# Almost Periodic Synchronization for Quaternion-Valued Neural Networks with Time-Varying Delays

Authors: Yongkun Li; Xiaofang Meng; Yuan Ye
Journal: Complexity
 (2018)

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## Abstract

This paper focuses on the global exponential almost periodic synchronization of quaternion-valued neural networks with time-varying delays. By virtue of the exponential dichotomy of linear differential equations, Banach’s fixed point theorem, Lyapunov functional method, and differential inequality technique, some sufficient conditions are established for assuring the existence and global exponential synchronization of almost periodic solutions of the delayed quaternion-valued neural networks, which are completely new. Finally, we give one example with simulation to show the applicability and effectiveness of our main results.


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## Body

## 1. Introduction

It is well known that the quaternion is a mathematical concept discovered by Hamilton in 1843, which did not arouse much attention for quite a long time, let alone real world applications. Due to the noncommutativity of quaternion multiplication, the study of quaternion is much more difficult than that of plurality, which is one of the reasons for the slow development of quaternion. Fortunately, over the past two decades, the quaternion theory has achieved a rapid development, especially in algebra, and found many applications in the real world, like attitude control, quantum mechanics, robotics, computer graphics, and so on [1–5]. For example, in the application of color image compression technology, one can apply the quaternion theory to encode and improve the color image; see [5]. In recent years, the quaternion-valued neural networks, as an extension of the real-valued neural networks and the complex-valued neural networks [6, 7], research has become a hot topic. It should be pointed out that, at present, almost all the investigations on quaternion-valued neural networks are mainly dealing with the stability, robustness, or dissipation of the equilibrium of the neural networks; see [8–17].On the one hand, since the concept of drive-response synchronization for coupled chaotic systems was proposed by Pecora and Carroll in [18], chaos synchronization has become a hot research topic due to its potential applications in secure communication, automatic control, biological systems, and information science; see, for instance, [19–26].On the other hand, the almost periodicity is a generalization of periodicity and is much more common than the periodicity, and so, almost periodic oscillation is a very important dynamic phenomenon for nonautonomous neural networks [27–30]. However, so far, few results have been available for the almost periodicity of complex-valued neural networks [31] and quaternion-valued neural networks. Moreover, in the past few years, although the problem of periodic synchronization for real-valued systems has been extensively studied by many scholars (see, for instance, [32–43]), there are few articles dealing with the almost periodic synchronization for real-valued systems, let alone dealing with the almost periodic synchronization for complex-valued or quaternion-valued neural networks. Very recently, some authors study the synchronization for complex-valued neural networks with time delays [44, 45]. Based on the LMI method and mathematical analysis technique, they obtain some sufficient conditions for synchronization of complex-valued neural networks. However, to the best of our knowledge, there has been no paper published on the almost periodic synchronization of the quaternion-valued neural networks, which remains as an open challenge. So, it is necessary to investigate the synchronization of quaternion-valued neural networks.Motivated by the above statement, the main purpose of this paper is to study the synchronization of quaternion-valued neural networks. The main contributions of this paper can be summarized as follows:(1) this is the first time to investigate the almost periodic synchronization of neural networks. (2) Compared with other results, the results of this paper are the ones about quaternion value and with time delays. Therefore, the results are less conservative and more general. (3) Our methods used in this paper can easily apply to study the almost periodic synchronization for other types of neural networks with delays.This paper is organized as follows: In Section2, we give the model description and introduce some definitions and preliminary lemmas and transform the quaternion-valued system (9) into an equivalent real-valued system. In Section 3, we establish some sufficient conditions for the existence and global exponential synchronization of almost periodic solutions of (9). In Section 4, we give an example to demonstrate the feasibility of our results. This paper ends with a brief conclusion in Section 5.


## 2. Model Description and Preliminaries

In this section, we shall first recall some fundamental definitions and lemmas which are used in what follows.First, we give some notations of this paper. LetR and Q stand for the real field and the skew field of quaternions, respectively. Rn×n, Qn×n denote the set of all n×n real-valued and quaternion-valued matrices, respectively. The skew field of quaternion is denoted by (1)Q≔x=xR+ixI+jxJ+kxK,where xR, xI, xJ, xK are real numbers and the elements i, j and k obey Hamilton’s multiplication rules: (2)ij=-ji=k,jk=-kj=i,ki=-ik=j,i2=j2=k2=ijk=-1.Definition 1 (see [46], let f∈BCR,Rn).
Functionf is said to be almost periodic if, for any ϵ>0, it is possible to find a real number l=l(ϵ)>0, for any interval with length lϵ; there exists a number τ=τ(ϵ) in this interval such that ft+τ-ft<ϵ, for all t∈R.We denote byAPR,Rn the set of all almost periodic functions from R to Rn.Definition 2.
Quaternion-valued functionf=fR+ifI+jfJ+kfK is an almost periodic function if and only if fR, fI, fJ and fK are almost periodic functions.Consider the following linear homogenous system(3)x′t=Atxtand linear nonhomogenous system(4)x′t=Atxt+ft,where At is an almost periodic matrix function and ft is an almost periodic vector function.Definition 3 (see [46]).
System (3) is said to admit an exponential dichotomy if there exist a projection P and positive constants α,β such that the fundamental solution matrix X(t) satisfies (5)XtPX-1s≤βe-αt-s,t≥s,XtI-PX-1s≤βe-αs-t,t≤s.Lemma 4 (see [46]).
If the linear system (3) admits an exponential dichotomy, then system (4) has a unique almost periodic solution that can be expressed as(6)xt=∫-∞tXtPX-1sfsds-∫t+∞XtI-PX-1sfsds,where Xt is the fundamental solution matrix of (3).Lemma 5 (see [46]).
Letcp be an almost periodic function on R and(7)Mcp≔limT→∞1T∫tt+Tcpsds>0,p=1,2,…,n.Then the linear system (8)x′t=diag-c1t,-c2t,…,-cntxtadmits an exponential dichotomy on R.Consider the following quaternion-valued neural networks with time-varying delays:(9)xp′t=-dptxpt+∑q=1napqtfqxqt+∑q=1nbpqtgqxqt-τpqt+upt,where p∈1,2,…,n≔T and xp(t)∈Q is the state of the pth neuron at time t; dpt>0 is the self-feedback connection weight; apqt∈Q and bpqt∈Q are the connection weight and the delay connection weight from neuron q to neuron p at time t, respectively; upt∈Q is an external input on the pth unit at time t; fqxqt∈Q and gqxqt-τpqt∈Q denote the activation functions without and with time-varying delays, respectively; τpqt represents the time-varying delay and satisfies 0≤τpqt≤τ.The initial value of (9) is given by (10)xps=ϕps,s∈-τ,0,p∈T,where ϕp∈C-τ,0,Q.The response system of (9) is designed as(11)yp′t=-dptypt+∑q=1napqtfqyqt+∑q=1nbpqtgqyqt-τpqt+upt+θpt,where p∈T,ypt∈Q denotes the state of the pth neuron at time t of the response system and θpt∈Q is a controller.The initial condition associated with (11) is of the form (12)yps=ψps,s∈-τ,0,p∈T,where ψp∈C-τ,0,Q.In order to overcome the inconvenience of the noncommutativity of quaternion multiplication, in the following, we always assume the following:(H1)
Letxq=xqR+ixqI+jxqJ+kxqK,xqR,xqI,xqJ,xqK∈R. Then fqxq and gqxq can be expressed as (13)fqxq=fqRxqR,xqI,xqJ,xqK+ifqIxqR,xqI,xqJ,xqK+jfqJxqR,xqI,xqJ,xqK+kfqKxqR,xqI,xqJ,xqK,gqxq=gqRxqR,xqI,xqJ,xqK+igqIxqR,xqI,xqJ,xqK+jgqJxqR,xqI,xqJ,xqK+kgqKxqR,xqI,xqJ,xqK,where fqν,gqν:R4→R, q∈T, ν∈R,I,J,K≔E.According to(H1), system (9) can be transformed into the following real-valued system:(14)xpR′t=-dptxpRt+∑q=1napqRtfqRt,x-apqItfqIt,x-apqJtfqJt,x-apqKtfqKt,x+∑q=1nbpqRtgqRt-τpqt,x-bpqItgqIt-τpqt,x-bpqJtgqJt-τpqt,x-bpqKtgqKt-τpqt,x+upRt,p∈T,xpI′t=-dptxpIt+∑q=1napqRtfqIt,x+apqItfqRt,x+apqJtfqKt,x-apqKtfqJt,x+∑q=1nbpqRtgqIt-τpqt,x+bpqItgqRt-τpqt,x+bpqJtgqKt-τpqt,x-bpqKtgqJt-τpqt,x+upIt,p∈T,xpJ′t=-dptxpJt+∑q=1napqRtfqJt,x+apqJtfqRt,x-apqItfqKt,x+apqKtfqIt,x+∑q=1nbpqRtgqJt-τpqt,x+bpqJtgqRt-τpqt,x-bpqItgqKt-τpqt,x+bpqKtgqIt-τpqt,x+upJt,p∈T,xpK′t=-dptxpKt+∑q=1napqRtfqKt,x+apqKtfqRt,x+apqItfqJt,x-apqJtfqIt,x+∑q=1nbpqRtgqKt-τpqt,x+bpqKtgqRt-τpqt,x+bpqItgqJt-τpqt,x-bpqJtgqIt-τpqt,x+upKt,p∈T,where fqνt,x≜fqνxqRt,xqIt,xqJt,xqKt, gqνt-τpqt,x≜gqνxqRt-τpqt,xqIt-τpqt,xqJt-τpqt,xqKt-τpqt,p,q∈T,ν∈E, and (15)apqt=apqRt+iapqIt+japqJt+kapqKt,p,q∈T,bpqt=bpqRt+ibpqIt+jbpqJt+kbpqKt,p,q∈T,upt=upRt+iupIt+jupJt+kupKt,p∈T.It follows from (14) that(16)Xp′t=-dpXpt+∑q=1nApqtFqt,x+∑q=1nBpqtGqt-τpqt,x+Upt,p∈T,where (17)Apqt=apqRt-apqIt-apqJt-apqKtapqItapqRt-apqKtapqJtapqJtapqKtapqRt-apqItapqKt-apqJtapqItapqRt,Bpqt=bpqRt-bpqIt-bpqJt-bpqKtbpqItbpqRt-bpqKtbpqJtbpqJtbpqKtbpqRt-bpqItbpqKt-bpqJtbpqItbpqRt,Fqt,x=fqRt,xfqIt,xfqJt,xfqKt,x,Gqt-τpqt,x=gqRt-τpqt,xgqIt-τpqt,xgqJt-τpqt,xgqKt-τpqt,x,Xpt=xpRt,xpIt,xpJt,xpKtT,Upt=upRt,upIt,upJt,upKtT.The initial condition associated with (16) is of the form (18)Xps=Φps,s∈-τ,0,p∈T,where Φp(s)=φpRs,φpIs,φpJs,φpKsT,φpν(s)∈C-τ,0,R,ν∈E.Similarly, the response system (11) can be transformed into the following real-valued system:(19)Yp′t=-dptYpt+∑q=1nApqtFqt,y+∑q=1nBpqtGqt-τpqt,y+Upt+Θpt,p∈T,where Yp(t)=ypRt,ypIt,ypJt,ypKtT,Θp(t)=θpRt,θpIt,θpJt,θpKtT.The initial condition associated with (19) is of the form (20)Yps=Ψps,s∈-τ,0,p∈T,where Ψp=ψpR,ψpI,ψpJ,ψpKT,ψpν(s)∈C-τ,0,R,ν∈E.LetZpt=Ypt-Xpt, Zp(t)=zpRt,zpIt,zpJt,zpKtT,p∈T. Then, from (16) and (19), we obtain the following error system:(21)Zp′t=-dptZpt+∑q=1nApqtFqt,y-Fqt,x+∑q=1nBpqtGqt-τpqt,y-Gqt-τpqt,x+Θpt,p∈T.In order to realize the almost periodic synchronization of the drive-response system, we choose the controller(22)θpRt=ϵpzpRt,θpIt=εpzpIt,θpJt=ρpzpJt,θpKt=ϱpzpKt,where ϵp,εp,ρp,ϱp∈R+.Definition 6.
The response system (19) and the drive system (16) can be globally exponentially synchronized, if there exist positive constants M≥1, λ>0 such that (23)Zt≤ψ-φMe-λt,t≥0,where Zt=max1≤p≤nmaxν∈Ezpνt, ψ-φ=max1≤p≤nmaxν∈Esups∈-τ,0ψpνs-φpνs.LetX=x∣x=x1R,x1I,x1J,x1K,x2R,x2I,x2J,x2K,…,xnR,xnI,xnJ,xnKT≔x1,x2,…,xnT∈APR,R4n with the norm xX=maxp∈Tmaxν∈Esupt∈Rxpνt; then X is a Banach space.Throughout this paper, we assume that the following conditions hold:(H2)
Functiondp∈CR,R+ with Mdp>0, Up∈CR,R4×1, Apq,Bpq∈CR,R4×4 and τpq∈C1R,R+ with supt∈Rτpq′t=β<1 are almost periodic, where p,q∈T.(H3)
There exist positive constantslqν,mqν such that for all xν,yν∈R, (24)fqνxqR,xqI,xqJ,xqK-fqνyqR,yqI,yqJ,yqK≤lqRxqR-yqR+lqIxqI-yqI+lqJxqJ-yqJ+lqKxqK-yqK,gqνxqR,xqI,xqJ,xqK-gqνyqR,yqI,yqJ,yqK≤mqRxqR-yqR+mqIxqI-yqI+mqJxqJ-yqJ+mqKxqK-yqK,and fqν0,0,0,0=gqν0,0,0,0=0, where q∈T, ν∈E.(H4)
There exists a positive constantκ such that (25)max⁡Ωpκ+upν+dp-≤κ,maxΩpdp-≔r<1,p∈T,ν∈E,where (26)Ωp=Λp+Δp,p∈T,Λp=∑q=1napqR++apqI++apqJ++apqK+lqR+lqI+lqJ+lqK,p∈T,Δp=∑q=1nbpqR++bpqI++bpqJ++bpqK+mqR+mqI+mqJ+mqK,p∈T.(H5)
There exists a positive constantλ such that for p∈T, (27)λ-dp-+ϵp+Λp+Δp1-βeλτ<0,λ-dp-+εp+Λp+Δp1-βeλτ<0,λ-dp-+ρp+Λp+Δp1-βeλτ<0,λ-dp-+ϱp+Λp+Δp1-βeλτ<0.


## 3. Main Results

In this section, we will study the existence of almost periodic solutions of system (16).Theorem 7.
LetH1–H4 hold. Then system (16) has a unique almost periodic solution in the region X∗=φ∣φ∈X,φX≤κ.Proof.
For anyφ∈X, we consider the linear almost periodic differential equation system(28)Xp′t=-dpXpt+∑q=1nApqtFqt,φ+∑q=1nBpqtGqt-τpqt,φ+Upt,p∈T.Combining H2 and Lemma 5, we can conclude that the linear system (29)Xp′t=-dptXpt,p∈Tadmits an exponential dichotomy on R; furthermore, by Lemma 4, we conclude that system (28) has a unique almost periodic solution that can be expressed as (30)Xφt=X1φt,X2φt,…,XnφtT,where (31)Xpφt=∫-∞te-∫stdpudu∑q=1nApqsFqs,φ+∑q=1nBpqsGqs-τpqs,φ+Upsds,p∈T.Now, we define an operator Φ:X∗→X as follows:(32)Φφ=Xφ.First, we show that, for any φ∈X∗, we have Φφ∈X∗. From (32), we have (33)ΦφpRt=∫-∞te-∫stdpudu∑q=1napqRsfqRs,φ-apqIsfqIs,φ-apqJsfqJs,φ-apqKsfqKs,φ+∑q=1nbpqRsgqRs-τpqs,φ-bpqIsgqIs-τpqs,φ-bpqJsgqJs-τpqs,φ-bpqKsgqKs-τpqs,φ+upRsds≤∫-∞te-∫stdpudu∑q=1napqR+fqRs,φ+apqI+fqIs,φ+apqJ+fqJs,φ+apqK+fqKs,φ+∑q=1nbpqR+gqRs-τpqs,φ+bpqI+gqIs-τpqs,φ+bpqJ+gqJs-τpqs,φ+bpqK+gqKs-τpqs,φ+upR+ds≤∫-∞te-∫stdpudu∑q=1napqR++apqI++apqJ++apqK+×lqR+lqI+lqJ+lqKφX+∑q=1nbpqR++bpqI++bpqJ++bpqK+mqR+mqI+mqJ+mqKφX+upR+ds≤1dp-Λpκ+Δpκ+upR+=Ωpκ+upR+dp-,p∈T.In a similar way, we can get (34)Φφpνt≤Ωpκ+upν+dp-,p∈T,ν=I,J,K.By (H4), we have (35)ΦφX≤κ,which implies that Φφ∈X∗, so the mapping Φ is a self-mapping from X∗ to X∗. Next, we shall prove that Φ is a contraction mapping. In fact, for any φ,ψ∈X∗, we have (36)Φφ-ΦψpRt=∫-∞te-∫stdpudu∑q=1napqRsfqRs,φ-fqRs,ψ-apqIsfqIs,φ-fqIs,ψ-apqJsfqJs,φ-fqJs,ψ-apqKsfqKs,φ-fqKs,ψ+∑q=1nbpqRsgqRs-τpqs,φ-gqRs-τpqs,ψ-bpqIsgqIs-τpqs,φ-gqIs-τpqs,ψ-bpqJsgqJs-τpqs,φ-gqJs-τpqs,ψ-bpqKsgqKs-τpqs,φ-gqKs-τpqs,ψds≤∫-∞te-∫stdpudu∑q=1napqR++apqI++apqJ++apqK+×lqR+lqI+lqJ+lqKφ-ψX+∑q=1nbpqR++bpqI++bpqJ++bpqK+mqR+mqI+mqJ+mqKφ-ψXds≤1dp-Λp+Δpφ-ψX=Ωpdp-φ-ψX,p∈T.Similarly, we can get (37)Φφ-Φψpνt≤Ωpdp-φ-ψX,p∈T,ν=I,J,K.By (H4), we have (38)Φφ-ΦψX≤rφ-ψX.Hence, Φ is a contraction mapping. Therefore, system (16) has a unique almost periodic solution in the region X∗=φ∈X:φX≤κ. The proof is complete.Theorem 8.
Suppose that(H1)–(H5) hold. Then the drive system (16) and the response system (19) are globally exponentially synchronized based on the controller (22).Proof.
In view of the error system (21), we have (39)zpRt′≤-dp-zpRt+∑q=1napqR+fqRt,y-fqRt,x+apqI+fqIt,y-fqIt,x+apqJ+fqJt,y-fqJt,x+apqK+fqKt,y-fqKt,x+∑q=1nbpqR+gqRt-τpqt,y-gqRt-τpqt,x+bpqI+gqIt-τpqt,y-gqIt-τpqt,x+bpqJ+gqJt-τpqt,y-gqJt-τpqt,x+bpqK+gqKt-τpqt,y-gqKt-τpqt,x+ϵpzpRt≤ϵp-dp-zpRt+∑q=1napqR++apqI++apqJ++apqK+lqRzqRt+lqIzqIt+lqJzqJt+lqKzqKt+∑q=1nbpqR++bpqI++bpqJ++bpqK+mqRzqRt-τpqt+mqIzqIt-τpqt+mqJzqJt-τpqt+mqKzqKt-τpqt,p∈T.In a similar way, we can obtain (40)zpνt′≤ϵp-dp-zpνt+∑q=1napqR++apqI++apqJ++apqK+lqRzqRt+lqIzqIt+lqJzqJt+lqKzqKt+∑q=1nbpqR++bpqI++bpqJ++bpqK+mqRzqRt-τpqt+mqIzqIt-τpqt+mqJzqJt-τpqt+mqKzqKt-τpqt,p∈T,ν=I,J,K.Now, we consider the Lyapunov function as follows: (41)Vt=V1t+V2t+V3t+V4twhere (42)V1t=∑p=1nzpRteλt+Wp,V2t=∑p=1nzpIteλt+Wp,V3t=∑p=1nzpJteλt+Wp,V4t=∑p=1nzpKteλt+Wp,Wp=11-β∑q=1nbpqR++bpqI++bpqJ++bpqK+∫t-τpqttmqRzqRs+mqIzqIs+mqJzqJs+mqKzqKseλs+τds,Computing the derivative of V(t) along the solutions of the error system (21), we can get (43)V1′t=∑p=1nzpRt′eλt+λeλtzpRt+11-β∑q=1nbpqR++bpqI++bpqJ++bpqK+mqRzqRt+mqIzqIt+mqJzqJt+mqKzqKt×eλt+τ-11-β∑q=1nbpqR++bpqI++bpqJ++bpqK+mqRzqRt-τpqt+mqIzqIt-τpqt+mqJzqJt-τpqt+mqKzqKt-τpqt×eλt-τpqt+τ1-τpq′t≤∑p=1nλ-dp-+ϵpzpRt+∑q=1napqR++apqI++apqJ++apqK+×lqRzqRt+lqIzqIt+lqJzqJt+lqKzqKt+bpqR++bpqI++bpqJ++bpqK+mqRzqRt-τpqt+mqIzqIt-τpqt+mqJzqJt-τpqt+mqKzqKt-τpqt+11-β∑q=1nbpqR++bpqI++bpqJ++bpqK+mqRzqRt+mqIzqIt+mqJzqJt+mqKzqKteλτ-∑q=1nbpqR++bpqI++bpqJ++bpqK+mqRzqRt-τpqt+mqIzqIt-τpqt+mqJzqJt-τpqt+mqKzqKt-τpqteλt≤∑p=1nλ-dp-+ϵpzpRt+∑q=1napqR++apqI++apqJ++apqK+×lqRzqRt+lqIzqIt+lqJzqJt+lqKzqKt+11-β∑q=1nbpqR++bpqI++bpqJ++bpqK+mqRzqRt+mqIzqIt+mqJzqJt+mqKzqKteλτeλt≤∑p=1nλ-dp-+ϵp+∑q=1napqR++apqI++apqJ++apqK+×lqR+lqI+lqJ+lqK+11-β∑q=1nbpqR++bpqI++bpqJ++bpqK+×mqR+mqI+mqJ+mqKeλτeλtZt=∑p=1nλ-dp-+ϵp+Λp+Δp1-βeλτeλtZt.Similarly, we can get (44)V2′t≤∑p=1nλ-dp-+εp+Λp+Δp1-βeλτeλtZt,V3′t≤∑p=1nλ-dp-+ρp+Λp+Δp1-βeλτeλtZt,V4′t≤∑p=1nλ-dp-+ϱp+Λp+Δp1-βeλτeλtZt.From (H5), we obtain (45)V′t≤0,which implies that Vt≤V0 for all t≥0.
On the other hand, we have(46)V10≤∑p=1nzpR0+11-β∑q=1nbpqR++bpqI++bpqJ++bpqK+×∫0-τpq00mqRzqRs+mqIzqIs+mqJzqJs+mqKzqKseλs+τds≤1+11-β∑p=1n∑q=1neλτλΔpψ-ϕ.Similarly, we can get (47)V20≤1+11-β∑p=1n∑q=1neλτλΔpψ-ϕ,V30≤1+11-β∑p=1n∑q=1neλτλΔpψ-ϕ,V40≤1+11-β∑p=1n∑q=1neλτλΔpψ-ϕ.It is obvious that Zteλt≤Vt; thus, for t≥0, we have (48)Zt≤Vte-λt≤V0e-λt≤Mψ-ϕe-λt,where (49)M=1+11-β∑p=1n∑q=1neλτλΔp>1.Therefore, the drive system (16) and the response system (19) are globally exponentially synchronized based on the controller (22). The proof is complete.


## 4. An Example

In this section, we give an example to illustrate the feasibility and effectiveness of our results obtained in Section3.Example 1.
Consider the following quaternion-valued neural networks with time-varying delay:(50)xp′t=-dptxpt+∑q=12apqtfqxqt+∑q=12bpqtgqxqt-τpqt+upt,and its response system(51)yp′t=-dptypt+∑q=12apqtfqyqt+∑q=12bpqtgqyqt-τpqt+upt+θpt.The coefficients are taken as follows:(52)d1t=4+cos⁡2t,d2t=6-sin⁡3t,a11t=0.1sin⁡2t+i0.3cos⁡t+j0.2sin⁡t+k0.4cos⁡2t,a12t=0.3cos⁡3t+i0.1sin⁡2t+j0.5sin⁡2t+k0.2sin⁡t,a21t=0.4sin⁡2t+i0.5cos⁡t+j0.1cos⁡3t+k0.2sin⁡t,a22t=0.3sin⁡t+i0.4sin⁡3t+j0.1cos⁡t-k0.2cos⁡3t,b11t=0.2cos⁡t+i0.4sin⁡t-j0.3sin⁡2t+k0.1sin⁡2t,b12t=0.5sin⁡t-i0.2cos⁡t-j0.3cos⁡2t-k0.1cos⁡t,b21t=0.2cos⁡2t-i0.5sin⁡2t+j0.3sin⁡3t-k0.4cos⁡3t,b22t=-0.6cos⁡t-i0.1cos⁡3t+j0.2cos⁡t-k0.5sin⁡3t,fqxq=132tan⁡xqR+xqI+i132sin⁡xqI+j132xqJ+k132sin⁡xqK,gqxq=124sin⁡xqR+i124xqI+xqK+j124sin⁡xqJ+k124xqK,τ11t=15sin⁡2t,τ12t=τ21t=16cos⁡t,τ22t=14sin⁡t,u1t=2sin⁡2t-icos⁡t+jsin⁡t+k3cos⁡t,u2t=2cos⁡t+i3sin⁡3t-5jcos⁡t-k4sin⁡t.By a simple calculation, we have(53)d1-=4,d2-=5,a11R+=0.1,a11I+=0.3,a11J+=0.2,a11K+=0.4,a12R+=0.3,a12I+=0.1,a12J+=0.5,a12K+=0.2,a21R+=0.4,a21I+=0.5,a21J+=0.1,a21K+=0.2,a22R+=0.3,a22I+=0.4,a22J+=0.1,a22K+=0.2,b11R+=0.2,b11I+=0.4,b11J+=0.3,b11K+=0.1,b12R+=0.5,b12I+=0.2,b12J+=0.3,b12K+=0.1,b21R+=0.2,b21I+=0.5,b21J+=0.3,b21K+=0.4,b22R+=0.6,b22I+=0.1b22J+=0.2,b22K+=0.5,u1R+=2,u1I+=1,u1J+=1,u1K+=3,u2R+=2,u2I+=3,u2J+=5,u2K+=4,τ=14,lqν=132,mqν=124,q=1,2,ν∈E,Λ1=0.525,Λ2=0.55,Δ1=0.7,Δ2≈0.9333.Take κ=2; then we have Ω1=1.225, Ω2=1.4833, (54)maxν∈EΩ1κ+u1ν+d1-,Ω2κ+u2ν+d2-≈1.3625,1.5933=1.5933<κ=2,maxΩ1d1-,Ω2d2-≈0.30625,0.29666=0.30625<1.Moreover, take ϵ1=1, ϵ2=1.5, ε1=0.8, ε2=1.8, ρ1=1.2, ρ2=1.6, ϱ1=0.5, ϱ2=1.9, λ=1, β=1/5, (55)λ-d1-+ϵ1+Λ1+Δ11-βeλτ≈-0.3515<0,λ-d2-+ϵ2+Λ2+Δ21-βeλτ≈-0.4520<0,λ-d1-+ε1+Λ1+Δ11-βeλτ≈-0.5515<0,λ-d2-+ε2+Λ2+Δ21-βeλτ≈-0.1520<0,λ-d1-+ρ1+Λ1+Δ11-βeλτ≈-0.1515<0,λ-d2-+ρ2+Λ2+Δ21-βeλτ≈-0.3520<0,λ-d1-+ϱ1+Λ1+Δ11-βeλτ≈-0.8515<0,λ-d2-+ϱ2+Λ2+Δ21-βeλτ≈-0.0520<0. We can verify that all the assumptions of Theorems 7 and 8 are satisfied. Therefore, the drive system (50) and its response system (51) are globally exponential synchronized (see Figures 1–3).Figure 1
The states of four parts ofx1 and x2.Figure 2
The states of four parts ofy1 and y2.Figure 3
Synchronization.


## 5. Conclusion

In this paper, we consider the global exponential almost periodic synchronization of quaternion-valued neural networks with time-varying delays. By means of the exponential dichotomy of linear differential equations, Banach’s fixed point theorem, Lyapunov functional method, and differential inequality technique, we establish the existence and global exponential synchronization of almost periodic solutions for system (9). To the best of our knowledge, this is the first time to study the almost periodic synchronization problem for neural networks. Furthermore, the method of this paper can be used to study other types of neural networks.



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