# Global Asymptotic Almost Periodic Synchronization of Clifford-Valued CNNs with Discrete Delays

Authors: Yongkun Li; Jianglian Xiang
Journal: Complexity
 (2019)

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## Abstract

In this paper, we are concerned with Clifford-valued cellular neural networks (CNNs) with discrete delays. Since Clifford algebra is a unital associative algebra and its multiplication is noncommutative, to overcome the difficulty of the noncommutativity of the multiplication of Clifford numbers, we first decompose the considered Clifford-valued neural network into2m2n real-valued systems. Second, based on the Banach fixed point theorem, we establish the existence and uniqueness of almost periodic solutions of the considered neural networks. Then, by designing a novel state-feedback controller and constructing a proper Lyapunov function, we study the global asymptotic synchronization of the considered neural networks. Finally, a numerical example is presented to show the effectiveness and feasibility of our results.


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## Body

## 1. Introduction

Clifford algebras, which were invented by mathematician W. K. Clifford, are algebras generated from vector spaces with quadratic forms and are unitary associative algebras. They contain real numbers, complex numbers, quaternions, and several other hypercomplex systems as special cases. Clifford algebras have important applications in a variety of fields, including geometry, theoretical physics, and digital image processing [1]. Clifford-valued neural networks generalize real-valued, complex-valued, quaternion-valued, and octonion-valued neural networks and so on. Since the Clifford-valued neural networks can use multistate activation functions to process multilevel information and require much fewer connection weight parameters of the networks, they have recently been an active research area [2–5]. Due to the noncommutativity of Clifford numbers’ multiplication, it brings great difficulties to the research of Clifford-valued neural networks. At present, the results of research on Clifford-valued neural networks are still rare, especially, for the nonautonomous ones.On the one hand, it is well known that, in the design and implementation of neural networks, the existence of periodic solutions or almost periodic solutions of nonautonomous neural networks is as important as the existence of equilibrium points of autonomous neural networks. At the same time, we know that, even if all the time-varying coefficients in a neural network are periodic functions (such as connection coefficients, connection weight functions, external inputs, and time delays, etc.), if their periods are incommensurable, then, for such a neural network, it is also impossible to have a periodic solution. Therefore, studying the almost periodicity of neural networks is more practical and important than studying the periodicity of neural networks. At present, the periodicity has been extensively studied as an important dynamic property of various neural networks. However, there has been no paper published on the almost periodicity of Clifford-valued neural networks yet.On the other hand, the synchronization of nonlinear systems has become an important research topic due to its potential applications in various fields such as secure communication, image encryption, information science, and so on. Particularly, recently, many authors have studied the synchronization problem for various neural network systems [6–15]. For example, the synchronization problem for chaotic memristor-based neural networks with time-varying delays was studied in [11]. The global asymptotic synchronization problem of nonidentical fractional-order neural networks with Riemann-Liouville derivative was investigated in [12]. The synchronization of an inertial neural network with time-varying delays was investigated in [13]. The finite-time cluster synchronization of coupled fuzzy cellular neural networks with Markovian switching topology, discontinuous activation functions, proportional leakage, and time-varying unbounded delays was studied in [14]. The global exponential almost periodic synchronization of quaternion-valued neural networks with time-varying delays was investigated in [15]. But until now, the results of the synchronization of Clifford-valued neural networks have not been reported.In summary, it is meaningful to study the almost periodicity and the synchronization problem of Clifford-valued neural networks. Therefore, our main purpose of this paper is to investigate the problem of the existence of almost periodic solutions and global asymptotic synchronization of Clifford-valued CNNs with discrete delays. To the best of our knowledge, this is the first paper to study such a problem for Clifford-valued neural networks with discrete delays. Our methods of this paper can be used to study other types of Clifford-valued neural networks.The rest of this paper is organized as follows: in Section2, we introduce some basic concepts, notations, and lemmas and give a model description. In Section 3, we study the existence of almost periodic solutions of Clifford-valued CNNs with discrete delays. In Section 4, we investigate the global asymptotic almost periodic synchronization of Clifford-valued CNNs with discrete delays. In Section 5, we give an example to demonstrate the effectiveness and feasibility of our results. In Section 5, we give a conclusion.


## 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={h1,h2,…,hν},1≤h1<h2<⋯<hν≤m. Moreover, e∅=e0=1 and e{h}=eh,h=1,2,…,m are called Clifford generators which satisfy the relations ei2=-1,i=1,2,…,m, and eiej+ejei=0,i≠j,i,j=1,2,…,m.For simplicity, when one element is the product of multiple Clifford generators, we will write its subscripts together. For example,e4e5e7e9=e4579. Denote Λ={∅,1,2,…,A,…,12⋯m}, and then we have(2)A=∑AaAeA,aA∈R,where ∑A‍ is short for ∑A∈Λ‍ and A is isomorphic to R2m.For anyx=∑A‍xA∈A, the involution of x is defined as(3)x¯=∑AxAe¯A,where e¯A=(-1)μ[A](μ[A]+1)/2eA, and if A=∅, then μ[A]=0; if A=h1h2…hν∈Λ, then μ[A]=ν. It is easy to see that eAe¯A=e¯AeA=1 and for every x=∑A‍xAeA∈A, x¯=∑A‍xAe¯A. Moreover, we have xy¯=y¯x¯,∀x,y∈A.For a Clifford-valued functionz=∑A‍zAeA:R→A, where zA:R→R,A∈Λ, its derivative is given by(4)z˙t=∑Az˙AteA.In view of eBe¯A=(-1)μ[A](μ[A]+1)/2eBeA, we can write eBe¯A=eC or eBe¯A=-eC, where eC is a Clifford generator of Clifford algebra A. Hence it is possible to find a unique corresponding basis eC for the given eBe¯A. Define (5)μB·A¯=0,ifeBe¯A=eC,1,ifeBe¯A=-eC,and then eBe¯A=(-1)μ[B·A¯]eC. In addition, for any E∈A, we can find a unique EC satisfying EB·A¯=(-1)μ[B·A¯]EC for eBe¯A=(-1)μ[B·A¯]eC. Therefore, (6)EB·A¯eBe¯A=EB·A¯-1μB·A¯eC=-1μB·A¯EC-1μB·A¯eC=ECeCand(7)E=∑CECeC∈A.Throughout this paper,An,Rm×n,Am×n represent the n-dimensional real Clifford vector space, the set of all m×n real matrices, and the set of all m×n real Clifford matrices. We define the norm of Rn as x=∑i=1n‍xi. For A=(aij)n×n, denote A=max1≤i≤n{∑j=1n‍aij} and, for x=∑A‍xAeA∈A, denote xA=∑A‍xA.In this paper, we consider the following Clifford-valued cellular neural network with discrete delays:(8)X˙it=-δitXit+∑j=1nγijtfjXjt+∑j=1nωijtgjXjt-τjt+Iit,t>0,where i=1,2,…,n, and n corresponds to the number of units in the neural network; Xi(t)∈A denotes the activation of the ith neuron at time t; δi(t)>0 represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs at time t; γij(t),ωij(t)∈A represent the strengths of connectivity without and with time delay between cells i and j at time t, respectively; fj,gj:A→A are the activation functions of the signal transmission; Ii(t)∈A is an external input on the ith unit at time t; τj(t)≥0 is the transmission delay at time t.The initial values of (8) are(9)Xit=φit,t∈-τ,0,i=1,2,…,n,where φi∈C([-τ,0],A),τ=max1≤j≤nsupt∈R{τjt}.LetX(t)=(X1(t),X2(t),…,Xn(t))T∈An, Δ(t)=diag(δ1(t),δ2(t),…,δn(t))∈Rn×n, Γ(t)=(γij(t))n×n,Ω(t)=(ωij(t))n×n∈An×n, I(t)=(I1(t),I2(t),…,In(t))T∈An, f(X(t))=(f1(X1(t)),f2(X2(t)),…,fn(Xn(t)))T∈An, and g(X(t-τ(t)))=(g1(X1(t-τ1(t))),g2(X2(t-τ2(t))),…,gn(Xn(t-τn(t))))T∈An. Then (8) can be transformed into the following equation:(10)X˙t=-ΔtXt+ΓtfXt+ΩtgXt-τt+It,t>0with the initial value:(11)Xt=φt,t∈-τ,0,φt=φ1t,φ2t,…,φntT.Definition 1 (see [16]).
A functiony∈C(R,Rn) is said to be almost periodic on R, if, for any ε>0, it is possible to find a real number l=l(ε)>0 such that, for any interval with length l(ε), there is a number τ=τ(ε) in this interval such that yt+τ-yt<ε.Denote byAP(R,Rn) the set of all almost periodic functions on R; then AP(R,Rn) is a Banach space with the supremum norm u=∑i=1n‍supt∈Ruit.Definition 2.
A functionf=(f1,f2,…,fn)T:R→An, where fi=∑A‍fiAeA,fiA:R→R, is called almost periodic if every fiA∈AP(R,R),i=1,2,…,n. Denote by AP(R,An) the set of all such functions.Consider the following linear system(12)x˙t=Qtxt+ft,where Q(t)=diag-q1t,-q2t,…,qnt,qi∈AP(R,R),f∈AP(R,Rn).According to [16], one can easily get the following lemma.Lemma 3.
Letqi(t) satisfy min1≤i≤ninft∈R{qi(t)}>0, and then system (12) has a unique almost periodic solution (13)xt=∫-∞te∫stQudufsds.


## 3. The Existence of Almost Periodic Solutions

In this section, we study the existence and uniqueness of almost periodic solutions by the contracting mapping principle.Firstly, since, for anyW∈A, we can find a unique WC satisfying WB·A¯=(-1)n(B·A¯)WC for eBe¯A=(-1)n(B·A¯)eC, then, based on eAe¯A=e¯AeA=1 and eBe¯AeA=eB, we can transform Clifford-valued system (10) into the following real-valued neural network:(14)x˙At=-ΔtxAt+∑BΓA·B¯tfBxt+∑BΩA·B¯tgBxτt+IAt,A∈Λ,where(15)xA=x1A,x2A,…,xnAT,∑AxAeA=X,-1nA·B¯ΓC=ΓA·B¯,-1nA·B¯ΩC=ΩA·B¯,∑CΓCeC=Γ,∑CΩCeC=Ω,eAeB¯=-1nA·B¯eC,ΓA·B¯=γijA·B¯n×n,ΩA·B¯=ωijA·B¯n×n,IA=I1A,I2A,…,InAT,fBx=f1Bx1C1,x1C2,…,x1C2m,f2Bx2C1,x2C2,…,x2C2m,…,fnBxnC1,xnC2,…,xnC2mT,gBxτt=g1Bx1C1t-τ1t,x1C2t-τ1t,…,x1C2mt-τ1t,g2Bx2C1t-τ2t,x2C2t-τ2t,…,x2C2mt-τ2t,…,gnBxnC1t-τnt,xnC2t-τnt,…,xnC2mt-τntT.Remark 4.
Ifx(t)=(x0(t),x1(t),…,xA(t),…,x1·2·⋯m(t))T is a solution of (14), then X(t)=(X1,X2,…,Xn)T=(∑A‍x1A(t)eA,∑A‍x2A(t)eA,…,∑A‍xnA(t)eA)T is a solution of (8) and vice versa.Denote(16)S=x0T,x1T,x2T,…,xAT,…,x12…mTT∈R2mn,f~S=f0xT,f1xT,f2xT,…,fAxT,…,f12…mxTT,g~Sτ=g0xτT,g1xτT,g2xτT,…,gAxτT,…,g12…mxτTT,I~t=I0T,I1T,I2T,…,IAT,…,I12…mTT,Δ~=Δ0⋯00Δ⋯0⋮⋮⋱⋮00⋯Δ2mn×2mn,Γ~=Γ0⋯ΓA¯⋯Γ12…m¯Γ1⋯Γ1·A¯⋯Γ1·12…m¯⋮⋯⋮⋯⋮Γ12…m⋯Γ12…m·A¯⋯Γ12…m·12…m¯2mn×2mn,Ω~=Ω0⋯ΩA¯⋯Ω12…m¯Ω1⋯Ω1·A¯⋯Ω1·12…m¯⋮⋯⋮⋯⋮Ω12…m⋯Ω12…m·A¯⋯Ω12…m·12…m¯2mn×2mn,and then (14) can be written as(17)S˙t=-Δ~tSt+Γ~tf~St+Ω~tg~Sτt+I~t,t>0with the initial value:(18)St=ϕt,t∈-τ,0.SetX={Φ∣Φ=((Φ0)T,(Φ1)T,…,(Φ1·2⋯m)T)T∈AP(R,R2mn)} with the norm(19)Φ0=∑i=1n∑AΦiA,ΦiA=supt∈RΦiAt,i=1,2,…,n,which is a Banach space.In order to obtain our results, we introduce the following assumptions.(H1)
Fori,j=1,2,…,n, δi,τj∈AP(R,R+),γij,ωij,Iij∈AP(R,A) and for each 1≤i≤n, min1≤i≤n{inft∈Rci(t)}>0.(H2)
Fori=1,2,…,n, fi,gi∈C(A,A) and there exist constants Lif,Lig such that (20)fix-fiyA≤Lifx-yA,gix-giyA≤Ligx-yA,and there exist constantsF,G>0 such that f(x)A≤F,g(x)A≤G, for any x,y∈A.Remark 5.
By(H2), we can obtain (21)∑AfiAyiB1,yiB2,…,yiB2m-fiAxiB1,xiB2,…,xiB2m≤Lif∑ByiB-xiB,∑AgiAyiB1,yiB2,…,yiB2m-giAxiB1,xiB2,…,xiB2m≤Lig∑ByiB-xiB.Theorem 6.
Let(H1) and (H2) hold. Suppose that (H3)
(Γ~+Lf+L¯+Lg)1/Δ~-<1, where(22)Γ~+=supt∈RΓ~t,Ω~+=supt∈RΩ~t,Δ~-=inft∈RΔ~t,I~+=supt∈RI~t,Lf=max1≤i≤nLif,Lg=max1≤i≤nLig.Then system of (8) has a unique almost periodic solution in the region (23)X0=Φ∈X∣Φ0≤Γ~+F+Ω~+G+I~+1Δ~-≔R.Proof.
For∀Φ∈X, we consider the following system:(24)S˙t=-Δ~tSt+Γ~tf~Φt+Ω~tg~Φt-τt+I~t.Since (H1), it follows from Lemma 3 that system (24) has a unique almost periodic solution (25)St=∫-∞te∫st-Δ~uduΓ~sf~Φs+Ω~sg~Φs-τs+I~sds. We define a mapping T:X0→X as follows: (26)TΦt=∫-∞te∫st-Δ~uduΓ~sf~Φs+Ω~sg~Φs-τs+I~sds.
We first prove thatT(X0)⊂X0. To this end, for each Φ∈X0, we have (27)TΦt=∫-∞te∫st-Δ~uduΓ~sf~Φs+Ω~sg~Φs-τs+I~sds≤∫-∞te∫st-Δ~uduΓ~sf~Φsds+∫-∞te∫st-Δ~uduΩ~sg~Φs-τsds+∫-∞te∫st-Δ~uduI~sds≤∫-∞te-Δ~-t-sΓ~+f~Φsds+∫-∞te-Δ~-t-sΩ~+g~Φs-τsds+∫-∞te-Δ~-t-sI~+ds≤Γ~+F∫-∞te-Δ~-t-sds+Ω~+G∫-∞te-Δ~-t-sds+I~+∫-∞te-Δ~-t-sds≤Γ~+F+Ω~+G+I~+1Δ~-=R,which implies TΦ∈X0. Hence, T(X0)⊂X0.
Then, we prove thatT is a contracting mapping. In fact, for every Φ,Ψ∈X0, we have (28)TΦ-TΨ≤∫-∞te∫st-Δ~uduΓ~sf~Φs-f~Ψsds+∫-∞te∫st-Δ~uduΩ~sg~Φs-τs-g~Ψs-τsds≤∫-∞te-Δ~-t-sΓ~+f~Φs-f~Ψsds+∫-∞te-Δ~-t-sΩ~+g~Φs-τs-g~Ψs-τsds≤Γ~+Lf∑i=1n∑BΦiB-ΨiB∫-∞te-Δ~-t-sds+Ω~+Lg∑i=1n∑BΦiB-ΨiB∫-∞te-Δ~-t-sds≤Γ~+Lf+Ω~+Lg1Δ~-Φ-Ψ.Hence, T:X0→X0 is a contracting mapping. Therefore, by the Banach fixed point theorem, there exists a unique point Φ∗∈X0 such that TΦ∗=Φ∗; that is, system (14) has a unique almost periodic solution. In view of Remark 4, we know that system (8) has a unique almost periodic solution. The proof is complete.


## 4. Almost Periodic Synchronization

In this section, we will investigate the global asymptotic synchronization problem of Clifford-valued CNNs with discrete delays. To this end, we consider the system (8) as the drive system and design the response system as(29)y˙it=-δityit+∑j=1nγijtfjyjt+∑j=1nωijtgjyjt-τjt+Iit+Uit,yit=ρit,t∈-τ,0,i=1,2,…,n,where ρi∈C([-τ,0],A), Yi(t)∈A represents the state of the response system, Ui(t) is a state-feedback controller, and the rest notation is the same as those in system (8).LetziA(t)=yiA(t)-xiA(t); then, from (8) and (29), we obtain the following error system:(30)z˙it=-δitzit+∑j=1nγijtfjyjt-fjxjt+∑j=1nωijtgjyjt-τjt-gjxjt-τjt+Uit,t>0,zit=ϱit,t∈-τ,0,i=1,2,…,n,where ϱi(t)=ρi(t)-φi(t). To realize the global asymptotic synchronization of the drive-response system, we choose the following state-feedback controller (31)Uit=θitzit+∑j=1nπijthjzjt,where i=1,2,…,n, θi:R→R+,πij:R→A,hj:A→A. By using a similar approach of transforming system (8) into system (14) and adopting the similar notation there, system (30) can be transformed into the following real-valued system:(32)z˙iAt=-δitziAt+∑j=1n∑BγijA·B¯tfjByjt-fjBxjt+∑j=1n∑BωijA·B¯tgjByjt-τijt-gjBxjt-τijt,t>0,+θitziAt+∑j=1n∑BπijA·B¯thjBzjt,ziAt=ϱiAt,t∈-τ,0,t∈-τ,0,i=1,2,…,n.Remark 7.
Under the premise that system (8) has an almost periodic solution, the synchronization of almost periodic system (8) and almost periodic system (29) is called the almost periodic synchronization.Theorem 8.
Let(H1)-(H3) hold. Suppose the following. (H4)
Fori,j=1,2,…,n, θi∈AP(R,R) and πij∈AP(R,A).(H5)
Fori=1,2,…,n, hi satisfy the idea that there exist constants Lih such that hi(x)-hi(y)A≤Lihx-yA for all x,y∈A, and hi(0)=0.(H6)
Fort∈R,i=1,2,…,n,τi∈C1(R,R+) with 1-τi′(t)>0 and (33)Θi≔δi--θi^-∑A∑B∑j=1nγjiA·B¯Lif-∑A∑B∑j=1nωjiA·B¯Lig1-τ^i′-∑A∑B∑j=1nπjiA·B¯Lih>0,where(34)θi^=supt∈Rθit,M^=supt∈RM¯t,γ^jiA·B¯=supt∈RγjiA·B¯t,τ^i′=supt∈Rτi′t,m^jiA·B¯=supt∈RmjiA·B¯t,ω^jiA·B¯=supt∈RωjiA·B¯t,δi-=inft∈Rδit,Lh=max1≤i≤nLih.Then the drive system (8) and the response system (29) implement global asymptotical almost periodic synchronization.Proof.
By Theorem6, system (8) has an almost periodic solution. In order to show that system (8) and system (29) are globally asymptotically synchronized, we consider a Lyapunov function (35)Vt=V1t+V2t, where (36)V1t=∑i=1n∑AziAt,V2=∑A∑B∑i=1n∑j=1nω^jiA·B¯Lig1-τ^i′∫t-τitt∑QziQsds. Calculating the derivatives of V1 and V2 along the solutions of system (32), we have (37)D+V1t=∑i=1n∑Az˙iAtsignziAt≤-∑i=1n∑AδitziAt+∑i=1n∑A∑B∑j=1nγijA·B¯tfjByjt-fjBxjt+∑i=1n∑A∑B∑j=1nωijA·B¯tgjByjt-τjt-gjBxjt-τjt+∑i=1n∑AθitziAt+∑i=1n∑A∑B∑j=1nπijA·B¯thjBzjt≤∑i=1n∑AziAt-δi-+θi^+∑A∑B∑i=1n∑j=1nγ^jiA·B¯Lif∑QziQt+∑A∑B∑i=1n∑j=1nω^jiA·B¯Lig∑QziQt-τit+∑A∑B∑i=1n∑j=1nπ^jiA·B¯Lih∑QziQt and (38)D+V2t=∑A∑B∑i=1n∑j=1nω^jiA·B¯Lig1-τ^i′∑QziQt-∑A∑B∑i=1n∑j=1nω^jiA·B¯Lig1-τi′t1-τ^i′∑QziQt-τit,≤∑A∑B∑i=1n∑j=1nω^jiA·B¯Lig1-τ^i′∑QziQt-∑A∑B∑i=1n∑j=1nω^jiA·B¯Lig∑QziQt-τit. Hence, (39)D+Vt=D+V1t+D+V2t≤∑i=1n∑AziAt-δi-+θi^+∑A∑B∑i=1n∑j=1nγ^jiA·B¯Lif+ω^jiA·B¯Lig1-τ^i′∑QziQt+∑A∑B∑i=1n∑j=1nπ^jiA·B¯Lih∑QziQt=∑i=1n∑QziQt-δi-+θi^+∑A∑B∑j=1nγ^jiA·B¯Lif+∑A∑B∑j=1nωjiA·B¯Lig1-τ^i′+∑A∑B∑j=1nπ^jiA·B¯Lih≤-V1tmin1≤i≤nΘi. Integrating the above inequality over interval [t0,t], we get(40)Vt+min1≤i≤nΘi∫t0tV1sds≤Vt0<+∞,t≥t0. Thus,(41)limsupt→+∞∫t0tV1sds≤Vt0≤Vt0min1≤i≤nΘi<+∞.Therefore, limt→+∞V1(t)=0. By the definition of V1(t), we obtain that limt→+∞yiBt-xiBt=0, for all B∈Λ. Hence, the drive system (8) and the response system (29) are globally asymptotically synchronized. The proof is complete.


## 5. A Numerical Example

In this section, we present an example to illustrate our results.Example 1.
Form=2,n=2, take the following Clifford-valued CNN as the drive system:(42)x˙t=-Δtxt+Γtfxt+Ωtgxt-τt+It,t>0,xt=φt,t∈-τ,0and the response system is designed as(43)y˙t=-Δtyt+Γtfyt+Ωtgyt-τt+It+Ut,t>0,xt=ϕt,t∈-τ,0,where(44)Δt=2+sin⁡2t004+sin⁡t,Γt=0.2sin⁡t+sin⁡te10.1+0.3cos⁡te2+0.6e120.05-0.2cos⁡5te2+0.4e120.1+0.2e1+0.05sin⁡3te12,Ωt=0.3sin⁡3t+0.01sin⁡te10.1+0.02cos⁡3te2+0.3e120.05-0.2sin⁡te2+0.05e120.2+0.2e1+0.05sin⁡3te12,fjxj=165sin⁡xj0-170sin⁡2xj2e2,gjxj=158cos⁡3xj2e2-155sin⁡6xj12e12,Ut=θitet+Πthzt-σt,hjzj=175sin⁡zj0-168sin⁡2zj2e2,θt=0.01+0.03sin⁡2t000.1+0.05cos⁡2t,I1tI2t=110sin⁡x10t+115sin⁡3x112te12110sin⁡x20t+115sin⁡3x212te12,τ1tτ2t=0.1sin⁡3.2t+0.20.1sin⁡3.2t+0.3,Πt=0.2sin⁡2t+0.01sin⁡te10.1+0.3cos⁡te2+0.6e120.05-0.2cos⁡5te2+0.3e120.1+0.2e1+0.05sin⁡3te12.According to their definitions, we have(45)Γ0t=0.2sin⁡t0.10.050.1,Γ1t=sin⁡t000.2,Γ2t=00.3cos⁡t-0.2cos⁡5t0,Γ12t=00.60.40.05sin⁡3t,Ω0t=0.3sin⁡3t0.10.050.2,Ω1t=0.01sin⁡t000.2,Ω2t=00.02cos⁡3t-0.2sin⁡t0,Ω12t=00.30.050.05sin⁡3t,Π0t=0.2sin⁡2t0.10.050.1,Π1t=0.01sin⁡t000.2,Π2t=00.3cos⁡t-0.2cos⁡5t0,Π12t=00.60.30.05sin⁡3t, and(46)Γ~=Γ0Γ1¯Γ2¯Γ12¯Γ1Γ1·1¯Γ1·2¯Γ1·12¯Γ2Γ2·1¯Γ2·2¯Γ2·12¯Γ12Γ12·1¯Γ12·2¯Γ12·12¯=Γ0-Γ1-Γ2-Γ12Γ1Γ0-Γ12Γ2Γ2Γ12Γ0-Γ1Γ12-Γ2Γ1Γ0,Ω~=Ω0Ω1¯Ω2¯Ω12¯Ω1Ω1·1¯Ω1·2¯Ω1·12¯Ω2Ω2·1¯Ω2·2¯Ω2·12¯Ω12Ω12·1¯Ω12·2¯Ω12·12¯=Ω0-Ω1-Ω2-Ω12Ω1Ω0-Ω12Ω2Ω2Ω12Ω0-Ω1Ω12-Ω2Ω1Ω0,Π~=Π0Π1¯Π2¯Π12¯Π1Π1·1¯Π1·2¯Π1·12¯Π2Π2·1¯Π2·2¯Π2·12¯Π12Π12·1¯Π12·2¯Π12·12¯=Π0-Π1-Π2-Π12Π1Π0-Π12Π2Π2Π12Π0-Π1Π12-Π2Π1Π0.By calculating, we obtainΔ~-=1, Γ~+=1.6, L1f=L2f=1/65, L1g=L2g=1/55, Ω~+=0.75, L1h=L2h=1/68, (γjiA·B¯)+=0.6, (ωjiA·B¯)+=0.3, (πjiA·B¯)+=0.6, θ1^=0.04, θ2^=0.15, c1-=1, c2-=3, τ^i′=0.32, i,j=1,2.It is easy to get the following inequalities:(47)Γ~+Lf+Ω~+Lg1Δ~-<1,min1≤i≤2-δi-+θi^+∑A∑B∑j=12γjiA·B¯+Lif+∑A∑B∑j=12ωjiA·B¯+Lig1-τ^i′+∑A∑B∑j=12πjiA·B¯+Lih≈-0.015<0.Therefore, all the conditions of Theorem 8 are satisfied. Hence, by Theorem 8, the drive system (42) and the response system (43) implement global asymptotic almost periodic synchronization (see Figures 1–5).Figure 1
The states ofx0(t), x1(t), x2(t), and x12(t) in (42) without any controller.Figure 2
The states of four parts ofx1(t) and x2(t) in 2-dimensional space without control.Figure 3
The states ofy0(t), y1(t), y2(t), and y12(t) in (43) under controller U(t).Figure 4
The states of four parts ofy1(t) and y2(t) in 2-dimensional space under controller U(t).Figure 5
Synchronization errorsz(t)=y(t)-x(t).


## 6. Conclusion

In this paper, we studied the problem of the global asymptotic almost periodic synchronization for a class of Clifford-valued CNNs with discrete delays by decomposing the considered neural networks into2m2n real-valued systems and by designing a new state-feedback controller. Our results are new and our methods can be used to study other types of Clifford-valued neural networks.



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