# Existence and Global Exponential Stability of Almost Automorphic Solution for Clifford-Valued High-Order Hopfield Neural Networks with Leakage Delays

Authors: Bing Li; Yongkun Li
Journal: Complexity
 (2019)

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## Abstract

In this paper, we study the existence and global exponential stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by direct method. That is to say, we do not decompose the systems under consideration into real-valued systems, but we directly study Clifford-valued systems. Our methods and results are new. Finally, an example is given to illustrate our main results.


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## Body

## 1. Introduction

In recent years, high-order Hopfield neural networks have become the object of intensive analysis by many scholars because of their stronger approximation characteristics, faster convergence speed, larger storage capacity, and higher fault tolerance than low-order Hopfield neural networks. A lot of excellent research results about their dynamic characteristics have been obtained [1–14].On the one hand, the Clifford algebra was proposed by the British mathematician William K. Clifford [15] in 1878 and is a generalization of the plural, quaternion, and Glassman algebra. Currently, Clifford algebra has been widely used in various fields such as neural computing, computer and robot vision, image and signal processing, and control problems. Studies have shown that Clifford-valued neural networks are superior to commonly used real-valued neural networks [16, 17], so they have become an active research field in recent years. However, because the multiplication of Clifford numbers does not satisfy the commutative law, it has brought great difficulties to the research of Clifford-valued neural networks. Therefore, the current results on the dynamics of Clifford-valued neural networks are still very rare. At present, only a few papers have been published on the dynamics of Clifford-valued neural networks [18–22]. It is worth mentioning that the results of these mentioned papers are obtained by decomposing Clifford-valued neural networks into real-valued networks. Therefore, it is meaningful to study the dynamics of Clifford-valued neural networks by direct method.On the other hand, it is well known that periodic and almost periodic oscillations are important dynamic behaviors of neural networks. Almost automorphy is an extension of almost periodicity and plays an important role in better understanding of almost periodicity. Therefore, almost automorphic oscillation is more complex than almost periodic oscillation. Considering the interaction between neurons in a neural network is very complex, it is meaningful to study the almost automorphic oscillation of neural networks.In addition, time delays are inevitable and may affect and change the dynamic behavior of dynamic systems [23, 24]. Therefore, neural networks with various delays have been extensively studied. In particular, since K. Gopalsamy [25] first studied the stability of neural networks with leakage delays, a lot of research has been done on neural networks with leakage delays [6, 11, 12, 26]. However, there is no research on the Clifford-valued neural network with leakage delays.Inspired by the above analysis and discussion, the main purpose of this paper is to study the existence and global exponential stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks with leakage delays by direct method; that is, we will study the considered Clifford-valued neural networks directly, instead of converting them to real-valued ones. As far as we know, this is the first paper to study the almost automorphic solutions of Clifford-valued high-order Hopfield neural networks with leakage delays. In addition, this is the first paper to study almost automorphic solutions of Clifford neural networks by direct method. So, our methods and results of this paper are new. Besides, our methods proposed in this paper can be used to study the problem of almost automorphic solutions for other types of Clifford-valued neural networks.This paper is organized as follows. In Section2, we introduce some basic definitions and lemmas and give a model description. In Section 3, we study the existence of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks with leakage delays. In Section 4, we investigate the global exponential stability of almost automorphic solutions of the neural networks. In Section 5, an example is given to demonstrate the proposed results. Finally, we draw a brief conclusion in Section 6.


## 2. Preliminaries and Model Description

The real Clifford algebra overRm is defined as(1)A=∑A⊆1,2,…,maAeA,aA∈R, where eA=eh1eh2⋯ehν with A=h1h2⋯hν, 1≤h1<h2<⋯<hν≤m. Moreover, e∅=e0=1 and eh, h=1,2,…,m are said to be Clifford generators and satisfy ei2=-1, i=1,2,…,m, and eiej+ejei=0, i≠j, i,j=1,2,…,m. For convenience, we will denote the product of Clifford generators eh1,eh2,…,ehν as eh1h2⋯hν. Let Π={∅,1,2,…,A,…,12⋯m}; then it is easy to see that A={∑AaAeA,aA∈R}, where ∑A is short for ∑A∈Π and dimA=2m.Forx=∑AxAeA∈A, we define xA=maxA∈Π{xA} and for x=(x1,x2,…,xn)T∈An, we define xAn=max1≤p≤n{xpA}.The derivative ofx(t)=∑AxA(t)eA is given by x˙(t)=∑Ax˙A(t)eA. For more knowledge about Clifford algebra, we refer the reader to [27].In this paper, we are concerned with the following Clifford-valued high-order Hopfield neural network with delays in the leakage term:(2)x˙pt=-aptxpt-ηpt+∑q=1nbpqtfqxqt+∑q=1n∑l=1ncpqltgqxqt-σpqltglxlt-νpqlt+Ipt,p=1,2,…,n,where xp(t)∈A corresponds to the state of the pth unit at time t, ap(t)∈R+ represents the rate with which the pth unit will reset its potential to the resting state when disconnected from the network and external inputs at time t, bpq(t)∈A denotes the strength of qth unit on pth unit at time t, cpql(t)∈A is the second-order synaptic weight of the neural networks, fq(t),gq(t)∈A denote the activation functions, Ip(t)∈A is the external input on the pth at time t, and ηp(t),σpql(t),νpql(t)∈R+ denote the transmission delays.Remark 1.
Whenm, the number of the generators of A, equals 0,1, and 2, system (2) degenerates into real-valued, complex-valued, and quaternion-valued systems, respectively.We will adopt the following notation:(3)ap-=inft∈Rapt,bpq+=supt∈RbpqtA,cpql+=supt∈RcpqltA,ηp+=supt∈Rηpt,σpql+=supt∈Rσpqlt,νpql+=supt∈Rσpqlt,ξ=max1≤p,q,l≤nηp+,σpql+,νpql+.The initial value of system (2) is given by (4)xps=φps∈A,x˙ps=φ˙ps∈A,s∈-ξ,0,where φp∈C1([-ξ,0],A).Denote byUC(R,R) the set of all uniformly continuous functions from R to R. Let BC(R,An) denote the set of all bounded continuous functions from R to An. Then (BC(R,An),·0) is a Banach space with the norm f0=max1≤p≤nsupt∈R{fptA}, where f=(f1,f2,…,fp)T∈BC(R,An).Definition 2.
A functionf∈BC(R,An) is said to be almost automorphic, if for every sequence of real numbers (sn′)n∈N there exists a subsequence (sn)n∈N such that (5)gt≔limn→∞ft+snis well defined for each t∈R, and (6)limn→∞gt-sn=ftfor each t∈R.From the above definition, similar to the proofs of the corresponding results in [28], it is not difficult to prove the following two lemmas.Lemma 3.
Ifα∈R,f,g∈AA(R,An), then αf,f+g,fg∈AA(R,An).Lemma 4.
Letf∈C(A,An) satisfy the Lipschitz condition and φ∈AA(R,A); then f(φ(·))∈AA(R,An).Lemma 5.
Ifx∈C1(R,A),x,x′∈AA(R,A), η∈AA(R,R)∩UC(R,R), then x(·-η(·))∈AA(R,A).Proof.
Sincex′∈AA(R,A) and η∈UC(R,R), x(t-η(t)) is uniformly continuous for t∈R. Hence, for each ε>0, there exists a positive number δ=ε/2 such that t1-t2<δ implies (7)xt1-xt2A<ε.Since x,η∈AA(R,A), for every sequence of real numbers (sn′)n∈N there exists a subsequence (sn)n∈N such that (8)limn→∞xt+sn≔x¯t,limn→∞x¯t-sn=x¯t,limn→∞ηt+sn≔η¯t,limn→∞η¯t-sn=η¯t for every t∈R. Therefore, there exists a positive integer N such that (9)limn→∞xt+sn-x¯tA<ε2,limn→∞ηt+sn-η¯t<ε2 for n>N and t∈R. Hence, we have (10)xt+sn-ηt+sn-x¯t-η¯tA≤xt+sn-ηt+sn-xt+sn-η¯tA+xt+sn-η¯t-x¯t-η¯tA<ε2+ε2=ε. Consequently, {x(t+sn-η(t+sn))} converges to x¯(t-η¯(t)) for each t∈R. Similarly, we can obtain that {x¯1(t-sn-η¯(t-sn))} converges to x(t-η(t)) for each t∈R. Therefore, x(·-η(·))∈AA(R,A). The proof is complete.Throughout this paper, we make the following assumptions:(H1)
Forp,q,l=1,2,…,n, ap,ηp,σpql,νpql∈AA(R,R+)∩UC(R,R) and min1≤p≤n{inft∈Rap(t)}>0, bpq,cpql,Ip∈AA(R,A).(H2)
Forq=1,2,…,n, fq,gq,hq∈C(A,A) and there exist positive constants Lqf,Lqg,Lqh,Gp such that (11)fqx-fqyA≤Lqfx-yA,gqx-gqyA≤Lqgx-yA,gpxA≤Gp.Moreover, forq=1,2,…,n,fp(0)=gp(0)=0.(H3)
max1≤p≤n{(1/ap-)Mp,(1+ap+/ap-)Mp}≔r<1, where (12)Mp=ap+ηp++∑q=1nbpq+Lqf+∑q=1n∑l=1ncpql+GlLqg+GqLlg,p=1,2,…,n.


## 3. The Existence of Almost Automorphic Solutions

In this section, we study the existence of almost automorphic solutions by the contracting mapping principle.Let(13)Y=x∈C1R,An:x,x′∈AAR,An,X=φ=φ1,φ2,…,φnT∣φ∈Y,p=1,2,…,n.For any φ=(φ1,φ2,…,φn)T∈X, we define the norm of φ as φX=max⁡{φ0,φ′0}, where φ0=max1≤p≤n{supt∈RφptA}; then X is a Banach space.Letφ0(t)=(∫-∞te-∫sta1(u)duI1(s)ds, ∫-∞te-∫sta2(u)duI2(s)ds,…,∫-∞te-∫stan(u)duIn(s)ds)T and take a positive number R>φ0X.Set(14)X0=φ=φ1,φ2,…,φnT∈X:φ-φ0X≤rR1-r,and then, for every φ∈X0, we have φX≤φ-φ0X+φ0X≤rR/(1-r)+R=R/(1-r).Theorem 6.
Assume that(H1)-(H3) hold. Then system (2) has at least one almost automorphic solution in X0.Proof.
Firstly, it is easy to check that ifx=(x1,x2,…,xn)T∈X is a solution of the integral equation(15)xpt=∫-∞te-∫stapuduaps∫s-ηpssx˙pudu+∑q=1nbpqsfqxqs+∑q=1n∑l=1ncpqlsgqxqs-σpqlsglxls-νpqls+Ipsds,where p=1,2,…,n, then x is also a solution of system (2).
Secondly, we define an operatorT:X→BC(R,An) by (16)Tφ=T1φ,T2φ,…,TnφT,where φ∈X,p=1,2,…,n, (17)Tpφt=∫-∞te-∫stapuduaps∫s-ηpssφ˙pudu+∑q=1nbpqsfqφqs+∑q=1n∑l=1ncpqlsgqφqs-σpqlsglφls-νpqls+Ipsds.We will prove that T maps X into itself. To this end, let (18)Γps=aps∫s-ηpssφ˙pudu+∑q=1nbpqsfqφqs+∑q=1n∑l=1ncpqlsgqφqs-σpqlsglφls-νpqls+Ips,p=1,2,…,n. Then by Lemmas 3–5, Γp∈AA(R,A). We will prove that TΓp∈AA(R,A).
Let(sn′)n∈N be a sequence of real numbers; since ap∈AA(R,R) and Γp∈AA(R,A), we can extract a subsequence (sn)n∈N of (sn′)n∈N such that, for each t∈R, (19)limn→+∞apt+sn≔a¯pt,limn→+∞a¯pt-sn=apt,p=1,2,…,nand (20)limn→+∞Γpt+sn≔Γ¯pt,limn→+∞Γ¯pt-sn=Γpt,p=1,2,…,n.Set (21)TpΓp¯t=∫-∞te-∫sta¯puduΓ¯psds,p=1,2,…,n,and then we have (22)TpΓpt+sn-TpΓp¯tA=∫-∞t+sne-∫st+snapuduΓpsds-∫-∞te-∫sta¯puduΓ¯psdsA=∫-∞t+sne-∫s-sntapδ+sndδΓpsds-∫-∞te-∫sta¯puduΓ¯psdsA=∫-∞te-∫utapδ+sndδΓpu+sndu-∫-∞te-∫sta¯pδdδΓ¯psdsA≤∫-∞te-∫utapδ+sndδΓpu+sndu-∫-∞te-∫utapδ+sndδΓ¯puduA+∫-∞te-∫utapδ+sndδΓ¯pudu-∫-∞te-∫uta¯pδdδΓ¯puduA≤∫-∞te-∫utapδ+sndδΓp1s+sn-Γ¯psAds+∫-∞te-∫utapδ+sndδ-e-∫uta¯pδdδΓ¯psdsA,p=1,2,…,n. By the Lebesgue dominated convergence theorem, we obtain that limn→+∞(TpΓp)(t+sn)=(TpΓp¯)(t) for each t∈R,p=1,2,…,n. Similarly, one can prove that limn→+∞(TpΓp¯)(t-sn)=(TpΓp)(t) for each t∈R,p=1,2,…,n. Hence, TpΓp∈AA(R,A),p=1,2,…,n.
Noticing that(23)Tpφ′t=Γpt-apt∫-∞te-∫stapuduΓpsds,p=1,2,…,n, similar to the above, we can prove that Tpφ′∈AA(R,A),p=1,2,…,n. Therefore, T maps X into itself.
Thirdly, we will prove thatT is a self-mapping from X0 to X0. In fact, for each φ∈X0, we have (24)Tφ-φ00=max1≤p≤nsupt∈RTpφt-φ0tA≤max1≤p≤nsupt∈R∫-∞te-∫stapuduaps∫s-ηpssφ˙puduA+∑q=1nbpqsfqφqsA+∑q=1n∑l=1ncpqlsgqφqs-σpqls×glφls-νpqlsAds≤max1≤p≤nsupt∈R∫-∞te-∫stapuduap+φ′0ηp++∑q=1nbpq+Lqfφ0+∑q=1n∑l=1ncpql+GlLqgφ0≤max1≤p≤n1ap-ap+ηp++∑q=1nbpq+Lqf+∑q=1n∑l=1ncpql+GlLqgφX≤max1≤p≤n1ap-MpφXand (25)Ψφ-φ0′0≤max1≤p≤nsupt∈Rapt∫t-ηpt,φpttφ˙pudu+∑q=1nbpqtfqφqt+∑q=1n∑l=1ncpqltgqφqt-σpqlt,φqtglφls-νpqlt,φltA+∫-∞tapte-∫stapuduaps∫s-ηps,φpssφ˙pudu+∑q=1nbpqsfqφqs+∑q=1n∑l=1ncpqlsgqφqs-σpqls,φqsglφls-νpqls,φlsdsA≤max1≤p≤nap+ηp++∑q=1nbpq+Lqf+∑q=1n∑l=1ncpql+GlLqgφX+ap+ap-ap+ηp++∑q=1nbpq+Lqf+∑q=1n∑l=1ncpql+GlLqgφX≤max1≤p≤n1+ap+ap-MpφX.Thus, we obtain (26)Tφ-φ0X=max1≤p≤nsupt∈RTφ-φ00,supt∈RTφ-φ0′0≤max1≤p≤n1ap-Mp,1+ap+ap-MpφX=rφX≤rR1-r.
Fourthly, we will prove thatT is a contracting mapping. In fact, for any φ,ψ∈X0, we have that (27)Tφ-Tψ0≤max1≤p≤n∫-∞te-∫stapuduaps∫s-ηpssφ˙pu-ψ˙puduA+∑q=1nbpqsfqφqs-fqψqsA+∑q=1n∑l=1ncpqlsgqφqs-σpqlsglφls-νpqls-gqφqs-σpqlsglφls-νpqlsAds≤max1≤p≤n∫-∞te-ap-t-sdsap+ηp+φ˙q-ψ˙q0+∑q=1nbpq+Lqfφq-ψq0+∑q=1n∑l=1ncpql+GqLlgφ-ψ0+GlLqgφ-ψ0≤max1≤p≤n1ap-Mpφ-ψX≤rφ-ψXand (28)Tφ-Tψ′0≤max1≤p≤n1+ap+ap-Mpφ-ψX≤rφ-ψX.Hence, by (H3), T is a contracting mapping principle. Therefore, there exists a unique fixed-point φ∗∈X0 such that Tφ∗=φ∗, which implies that system (2) has an almost automorphic solution in X0. The proof is complete.


## 4. Global Exponential Stability

In this section, we investigate the global exponential stability of almost automorphic solutions by reduction to absurdity.Definition 7.
Letx=(x1,x2,…,xn)T be an almost automorphic solution of system (2) with the initial value φ=(φ1,φ2,…,φn)T∈C([-ξ,0],An) and let y=(y1,y2,…,yn)T be an arbitrary solution of system (2) with the initial value ψ=(ψ1,ψ2,…,ψn)T∈C([-ξ,0],An), respectively. If there exist positive constants λ and M such that (29)xt-yt1≤Mφ-ψξe-λt,∀t>0,where (30)xt-yt1=max⁡xt-ytAn,xt-yt′An,φ-ψξ=max⁡supt∈-ξ,0max1≤p≤nφpt-ψptA,supt∈-ξ,0max1≤p≤nφpt-ψpt′A.Then the almost automorphic solution of system (2) is said to be globally exponentially stable.Theorem 8.
Assume that(H1)-(H3) hold. Then system (2) has a unique pseudo almost automorphic solution that is globally exponentially stable.Proof.
By Theorem6, system (2) has a pseudo almost periodic solution. Let x(t) be an almost automorphic solution of (2) with the initial value φ(t) and let y(t) be an arbitrary solution with the initial value ψ(t). Set zp(t)=xp(t)-yp(t), ϕp(t)=φp(t)-ψp(t), and we have(31)z˙pt=-aptzpt-ηpt+∑q=1nbpqtf~qzqt+∑q=1n∑l=1ncpqltg~qzqt-σpqltg~qzqt-νpqlt,p=1,2,…,n,where (32)f~qzqt=fqxqt-fqyqt,g~qzqt-σpqltg~qzqt-νpqlt=gqxqt-σpqltgqxqt-νpqlt-gqyqt-σpqltgqyqt-νpqlt. Let Θp and Δp be defined by (33)Θpω=ap--ω-ap+ηp+eωηp++∑q=1nbpq+Lqfeω+∑q=1n∑l=1ncpql+Lqgeωσpql++Llgeωνpql+ and (34)Δpω=ap--ω-ap++ap--ωap+ηp+eωηp++∑q=1nbpq+Lqfeω+∑q=1n∑l=1ncpql+Lqgeωσpql++Llgeωνpql+, where p=1,2,…,n. When ω=0, we get (35)Θp0>0andΔp0>0,p=1,2,…,n. Since Θp(ω),Δp(ω) are continuous on [0,+∞) and Θp(ω),Δp(ω)→-∞ as ω→+∞, there exist εp,εp∗>0 such that Θp(εp)=Δ(εp∗)=0 and Θp(ε)>0 for ε∈(0,εp) and Δ(εp∗)>0 for ε∈(0,εp∗), p=1,2,…,n. Let α=min1≤p≤n{ωp,εp∗}; then we have (36)Θpα≥0,Δpα≥0,p=1,2,…,n.So we can take a positive constant λ∈(0,min{α,a1-,a2-,…,an-}). Obviously, we have (37)Θpλ>0,Θpλ>0,p=1,2,…,n,which implies that(38)1ap--λap+ηp+eληp++∑q=1nbpq+Lqfeλ+∑q=1n∑l=1ncpql+GlLqgeωσpql++GqLlgeωνpql+<1,(39)1+ap+ap--λap+ηp+eληp++∑q=1nbpq+Lqfeλ+∑q=1n∑l=1ncpql+GlLqgeωσpql++GqLlgeωνpql+<1,where p=1,2,…,n. Let M=max1≤p≤n{ap-/Mp}. From (H3), we have M>1. Hence, for p=1,2,…,n, we can obtain (40)1M≤1ap--λ×ap+ηp+eληp++∑q=1nbpq+Lqfeλ+∑q=1n∑l=1ncpql+GlLqgeωσpql++GqLlgeωνpql+. By (31), we have(41)z˙pt+aptzpt=apt∫t-ηpttz˙psds+∑q=1nbpqtf~qzqt+∑q=1n∑l=1ncpqltg~qzqt-σpqltg~qzqt-νpqlt.Multiplying (41) by e∫0sap(u)du and integrating on [0,t], we have(42)zpt=ϕp0e-∫0tapudu+∫0te-∫stapuduaps∫s-ηpssz˙pudu+∑q=1nbpqsf~qzqs+∑q=1n∑l=1ncpqlsg~qzqs-σpqlsg~qzqs-νpqlsds,p=1,2,…,n.It is easy to see that (43)zt1=ϕt1≤ϕξ≤Mϕξe-λt,t∈-ξ,0. We claim that(44)zt1≤Mϕξe-λt,t∈0,+∞.
To prove (44), we show that, for any β>1, the following inequality holds:(45)zt1≤βMϕξe-λt,t>0.If (45) is false, then there must be some t1>0 such that(46)zt11=max1≤p≤nzpt1A,z˙pt1A=βMϕξe-λt1and(47)zt1≤βMϕξe-λt,t∈-ξ,t1.By (39), (42), (46), and (47), we have(48)zpt1A≤ϕξe-t1ap-+βMϕξe-λs∫0t1e-t1-sap-ap+ηp+eληp++∑q=1nbpq+Lqf+∑q=1n∑l=1ncpql+GlLqgeλσpql++GqLlgeλνpql+ds≤βMϕξe-λt1eλ-ap-t1βM+1ap--λap+ηp+eληp++∑q=1nbpq+Lqf+∑q=1n∑l=1ncpql+GlLqgeλσpql++GqLlgeλνpql+1-eλ-ap-t1≤βMϕξe-λt1eλ-ap-t1(1M-1ap--λ∑q=1nap+ηp+eληp++∑q=1nbpq+Lqf+∑q=1n∑l=1ncpql+GlLqgeλσpql++GqLlgeλνpql++1ap--λ∑q=1nap+ηp+eληp++∑q=1nbpq+Lqf+∑q=1n∑l=1ncpql+GlLqgeλσpql++GqLlgeλνpql+≤βMϕξe-λt11ap--λ∑q=1nap+ηp+eληp++∑q=1nbpq+Lqf+∑q=1n∑l=1ncpql+GlLqgeλσpql++GqLlgeλνpql+<βMϕξe-λt1,p=1,2,…,n.Finding the derivative of (42), then by (39), (46), and (47), we have(49)z˙pt1A≤ap+ϕξe-apt1+ap+ηp+e-λt1-ηp+βMϕξ+e-λt1βMϕξ∑q=1nbpq+Lqf+βMϕξ∑q=1n∑l=1ncpql+GlLqge-λt1-σpql++GqLlge-λt1-νpql++βMϕξ∫0t1ap+e-t1-sap-ap+ηp+e-λs-ηp++e-λs∑q=1nbpq+Lqf+∑q=1n∑l=1ncpql+GlLqge-λt1-σpql++GqLlge-λt1-νpql+ds≤βMϕξe-λt1ap+eλ-ap-t1βM+ap+ap--λap+ηp+eληp++∑q=1nbpq+Lqf+∑q=1n∑l=1ncpql+GlLqgeλσpql++GqLlgeλνpql+1-eλ-ap-t1≤βMϕξe-λt1ap+eλ-ap-t11M-1ap--λ∑q=1n(ap+ηp+eληp++∑q=1nbpq+Lqf+∑q=1n∑l=1ncpql+GlLqgeλσpql++GqLlgeλνpql++1+ap+ap--λap+ηp+eληp++∑q=1nbpq+Lqf+∑q=1n∑l=1ncpql+GlLqgeλσpql++GqLlgeλνpql+≤βMϕξe-λt11+ap+ap--λap+ηp+eληp++∑q=1nbpq+Lqf+∑q=1n∑l=1ncpql+GlLqgeλσpql++GqLlgeλνpql+<βMϕξe-λt1,p=1,2,…,n.From (48) and (49), we have zt11<βMϕξe-λt1, which contradicts equality (46). Hence, (45) holds. Letting β→1, then (44) holds. Therefore, the almost automorphic solution of (2) is globally exponentially stable. The proof is complete.


## 5. Example

In this section, we give an example to show the feasibility of our results obtained in this paper.Example 1.
In system (2), let m=n=2, and take (50)fqxq=122e0sin⁡xq0+120e1sin⁡xq1+121e2sin⁡xq2+127e12sin⁡xq12,q=1,2,gqxq=116e0sin⁡xq0+120e1sin⁡xq1+115e2sin⁡xq2+118e12sin⁡xq12,q=1,2,a1ta2t=1+0.1sin⁡2t1.2+0.2cos⁡3t,η1tη2t=0.15+0.02sin⁡2t0.16+0.012cos⁡3t,b11tb12tb21tb22t=0.1e0sin⁡6t+0.2e1sin⁡6t0.13e0+0.1e12sin⁡7t0.15e0+0.1e1cos⁡5t+0.2e12cos⁡2t0.11e0+0.2e2sin⁡3t,c111tc112tc121tc122t=0.21e0sin⁡6t+0.12e1sin⁡3t0.11e0+0.12e12sin⁡2t0.14e0+0.11e1cos⁡5t+0.12e12cos⁡25t0.12e0+0.12e2sin⁡3t,c211tc212tc221tc222t=0.11e0sin⁡6t+0.12e1sin⁡5t0.11e0+0.12e12sin⁡7t0.14e0+0.11e1cos⁡5t+0.12e12cos⁡23t0.12e0+0.12e2sin⁡3t,σ111tσ112tσ121tσ122t=0.002sin⁡6t+0.012-sin⁡t1-0.1cos⁡7t0.001sin⁡2t+0.01,σ211tσ212tσ221tσ222t=0.012sin⁡3t+0.020.1-0.03cos⁡t2-sin⁡t20.01sin⁡6t+0.011,ν111tν112tν121tν122t=0.02sin⁡5t+0.031-0.7sin⁡t3-2cos⁡2t0.001sin⁡7t+0.01,ν211tν212tν221tν222t=0.002sin⁡3t+0.025-3cos⁡t0.1-0.07sin⁡t0.01sin⁡8t+0.015,Q1tQ2t=115e0sin⁡25t+120e1sin⁡3t+112e2cos⁡3t+115e12sin⁡7t115e0sin⁡23t+112e1sin⁡6t+110e2cos⁡7t+120e12sin23t.By computing, we get Lqf=1/20, Lqg=1/15, Gq=0.067, q=1,2, a1-=0.9, a2-=1, a1+=1.1, a2+=1.4, η1+=0.17, η2+=0.172, b11+=0.2, b12+=0.13, b21+=0.2, b22+=0.2, c111+=0.21, c112+=0.12, c121+=0.14, c122+=0.12, c211+=0.12, c212+=0.12, c221+=0.14, c222+=0.12, M1=0.2088, M2=0.2653, and (51)r=max1≤p≤21ap-Mp,1+ap+ap-Mp≈0.6366<1. Therefore, all of the conditions of Theorem 8 are satisfied. Hence, system (2) has one almost automorphic solution that is globally exponentially stable (see Figures 1–3).Figure 1
Curves ofxp0(t) and xp1(t), p=1,2.Figure 2
Curves ofxp2(t) and xp12(t), p=1,2.Figure 3
Curves ofx0(t), x1(t), x2(t), and x12(t) in 3-dimensional space for stable case.


## 6. Conclusion

In this paper, we obtained the existence and global exponential stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by direct method. Our methods and results are new. The methods proposed in this paper can be used to study the problem of almost automorphic solutions of other types of Clifford-valued neural networks with or without leakage delays such as Clifford-valued BAM networks, Clifford-valued cellular neural networks, and Clifford-valued shunting inhibitory cellular neural networks. Studying the dynamics of the Clifford-valued neural networks with impulsive effects is our future work.



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