xi∗v−η˜jivΔw˜iv,where i=1,2,…,n, j=1,2,…,m, and k∈−μ0,+∞.
For convenience, let(49)a0=1−e−aual,a˜0=1−e−a˜ua˜l,Z=zi, W=wj, zik=xik−xi∗k, and wjk=yjk−yj∗k, where i=1,2,…,n, j=1,2,…,m, and k∈ℤ.
Similar to the argument as that in (37), it gets from (47) that(50)Zk2p≤e−alkγ2p+max1≤i≤n∑j=1ma0bijuLjf∑s=0k−1e−alk−s−12p−1∑s=0k−1e−alk−s−1Eyjs−yj∗s2p1/2p+max1≤i≤n∑j=1ma0ciju+αij+βijLjf∑s=0k−1e−alk−s−12p−1×∑s=0k−1e−alk−s−1Eyjs−τijs−yj∗s−τijs2p1/2p+max1≤i≤n∑j=1ma0dijuLjσ∑s=0k−1e−alk−s−12p−1×∑s=0k−1e−alk−s−1Eyjs−ηijs−yj∗s−ηijsΔwjs2p1/2p≤e−alkγ2p+max1≤i≤n∑j=1ma0bijuLjf∑s=0k−1e−alk−s−12p−1∑s=0k−1e−alk−s−1Ws2p2p1/2p+max1≤i≤n∑j=1ma0ciju+αij+βijLjf∑s=0k−1e−alk−s−12p−1×∑s=0k−1e−alk−s−1Ws−τijs2p2p1/2p+max1≤i≤n∑j=1ma0C2p1/2pdijuLjσ∑s=0k−1e−alk−s−12p−1∑s=0k−1e−alk−s−1Ws−ηijs2p2p1/2p.
Similarly, it follows from (48) that(51)Wk2p≤e−a˜lkγ2p+max1≤j≤m∑i=1na˜0b˜jiuLif˜∑s=0k−1e−a˜lk−s−12p−1∑s=0k−1e−a˜lk−s−1Zs2p2p1/2p+max1≤j≤m∑i=1na˜0c˜jiu+α˜ji+β˜jiLif˜∑s=0k−1e−a˜lk−s−12p−1×∑s=0k−1e−a˜lk−s−1Zs−τ˜jis2p2p1/2p+max1≤j≤m∑i=1na˜0C2p1/2pd˜jiuLiσ˜∑s=0k−1e−a˜lk−s−12p−1∑s=0k−1e−a˜lk−s−1Zs−η˜jis2p2p1/2p.
Be aware ofH3 in Theorem 1, there exists a constant λ>0 small enough such that(52)max1≤i≤n∑j=1meλa01−e−al−2pλbijuLjf+eμ0λciju+αij+βijLjf+eμ0λC2p1/2pdijuLjσ=defρ≤1,max1≤j≤m∑i=1neλa˜01−e−a˜l−2pλb˜jiuLif˜+eμ0λc˜jiu+α˜ji+β˜jiLif˜+eμ0λC2p1/2pd˜jiuLiσ˜=defρ˜≤1.
DefineVk=maxZk2p,Wk2p, k∈ℤ.
Next, we claim that there exists a constantM0>1 such that(53)Vk≤M0γ2pe−λk,∀k∈−μ0,+∞.
If (53) is invalid, then there must exist k0∈0,+∞ℤ such that one of the following two cases holds:C1 Zk02p>M0γ2pe−λk0 and Vk≤M0γ2pe−λk,∀k∈−μ0,k0C2 Wk02p>M0γ2pe−λk0 and Vk≤M0γ2pe−λk,∀k∈−μ0,k0
IfC1 is valid, it follows from (50) that(54)Zk02p≤e−alk0γ2p+max1≤i≤n∑j=1ma0bijuLjfM0γ2p∑s=0k−1e−alk−s−12p−1∑s=0k−1e−alk−s−1e−2pλs1/2p+max1≤i≤n∑j=1ma0M0γ2pciju+αij+βijLjf+C2p1/2pdijuLjσ×∑s=0k−1e−alk−s−12p−1∑s=0k−1e−alk−s−1e−2pλs−μ01/2p≤e−alk0γ2p+max1≤i≤n∑j=1ma0bijuLjfM0γ2pe−λk0eλ1−e−alk01−e−al1−1/2p∑s=0k0−1e−al−pλk0−s−11/2p+max1≤i≤n∑j=1ma0M0γ2pciju+αij+βijLjf+C2p1/2pdijuLjσ×e−λk0eμ0+1λ1−e−alk01−e−al1−1/2p∑s=0k0−1e−al−pλk0−s−11/2p≤e−alk0γ2p+max1≤i≤n∑j=1ma0M0γ2pe−λk0bijuLjf+eμ0λciju+αij+βijLjf+eμ0λC2p1/2pdijuLjσ×eλ1−e−alk01−e−al1−1/2p1−e−al−2pλk01−e−al−2pλ1/2p≤M0γ2pe−λk01M0e−al−λk0+max1≤i≤n∑j=1ma0bijuLjf+eμ0λciju+αij+βijLjf+eμ0λC2p1/2pdijuLjσeλ1−e−al−λk01−e−al−2pλ≤M0γ2pe−λk0e−al−λk0+ρ1−e−al−λk0≤M0γ2pe−λk0.
In the fourth inequality from the bottom of (54), we use the fact 1−e−alk01−1/2p1−e−al−pλk01/2p≤1−e−al−λk0 and 1−e−al1/2p≥1−e−al−pλ1/2p. (54) contradicts C1. So, C1 is invalid. Similarly, we can easily conclude C2 is invalid by using (51). Therefore, (53) is satisfied. By (53), system (8) is 2p-th moment globally exponentially stable. This completes the proof.According to Theorems 1 and 2, we can easily obtain the following theorem.Theorem 4.
Assume that all conditions in Theorem 1 hold, and then system (8) admits a 2p-th mean almost periodic sequence solution, which is 2p-th moment globally exponentially stable. Further, if all coefficients in system (8) are periodic sequences, then system (8) admits at least one 2p-th mean periodic sequence solution, which is globally exponentially stable.Proof.
The result can be easily obtained by Theorem 3, so we omit it. This completes the proof.Remark 2.
Assume thatXk=x1k,x2k,…,xnk is a solution of (1), and then the length of Xk is usually measured by X∞=supk∈ℝmax1≤i≤nxik. However, if Xk is a solution of stochastic system, its length should not be measured by X∞ because Xk is a random variable. In this paper, we use norm X2p=max1≤i≤nsupk∈ℤExik2p1/2p,2p>1 for random variable Xk. Owing to the expectation E and order p in X2p, the computing processes of this paper are more complicated than those in literatures [41–47]. It is worth mentioning that Minkoswki inequality in Lemma 2 and Hölder inequality in Lemma 3 are crucial to the computing processes. The facts above are obvious from the computations of (32), (42), (50), and (54) in the proofs of Theorems 1 and 3. Further, the stochastic terms dijσjΔwj and d˜jiσ˜iΔw˜ii=1,2,…,n;j=1,2,…,m in system (8) also increase the complexity of computing. This point is also clear from the computations of (42) and (50).


## 5. Examples and Computer Simulations

Example 5.1. Consider the following continuous-time BAM neural networks with random perturbation:(55)dxt=−xt+0.01sin5tsinyt+0.05cos7tcosyt−1dt+0.01dwt,dyt=−0.2yk+0.015sin21tcosxt−1dt+0.03cos23tdwt,∀t∈ℝ.


### 5.1. Semidiscrete Model

Based on model (55), we obtain the following semidiscrete model by using the semidiscretization technique:(56)xk+1=e−1xk+1−e−10.01sin5ksinyk+0.05cos7kcosyk−1+0.01Δwk,yk+1=e−0.2yk+1−e−0.20.20.15sin21kcosxk−1+0.03cos23kΔwk,where k∈ℤ.Remark 3.
In literature [44], Huang et al. studied model (1) and obtained some sufficient conditions for the existence of a unique almost periodic sequence solution which is globally attractive. In [45], they considered system (2) and studied the dynamics of 2N almost periodic sequence solutions. But neither of them considered the random factors. Such as, the obtained results in [44, 45] cannot be applied to the study for stochastic model (56). Therefore, the work in this paper complements the corresponding results in [44, 45].


### 5.2. Discrete Model Formulated by the Euler Scheme

Based on model (55), we obtain the following discrete-time model by using the Euler method:(57)xk+1=0.01sin5ksinyk+0.05cos7kcosyk−1+0.01Δwk,yk+1=0.8yk+0.15sin21kcosxk−1+0.03cos23kΔwk,where k∈ℤ.In Figures1 and 2, we give two plots of numerical solutions which are produced by continuous-time model (55), semidiscrete model (56), and Euler-discretization model (57), respectively. Compared with Euler-discretization model (57), semidiscrete model (56) gives a more accurate characterization for continuous-time model (55).Figure 1
Trajectories of state variablex in models (55), (56), and (57), respectively.Figure 2
Trajectories of state variabley in models (55), (56), and (57), respectively.Remark 4.
In literature [51–54], the authors discussed the dynamics of periodic solutions of discrete-time cellular neural networks formulated by the Euler scheme. From the above discussion, semidiscrete stochastic system (8) gives a more accurate and realistic formulation for studying the dynamics of discrete-time cellular neural networks. In a way, the work of this paper complements and improves some corresponding results in [51–54].
Corresponding to system (8), we have al=au=1, a˜l=a˜u=1, Ljf=Ljσ=Ljf˜=Ljσ˜=1, b¯ij=0.01, c¯ij=0.05, α11=α12=0.1, β11=β12=0.02, α21=α22=0.04, β21=β22=0.2, d¯ij=0.02, and i,j=1,2.
Takingp=2, by simple calculation,(58)C41/4=4,D∗≈0.72,K∗≈0.04,r4≈0.88<1.
According to Theorems 1 and 3, system (56) admits a 4-th mean almost periodic sequence solution, which is 4-th moment globally exponentially stable (see Figures 3–5).
Figure3 depicts a numerical solution x,y of semidiscrete stochastic model (56). Observe that the trajectories of x,y demonstrate almost periodic oscillations. Figures 4 and 5 display three numerical solutions of semidiscrete stochastic model (56) at different initial values 1.5,2.5, 2.5,1.5, and 0.2,0.3, respectively. They are shown that semidiscrete stochastic model (56) is 4-th moment globally exponentially stable.
Example 5.2. Consider the corresponding determinant model of system (56) as follows:(59)xk+1=e−1xk+1−e−10.01sin5ksinyk+0.05cos7kcosyk−1+0.01,yk+1=e−0.2yk+1−e−0.20.20.15sin21kcosxk−1+0.03cos23k,where k∈ℤ. In Figures 6 and 7, we give the results of contrast between stochastic model (56) and determinant model (59). Figures 6 and 7 indicate that the effects of stochastic perturbation on state variables x and y are significant. And the stochastic influence on state variable y is more obvious than that on state variable x. In Figure 8, we give a result of globally exponentially stable contrast between stochastic model (56) and determinant model (59). Figure 8 reveals that the convergence speed of stochastic model (56) is faster than determinant model (59).Figure 3
4-th mean almost periodicity of state variablesx,yT in model (56).Figure 4
4-th moment global exponential stability of state variablex of model (56).Figure 5
4-th moment global exponential stability of state variabley of model (56).Figure 6
Comparison of state variablex between stochastic model (56) and determinant model (59).Figure 7
Comparison of state variabley1 between stochastic model (56) and determinant model (59).Figure 8
Convergence speed comparison between stochastic model (56) and determinant model (59).
(a)
(b)Remark 5.
From example 5.2, stochastic disturbance brings a positive effect on the global exponential stability of the models.


## 5.1. Semidiscrete Model

Based on model (55), we obtain the following semidiscrete model by using the semidiscretization technique:(56)xk+1=e−1xk+1−e−10.01sin5ksinyk+0.05cos7kcosyk−1+0.01Δwk,yk+1=e−0.2yk+1−e−0.20.20.15sin21kcosxk−1+0.03cos23kΔwk,where k∈ℤ.Remark 3.
In literature [44], Huang et al. studied model (1) and obtained some sufficient conditions for the existence of a unique almost periodic sequence solution which is globally attractive. In [45], they considered system (2) and studied the dynamics of 2N almost periodic sequence solutions. But neither of them considered the random factors. Such as, the obtained results in [44, 45] cannot be applied to the study for stochastic model (56). Therefore, the work in this paper complements the corresponding results in [44, 45].


## 5.2. Discrete Model Formulated by the Euler Scheme

Based on model (55), we obtain the following discrete-time model by using the Euler method:(57)xk+1=0.01sin5ksinyk+0.05cos7kcosyk−1+0.01Δwk,yk+1=0.8yk+0.15sin21kcosxk−1+0.03cos23kΔwk,where k∈ℤ.In Figures1 and 2, we give two plots of numerical solutions which are produced by continuous-time model (55), semidiscrete model (56), and Euler-discretization model (57), respectively. Compared with Euler-discretization model (57), semidiscrete model (56) gives a more accurate characterization for continuous-time model (55).Figure 1
Trajectories of state variablex in models (55), (56), and (57), respectively.Figure 2
Trajectories of state variabley in models (55), (56), and (57), respectively.Remark 4.
In literature [51–54], the authors discussed the dynamics of periodic solutions of discrete-time cellular neural networks formulated by the Euler scheme. From the above discussion, semidiscrete stochastic system (8) gives a more accurate and realistic formulation for studying the dynamics of discrete-time cellular neural networks. In a way, the work of this paper complements and improves some corresponding results in [51–54].
Corresponding to system (8), we have al=au=1, a˜l=a˜u=1, Ljf=Ljσ=Ljf˜=Ljσ˜=1, b¯ij=0.01, c¯ij=0.05, α11=α12=0.1, β11=β12=0.02, α21=α22=0.04, β21=β22=0.2, d¯ij=0.02, and i,j=1,2.
Takingp=2, by simple calculation,(58)C41/4=4,D∗≈0.72,K∗≈0.04,r4≈0.88<1.
According to Theorems 1 and 3, system (56) admits a 4-th mean almost periodic sequence solution, which is 4-th moment globally exponentially stable (see Figures 3–5).
Figure3 depicts a numerical solution x,y of semidiscrete stochastic model (56). Observe that the trajectories of x,y demonstrate almost periodic oscillations. Figures 4 and 5 display three numerical solutions of semidiscrete stochastic model (56) at different initial values 1.5,2.5, 2.5,1.5, and 0.2,0.3, respectively. They are shown that semidiscrete stochastic model (56) is 4-th moment globally exponentially stable.
Example 5.2. Consider the corresponding determinant model of system (56) as follows:(59)xk+1=e−1xk+1−e−10.01sin5ksinyk+0.05cos7kcosyk−1+0.01,yk+1=e−0.2yk+1−e−0.20.20.15sin21kcosxk−1+0.03cos23k,where k∈ℤ. In Figures 6 and 7, we give the results of contrast between stochastic model (56) and determinant model (59). Figures 6 and 7 indicate that the effects of stochastic perturbation on state variables x and y are significant. And the stochastic influence on state variable y is more obvious than that on state variable x. In Figure 8, we give a result of globally exponentially stable contrast between stochastic model (56) and determinant model (59). Figure 8 reveals that the convergence speed of stochastic model (56) is faster than determinant model (59).Figure 3
4-th mean almost periodicity of state variablesx,yT in model (56).Figure 4
4-th moment global exponential stability of state variablex of model (56).Figure 5
4-th moment global exponential stability of state variabley of model (56).Figure 6
Comparison of state variablex between stochastic model (56) and determinant model (59).Figure 7
Comparison of state variabley1 between stochastic model (56) and determinant model (59).Figure 8
Convergence speed comparison between stochastic model (56) and determinant model (59).
(a)
(b)Remark 5.
From example 5.2, stochastic disturbance brings a positive effect on the global exponential stability of the models.


## 6. Conclusions and Future Works

In this paper, we formulate a discrete analogue of BAM neural networks with stochastic perturbations and fuzzy operations by using semidiscretization technique. The existence of2p-th mean almost periodic sequence solutions and 2p-th moment global exponential stability for the above models are investigated with the help of Krasnoselskii’s fixed point theorem and stochastic theory. The main results obtained in this paper are completely new, and the methods used in this paper provide a possible technique to study 2p-th mean almost periodic sequence solution and 2p-th moment global exponential stability of semidiscrete models with stochastic perturbations and fuzzy operations.With a careful observation of Theorems 1 and 3, it is not difficult to discover that(1)
2p>1 is crucial to the 2p-th mean almost periodicity and moment global exponential stability of system (8)(2)
The time delays have no effect on the existence of2p-th mean almost periodicity and 2p-th moment global exponential stability of system (8)(3)
Stochastic disturbance may bring a positive effect on the global exponential stability for determinant modelIn the future, the following aspects can be explored further:(1)
The methods used in this paper can be applied to study other types of neural networks, such as impulsive neural networks, high-order neural networks, neural networks on time scales, etc.(2)
Other types of fuzzy neural networks could be investigated, such as Takagi-Sugeno fuzzy neural networks(3)
Other dynamic behaviours of system (8) should be further discussed(4)
The case of2p∈0,1 could be further explored



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