# New Stability Criterion for Fractional-Order Quaternion-Valued Neural Networks Involving Discrete and Leakage Delays

Authors: Bingjun Li; Bingnan Tang
Journal: Journal of Mathematics
 (2021)

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## Abstract

In the current work, we are devoted to the issue of uniform stability of fractional-order quaternion-valued neural networks involving discrete and leakage delays. Making use of the contracting mapping theory, we prove that the equilibrium point of the involved fractional-order quaternion-valued neural networks exists and is unique. Taking advantage of mathematical analysis strategy, a sufficient criterion involving delay to verify the global uniform stability for the considered fractional-order quaternion-valued neural networks is set up. Computer simulation figures are displayed to sustain the rationality of the established conclusions. This study generalizes and supplements the research of Xiu et al. (2020).


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## Body

## 1. Introduction

It is public knowledge that that neural networks own broad application prospects in numerous aspects and many demesnes such as pattern recognition, artificial intelligence, graph manipulation, psychophysics, control engineering, bioscience, and so on [1–3]. Generally speaking, time delay usually arises in artificial neural networks and biological systems due to the time lag of signal transmission. The study shows that the time delay often brings about some unexpected dynamical phenomena such as loss of stability, periodic vibration, chaos, and so on [4]. Thus, it is an important task for us to reveal the influence of time delay on various dynamical phenomena in delayed neural networks. During the past several decades, plenty of scholars made great effort to investigate a great variety of dynamical behaviors of delayed neural networks and fruitful results have been reported. For instance, Kong et al. [5] discussed the periodic and homoclinic solutions of discontinuous delayed neural networks. In 2019, Aouiti et al. [6] obtained the sufficient condition to ensure the existence and exponential stability of piecewise pseudo almost periodic solution to neutral-type inertial neural networks involving delays and impulses. In 2019, Huang et al. [7] set up a sufficient criterion to guarantee the existence of anti-periodic solutions and exponential stability for shunting inhibitory cellular neural networks involving proportional time delays by applying Lyapunov functional, inequality shills, and some mathematical analyses. In 2020, Abdelaziz and Chérif [8] carried out the study on the piecewise asymptotic almost periodic solutions of fuzzy Cohen–Grossberg neural networks involving impulsive effect. Xu and Li [9] did a valuable and novel work on anti-periodic solution to delayed cellular neural networks involving D operator. For more detailed publications, one can refer to [10–12].All the above works are only concerned with real-valued neural networks. Here we would like to point out that there are other types of multidimensional valued neural networks. As the extension of real-valued neural networks (RVNNs), complex-valued neural networks (CVNNs) occupy an important position in handling signal and intrinsic information of neural networks. Especially, they are often applied in different physical waves such as sound wave, elastic wave, electronic wave, optical wave, and so on. In 1843, Hamilton [13] proposed quaternion-valued neural networks (QVNNs), which are extension version of RVNNs and CVNNs. The skew of quaternion is given by Q:=y=yR+iyI+jyJ+kyK, where yR,yI,yJ,yK∈R and i,j,k obey the following operation:(1)ij=−ji=k,jk=−kj=i,ki=−ik=j,i2=j2=k2=ijk=−1.The investigation on QVNNs has attracted great attention from a lot of scholars since they have been found to have tremendous application in numerous areas such as color night vision, image impression, spatial rotation, three dimension geometrical affine transformation, and so on [14–16]. At present, some fruits on the dynamics of QVNNs have been reported. For example, Lin et al. [17] dealt with the global exponential synchronization problem of inertial memristor-based QVNNs with delays. Jiang and Wang [18] studied the almost periodic solutions of delayed QVNNs. You et al. [19] made a detailed analysis on the exponential stability of discrete-time quaternion-valued neural networks with leakage delay and discrete delays. For more concrete literatures, we refer the readers to [20–24].It is worth mentioning that all the above works on quaternion-valued neural networks mainly focus on the integer-order case and does not involve the fractional-order ones. The study on fractional-order neural networks has been keeping a very slow level due to the lack of actual background and fractional calculus theories. With the development of the research on fractional calculus, it is recognized that fractional-order differential system has greater advantages than classical integer-order one since it can give a description of the hereditary trait and memory nature for many materials and dynamic processes [14, 15]. Recently, lots of fractional-order neural networks have already aroused high attention from academic circles and a great deal of excellent fruits on fractional-order neural networks have been reported constantly. For instance, Udhayakumar et al. [25] focused on the multiple ψ-type stability issue for fractional-order quaternion-valued neural networks. Liu et al. [26] set up a set of sufficient conditions to guarantee the asymptotic synchronization of fractional-order neural networks involving delays. Du and Lu [27] investigated the finite-time synchronization problem for fractional-order delayed memristor-based neural networks. For more detailed studies, we refer the readers to [28–37]. However, there are few publications on fractional-order quaternion-valued neural networks. Stimulated by the discussion above, in this work, we will explore the research on the stability for fractional-order quaternion-valued neural networks involving delays. In a word, this work will mainly focus on the following issues: (a) prove the existence and uniqueness of equilibrium point of fractional-order quaternion-valued neural networks; (b) set up the sufficient criterion to ensure global uniform stability of fractional-order quaternion-valued neural networks.In 2017, Zhang et al. [4] studied the following complex-valued neural networks:(2)dρuitdtρ=−γiuit−+∑h=1mαihghuht+∑h=1mβihghuht−ϑ+Li,where i=1,2,…,m, ρ∈0,1, uit∈C (C denotes the set of complex numbers) stands for the state of the ith neuron at time t, αih,βih∈C stand for the connection weight without and with time delays, respectively, Li∈C denotes the external input, ϑ,ϱ≥0 stand for the transmission delay and the leakage delay, respectively, and gh∈C denotes the activation function. For details, one can see [4]. Making use of contraction mapping principle, a sufficient condition to ensure the existence and uniqueness of the equilibrium point for system (2) is set up. Applying mathematical analysis skills, a set of delay-dependent criteria to check the global uniform stability of system (2) is established.In this present work, we modify system (2) as the following fractional-order quaternion-valued neural networks:(3)dρuitdtρ=−γiuit−+∑h=1mαihghuht+∑h=1mβihghuht−ϑ+Li,where i=1,2,…,m, ρ∈0,1, uit∈Q stands for the state of the ith neuron at time t, αih,βih∈Q stand for the connection weight without and with time delays, respectively, Li∈Q denotes the external input, ϑ,ϱ≥0 stand for the transmission delay and the leakage delay, respectively, and gh∈Q denotes the activation function.This paper is organized as follows. In Section2, the basic definitions, lemmas, and essential theories on fractional calculus and quaternion algebra are given. In Section 3, the existence and uniqueness of solution of model (3) are stated. In Section 4, a new delay-dependent criterion to check the global uniform stability of model (3) is derived. In Section 5, software simulation plots are presented to support the derived chief conclusions of this study. Section 6 ends this paper.


## 2. Preliminaries and Assumptions

Now we give some related notations.Z+ stands for the set of positive integer numbers, and i,j,k are imaginary units. Qn,ℛm×n,Qm×n stand for the set of n-dimensional quaternion-valued vectors, m×n real-valued matrices, and quaternion-valued matrices. The norm of quaternion-valued matrices M=mijn×n∈Qn×n is given by M=∑i=1n∑j=1nmij2. Denote C−T,0,Rm the Banach space of continuous m-real vector functions defined on −T,0 with the norm ϕt=maxi=1,2,…,msupt∈−T,0etϕit, where ϕ∈C−T,0,Rm. For ut=u1t,u2t,…,umtT∈C0,+∞,Rm, the norm of ut is defined by ut=maxi=1,2,…,msupt∈0,+∞etuit.Definition 1.
(see [38]). The Caputo fractional-order derivative with order ρ for the function uα is given by(4)Dρuβ=1Γl−ρ∫β0βulvβ−νρ−l+1dν,where uβ∈β0,∞,R,Γs=∫0∞βs−1e−βdβ, β≥β0, and l∈Z+,l−1≤ρ<l; Dρuβ implies that dρuβ/dβ is a Caputo fractional-order derivative operator.
For eachy=yR+iyI+jyJ+kyK∈Q and each activation g:Q⟶Q, gh has the form:(5)gy=gRyR,yI,yJ,yK+gIyR,yI,yJ,yK+gJyR,yI,yJ,yK+gKyR,yI,yJ,yK.
Thus,ghu in system (3) has the following form:(6)ghuh=ghRyhR,yhI,yhJ,yhK+ghIyhR,yhI,yhJ,yhK+ghJyhR,yhI,yhJ,yhK+ghKyhR,yhI,yhJ,yhK,where h=1,2,…,m.
System (3) can be rewritten as the following equivalent form:(7)dρutdtρ=−Aut−+ℬgut+Cgut−ϑ+ℒ,where ut=u1t,u2t,…,umtT∈Qm, A=γ1,γ2,…,γmT∈Rm×m, ℬ=αihm×m∈Qm×m, C=βihm×m∈Qm×m, ℒ=Lim×m∈Qm×m, gut=g1u1t,g2u2t,…,gmumt∈Qm×m, ρ∈0,1, and ϑ,ϱ≥0.
The initial values of (3) can be written as follows:(8)uhs=ϕhRs+iϕhIs+jϕhJs+kϕhKs,s∈−τ,0,h=1,2,…,m,where τ=max,ϑ,ϕhRs,ϕhIs,ϕhJs,ϕhKs∈C−τ,0,R.Definition 2.
(see [39]). We say that the equilibrium point u∗ of model (3) is stable provided that for every ϵ>0, ∃ζ=ηt0,ε>0 which satisfies t≥t0≥0,ϕt−u∗≤ζ which implies ut,t0,ϕ−u∗≤ε for every solution ut,t0,ϕ of model (1). The equilibrium point u∗ of model (3) is uniformly stable provided that η has nothing to do with t0.Lemma 1.
(see [40]). Let ut∈Cn0,∞ and n−1<ρ<n∈Z+; then, the following equalities hold.(i)
D−r1D−r2ut=D−r1+r2ut,r1,r2≥0.(ii)
DrD−rut=ut,r≥0.(iii)
D−rDrut=ut−∑l=0n−1tl/l!ul0,r≥0.
In order to obtain the key results of this study, we give the hypotheses as follows.
ℋ1gh0,0,0,0=0 and ∃ positive constants GhRR,GhRI,GhRJ,GhRK,GhIR,GhII,GhIJ,GhIK,GhJR,GhJI,GhJJ,GhJK, GhKR,GhKI,GhKJ,GhKK such that(9)ghRyhR,yhI,yhJ,yhK−ghRxhR,xhI,xhJ,xhK≤GhRRyhR−xhR+GhRIyhI−xhI+GhRJyhJ−xhJ+GhRKyhK−xhK,ghIyhR,yhI,yhJ,yhK−ghIxhR,xhI,xhJ,xhK≤GhIRyhR−xhR+GhIIyhI−xhI+GhIJyhJ−xhJ+GhIKyhK−xhK,ghJyhR,yhI,yhJ,yhK−ghJxhR,xhI,xhJ,xhK≤GhJRyhR−xhR+GhJIyhI−xhI+GhJJyhJ−xhJ+GhJKyhK−xhK,ghKyhR,yhI,yhJ,yhK−ghKxhR,xhI,xhJ,xhK≤GhKRyhR−xhR+GhKIyhI−xhI+GhKJyhJ−xhJ+GhKKyhK−xhK,for every xhR,xhI,xhJ,xhK∈R and h=1,2,…,m.


## 3. Existence and Uniqueness

In this part, we investigate the existence and uniqueness of the equilibrium point for model (3) (i.e., (7)). Let u∗∈Qm be the equilibrium point for model (7); then, we have(10)−Au∗+ℬgu∗+Cgu∗+ℒ=0.Then,(11)u∗=A−1ℬ+Cgu∗+A−1ℒ.From (11), we know that if we can prove the following map: Γ:Qm⟶Qm,(12)Γu=A−1ℬ+Cgu∗+A−1ℒhas a unique fixed point, then we can conclude that the equilibrium point of model (7) exists and is unique.ℋ2θU<1, where θ=maxh=1,2,…,mθh1,θh2,θh3,θh4,U=A−1ℬ+C and(13)θh1=GhRR2+GhRRGhRIGhRRGhRJ+GhRRGhRK+GhIR2+GhIRGhII+GhIRGhIJ+GhIRGhIK+GhJR2+GhJRGhJI+GhJRGhJJ+GhJRGhJK+GhKR2+GhKRGhKI+GhKRGhKJ+GhKRGhKK,θh1=GhRI2+GhRRGhRI+GhRIGhRJ+GhII2+GhIRGhII+GhIIGhIJ+GhJI2+GhJRGhJI+GhJIGhJJ+GhKI2+GhKRGhKI+GhKIGhKJ,θh1=GhRJ2GhRRGhRJ+GhRIGhRJ+GhRJGhRK+GhRJGhRK+GhIJ2+GhIRGhIJ+GhIIGhIJ+GhIJGhIK+GhIJGhIK+GhJJ2+GhJRGhJJ+GhJIGhJJ+GhJJGhJK+GhJJGhJK+GhKJ2+GhKRGhKJ+GhKIGhKJ+GhKJGhKK+GhKJGhKK,θh1=GhRK2+GhRRGhRK+GhRJGhRK+GhRJGhRK+GhIK2+GhIRGhIK+GhIJGhIK+GhIJGhIK+GhJK2+GhJRGhJK+GhJJGhJK+GhJJGhJK+GhKK2+GhKRGhKK+GhKJGhKK+GhKJGhKK.Theorem 1.
Ifℋ1 and ℋ2 hold, then model (3) owns a unique equilibrium point.Proof.
Letu1=u1R+iu1I+ju1J+ku1K and u2=u2R+iu2I+ju2J+ku2K. Then,(14)ghu1−ghu22=ghRu1+ighIu1+jghJu1+kghKu1−ghRu2+ighIu2+jghJu2+kghKu22=ghRu1−ghRu22+ghIu1−ghIu22+ghJu1−ghJu22+ghKu1−ghKu22≤GhRRu1hR−u2hR+GhRIu1hI−u2hI+GhRJu1hJ−u2hJ+GhRKu1hK−u2hK2+GhIRu1hR−u2hR+GhIIu1hI−u2hI+GhIJu1hJ−u2hJ+GhIKu1hK−u2hK2+GhJRu1hR−u2hR+GhJIu1hI−u2hI+GhJJu1hJ−u2hJ+GhJKu1hK−u2hK2+GhKRu1hR−u2hR+GhKIu1hI−u2hI+GhKJu1hJ−u2hJ+GhKKu1hK−u2hK2=GhRR2u1hR−u2hR2+GhRI2u1hI−u2hI2+GhRJ2u1hJ−u2hJ2+GhRK2u1hK−u2hK2+2GhRRGhRIu1hR−xhRu1hI−u2hI+2GhRRGhRJu1hR−u2hRu1hJ−u2hJ+2GhRRGhRKu1hR−xhRu1hK−u2hK+2GhRIGhRJu1hI−u2hIu1hJ−u2hJ+2GhRJGhRKu1hJ−xhJu1hK−u2hK+2GhRJGhRKu1hJ−u2hJu1hK−u2hK+GhIR2u1hR−u2hR2+GhII2u1hI−u2hI2+GhIJ2u1hJ−u2hJ2+GhIK2u1hK−u2hK2+2GhIRGhIIu1hR−xhRu1hI−u2hI+2GhIRGhIJu1hR−u2hRu1hJ−u2hJ+2GhIRGhIKu1hR−xhRu1hK−u2hK+2GhIIGhIJu1hI−u2hIu1hJ−u2hJ+2GhIJGhIKu1hJ−xhJu1hK−u2hK+2GhIJGhIKu1hJ−u2hJu1hK−u2hK+GhJR2u1hR−u2hR2+GhJI2u1hI−u2hI2+GhJJ2u1hJ−u2hJ2+GhJK2u1hK−u2hK2+2GhJRGhJIu1hR−xhRu1hI−u2hI+2GhJRGhJJu1hR−u2hRu1hJ−u2hJ+2GhJRGhJKu1hR−xhRu1hK−u2hK+2GhJIGhJJu1hI−u2hIu1hJ−u2hJ+2GhJJGhJKu1hJ−xhJu1hK−u2hK+2GhJJGhJKu1hJ−u2hJu1hK−u2hK+GhKR2u1hR−u2hR2+GhKI2u1hI−u2hI2+GhKJ2u1hJ−u2hJ2+GhKK2u1hK−u2hK2+2GhKRGhKIu1hR−xhRu1hI−u2hI+2GhKRGhKJu1hR−u2hRu1hJ−u2hJ+2GhKRGhKKu1hR−xhRu1hK−u2hK+2GhKIGhKJu1hI−u2hIu1hJ−u2hJ+2GhKJGhKKu1hJ−xhJu1hK−u2hK+2GhKJGhKKu1hJ−u2hJu1hK−u2hK≤GhRR2u1hR−u2hR2+GhRI2u1hI−u2hI2+GhRJ2u1hJ−u2hJ2+GhRK2u1hK−u2hK2+GhRRGhRIu1hR−xhR2+u1hI−u2hI2+GhRRGhRJu1hR−u2hR2+u1hJ−u2hJ2+GhRRGhRKu1hR−xhR2+u1hK−u2hK2+GhRIGhRJu1hI−u2hI2+u1hJ−u2hJ2+GhRJGhRKu1hJ−xhJ2+u1hK−u2hK2+GhRJGhRKu1hJ−u2hJ2+u1hK−u2hK2+GhIR2u1hR−u2hR2+GhII2u1hI−u2hI2+GhIJ2u1hJ−u2hJ2+GhIK2u1hK−u2hK2+GhIRGhIIu1hR−xhR2+u1hI−u2hI2+GhIRGhIJu1hR−u2hR2+u1hJ−u2hJ2+GhIRGhIKu1hR−xhR2+u1hK−u2hK2+GhIIGhIJu1hI−u2hI2+u1hJ−u2hJ2+GhIJGhIKu1hJ−xhJ2+u1hK−u2hK2+GhIJGhIKu1hJ−u2hJ2+u1hK−u2hK2+GhJR2u1hR−u2hR2+GhJI2u1hI−u2hI2+GhJJ2u1hJ−u2hJ2+GhJK2u1hK−u2hK2+GhJRGhJIu1hR−xhR2+u1hI−u2hI2+GhJRGhJJu1hR−u2hR2+u1hJ−u2hJ2+GhJRGhJKu1hR−xhR2+u1hK−u2hK2+GhJIGhJJu1hI−u2hI2+u1hJ−u2hJ2+GhJJGhJKu1hJ−xhJ2+u1hK−u2hK2+GhJJGhJKu1hJ−u2hJ2+u1hK−u2hK2+GhKR2u1hR−u2hR2+GhKI2u1hI−u2hI2+GhKJ2u1hJ−u2hJ2+GhKK2u1hK−u2hK2+GhKRGhKIu1hR−xhR2+u1hI−u2hI2+GhKRGhKJu1hR−u2hR2+u1hJ−u2hJ2+GhJRGhJIu1hR−xhR2+u1hI−u2hI2+GhJRGhJJu1hR−u2hR2+u1hJ−u2hJ2+GhKJGhKKu1hJ−xhJ2+u1hK−u2hK2+GhKJGhKKu1hJ−u2hJ2+u1hK−u2hK2=θh1u1hR−u2hR2+θh2u1hI−u2hI2+θh3u1hJ−u2hJ2+θh4u1hK−u2hK2.
Then, for anyu1,u2∈Qm, one has(15)Γu1−Γu2=Ugu1−gu2≤θUu1−u2.
In view ofℋ2, one can easily know that Γu is contractive map. Thus, Γu owns a unique fixed point, which implies that model (3) owns a unique equilibrium point. The proof finishes.


## 4. Uniform Stability

In the current part, we explore the global uniform stability issue of the equilibrium point for model (3). Let(16)A1p1=1−e−γ+∑h=1mαp1hRGhRR+αp1hIGhIR+αp1hJGhJR+αp1hKGhKR+e−ϑ∑h=1mβp1hRGhRR+βp1hIGhIR+βp1hJGhJR+βp1hKGhKRA2p1=∑h=1mαp1hRGhRI+αp1hIGhII+αp1hJGhJI+αp1hKGhKI+e−ϑ∑h=1mβp1hRGhRI+βp1hIGhII+βp1hJGhJI+βp1hKGhKI,(17)A3p1=∑h=1mαp1hRGhRJ+αp1hIGhIJ+αp1hJGhJJ+αp1hKGhKJ+e−ϑ∑h=1mβp1hRGhRJ+βp1hIGhIJ+βp1hJGhJJ+βp1hKGhKJ,A4p1=∑h=1mαp1hRGhRK+αp1hIGhIK+αp1hJGhJK+αp1hKGhKK+e−ϑ∑h=1mβp1hRGhRK+βp1hIGhIK+βp1hJGhJK+βp1hKGhKK,A5p1=1+e−γ+e−ϑ∑h=1mβp1hRGhRR+βp1hIGhIR+βp1hJGhJR+βp1hKGhKR,A6p1=e−ϑ∑h=1mβp1hRGhRI+βp1hIGhII+βp1hJGhJI+βp1hKGhKI,A7p1=e−ϑ∑h=1mβp1hRGhRJ+βp1hIGhIJ+βp1hJGhJJ+βp1hKGhKJ,A8p1=e−ϑ∑h=1mβp1hRGhRK+βp1hIGhIK+βp1hJGhJK+βp1hKGhKK,ℬ1p2=1−e−γ+∑h=1mαp2hIGhII+αp2hRGhRI+αp2hJGhJI+αp2hKGhKI+e−ϑ∑h=1mβp2hIGhII+βp2hRGhRI+βp2hJGhJI+βp2hKGhKI,ℬ2p2=∑h=1mαp2hIGhIR+αp2hRGhRR+αp2hJGhJR+αp2hKGhKR+e−ϑ∑h=1mβp2hIGhIR+βp2hRGhRR+βp2hJGhJR+βp2hKGhKR,ℬ3p2=∑h=1mαp2hIGhIJ+αp2hRGhRJ+αp2hJGhJJ+αp2hKGhKJ+e−ϑ∑h=1mβp2hIGhIJ+βp2hIGhRJ+βp2hJGhJJ+βp2hKGhKJ,ℬ4p2=∑h=1mαp2hIGhIK+αp2hIGhRK+αp2hJGhJK+αp2hKGhKK+e−ϑ∑h=1mβp2hIGhIK+βp2hIGhRK+βp2hJGhJK+βp2hKGhKK,ℬ5p2=1+e−γ+e−ϑ∑h=1mβp2hIGhII+βp2hIGhRI+βp2hJGhJI+βp2hKGhKI,ℬ6p2=e−ϑ∑h=1mβp2hIGhIR+βp2hIGhRR+βp2hJGhJR+βp2hKGhKR,ℬ7p2=e−ϑ∑h=1mβp2hIGhIJ+βp2hIGhRJ+βp2hJGhJJ+βp2hKGhKJ,ℬ8p2=e−ϑ∑h=1mβp2hIGhIK+βp2hRGhRK+βp2hJGhJK+βp2hKGhKK,C1p3=1−e−γ+∑h=1mαp3hJGhJJ+αp3hIGhIJ+αp3hRGhRJ+αp3hKGhKJ+e−ϑ∑h=1mβp3hJGhJJ+βp3hIGhIJ+βp3hRGhRJ+βp3hKGhKJ,(18)C2p3=∑h=1mαp3hJGhJI+αp3hIGhII+αp3hRGhRI+αp3hKGhKI+e−ϑ∑h=1mβp3hJGhJI+βp3hIGhII+βp3hRGhRI+βp3hKGhKI,C3p3=∑h=1mαp3hJGhJR+αp3hIGhIR+αp3hRGhRR+αp3hKGhKR+e−ϑ∑h=1mβp3hJGhJR+βp3hIGhIR+βp3hRGhRR+βp3hKGhKR,C4p3=∑h=1mαp3hJGhJK+αp3hIGhIK+αp3hRGhRK+αp3hKGhKK+e−ϑ∑h=1mβp1hJGhJK+βp3hIGhIK+βp3hJGhRK+βp3hKGhKK,C5p3=1+e−γ+e−ϑ∑h=1mβp3hJGhJJ+βp3hIGhIJ+βp3hRGhRJ+βp3hKGhKJ,C6p3=e−ϑ∑h=1mβp3hJGhJI+βp3hIGhII+βp3hRGhRI+βp3hKGhKI,C7p3=e−ϑ∑h=1mβp3hJGhJR+βp3hIGhIR+βp3hJGhRR+βp3hKGhKR,C8p3=e−ϑ∑h=1mβp3hJGhJK+βp3hIGhIK+βp3hRGhRK+βp3hKGhKK,D1p4=1−e−γ+∑h=1mαp4hKGhKK+αp4hIGhIK+αp4hJGhJK+αp4hRGhRK+e−ϑ∑h=1mβp4hKGhKK+βp4hIGhIK+βp4hJGhJK+βp4hRGhRK,D4p4=∑h=1mαp4hKGhKI+αp4hIGhII+αp4hJGhJI+αp4hRGhRI+e−ϑ∑h=1mβp4hKGhKI+βp4hIGhII+βp4hJGhJI+βp4hRGhRI,D4p4=∑h=1mαp4hKGhKJ+αp4hIGhIJ+αp4hJGhJJ+αp4hRGhRJ+e−ϑ∑h=1mβp4hKGhKJ+βp4hIGhIJ+βp4hJGhJJ+βp4hRGhRJ,D4p4=∑h=1mαp4hKGhKR+αp4hIGhIR+αp4hJGhJR+αp4hRGhRR+e−ϑ∑h=1mβp4hKGhKR+βp4hIGhIR+βp4hJGhJR+βp4hRGhRR,D5p4=1+e−γ+e−ϑ∑h=1mβp1hKGhKK+βp4hIGhIK+βp4hJGhJK+βp4hRGhRK,D6p4=e−ϑ∑h=1mβp4hKGhKI+βp4hIGhII+βp4hJGhJI+βp4hRGhRI,D7p4=e−ϑ∑h=1mβp4hKGhKJ+βp4hIGhIJ+βp4hJGhJJ+βp4hRGhRJ,D8p4=e−ϑ∑h=1mβp4hKGhKR+βp4hIGhIR+βp4hJGhJR+βp4hRGhRR,where γ=maxi=1,2,…,mγi and p1,p2,p3,p4∈1,2,…,m. Assume that ℋ3W=minW1,W2,W3,W4>0, where(19)W1=A1p1−ℬ2p2−C2p3−D2p4,W2=ℬ1p1−A2p1−C3p3−D3p4,W3=C1p3−A1p1−ℬ3p2−D4p4,W4=D1p4−A4p1−ℬ4p2−C3p3.Theorem 2.
Ifℋ1-ℋ3 hold, then equilibrium point of system (3) is uniformly stable.Proof.
In view of Theorem1, we know that system (3) has a unique equilibrium point u∗=u1∗,u2∗,…,um∗T. Let u¯t=ut−u∗; then, system (3) becomes(20)dρu¯tdtρ=−Au¯t−+ℬfu¯+Cfu¯t−ϑ,where fu¯t=gut−gu∗. The initial value(21)u¯hs=ϕ¯hRs+iϕ¯hIs+jϕ¯hJs+kϕ¯hKs,s∈−τ,0,h=1,2,…,m,where τ=max,ϑ,ϕhRs,ϕ¯hIs,ϕ¯hJs,ϕ¯hKs∈C−τ,0,R, and ϕ¯hRs=ϕhRs−uh∗,ϕ¯hIs=ϕhIs−uh∗,ϕ¯hJs=ϕhJs−uh∗,ϕ¯hKs=ϕhKs−uh∗ and u¯t=u¯1t,u¯2t,…,u¯mtT. Let u¯t=u¯Rt+iu¯It+ju¯Jt+ku¯Kt, where u¯Rt=u¯1Rt,u¯2Rt,…,u¯mRtT, u¯It=u¯1It,u¯2It,…,u¯mItT, u¯Jt=u¯1Jt,u¯2Jt,…,u¯mJtT, u¯Kt=u¯1Kt,u¯2Kt,…,u¯mKtT. It follows from (20) that(22)dρu¯iRtdtρ+idρu¯iItdtρ+jdρu¯iJtdtρ+kdρu¯iKtdtρ=−γiu¯iRt−+∑h=1mαihRghR−αihIghI−αihJghJ−αihKghK+∑h=1mβihRg˜hR−βihIg˜hI−βihJg˜hJ−βihKg˜hK+i−γiu¯iIt−+∑h=1mαihIghR+αihRtghI−αihKtghJ+αihJghK+∑h=1mβihIg˜hR+βihRg˜hI−βihKg˜hJ+βihJg˜hK+j−γiu¯iJt−+∑h=1mαihJghR+αihKghI+αihRghJ−αihIghK+∑h=1mβihJg˜hR+βihKg˜hI+βihRg˜hJ−βihIg˜hK+k−γiu¯iKt−+∑h=1mαihKghR−αihJghI+αihIghJ+αihRghK+∑h=1mβihKg˜hR−βihJg˜hI+βihIg˜hJ+βihRg˜hK,where(23)ghR=ghRu¯hRt,u¯hit,u¯hJt,u¯hKt,ghI=ghIu¯hRt,u¯hit,u¯hJt,u¯hKt,ghJ=ghJu¯hRt,u¯hit,u¯hJt,u¯hKt,ghK=ghKu¯hRt,u¯hit,u¯hJt,u¯hKt,g˜hR=g˜hRu¯hRt−ϑ,u¯hit−ϑ,u¯hJt−ϑ,u¯hKt−ϑ,g˜hI=g˜hIu¯hRt−ϑ,u¯hit−ϑ,u¯hJt−ϑ,u¯hKt−ϑ,g˜hJ=g˜hJu¯hRt−ϑ,u¯hit−ϑ,u¯hJt−ϑ,u¯hKt−ϑ,g˜hK=g˜hKu¯hRt−ϑ,u¯hit−ϑ,u¯hJt−ϑ,u¯hKt−ϑ.
It follows from (22) that(24)dρu¯iRtdtρ=−γiu¯iRt−+∑h=1mαihRghR−αihIghI−αihJghJ−αihKghK+∑h=1mβihRg˜hR−βihIg˜hI−βihJg˜hJ−βihKg˜hK,dρu¯iItdtρ=−γiu¯iIt−+∑h=1mαihIghR+αihRghI−αihKghJ+αihJghK+∑h=1mβihIg˜hR+βihRg˜hI−βihKg˜hJ+βihJg˜hKdρu¯iJtdtρ=−γiu¯iJt−+∑h=1mαihJghR+αihKghI+αihRghJ−αihIghK+∑h=1mβihJg˜hR+βihKg˜hI+βihRg˜hJ−βihIg˜hK,dρu¯iKtdtρ=−γiu¯iKt−+∑h=1mαihKghR−αihJghI+αihIghJ+αihRghK+∑h=1mβihKg˜hR−βihJg˜hI+βihIg˜hJ+βihRg˜hK,where i=1,2,…,m. In view of Lemma 1, one gets(25)u¯iRt−u¯iR0=d−ρu¯iRtdt−ρ−γiu¯iRt−+∑h=1mαihRghR−αihIghI−αihJghJ−αihKghK+∑h=1mβihRg˜hR−βihIg˜hI−βihJtg˜hJ−βihKg˜hK,u¯iIt−u¯iI0=d−ρu¯iItdt−ρ−γiu¯iIt−+∑h=1mαihIghR+αihRghI−αihKghJ+αihJghK+∑h=1mβihIg˜hR+βihRg˜hI−βihKg˜hJ+βihJg˜hK,u¯iJt−u¯iJ0=d−ρu¯iJtdt−ρ−γiu¯iJt−+∑h=1mαihJghR+αihKghI+αihRghJ−αihIghK+∑h=1mβihJg˜hR+βihKg˜hI+βihRg˜hJ−βihIg˜hK.
From the first equation of (25), one has(26)u¯iRt=u¯iR0+1Γρ∫0tt−sρ−1−γiu¯iRt−ϱ+∑h=1mαihRghR−αihIghI−αihJghJ−αihKghK+∑h=1mβihRg˜hR−βihIg˜hI−βihJg˜hJ−βihKg˜hKds,which leads to(27)u¯iRt≤u¯iR0+1Γρ∫0tt−sρ−1γiu¯iRt−+∑h=1m|αihRghR+αihIghI+αihJghJ+αihKghK+∑h=1mβihRg˜hR+βihIg˜hI+βihJg˜hJ+βihKg˜hK≤u¯iR0+γi1Γρ∫0tt−sρ−1u¯iRt−ds+∑h=1mαihR1Γρ∫0tt−sρ−1ghRds+∑h=1mαihI1Γρ∫0tt−sρ−1ghIds+∑h=1mαihJ1Γρ∫0tt−sρ−1ghJds+∑h=1mαihK1Γρ∫0tt−sρ−1ghKds+∑h=1mβihR1Γρ∫0tt−sρ−1g˜hRds+∑h=1mβihI1Γρ∫0tt−sρ−1g˜hIds+∑h=1mβihJ1Γρ∫0tt−sρ−1g˜hJds+∑h=1mβihK1Γρ∫0tt−sρ−1g˜hKds≤u¯iR0+γi1Γρ∫0tt−sρ−1u¯iRt−ds+∑h=1mαihR1Γρ∫0tt−sρ−1GhRRu¯hRs+GhRIu¯hIs+GhRJu¯hJs+GhRKu¯hKsds+∑h=1mαihI1Γρ∫0tt−sρ−1GhIRu¯hRs+GhIIu¯hIs+GhIJu¯hJs+GhIKu¯hKsds+∑h=1mαihJ1Γρ∫0tt−sρ−1GhJRu¯hRs+GhJIu¯hIs+GhJJu¯hJs+GhJKu¯hKsds+∑h=1mαihK1Γρ∫0tt−sρ−1GhKRu¯hRs+GhKIu¯hIs+GhKJu¯hJs+GhKKu¯hKsds+∑h=1mβihR1Γρ∫0tt−sρ−1GhRRu¯hRs−ϑ+GhRIu¯hIs−ϑ+GhRJu¯hJs−ϑ+GhRKu¯hKs−ϑds+∑h=1mβihI1Γρ∫0tt−sρ−1GhIRu¯hRs−ϑ+GhIIu¯hIs−ϑ+GhIJu¯hJs−ϑ+GhIKu¯hKs−ϑds+∑h=1mβihJ1Γρ∫0tt−sρ−1GhJRu¯hRs−ϑ+GhJIu¯hIs−ϑ+GhJJu¯hJs−ϑ+GhJKu¯hKs−ϑds+∑h=1mβihK1Γρ∫0tt−sρ−1GhKRu¯hRs−ϑ+GhKIu¯hIs−ϑ+GhKJu¯hJs−ϑ+GhKKu¯hKs−ϑds=u¯iR0+γi1Γρ∫0tt−sρ−1u¯iRt−ds+∑h=1mαihRGhRR+αihIGhIR+αihJGhJR+αihKGhKR1Γρ∫0tt−sρ−1u¯hRsds+∑h=1mαihRGhRI+αihIGhII+αihJGhJI+αihKGhKI1Γρ∫0tt−sρ−1u¯hIsds+∑h=1mαihRGhRJ+αihIGhIJ+αihJGhJJ+αihKGhKJ1Γρ∫0tt−sρ−1u¯hJsds+∑h=1mαihRGhRK+αihIGhIK+αihJGhJK+αihKGhKK1Γρ∫0tt−sρ−1u¯hKsds+∑h=1mβihRGhRR+βihIGhIR+βihJGhJR+βihKGhKR1Γρ∫0tt−sρ−1u¯hRs−ϑds+∑h=1mβihRGhRI+βihIGhII+βihJGhJI+βihKGhKI1Γρ∫0tt−sρ−1u¯hIs−ϑds+∑h=1mβihRGhRJ+βihIGhIJ+βihJGhJJ+βihKGhKJ1Γρ∫0tt−sρ−1u¯hJs−ϑds+∑h=1mβihRGhRK+βihIGhIK+βihJGhJK+βihKGhKK1Γρ∫0tt−sρ−1u¯hKs−ϑds.
Multiplying bye−t by (23) leads to(28)e−tu¯iRt=e−tu¯iR0+γi1Γρ∫0tt−sρ−1e−tu¯iRt−ϱds+∑h=1mαihRGhRR+αihIGhIR+αihJGhJR+αihKGhKR1Γρ∫0tt−sρ−1e−tu¯hRsds+∑h=1mαihRGhRI+αihIGhII+αihJGhJI+αihKGhKI1Γρ∫0tt−sρ−1e−tu¯hIsds+∑h=1mαihRGhRJ+αihIGhIJ+αihJGhJJ+αihKGhKJ1Γρ∫0tt−sρ−1e−tu¯hJsds+∑h=1mαihRGhRK+αihIGhIK+αihJGhJK+αihKGhKK1Γρ∫0tt−sρ−1e−tu¯hKsds+∑h=1mβihRGhRR+βihIGhIR+βihJGhJR+βihKGhKR1Γρ∫0tt−sρ−1e−tu¯hRs−ϑds+∑h=1mβihRGhRI+βihIGhII+βihJGhJI+βihKGhKI1Γρ∫0tt−sρ−1e−tu¯hIs−ϑds+∑h=1mβihRGhRJ+βihIGhIJ+βihJGhJJ+βihKGhKJ1Γρ∫0tt−sρ−1e−tu¯hJs−ϑds+∑h=1mβihRGhRK+βihIGhIK+βihJGhJK+βihKGhKK1Γρ∫0tt−sρ−1e−tu¯hKs−ϑds=e−tu¯iR0+γi1Γρ∫0tt−sρ−1e−t+se−ϱe−s+ϱu¯iRt−ϱds+∑h=1mαihRGhRR+αihIGhIR+αihJGhJR+αihKGhKR1Γρ∫0tt−sρ−1e−t+se−su¯hRsds+∑h=1mαihRGhRI+αihIGhII+αihJGhJI+αihKGhKI1Γρ∫0tt−sρ−1e−t+se−su¯hIsds+∑h=1mαihRGhRJ+αihIGhIJ+αihJGhJJ+αihKGhKJ1Γρ∫0tt−sρ−1e−t+se−su¯hJsds+∑h=1mαihRGhRK+αihIGhIK+αihJGhJK+αihKGhKK1Γρ∫0tt−sρ−1e−t+se−su¯hKsds+∑h=1mβihRGhRR+βihIGhIR+βihJGhJR+βihKGhKR1Γρ×∫0tt−sρ−1e−t+se−ϑe−s+ϑu¯hRs−ϑds+∑h=1mβihRGhRI+βihIGhII+βihJGhJI+βihKGhKI1Γρ×∫0tt−sρ−1e−te−t+se−ϑe−s+ϑu¯hIs−ϑds+∑h=1mβihRGhRJ+βihIGhIJ+βihJGhJJ+βihKGhKJ1Γρ×∫0tt−sρ−1e−te−t+se−ϑe−s+ϑu¯hJs−ϑds+∑h=1mβihRGhRK+βihIGhIK+βihJGhJK+βihKGhKK1Γρ×∫0tt−sρ−1e−te−t+se−ϑe−s+ϑu¯hKs−ϑds=e−tu¯iR0+e−ϱγi1Γρ∫0ϱ+∫ϱtt−sρ−1e−t+se−ϱe−s+ϱu¯iRt−ϱds+∑h=1mαihRGhRR+αihIGhIR+αihJGhJR+αihKGhKR1Γρ∫0tt−sρ−1e−t+se−su¯hRsds+∑h=1mαihRGhRI+αihIGhII+αihJGhJI+αihKGhKI1Γρ∫0tt−sρ−1e−t+se−su¯hIsds+∑h=1mαihRGhRJ+αihIGhIJ+αihJGhJJ+αihKGhKJ1Γρ∫0tt−sρ−1e−t+se−su¯hJsds+∑h=1mαihRGhRK+αihIGhIK+αihJGhJK+αihKGhKK1Γρ∫0tt−sρ−1e−t+se−su¯hKsds+∑h=1mβihRGhRR+βihIGhIR+βihJGhJR+βihKGhKR1Γρ×∫0ϑ+∫ϑtt−sρ−1e−t+se−ϑe−s+ϑu¯hRs−ϑds+∑h=1mβihRGhRI+βihIGhII+βihJGhJI+βihKGhKI1Γρ×∫0ϑ+∫ϑtt−sρ−1e−t+se−ϑe−s+ϑu¯hIs−ϑds+∑h=1mβihRGhRJ+βihIGhIJ+βihJGhJJ+βihKGhKJ1Γρ×∫0ϑ+∫ϑtt−sρ−1e−t+se−ϑe−s+ϑu¯hJs−ϑds+∑h=1mβihRGhRK+βihIGhIK+βihJGhJK+βihKGhKK1Γρ×∫0ϑ+∫ϑtt−se−t+se−ϑe−s+ϑu¯hKs−ϑds≤e−tu¯iR0+e−ϱγisups∈−ϱ,0e−sϕ¯iRs+sups∈0,t−ϱe−su¯iRs1Γρ∫0tvρ−1e−vdv+∑h=1mαihRGhRR+αihIGhIR+αihJGhJR+αihKGhKRsups∈0,te−tu¯iRs1Γρ∫0tvρ−1e−vdv+∑h=1mαihRGhRI+αihIGhII+αihJGhJI+αihKGhKIsups∈0,te−tu¯iIs1Γρ∫0tvρ−1e−vdv+∑h=1mαihRGhRJ+αihIGhIJ+αihJGhJJ+αihKGhKJsups∈0,te−tu¯iJs1Γρ∫0tvρ−1e−vdv+∑h=1mαihRGhRK+αihIGhIK+αihJGhJK+αihKGhKKsups∈0,te−tu¯iKs1Γρ∫0tvρ−1e−vdv+e−ϑ∑h=1mβihRGhRR+βihIGhIR+βihJGhJR+βihKGhKR×sups∈−ϑ,0e−sϕ¯hRs+sups∈0,t−ϑe−su¯hRs1Γρ∫0tvρ−1e−vdv+e−ϑ∑h=1mβihRGhRI+βihIGhII+βihJGhJI+βihKGhKI×sups∈−ϑ,0e−sϕ¯hRs+sups∈0,t−ϑe−su¯hIs1Γρ∫0tvρ−1e−vdv+e−ϑ∑h=1mβihRGhRJ+βihIGhIJ+βihJGhJJ+βihKGhKJ×sups∈−ϑ,0e−sϕ¯hRs+sups∈0,t−ϑe−su¯hJs+1Γρ∫0tvρ−1e−vdv+e−ϑ∑h=1mβihRGhRK+βihIGhIK+βihJGhJK+βihKGhKK×sups∈−ϑ,0e−sϕ¯hRs+sups∈0,t−ϑe−su¯hKs1Γρ∫0tvρ−1e−vdv≤ϕ¯t+e−γϕ¯t+e−γu¯iRs+∑h=1mαihRGhRR+αihIGhIR+αihJGhJR+αihKGhKRu¯iRs+∑h=1mαihRGhRI+αihIGhII+αihJGhJI+αihKGhKIu¯iIs+∑h=1mαihRGhRJ+αihIGhIJ+αihJGhJJ+αihKGhKJu¯iJs+∑h=1mαihRGhRK+αihIGhIK+αihJGhJK+αihKGhKKu¯iKs+e−ϑ∑h=1mβihRGhRR+βihIGhIR+βihJGhJR+βihKGhKRϕ¯hRt+u¯hRs+e−ϑ∑h=1mβihRGhRI+βihIGhII+βihJGhJI+βihKGhKIϕ¯hIt+u¯hIs+e−ϑ∑h=1mβihRGhRJ+βihIGhIJ+βihJGhJJ+βihKGhKJϕ¯hJt+u¯hJs+e−ϑ∑h=1mβihRGhRK+βihIGhIK+βihJGhJK+βihKGhKKϕ¯hKt+u¯hKs=ϕ¯Rt+e−γϕ¯Rt+e−γu¯iRs+∑h=1mαihRGhRR+αihIGhIR+αihJGhJR+αihKGhKRu¯iRs+∑h=1mαihRGhRI+αihIGhII+αihJGhJI+αihKGhKIu¯iIs+∑h=1mαihRGhRJ+αihIGhIJ+αihJGhJJ+αihKGhKJu¯iJs+∑h=1mαihRGhRK+αihIGhIK+αihJGhJK+αihKGhKKu¯iKs+e−ϑ∑h=1mβihRGhRR+βihIGhIR+βihJGhJR+βihKGhKRϕ¯hRt+u¯hRs+e−ϑ∑h=1mβihRGhRI+βihIGhII+βihJGhJI+βihKGhKIϕ¯hIt+u¯hIs+e−ϑ∑h=1mβihRGhRJ+βihIGhIJ+βihJGhJJ+βihKGhKJϕ¯hJt+u¯hJs+e−ϑ∑h=1mβihRGhRK+βihIGhIK+βihJGhJK+βihKGhKKϕ¯hKt+u¯hKs.
It is easy to see that∃p1∈1,2,…,m such that(29)maxisupte−tu¯iRt=supte−tu¯p1Rt.
Then, it follows from (28) that(30)u¯Rt=maxisupte−tu¯iRt=supte−tu¯p1Rt≤e−γ+∑h=1mαp1hRGhRR+αp1hIGhIR+αp1hJGhJR+αp1hKGhKR+e−ϑ∑h=1mβp1hRGhRR+βp1hIGhIR+βp1hJGhJR+βp1hKGhKRu¯Rs+∑h=1mαp1hRGhRI+αp1hIGhII+αp1hJGhJI+αp1hKGhKI+e−ϑ∑h=1mβp1hRGhRI+βp1hIGhII+βp1hJGhJI+βp1hKGhKIu¯Is+∑h=1mαp1hRGhRJ+αp1hIGhIJ+αp1hJGhJJ+αp1hKGhKJ+e−ϑ∑h=1mβp1hRGhRJ+βp1hIGhIJ+βp1hJGhJJ+βp1hKGhKJu¯Js+∑h=1mαp1hRGhRK+αp1hIGhIK+αp1hJGhJK+αp1hKGhKK+e−ϑ∑h=1mβp1hRGhRK+βp1hIGhIK+βp1hJGhJK+βp1hKGhKKu¯Ks+1+e−γ+e−ϑ∑h=1mβp1hRGhRR+βp1hIGhIR+βp1hJGhJR+βp1hKGhKRϕ¯Rt+e−ϑ∑h=1mβp1hRGhRI+βp1hIGhII+βp1hJGhJI+βp1hKGhKIϕ¯It+e−ϑ∑h=1mβp1hRGhRJ+βp1hIGhIJ+βp1hJGhJJ+βp1hKGhKJϕ¯Jt+e−ϑ∑h=1mβp1hRGhRK+βp1hIGhIK+βp1hJGhJK+βp1hKGhKKϕ¯Kt.
Then, we have(31)u¯Rt≤1A1p1A2p1u¯It+A3p1u¯Jt+A4p1u¯Kt+A5p1ϕ¯Rt+A6p1ϕ¯It+A7p1ϕ¯Jt+A8p1ϕ¯Kt.
In a similar way, we can obtain(32)u¯It≤1ℬ1p2ℬ2p2u¯Rt+ℬ3p2u¯Jt+ℬ4p2u¯Kt+ℬ5p2ϕ¯Rt+ℬ6p2ϕ¯It+ℬ7p2ϕ¯Jt+ℬ8p2ϕ¯Kt,(33)u¯Jt≤1C1p3C2p3u¯Rt+C3p3u¯It+C4p3u¯Kt+C5p3ϕ¯Rt+C6p3ϕ¯It+C7p3ϕ¯Jt+C8p3ϕ¯Kt,(34)u¯Kt≤1D1p4C2p4u¯Rt+D3p4u¯It+D4p4u¯Jt+D5p4ϕ¯Rt+D6p4ϕ¯It+D7p4ϕ¯Jt+D8p4ϕ¯Kt.
By (31)–(34), we obtain(35)A1p1−ℬ2p2−C2p3−D2p4u¯Rt+ℬ1p1−A2p1−C3p3−D3p4u¯It+C1p3−A1p1−ℬ3p2−D4p4u¯Jt+D1p4−A4p1−ℬ4p2−C3p3u¯Kt≤A5p1+ℬ5p2+C5p3+D5p4ϕ¯Rt+A6p1+ℬ6p2+C6p3+D6p4ϕ¯It+A7p1+ℬ7p2+C7p3+D7p4ϕ¯Jt+A8p1+ℬ8p2+C8p3+D8p4ϕ¯Kt.
It follows from (35) that(36)u¯Rt+u¯It+u¯Jt+u¯Kt≤Π1ϕ¯Rt+Π2ϕ¯It+Π3ϕ¯Jt+Π4ϕ¯Kt,where(37)Π1=A5p1+ℬ5p2+C5p3+D5p4W>0,Π2=A6p1+ℬ6p2+C6p3+D6p4W>0,Π3=A7p1+ℬ7p2+C7p3+D7p4W>0,Π4=A8p1+ℬ8p2+C8p3+D8p4W>0.
Thus, we can come to the conclusion that the unique equilibrium pointu∗ of system (3) is uniformly stable. This ends the proof.Remark 1.
In 2017, Zhang et al. [4] analyzed the stability issue of fractional-order complex-valued neural networks involving leakage and discrete delays. In this study, we have investigated the stability of fractional-order quaternion-valued neural networks involving discrete and leakage delays. The investigation on the stability of fractional-order quaternion-valued neural networks becomes more complex than that of Zhang et al. [4] since the fractional-order quaternion-valued neural networks has been decomposed into more multidimensional neural networks. This study replenishes the research of Zhang et al. [4].


## 5. Simulation Results

Consider the fractional-order quaternion-valued neural networks:(38)d0.72u1tdt0.72=−γ1u1t−0.52+∑h=12α1hghuht+∑h=12β1hghuht−0.44+L1,d0.72u2tdt0.72=−γ2u2t−0.52+∑h=12α2hghuht+∑h=12β2hghuht−0.44+L2,where u1t=u1Rt+iu1It+ju1Jt+ku1Kt,u2t=u2Rt+iu2It+ju2Jt+ku2Kt, and(39)α11=0.22+0.25i+0.34j+0.53k,α12=0.33+0.43i−0.67j+0.45k,α21=0.19+0.65i−0.67j−0.88k,α22=0.43+0.35i−0.23j−0.38k,β11=0.56−0.55i−0.76j+0.39k,β12=0.22+0.79i+0.81j−0.66k,β21=0.89+0.34i+0.73j−0.48k,β22=0.35+0.84i+0.67j−0.85k,L1=0.77+0.33i+0.28j−0.36k,L2=0.67+0.34i+0.57j−0.62k,ghuh=0.3cosuhR+0.4isinuhI+uhJ+uhK+0.5jsinuhJ+0.2ksinuhR+uhJ+uhK.It is easy to obtain that model (38) owns the zero equilibrium point. Let GhRR=GhRI=GhRJ=GhRK=GhIR=GhII=GhIJ=GhIK=GhJR,GhJI=GhJJ=GhJK=GhKR=GhKI=GhKJ=GhKK=1. Utilizing the Matlab software, we obtain that θ=0.7845 and U=0.6812, W=0.6008. Thus, we can check that all the hypotheses of Theorem 2 are fulfilled. Thus, one can conclude that the zero equilibrium point of model (38) is uniformly stable. Figure 1 shows that the state of four parts will be close to zero gradually.Figure 1
The simulation trajectories of system (38).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)


## 6. Conclusions

Based on earlier research studies, a class of fractional-order quaternion-valued neural networks involving discrete and leakage delays has been set up. A set of sufficient conditions guaranteeing the existence and uniqueness of the equilibrium point for the investigated fractional-order quaternion-valued neural networks has been derived. In addition, the global uniform stability of the involved fractional-order quaternion-valued neural networks has been systematically discussed. This work generalizes the work of Zhang et al. [4]. The research idea can also be utilized to investigate many other types of fractional-order quaternion-valued neural networks. In addition, we know that Clifford analysis is more general than quaternion one [36, 37]. We will study the global uniform stability of Clifford-valued neural networks in the near future.



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*Source: 9988073-2021-06-03.xml*