234 Udhayakumar et al. / Front Inform Technol Electron Eng 2020 21(2):234-246

Frontiers of Information Technology & Electronic Engineering

www.jzus.zju.edu.cn; engineering.cae.cn; www.springerlink.com

ISSN 2095-9184 (print); ISSN 2095-9230 (online)

E-mail: jzus@zju.edu.cn

Mittag-Leffler stability analysis ofmultiple

equilibriumpoints in impulsive fractional-order

quaternion-valued neural networks

K. UDHAYAKUMAR1, R. RAKKIYAPPAN1, Jin-de CAO‡2, Xue-gang TAN3

1Department of Mathematics, Bharathiar University, Coimbatore 641046, India2School of Mathematics, Southeast University, Nanjing 210096, China3School of Automation, Southeast University, Nanjing 210096, China

E-mail: udhai512@gmail.com; rakkigru@gmail.com; jdcao@seu.edu.cn; xgtan_sde@163.com

Received Aug. 14, 2019; Revision accepted Oct. 19, 2019; Crosschecked Dec. 4, 2019

Abstract: In this study, we investigate the problem of multiple Mittag-Leffler stability analysis for fractional-orderquaternion-valued neural networks (QVNNs) with impulses. Using the geometrical properties of activation functionsand the Lipschitz condition, the existence of the equilibrium points is analyzed. In addition, the global Mittag-Lefflerstability of multiple equilibrium points for the impulsive fractional-order QVNNs is investigated by employing theLyapunov direct method. Finally, simulation is performed to illustrate the effectiveness and validity of the mainresults obtained.

Key words: Mittag-Leffler stability; Fractional-order; Quaternion-valued neural networks; Impulsehttps://doi.org/10.1631/FITEE.1900409 CLC number: O175

1 Introduction

Recently, neural networks (NNs) have attractedattention from various fields due to their wide ap-plications in image processing, system identification,propagation, pattern recognition, associative mem-ory, combinational optimization, etc. Most of theseapplications depend on the dynamical properties ofNNs. Therefore, different kinds of stability analy-sis and properties including bifurcation and chaos inNNs have attracted much attention (Cao and Xiao,2007; Rakkiyappan et al., 2014, 2015a, 2015b, 2016;Stamova, 2014; Wang H et al., 2015; Li XD and Wu,2016; Li XD and Ding, 2017; Li XD et al., 2017; Wuand Zeng, 2017; Yang et al., 2018; Huang YJ and Li,

‡ Corresponding authorORCID: K. UDHAYAKUMAR, https://orcid.org/0000-0001-

5764-1990; Jin-de CAO, https://orcid.org/0000-0003-3133-7119c© Zhejiang University and Springer-Verlag GmbH Germany, partof Springer Nature 2020

2019; Khan et al., 2019; Li X et al., 2019; Nie et al.,2019; Pang et al., 2019; Qi et al., 2019; Wang JJand Jia, 2019). Quaternion-valued NNs (QVNNs)are a generic extension of real- and complex-valuedNNs (RVNNs and CVNNs), and they inhibit thenon-commutative property of the quaternion algebra(Chen XF et al., 2017; Hu et al., 2017; Liu Y et al.,2017, 2018; Song and Chen, 2018; Li N and Zheng,2020). The quaternion problems are more difficultthan those in real- or complex-valued systems, whichis the reason for the slow development in quaternionfields.

It is well known that two-dimensional data canbe processed well in CVNNs and many RVNNs. Ifthe data is three- or four-dimensional, such as inthe cases with body images, color images, and four-dimensional signals, it can be directly encoded interms of quaternions in quaternion networks, show-ing QVNNs to be more important than CVNNs and

Udhayakumar et al. / Front Inform Technol Electron Eng 2020 21(2):234-246 235

RVNNs. Moreover, quaternions have attracted at-tention in a wide range of applications, includingrotation, image comparison, and color night vision.Recently, a few researchers have considered theo-retical investigations of the dynamical properties ofQVNNs, concentrating mainly on the global behav-ior such as global stability. Decomposition of quater-nions is a useful method of dealing with the non-commutativity of quaternion fields. Using plural de-composition and the Lipschitz technique, the sta-bility results of QVNNs were obtained in terms ofcontinuous and discrete time cases by Liu Y et al.(2018). In the past decade, the incorporation offractional calculus into NNs has achieved better re-sults than the integer-order NNs investigated by Caoand Xiao (2007) and Abdurahman et al. (2015).This is because the fractional-order derivatives in-herently have excellent memory and hereditary prop-erties in representing the network model. As is wellknown, there are many advantages over the integer-order NNs and the corresponding fractional-orderNNs. However, the main difference is that fractional-order systems are more accurate than integer-ordersystems; i.e, there are more degrees of freedom infractional-order systems. Moreover, compared withclassical integer-order systems, fractional-order sys-tems are characterized by infinite memory. Con-sidering all the above-mentioned reasons, the incor-poration of a memory term into an NN model isunavoidable. On the other hand, some researchershave attempted to investigate the advantages of bothquaternions and fractional derivatives in NNs, andproposed fractional-order QVNNs.

It is well known that, many factors such astime delay, chaos, bifurcation, and system com-plexities influence the fractional-order NNs, result-ing in instability at certain time instants. There-fore, the integration of impulsivity into the pro-posed fractional-order QVNNs becomes essential.Combining the memory and hereditary properties offractional-order systems and the impulsive effect, theresulting impulsive fractional-order QVNNs guar-antee better outcomes when compared with usualinteger-order NNs. As one of the classical phenom-ena of dynamic NNs, multistability analysis has beenstudied extensively by Zeng et al. (2010), Huang Yet al. (2012), Zeng and Zheng (2012), Liu P et al.(2017, 2018), and Zhang FH and Zeng (2018). InPopa and Kaslik (2018), periodic solutions for the

Hopfield-type integer-order NNs were studied in thepresence of both time-dependent and distributed de-lays, taking the impulsive effects into account. Inthis study, we investigate the multistability problemof impulsive fractional-order QVNNs in the Mittag-Leffler sense. To the best of our knowledge, thisis the first time the Mittag-Leffler stability theoryhas been developed thoroughly for the case of im-pulsive fractional-order QVNNs. Mittag-Leffler sta-bility analysis in fractional-order systems is still anopen problem.

Motivated by the above discussions, we con-duct the study of multiple Mittag-Leffler stabil-ity results on impulsive fractional-order QVNNs.First, n-dimensional QVNNs are converted into a4n-dimensional RVNN system using the decomposi-tion and non-commutative properties of quaternions.Then, sufficient conditions for Mittage-Leffler stabil-ity are discussed for fractional-order nonlinear sys-tems. Finally, two numerical examples are givento demonstrate the effectiveness of the theoreticalresults.

2 Preliminaries

In this section, we present some important defi-nitions and lemmas of fractional calculus which helpprove the main results.Definition 1 (Podlubny, 1998; Kilbas et al., 2006)The Caputo fractional derivative of order 0 < α forf(t) ∈ Cn([t0,+∞],R) is

Ct0D

αt f(t) =

1

Γ(n− α)

t∫

t0

(t− s)n−(α+1)f (n)(s)ds,

where Γ(·) is a Gamma function defined as Γ(α) =∞∫t0

e−t

t1−α dt (n− 1 < α < n). If we choose 0 < α < 1,

Ct0D

αt f(t) =

1

Γ(1− α)

t∫

t0

(t− s)−α df(1)(s)

dsds.

For convenience, in the rest of this paper, we adoptDα to denote Caputo’s fractional derivative operatorCt0D

αt .

Definition 2 (Podlubny, 1998; Kilbas et al., 2006)For any α, β > 0 and real number ν, the Mittag-Leffler function Eα,β(ν) with two parameters is

236 Udhayakumar et al. / Front Inform Technol Electron Eng 2020 21(2):234-246

defined as

Eα,β(ν) =

∞∑q=0

νq

Γ(qα+ β).

If β = 1, we can obtain the one-parameter form of

the Mittag-Leffler function E(ν) =∞∑q=0

νq

Γ(qα+1) ; if

α = β = 1, then E1,1(ν) = eν .

Consider the following fractional-order QVNNswith impulses:⎧⎪⎪⎨⎪⎪⎩Dαhp(t) = −cphp(t) +

n∑q=1

apqfq (hq(t)) +Rp,

δhp(tk) = hp(t+k )− hp(t

−k ) = βkp (hp(tk)) ,

(1)where k = 1, 2, . . . ,m and p = 1, 2, . . . , n, or, equiv-alently, in the vector form{

Dαh(t) = −Ch(t) +Af(h(t))+R,δh(tk) = h(t+k )− h(t−k ) = βk

(h(tk)

),

(2)

where k = 1, 2, . . . ,m, 0 < α < 1, h(t) =(h1(t), h2(t), . . . , hn(t)

)T ∈ Qn is the state variable

at time t, C = diag(c1, c2, . . . , cn) with cp > 0,f(h(t)

)denotes the neuron activation functions,

A ∈ Qn×n is the interconnection matrix, R =

(R1,R2, . . . ,Rn)T ∈ Q

n is an external input, and βkdenotes the impulsive operator. The time sequenceis represented by {tk} for all k ∈ Z and satisfies0 = t0 < t1 < . . . < tk < . . . , limk→+∞ tk = +∞.

Definition 3 If a vector h∗ ∈ Rn satisfies{

− Ch∗ +Af(h∗) +R = 0,

βk(h∗) = 0,

where k = 1, 2, . . . , n, then h∗ is called an equilibriumof NNs (2).Definition 4 (Chen JJ et al., 2014) The equi-librium point h∗ = (h∗1, h

∗2, . . . , h

∗n)

T of system (2)is said to be globally Mittag-Leffler stable. Thereexist positive constants U and G, such that for anysolution h(t) of system (2) with initial value h0, wehave

||h(t)− h∗|| ≤ G ||h0 − h∗||Eq (−U(t− t0)q) ,

where t ≥ t0. If the equilibrium point h∗ ofsystem (2) is globally Mittag-Leffler stable, thenQVNNs (2) are globally Mittag-Leffler stable.

System (2) follows from the non-commutativeproperty of quaternion algebra, and uses the Hamil-ton rules: ij = k, ji = −k, jk = i, kj = −i,

ki = j, ik = −j, ijk = i2 = j2 = k2 = −1, andν ∈ {R, I, J,K}; thus, we can rewrite NNs (2) as thefollowing four real-valued NNs:

DαhR(t) =− ChR(t) +ARfR(hR(t)

) −AIf I(hI(t)

)−AJfJ(hJ (t))−AKfK(hK(t)) +RR,

DαhI(t) =− ChI(t) +ARf I(hI(t)) +AIfR(hR(t))

+AJfK(hK(t))−AKfJ(hJ (t)) +RI ,

DαhJ(t) =− ChJ(t) +ARfJ(hJ(t))

−AIfK(hK(t)) +AJfR(hR(t))

+AKf I(zI(t)) +RJ ,

DαhK(t) =− ChK(t) +ARfK(hK(t))

+AIfJ(hJ(t)) −AJf I(hI(t))

+AKfR(hR(t)) +RK ,

hν(t−k ) =hν(tk), h

ν(t+k )− hν(t−k ) = βνk (h(tk)),

(3)

where t �= tk and p = 1, 2, . . . , n.Assumption 1 The impulsive operator βk =

[βk1, βk2, . . . , βkn]T is defined on {ϕ : (−∞, tk] →

Ln|ϕ is piecewise continuous on (−∞, 0], left con-

tinuous on [0, tk], with a first-kind discontinuity attr, and differentiable on every interval (tr−1, tr), 1 ≤r ≤ k}. This will require βk(ϕ0) = 0 for any constantfunction ϕ0.

Assumption 2 G(1) = (1,∞), G(−1) =

(−∞,−1), and GL(h) = G(hR) + iG(hI)+ jG(hJ ) +

kG(hK) for every ς ∈ {±1,±i,±j,±k}n , and we de-fine the set Φς = GL(ς1)×GL(ς2)× . . .×GL(ςn). Forexample, we take GL(−1 + i− j + k) = (−∞,−1) +

(1,∞)i + (−∞,−1)j + (1,∞)k.

Assumption 3 The components of the activationfunctions fR

p , f Ip , fJ

p , and fKp are bounded and glob-

ally Lipschitz continuous; then, for any constantsϑνRq , ϑνIq , ϑνJq , and ϑνKq , ν = R, I, J,K, |fν

p (h)| ≤1, and |fν

q (hR1 , h

I1, h

J1 , h

K1 ) − fν

q (hR2 , h

I2, h

J2 , h

K2 )| ≤

ϑνRq |hR1 −hR2 |+ϑνIq |hI1−hI2|+ϑνJq |hJ1−hJ2 |+ϑνKq |hK1 −hK2 |.Assumption 4 There exists an l ∈ (0, 1) suchthat functions fν

q (hν) satisfy fν

q (hν) ≥ l if hν ≥ 1,

and fνq (h

ν) ≤ −l if hν ≤ −1, for q = 1, 2, . . . , n andν ∈ {R, I, J,K}.Assumption 5 The external input vector satisfies

||Rp(t)|| <aRppl − dp −(|aIpp|+ |aJpp|+ |aKpp|

)

−n∑

q �=p

(|aRpq|+ |aIpq|+ |aJpq|+ |aKpq|),

Udhayakumar et al. / Front Inform Technol Electron Eng 2020 21(2):234-246 237

where t ∈ R and p = 1, 2, . . . , n.

Assumption 6 For any k ∈ Z+ and ς ∈

{±1,±i,±j,±k}n , if σ(t) ∈ Φς , then σ(tk)+βk(σ) ∈Φς .

For theoretical investigations, we need the fol-lowing lemmas:Lemma 1 (Wu and Zeng, 2017) Let χ1 > 0, χ2 >

0, χ3 > 1, χ4 > 1, and 1χ3

+ 1χ4

= 1. Then for anyδ > 0, we have χ1χ2 ≤ 1

χ3(χ1δ)

χ3 + 1χ4

(χ21δ )

χ4 . Theinequality holds if and only if (χ1δ)

χ3 = (χ21δ )

χ4 .

Lemma 2 (Popa and Kaslik, 2018) Let Eq. (4)(at the bottom of this page) hold, where Mν

p =1

dp

(||Rν

p || +n∑

q=1

(|aRpq|+ |aIpq|+ |aJpq|+ |aKpq|) )

, p =

1, 2, . . . , n, and ν ∈ {R, I, J,K}.If Assumption 3 is satisfied, then the following

statements hold:1. There exists at least one equilibrium point of

QVNNs (1) corresponding to input vector R in setΦR.

2. Every equilibrium point of QVNNs (1) be-longs to set ΦR.Lemma 3 (Popa and Kaslik, 2018) Suppose thatAssumptions 1–5 hold. Then the following condi-tions are true:

1. In every set Φς (ς ∈ {±1,±i,±j,±k}n), thereexists at least one equilibrium point of QVNNs (1)corresponding to the external input vector R.

2. If Assumption 6 is satisfied, then set Φς (ς ∈{±1,±i,±j,±k}n) is a positively invariant set.

3. If Assumption 3 holds, then the equilibriumpoint of QVNNs (1) corresponding to input vectorR in set Φς (ς ∈ {±1,±i,±j,±k}n) is unique andexponentially stable.Lemma 4 (Zhang XX et al., 2017) For the followingfractional-order impulsive system:

{Dαh(t) = −Ch(t) +Af(t, h(t)) +R,δh(tk) = h(t+k )− h(t−k ) = βk(h(tk)),

(5)

where k = 1, 2, . . . ,m, assume that the following con-ditions hold: (1) f(t, 0) = 0 (t > 0); (2) βk = 0 (k =

1, 2, . . . ,m); (3) there exists a positive definite func-tion V (t) that satisfies DαV (t, e(t)) ≤ −ξV (t, e(t))

and V (t+, e(t) + Ek(h)) ≤ V (t, e(t)), where t = tkand k = 1, 2, . . . ,m. Then the equilibrium point ofQVNNs (1) is Mittag-Leffler stable.Remark 1 Based on Brouwer’s and Leray-Schauder’s fixed point theories (Schauder, 1930),Lemmas 1 and 2 can be easy to prove. For details,see Lemma 2 and Theorem 2 in Popa and Kaslik(2018).

3 Main results

In this section, the Mittag-Leffler stability anal-ysis of multiple equilibrium points for the impulsivefractional-order QVNNs is investigated. ConsiderQVNNs (1) have the initial value hν(0) = hν0 . Let h∗

be the equilibrium point of the impulsive fractional-order QVNNs (1) and thus make the transformationeν(t) = hν(t) − h∗. Then system (3) is transformedinto

DαeR(t) =− CeR(t) +ARfR(eR(t))−AIf I(eI(t))

−AJfJ(eJ(t))−AKfK(eK(t)),

DαeI(t) =− CeI(t) +ARf I(eI(t)) +AIfR(eR(t))

+AJfK(eK(t)) −AKfJ(eJ(t)),

DαeJ(t) =− CeJ(t) +ARfJ(eJ(t))−AIfK(eK(t))

+AJfR(eR(t)) +AKf I(eI(t)),

DαeK(t) =− CeK(t) +ARfK(eK(t)) +AIfJ(eJ (t))

−AJf I(eI(t)) +AKfR(eR(t)),

eν(t−k ) =eν(tk), e

ν(t+k )− eν(t−k ) = Γνk (e(tk)),

e(0) =e0, t �= tk, p = 1, 2, . . . , n, (6)

where e(t) = (e1(t), e2(t), . . . , en(t))T, f(t, e(t)) =

(f1(t, e1(t)), f2(t, e2(t)), . . . , fn(t, en(t)))T, and

fp(t, ep(t)) = fp(t, ep+h∗p)−fp(t, h∗p) (p = 1, 2, . . . , n

and e0 = h0 − h∗).Theorem 1 Assume that the conditions of Lemmas2 and 3 hold, and Γνk (h

ν(tk)) = −η(hν(tk) − h∗)(k = 1, 2, . . . ,m, ζ > 0, and ν ∈ {R, I, J,K}) whereh∗ is the steady state of the impulsive QVNNs (1).

ΦR =([−MR

1 ,MR1

]+ i

[−M I1 ,M

I1

]+ j

[−MJ1 ,M

J1

]+ k

[−MK1 ,M

K1

])· ([−MR

2 ,MR2

]+ i

[−M I2 ,M

I2

]+ j

[−MJ2 ,M

J2

]+ k

[−MK2 ,M

K2

])· . . . · ([−MR

n ,MRn

]+ i

[−M In,M

In

]+ j

[−MJn ,M

Jn

]+ k

[−MKn ,M

Kn

]). (4)

238 Udhayakumar et al. / Front Inform Technol Electron Eng 2020 21(2):234-246

If |1 − δνkp|ζ ≤ 1 and there exists inequality (7) (atthe bottom of this page), where

Aνpq = aRpqϑ

Rνq − aIpqϑ

Iνq − aJpqϑ

Jνq − aKpqϑ

Kνq ,

Aνpq = aRpqϑ

Iνq + aIpqϑ

Rνq + aJpqϑ

Kνq − aKpqϑ

Jνq ,

Aνpq = aRpqϑ

Jνq − aIpqϑ

Kνq + aJpqϑ

Rνq + aKpqϑ

Iνq ,

Aνpq = aRpqϑ

Kνq + aIpqϑ

Jνq − aJpqϑ

Iνq − aKpqϑ

Rνq ,

then the impulsive fractional-order QVNN (1) isMittag-Leffler stable.Proof Consider the following Lyapunov functioncandidate:

V (t, e(t)) = V1 + V2 + V3 + V4, (8)

where

V1 =

n∑p=1

ζ−1|eRp (t)|ζ , V2 =

n∑p=1

ζ−1|eIp(t)|ζ ,

V3 =n∑

p=1

ζ−1|eJp (t)|ζ , V4 =n∑

p=1

ζ−1|eKp (t)|ζ .

When t �= tk (k = 1, 2, . . . ,m), calculating the frac-tional derivatives of V (t, e(t)) along the trajectoriesof NNs (3), we can find from Lemma 1 and NNs (3)that inequality (9) (on the next page) is valid.

Using Lemma 1, we can obtain

|eRp (t)|ζ−1|eRq (t)| ≤ ζ−1

((ζ − 1)|eRp (t)|ζεR1

+ |eRq (t)|ζ1

εRζ−1

1

),

|eRp (t)|ζ−1|eIq(t)| ≤ ζ−1

((ζ − 1)|eRp (t)|ζεR2

+ |eIq(t)|ζ1

εRζ−1

2

),

|eRp (t)|ζ−1|eJq (t)| ≤ ζ−1

((ζ − 1)|eRp (t)|ζεR3

+ |eJq (t)|ζ1

εRζ−1

3

),

|eRp (t)|ζ−1|eKq (t)| ≤ ζ−1

((ζ − 1)|eRp (t)|ζεR4

+ |eKq (t)|ζ 1

εRζ−1

4

).

Substituting the above inequalities into inequal-ity (9), then we have inequality (10) (on the nextpage).

Similar to the proof of DαV1, we can estimateDαV2, D

αV3, and DαV4, which are omitted here tosave space. As for DαV1, we have inequalities (11)–(13), shown on page 240.

Adding inequalities (10)–(13), we obtain in-equality (14) (shown on page 241).

From inequality (7), we can select a positive

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

λR = min1≤p≤n

{cpζ −

n∑q=1

(AR

pq

((ζ − 1)εR1 + εR

1−ζ

1

)+ AI

pq(ζ − 1)εR2 + AJpq(ζ − 1)εR3

+AKpq(ζ − 1)εR4 + AR

pqεI1−ζ

1 + ARpqε

J1−ζ

1 + ARpqε

K1−ζ

1

)}> 0,

λI = min1≤p≤n

{cpζ −

n∑q=1

(AI

pq

((ζ − 1)εI2 + εI

1−ζ

2

)+ AR

pq

((ζ − 1)εI1

)+ AJ

pq(ζ − 1)εI3

+AKpq(ζ − 1)εI4 + AI

pqεR1−ζ

2 + AIpqε

J1−ζ

2 + AIpqε

K1−ζ

2

)}> 0,

λJ = min1≤p≤n

{cpζ −

n∑q=1

(AJ

pq

((ζ − 1)εJ3 + εJ

1−ζ

3

)+ AR

pq

((ζ − 1)εJ1

)+ AI

pq(ζ − 1)εJ2

+AKpq(ζ − 1)εJ4 + AJ

pqεR1−ζ

3 + AJpqε

I1−ζ

3 + AJpqε

K1−ζ

3

)}> 0,

λK = min1≤p≤n

{cpζ −

n∑q=1

(AK

pq

((ζ − 1)εK4 + εK

1−ζ

4

)+ AR

pq

((ζ − 1)εK1

)+ AI

pq(ζ − 1)εK2

+AJpq(ζ − 1)εK3 ++AK

pqεR1−ζ

4 + AKpqε

I1−ζ

4 + AJpqε

K1−ζ

4

)}> 0.

(7)

Udhayakumar et al. / Front Inform Technol Electron Eng 2020 21(2):234-246 239

DαV1 ≤n∑

p=1

|eRp (t)|ζ−1

[(−cp)|eRp (t)|ζ +

n∑q=1

aRpq

(ϑRRp |eRq (t)|+ ϑRI

p |eIq(t)|+ ϑRJp |eJq (t)|+ ϑRK

p |eKq (t)|)

−n∑

q=1

aIpq

(ϑIRp |eRq (t)|+ ϑIIp |eIq(t)|+ ϑIJp |eJq (t)|+ ϑIKp |eKq (t)|

)−

n∑q=1

aJpq

(ϑJRp |eRq (t)|+ ϑJIp |eIq(t)|

+ ϑJJp |eJq (t)|+ ϑJKp |eKq (t)|)−

n∑q=1

aKpq

(ϑKRp |eRq (t)|+ ϑKI

p |eIq(t)|+ ϑKJp |eJq (t)|+ ϑKK

p |eKq (t)|)]

=

n∑p=1

(−cp)|eRp (t)|ζ +n∑

p=1

n∑q=1

aRpq|eRp (t)|ζ−1

(ϑRRq |eRq (t)|+ ϑRI

q |eIq(t)|+ ϑRJq |eJq (t)|+ ϑRK

q |eKq (t)|)

−n∑

p=1

n∑q=1

aIpq|eRp (t)|ζ−1

(ϑIRq |eRq (t)|+ ϑIIq |eIq(t)|+ ϑIJq |eJq (t)|+ ϑIKq |eKq (t)|

)

−n∑

p=1

n∑q=1

aJpq|eRp (t)|ζ−1

(ϑJRq |eRq (t)|+ ϑJIq |eIq(t)|+ ϑJJq |eJq (t)|+ ϑJKq |eKq (t)|

)

−n∑

p=1

n∑q=1

aKpq|eRp (t)|ζ−1

(ϑKRq |eRq (t)|+ ϑKI

q |eIq(t)|+ ϑKJq |eJq (t)|+ ϑKK

q |eKq (t)|)

=n∑

p=1

(−cp)|eRp (t)|ζ +n∑

p=1

n∑q=1

(aRpqϑ

RRq |eRp (t)|ζ−1|eRq (t)|+ aRpqϑ

RIq |eRp (t)|ζ−1|eIq(t)|+ aRpqϑ

RJq |eRp (t)|ζ−1

· |eJq (t)|+ aRpqϑRKq |eRp (t)|ζ−1|eKq (t)|

)−

n∑p=1

n∑q=1

(aIpqϑ

IRq |eRp (t)|ζ−1|eRq (t)|+ aIpqϑ

IIq |eRp (t)|ζ−1|eIq(t)|

+ aIpqϑIJq |eRp (t)|ζ−1|eJq (t)|+ aIpqϑ

IKq |eRp (t)|ζ−1|eKq (t)|

)−

n∑p=1

n∑q=1

(aJpqϑ

JRq |eRp (t)|ζ−1|eRq (t)|

+ aJpqϑJIq |eRp (t)|ζ−1|eIq(t)|+ aJpqϑ

JJq |eRp (t)|ζ−1|eJq (t)|+ aJpqϑ

JKq |eRp (t)|ζ−1|eKq (t)|

)

−n∑

p=1

n∑q=1

(aKpqϑ

KRq |eRp (t)|ζ−1|eRq (t)|+ aKpqϑ

KIq |eRp (t)|ζ−1|eIq(t)|+ aKpqϑ

KJq |eRp (t)|ζ−1|eJq (t)|

+ aKpqϑKKq |eRp (t)|ζ−1|eKq (t)|

).

(9)

DαV1 ≤n∑

q=1

{(aRpqϑ

RRq − aIpqϑ

IRq − aJpqϑ

JRq − aKpqϑ

KRq

)(ζ − 1)εR1 +

(aRpqϑ

RIq − aIpqϑ

IIq − aJpqϑ

JIq − aKpqϑ

KIq

)

· (ζ − 1)εR2 +

(aRpqϑ

RJq − aIpqϑ

IJq − aJpqϑ

JJq − aKpqϑ

KJq

)(ζ − 1)εR3 +

(aRpqϑ

RKq − aIpqϑ

IKq − aJpqϑ

JKq

− aKpqϑKKq

)(ζ − 1)εR4 +

(aRpqϑ

RRq − aIpqϑ

IRq − aJpqϑ

JRq − aKpqϑ

KRq

)εR

1−ζ

1 − cpζ

}V1

+n∑

q=1

{(aRpqϑ

RIq − aIpqϑ

IIq − aJpqϑ

JIq − aKpqϑ

KIq

)εR

1−ζ

2

}V2 +

n∑q=1

{(aRpqϑ

RJq − aIpqϑ

IJq − aJpqϑ

JJq

− aKpqϑKJq

)εR

1−ζ

3

}V3 +

n∑q=1

{(aRpqϑ

RKq − aIpqϑ

IKq − aJpqϑ

JKq − aKpqϑ

KKq

)εR

1−ζ

4

}V4.

(10)

240 Udhayakumar et al. / Front Inform Technol Electron Eng 2020 21(2):234-246

DαV2 ≤n∑

q=1

{(aRpqϑ

IRq + aIpqϑ

RRq + aJpqϑ

KRq − aKpqϑ

JRq

)(ζ − 1)εI1 +

(aRpqϑ

IIq + aIpqϑ

RIq + aJpqϑ

KIq − aKpqϑ

JIq

)

· (ζ − 1)εI2 +

(aRpqϑ

IJq + aIpqϑ

RJq + aJpqϑ

KJq − aKpqϑ

JJq

)(ζ − 1)εI3 +

(aRpqϑ

IKq + aIpqϑ

RKq + aJpqϑ

KKq

− aKpqϑJKq

)(ζ − 1)εI4 +

(aRpqϑ

IIq + aIpqϑ

RIq + aJpqϑ

KIq − aKpqϑ

JIq

)εI

1−ζ

2 − cpζ

}V2 +

n∑q=1

{(aRpqϑ

IRq

+ aIpqϑRRq + aJpqϑ

KRq − aKpqϑ

JRq

)εI

1−ζ

1

}V1 +

n∑q=1

{(aRpqϑ

IJq + aIpqϑ

RJq + aJpqϑ

KJq − aKpqϑ

JJq )εI

1−ζ

3

}V3

+n∑

q=1

{(aRpqϑ

IKq + aIpqϑ

RKq + aJpqϑ

KKq − aKpqϑ

JKq )εR

1−ζ

4

}V4.

(11)

DαV3 ≤n∑

q=1

{(aRpqϑ

JRq − aIpqϑ

KRq + aJpqϑ

RRq + aKpqϑ

IRq

)(ζ − 1)εJ1 +

(aRpqϑ

JIq − aIpqϑ

KIq + aJpqϑ

RIq + aKpqϑ

IIq

)

· (ζ − 1)εJ2 +

(aRpqϑ

JJq − aIpqϑ

KJq + aJpqϑ

RJq + aJpqϑ

IJq

)(ζ − 1)εJ3 +

(aRpqϑ

JKq − aIpqϑ

KKq + aJpqϑ

RKq

+ aKpqϑIKq

)(ζ − 1)εJ4 +

(aRpqϑ

JJq − aIpqϑ

KJq + aJpqϑ

RJq + aKpqϑ

IJq

)εJ

1−ζ

3 − cpζ

}V3 +

n∑q=1

{(aRpqϑ

JRq

− aIpqϑKRq + aJpqϑ

RRq + aKpqϑ

IRq

)εJ

1−ζ

1

}V1 +

n∑q=1

{(aRpqϑ

JIq − aIpqϑ

KIq + aJpqϑ

RIq + aKpqϑ

IIq

)εJ

1−ζ

2

}V2

+

n∑q=1

{(aRpqϑ

JKq − aIpqϑ

KKq + aJpqϑ

RKq + aKpqϑ

IKq

)εJ

1−ζ

4

}V4.

(12)

DαV4 ≤n∑

q=1

{(aRpqϑ

KRq + aIpqϑ

JRq − aJpqϑ

IRq + aKpqϑ

RRq

)(ζ − 1)εK1 +

(aRpqϑ

KIq + aIpqϑ

JIq − aJpqϑ

IIq + aKpqϑ

RIq

)

· (ζ − 1)εK2 +

(aRpqϑ

KJq + aIpqϑ

JJq − aJpqϑ

IJq + aKpqϑ

RJq

)(ζ − 1)εK3 +

(aRpqϑ

KKq + aIpqϑ

JKq − aJpqϑ

IKq

+ aKpqϑRKq

)(ζ − 1)εK4 + (aRpqϑ

KKq + aIpqϑ

JKq − aJpqϑ

IKq + aKpqϑ

RKq )εK

1−ζ

4 − cpζ

}V4 +

n∑q=1

{(aRpqϑ

KRq

+ aIpqϑJRq − aJpqϑ

IRq + aKpqϑ

RRq

)εK

1−ζ

1

}V1 +

n∑q=1

{(aRpqϑ

KIq + aIpqϑ

JIq − aJpqϑ

IIq + aKpqϑ

RIq

)εK

1−ζ

2

}V2

+

n∑q=1

{(aRpqϑ

KJq + aIpqϑ

JJq − aJpqϑ

IJq + aKpqϑ

RJq

)εK

1−ζ

3

}V3.

(13)

constant ξ > 0 such that min(λR, λI , λJ , λK) ≥ ξ >

0. This implies that

DαV (t, e(t)) ≤ −ξV (t, e(t)), for t �= tk. (15)

When t = tk (k = 1, 2 . . . ,m), we obtain

inequality (16) (on the next page).

Therefore, as a consequence of Lemma 4 andinequalities (15) and (16), it can be concluded thatthe impulsive fractional-order QVNN as QVNNs (1)is Mittag-Leffler stable.

Udhayakumar et al. / Front Inform Technol Electron Eng 2020 21(2):234-246 241

DαV (t, e(t)) =

n∑q=1

{− cpζ + AR

pq

((ζ − 1)εR1 + εR

1−ζ

1

)+ AI

pq(ζ − 1)εR2 + AJpq(ζ − 1)εR3 + AK

pq(ζ − 1)εR4

+ ARpqε

I1−ζ

1 + ARpqε

J1−ζ

1 + ARpqε

K1−ζ

1

}V1 +

n∑q=1

{− cpζ + AR

pq

((ζ − 1)εI1

)+ AI

pq

((ζ − 1)εI2

+ εI1−ζ

2

)+ AJ

pq(ζ − 1)εI3 + AKpq(ζ − 1)εI4 + AI

pqεR1−ζ

2 + AIpqε

J1−ζ

2 + AIpqε

K1−ζ

2

}V2(t)

+n∑

q=1

{− cpζ + AR

pq

((ζ − 1)εJ1

)+ AI

pq(ζ − 1)εJ2 + AJpq

((ζ − 1)εJ3 + εJ

1−ζ

3

)+ AK

pq(ζ − 1)εJ4

+ AJpqε

R1−ζ

3 + AJpqε

I1−ζ

3 + AJpqε

K1−ζ

3

}V3 +

n∑q=1

{− cpζ + AR

pq

((ζ − 1)εK1

)+ AI

pq(ζ − 1)εK2

+ AJpq(ζ − 1)εK3 + AK

pq

((ζ − 1)εK4 + εK

1−ζ

4

)+ AK

pqεR1−ζ

4 + AKpqε

I1−ζ

4 + AJpqε

K1−ζ

4

}V4

≤− λRV1 − λIV2 − λJV3 − λKV4.

(14)

V (t+k , e(t+k )) =

n∑p=1

ζ−1

∣∣∣∣eRp (tk) + ηRkp(eRp (tk))

∣∣∣∣ζ

+

n∑p=1

ζ−1

∣∣∣∣eIp(tk) + ηIkp(eIp(tk))

∣∣∣∣ζ

+

n∑p=1

ζ−1

∣∣∣∣eJp (tk) + ηJkp(eJp (tk))

∣∣∣∣ζ

+

n∑p=1

ζ−1

∣∣∣∣eKp (tk) + ηKkp(eKp (tk))

∣∣∣∣ζ

=n∑

p=1

ζ−1

∣∣∣∣hRp (tk)− h∗ − δRkp

(hp(tk)− h∗

)∣∣∣∣ζ

+n∑

p=1

ζ−1

∣∣∣∣hIp(tk)− h∗ − δIkp

(hp(tk)− h∗

)∣∣∣∣ζ

+n∑

p=1

ζ−1

∣∣∣∣hJp (tk)− h∗ − δJkp

(hp(tk)− h∗

)∣∣∣∣ζ

+n∑

p=1

ζ−1

∣∣∣∣hKp (tk)− h∗ − δKkp

(hp(tk)− h∗

)∣∣∣∣ζ

=n∑

p=1

ζ−1

∣∣∣∣1− δRkp

∣∣∣∣ζ∣∣∣∣hRp (tk)− h∗

∣∣∣∣ζ

+n∑

p=1

ζ−1

∣∣∣∣1− δIkp

∣∣∣∣ζ ∣∣∣∣hIp(tk)− h∗

∣∣∣∣ζ

+n∑

p=1

ζ−1

∣∣∣∣1− δJkp

∣∣∣∣ζ∣∣∣∣hJp (tk)− h∗

∣∣∣∣ζ

+n∑

p=1

ζ−1

∣∣∣∣1− δKkp

∣∣∣∣ζ∣∣∣∣hKp (tk)− h∗

∣∣∣∣ζ

≤V (tk, e(tk)).

(16)

Remark 2 Theorem 1 gives the Mittag-Lefflerstability conditions for the impulsive fractional-orderQVNNs (1). This condition depends on the param-eters of NNs and the activation functions.Remark 3 In the past few decades, NNs havebeen studied extensively for their wide applica-tions. Mittag-Leffler stability analysis of RVNNsand CVNNs has attracted the attention of many re-searchers. Liu P et al. (2018) introduced a class ofinteger-order recurrent NNs with unbounded time-varying delays. Using the geometrical propertiesof non-monotonic activation functions, they showed

that the addressed system has exactly (2Ki + 1)n

equilibrium points, of which (Ki + 1) were locallyasymptotically stable while others were unstable.Tyagi et al. (2016) provided sufficient conditions forMittag-Leffler stability of the equilibrium points forCVNNs. In this study, sufficient conditions are an-alyzed to achieve the coexistence and Mittag-Lefflerstability of equilibrium points. The number of equi-librium points of QVNNs is larger than that ofCVNNs.Theorem 2 Assume that the conditions of Lemmas2 and 3 hold, and βν

k (hν(tk)) = −δ(hν(tk) − h∗)

242 Udhayakumar et al. / Front Inform Technol Electron Eng 2020 21(2):234-246

(k = 1, 2, . . . ,m), where h∗ is the steady state of theimpulsive NNs (1). If |1− δνkp| ≤ 1 and there exist

ψR = min1≤p≤n

[cp −

n∑q=1

(|aRpq|+ |aIpq|+ |aJpq|+ |aKpq|

)

·(ϑRRq + ϑIRq + ϑJRq + ϑKR

q

)]> 0,

ψI = min1≤p≤n

[cp −

n∑q=1

(|aRpq|+ |aIpq|+ |aJpq|+ |aKpq|

)

·(ϑRIq + ϑIIq + ϑJIq + ϑKI

q

)]> 0,

ψJ = min1≤p≤n

[cp −

n∑q=1

(|aRpq|+ |aIpq|+ |aJpq|+ |aKpq|

)

·(ϑRJq + ϑIJq + ϑJJq + ϑKJ

q

)]> 0,

ψK = min1≤p≤n

[cp −

n∑q=1

(|aRpq|+ |aIpq|+ |aJpq|+ |aKpq|

)

·(ϑRKq + ϑIKq + ϑJKq + ϑKK

q

)]> 0,

(17)

then the impulsive fractional-order QVNN (1) isMittag-Leffler stable.Proof Consider the following Lyapunov functioncandidate:

V (t, e(t)) = V1 + V2 + V3 + V4, (18)

where

V1 =

n∑p=1

|eRp (t)|, V2 =

n∑p=1

|eIp(t)|,

V3 =

n∑p=1

|eJp (t)|, V4 =

n∑p=1

|eKp (t)|.

When t �= tk (k = 1, 2, . . . ,m), calculating thefractional-order derivative of V (t, e(t)) along the tra-jectories of Eq. (3), we obtain inequality (19) (on thenext page).

Similarly, we can obtain inequalities (20)–(22)(on the next page). Adding inequalities (19)–(22),we obtain inequality (23) (on the next page).

From inequality (17), we can select a positiveconstant ξ > 0 such that min(ψR, ψI , ψJ , ψK) ≥ξ > 0. This implies that

DαV (t, e(t)) ≤ −ξV (t, e(t)), t �= tk. (24)

When t = tk (k = 1, 2, . . . ,m), we have

V (t+k , e(t+k )) =

n∑p=1

∣∣∣∣eRp (tk) + ηRkp(eRp (tk))

∣∣∣∣

+n∑

p=1

∣∣∣∣eIp(tk) + ηIkp(eIp(tk))

∣∣∣∣

+n∑

p=1

∣∣∣∣eJp (tk) + ηJkp(eJp (tk))

∣∣∣∣

+

n∑p=1

∣∣∣∣eKp (tk) + ηKkp(eKp (tk))

∣∣∣∣

=

n∑p=1

∣∣∣∣hRp (tk)− h∗ − δRkp(hp(tk)− h∗)∣∣∣∣

+

n∑p=1

∣∣∣∣hIp(tk)− h∗ − δIkp(hp(tk)− h∗)∣∣∣∣

+

n∑p=1

∣∣∣∣hJp (tk)− h∗ − δJkp(hp(tk)− h∗)∣∣∣∣

+

n∑p=1

∣∣∣∣hKp (tk)− h∗ − δKkp(hp(tk)− h∗)∣∣∣∣

=n∑

p=1

∣∣∣∣1− δRkp

∣∣∣∣ ·∣∣∣∣hRp (tk)− h∗

∣∣∣∣

+n∑

p=1

∣∣∣∣1− δIkp

∣∣∣∣ ·∣∣∣∣hIp(tk)− h∗

∣∣∣∣

+

n∑p=1

∣∣∣∣1− δJkp

∣∣∣∣ ·∣∣∣∣hJp (tk)− h∗

∣∣∣∣

+

n∑p=1

∣∣∣∣1− δKkp

∣∣∣∣ ·∣∣∣∣hKp (tk)− h∗

∣∣∣∣≤V (tk, e(tk)).

(25)

Therefore, from Lemma 4 and inequalities (24)and (25), it can be concluded that the impulsivefractional-order QVNN as QVNNs (1) is Mittag-Leffler stable.Remark 4 In reviewing the existing works(Wang F et al., 2015; Song et al., 2016a; WangLM et al., 2017), fractional-order CVNNs (WangF et al., 2015; Wang LM et al., 2017) andinteger-order systems (Song et al., 2016a, 2016b)have been studied extensively over the last fewdecades. Recently, the Mittag-Leffler stability anal-ysis of fractional-order QVNNs was investigatedby Yang et al. (2018). However, the results on

Udhayakumar et al. / Front Inform Technol Electron Eng 2020 21(2):234-246 243

DαV1 ≤n∑

p=1

sgn(eRp (t))DαeRp (t)

=

n∑p=1

[(− cp +

n∑q=1

(|aRpq|ϑRR

p + |aIpq|ϑIRp + |aJpq|ϑJRp + |aKpq|ϑKRp

))|eRq (t)|+

n∑q=1

(|aRpq|ϑRI

p + |aIpq|ϑIIp

+ |aJpq|ϑJIp + |aKpq|ϑKIp

)|eIq(t)|+

n∑q=1

(|aRpq|ϑRJ

p + |aIpq|ϑIJp + |aJpq|ϑJJp + |aKpq|ϑKJp

)|eJq (t)|

+

n∑q=1

(|aRpq|ϑRK

p + |aIpq|ϑIKp + |aJpq|ϑJKp + |aKpq|ϑKKp

)|eKq (t)|

]. (19)

DαV2 ≤n∑

p=1

[ n∑q=1

(|aRpq|ϑIRp + |aIpq|ϑRR

p + |aJpq|ϑKRp + |aKpq|ϑJRp

)|eRq (t)|+

n∑q=1

(− cp + |aRpq|ϑIIp + |aIpq|ϑRI

p

+ |aJpq|ϑKIp + |aKpq|ϑJIp

)|eIq(t)|+

n∑q=1

(|aRpq|ϑIJp + |aIpq|ϑRJ

p + |aJpq|ϑKJp + |aKpq|ϑJJp

)|eJq (t)|

+

n∑q=1

(|aRpq|ϑIKp + |aIpq|ϑRK

p + |aJpq|ϑKKp + |aKpq|ϑJKp

)|eKq (t)|

]. (20)

DαV3 ≤n∑

p=1

[ n∑q=1

(|aRpq|ϑJRp + |aIpq|ϑKR

p + |aJpq|ϑRRp + |aKpq|ϑIRp

)|eRq (t)|+

n∑q=1

(|aRpq|ϑJIp + |aIpq|ϑKI

p

+ |aJpq|ϑRIp + |aKpq|ϑIIp

)|eIq(t)|+

n∑q=1

(− cp + |aRpq|ϑJJp + |aIpq|ϑKJ

p + |aJpq|ϑRJp + |aKpq|ϑIJp

)|eJq (t)|

+

n∑q=1

(|aRpq|ϑJKp + |aIpq|ϑKK

p + |aJpq|ϑRKp + |aKpq|ϑIKp

)|eKq (t)|

]. (21)

lDαV4 ≤n∑

p=1

[ n∑q=1

(|aRpq|ϑKR

p + |aIpq|ϑJRp + |aJpq|ϑIRp + |aKpq|ϑRRp

)|eRq (t)|+

n∑q=1

(|aRpq|ϑKI

p + |aIpq|ϑJIp

+ |aJpq|ϑIIp + |aKpq|ϑRIp

)|eIq(t)|+

n∑q=1

(|aRpq|ϑKJ

p + |aIpq|ϑJJp + |aJpq|ϑIJp + |aKpq|ϑRJp

)|eJq (t)|

+n∑

q=1

(|aRpq|ϑKK

p + |aIpq|ϑJKp + |aJpq|ϑIKp + |aKpq|ϑrKp)|eKq (t)|

]. (22)

DαV (t, e(t)) ≤− min1≤p≤n

[cp −

n∑q=1

(|aRpq|+ |aIpq|+ |aJpq|+ |aKpq|

)(ϑRRq + ϑIRq + ϑJRq + ϑKR

q

)]V1

− min1≤p≤n

[cp −

n∑q=1

(|aRpq|+ |aIpq|+ |aJpq|+ |aKpq|

)(ϑRIq + ϑIIq + ϑJIq + ϑKI

q

)]V2

− min1≤p≤n

[cp −

n∑q=1

(|aRpq|+ |aIpq|+ |aJpq|+ |aKpq|

)(ϑRJq + ϑIJq + ϑJJq + ϑKJ

q

)]V3

− min1≤p≤n

[cp −

n∑q=1

(|aRpq|+ |aIpq|+ |aJpq|+ |aKpq|

)(ϑRKq + ϑIKq + ϑJKq + ϑKK

q

)]V4. (23)

244 Udhayakumar et al. / Front Inform Technol Electron Eng 2020 21(2):234-246

impulsive fractional-order QVNNs have not been dis-cussed to date. In light of this, we have analyzedthe Mittag-Leffler stability of multiple equilibriumpoints for the impulsive fractional-order QVNNs us-ing the Lyapunov direct method, which is much morecomplicated than that of CVNNs.

4 Numerical simulations

In this section, simulation results are given toillustrate the efficiency of the impulsive fractional-order QVNNs.Example 1 Consider the following impulsivefractional-order QVNN:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

D0.9h1(t) = −c1h1(t) + a11f1(h1(t))

+ a12f2(h2(t)) +R1,

Δhp(tk) = δk1(h1(tk)− h∗),

D0.9h2(t) = −c2h2(t) + a21f1(h1(t))

+ a22f2(h2(t)) +R2,

Δhp(tk) = δk2(h2(tk)− h∗),

(26)

where k = 1, 2, . . . ,m, p = 1, 2, . . . , n, f1(h1(t)) =

f2(h2(t)) = tanh(t), a11 = 8.6 + 0.12i− 0.5j + 1.2k,a12 = 0.4 + 0.36i + 0.5j + 0.04k, a21 = 0.3 − 0.1i +

0.25j + 0.42k, a22 = 6 + 0.25i + 0.1j + 0.2k, R1 =

0.5+ 0.4i− 0.6j+ 0.6k, R2 = 0.7− 1.1i+ 0.4j− 1.4k,c1 = c2 = 4.5, ϑν1ν21 = ϑν1ν22 = 0.01, and ν1, ν2 ∈{R, I, J,K}. Calculation shows that inequality (7)in Theorem 1 is satisfied with p = 1, 2 and q = 1, 2.Consequently, QVNN (26) is Mittag-Leffler stable.Fig. 1 shows the trajectories of QVNN (26) withdifferent initial values.Example 2 Consider the following impulsivefractional-order QVNN:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

D0.7h1(t) = −c1h1(t) + a11f1(h1(t))

+ a12f2(h2(t)) +R1,

Δhp(tk) = δk1(h1(tk)− h∗),

D0.7h2(t) = −c2h2(t) + a21f1(h1(t))

+ a22f2(h2(t)) +R2,

Δhp(tk) = δk2(h2(tk)− h∗),

(27)

where k = 1, 2, . . . ,m, p = 1, 2, . . . , n, f1(h1(t)) =

f2(h2(t)) = exp(−t2), a11 = 6.3 + 0.1i− 0.2j + 1.4k,a12 = 0.8 + 0.6i + 0.1j + 0.001k, a21 = 0.3 +

0.3i − 0.12j + 0.4k, a22 = 4.9 − 0.32i + 0j + 0.25k,R1 = 0.9 + 0.8i + 0.5j + 0.4k, R2 = 1.7 − 2.1i +

0 2 4 6 8 10 12 14 16 18 20t

10(a)8

6420

−10−8−6−4−2hR

(t)

−4

−2

0 1

2

4

0 1

0 2 4 6 8 10 12 14 16 18 20t

10

5

0

−10

(b)

−5

hI(t)

−4

−2

0 1

2

4

0 1

0 2 4 6 8 10 12 14 16 18 20t

15

10

5

0

−15

−10

−5

hJ(t)

−4

−2

0 1

2

4

0 1

3

−3

(c)

0 2 4 6 8 10 12 14 16 18 20t

10

5

0

−10

−5

hK(t)

−4

−2

0 1

2

4

0 1

(d)

Fig. 1 Trajectories of state variables hR(t) (a), hI(t)

(b), hJ(t) (c), and hK(t) (d) in Example 1 with α =

0.9 and tk = 0.02k

0.4j + 1.4k, c1 = c2 = 5.5, ϑν1ν21 = ϑν1ν22 = 0.01

and ν1, ν2 ∈ {R, I, J,K}. Calculation shows that in-equality (7) in Theorem 1 is satisfied with p = 1, 2

and q = 1, 2. Consequently, QVNN (27) is Mittag-Leffler stable. Fig. 2 shows that the trajectories ofQVNN (27) are Mittag-Leffler stable with differentinitial values.

5 Conclusions

In this study, we investigated the Mittag-Lefflerstability analysis of multiple equilibrium pointsfor fractional-order QVNNs with an impulse term.By employing the non-commutative property ofquaternion multiplication, QVNNs were convertedinto four RVNNs. According to the definition of theactivation functions, the existence of equilibriumpoints was also analyzed. Sufficient conditions were

Udhayakumar et al. / Front Inform Technol Electron Eng 2020 21(2):234-246 245

0 1 2 3 4 5 6 7 8 9 10t

1086420

−10−8−6−4−2h

R(t)

340 0.5

56

0 0.5−6

−4−3

−5

(a)

0 1 2 3 4 5 6 7 8 9 10t

10

5

0

−10

−5

hI(t)

−6

−3

0 0.5

3

6

0 0.545

−5−4

(b)

0 1 2 3 4 5 6 7 8 9 10t

15

10

5

0

−15

−10

−5hJ (t)

−6

−3

0 0.5

3

6

0 0.5

5

−4−5

4

(c)

0 1 2 3 4 5 6 7 8 9 10t

1086420

−10−8−6−4−2h

K(t)

340 0.5

56

0 0.5−6

−4−3

−5

(d)

Fig. 2 Trajectories of state variables hR(t) (a), hI(t)

(b), hJ(t) (c), and hK(t) (d) in Example 2 with α =

0.7 and tk = 0.02k

derived to assure the Mittag-Leffler stability ofmultiple equilibrium points for QVNNs. Simulationresults validated our theoretical solutions.

Contributors

K. UDHAYAKUMAR and R. RAKKIYAPPAN

designed the research. K. UDHAYAKUMAR drafted the

manuscript. Jin-de CAO helped organize the manuscript

and gave some suggestions to improve the results. Jin-

de CAO and Xue-gang TAN revised and finalized the paper.

Compliance with ethics guidelines

K. UDHAYAKUMAR, R. RAKKIYAPPAN, Jin-de

CAO, and Xue-gang TAN declare that they have no con-

flict of interest.

ReferencesAbdurahman A, Jiang HJ, Teng ZD, 2015. Finite-time syn-

chronization for memristor-based neural networks withtime-varying delays. Neur Netw, 69:20-28.https://doi.org/10.1016/j.neunet.2015.04.015

Cao JD, Xiao M, 2007. Stability and Hopf bifurcation in asimplified BAM neural network with two time delays.IEEE Trans Neur Netw, 18(2):416-430.https://doi.org/10.1109/TNN.2006.886358

Chen JJ, Zeng ZG, Jiang P, 2014. Global Mittag-Leffler stability and synchronization of memristor-basedfractional-order neural networks. Neur Netw, 51:1-8.https://doi.org/10.1016/j.neunet.2013.11.016

Chen XF, Song QK, Li ZS, et al., 2017. Stability analysisof continuous-time and discrete-time quaternion-valuedneural networks with linear threshold neurons. IEEETrans Neur Netw Learn Syst, 29(7):2769-2781.https://doi.org/10.1109/TNNLS.2017.2704286

Hu J, Zeng CN, Tan J, 2017. Boundedness and periodicity forlinear threshold discrete-time quaternion-valued neuralnetwork with time-delays. Neurocomputing, 267:417-425. https://doi.org/10.1016/j.neucom.2017.06.047

Huang Y, Zhang H, Wang Z, 2012. Multistability and multi-periodicity of delayed bidirectional associative memoryneural networks with discontinuous activation functions.Appl Math Comput, 219(3):899-910.https://doi.org/10.1016/j.amc.2012.06.068

Huang YJ, Li CH, 2019. Backward bifurcation and stabilityanalysis of a network-based SIS epidemic model withsaturated treatment function. Phys A, 527:121407.https://doi.org/10.1016/j.physa.2019.121407

Khan H, Gómez-Aguilar J, Khan A, et al., 2019. Stabilityanalysis for fractional order advection–reaction diffusionsystem. Phys A, 521:737-751.https://doi.org/10.1016/j.physa.2019.01.102

Kilbas AA, Srivastava HM, Trujillo JJ, 2006. Theory andApplications of Fractional Differential Equations. Else-vier, Amsterdam, the Netherlands.

Li N, Zheng WX, 2020. Passivity analysis for quaternion-valued memristor-based neural networks with time-varying delay. IEEE Trans Neur Netw Learn Syst,31(2):39-650.https://doi.org/10.1109/TNNLS.2019.2908755

Li X, Ho DWC, Cao JD, 2019. Finite-time stability andsettling-time estimation of nonlinear impulsive systems.Automatica, 99:361-368.https://doi.org/10.1016/j.automatica.2018.10.024

Li XD, Ding YH, 2017. Razumikhin-type theorems for time-delay systems with persistent impulses. Syst ContrLett, 107:22-27.https://doi.org/10.1016/j.sysconle.2017.06.007

Li XD, Wu JH, 2016. Stability of nonlinear differentialsystems with state-dependent delayed impulses. Auto-matica, 64:63-69.https://doi.org/10.1016/j.automatica.2015.10.002

Li XD, Zhang XL, Song SL, 2017. Effect of delayed impulseson input-to-state stability of nonlinear systems. Auto-matica, 76:378-382.https://doi.org/10.1016/j.automatica.2016.08.009

Liu P, Zeng Z, Wang J, 2017. Multiple Mittag-Leffler stabilityof fractional-order recurrent neural networks. IEEETrans Syst Man Cybern Syst, 47(8):2279-2288.https://doi.org/10.1109/TSMC.2017.2651059

246 Udhayakumar et al. / Front Inform Technol Electron Eng 2020 21(2):234-246

Liu P, Zeng Z, Wang J, 2018. Multistability of recurrentneural networks with nonmonotonic activation functionsand unbounded time-varying delays. IEEE Trans NeurNetw Learn Syst, 29(7):3000-3010.https://doi.org/10.1109/TNNLS.2017.2710299

Liu Y, Zhang D, Lu J, 2017. Global exponential stability forquaternion-valued recurrent neural networks with time-varying delays. Nonl Dynam, 87(1):553-565.https://doi.org/10.1007/s11071-016-3060-2

Liu Y, Zhang D, Lou J, et al., 2018. Stability analysis ofquaternion-valued neural networks: decomposition anddirect approaches. IEEE Trans Neur Netw Learn Syst,29(9):4201-4211.https://doi.org/10.1109/TNNLS.2017.2755697

Nie XB, Liang JL, Cao JD, 2019. Multistability analysisof competitive neural networks with Gaussian-wavelet-type activation functions and unbounded time-varyingdelays. Appl Math Comput, 356:449-468.https://doi.org/10.1016/j.amc.2019.03.026

Pang DH, Jiang W, Liu S, et al., 2019. Stability analysis fora single degree of freedom fractional oscillator. Phys A,523:498-506.https://doi.org/10.1016/j.physa.2019.02.016

Podlubny I, 1998. Fractional Differential Equations: anIntroduction to Fractional Derivatives, Fractional Dif-ferential Equations, to Methods of Their Solution andSome of Their Applications. Academic Press, SanDiego, USA.

Popa CA, Kaslik E, 2018. Multistability and multiperiodicityin impulsive hybrid quaternion-valued neural networkswith mixed delays. Neur Netw, 99:1-18.https://doi.org/10.1016/j.neunet.2017.12.006

Qi XN, Bao HB, Cao JD, 2019. Exponential input-to-statestability of quaternion-valued neural networks with timedelay. Appl Math Comput, 358:382-393.https://doi.org/10.1016/j.amc.2019.04.045

Rakkiyappan R, Velmurugan G, Cao JD, 2014. Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with time de-lays. Nonl Dynam, 78(4):2823-2836.https://doi.org/10.1007/s11071-014-1628-2

Rakkiyappan R, Cao JD, Velmurugan G, 2015a. Ex-istence and uniform stability analysis of fractional-order complex-valued neural networks with time delays.IEEE Trans Neur Netw Learn Syst, 26(1):84-97.https://doi.org/10.1109/TNNLS.2014.2311099

Rakkiyappan R, Velmurugan G, Cao J, 2015b. Stabilityanalysis of fractional-order complex-valued neural net-works with time delays. Chaos Sol Fract, 78:297-316.https://doi.org/10.1016/j.chaos.2015.08.003

Rakkiyappan R, Velmurugan G, Rihan FA, et al., 2016. Sta-bility analysis of memristor-based complex-valued re-current neural networks with time delays. Complexity,21(4):14-39. https://doi.org/10.1002/cplx.21618

Schauder J, 1930. Der fixpunktsatz in funktionalraümen.Stud Math, 2:171-180.

Song QK, Chen XF, 2018. Multistability analysis ofquaternion-valued neural networks with time delays.IEEE Trans Neur Netw Learn Syst, 29(1):5430-5440.https://doi.org/10.1109/TNNLS.2018.2801297

Song QK, Yan H, Zhao ZJ, et al., 2016a. Global exponential

stability of complex-valued neural networks with bothtime-varying delays and impulsive effects. Neur Netw,79:108-116.https://doi.org/10.1016/j.neunet.2016.03.007

Song QK, Yan H, Zhao ZJ, et al., 2016b. Global exponentialstability of impulsive complex-valued neural networkswith both asynchronous time-varying and continuouslydistributed delays. Neur Netw, 81:1-10.https://doi.org/10.1016/j.neunet.2016.04.012

Stamova I, 2014. Global Mittag-Leffler stability and synchro-nization of impulsive fractional-order neural networkswith time-varying delays. Nonl Dynam, 77(4):1251-1260. https://doi.org/10.1007/s11071-014-1375-4

Tyagi S, Abbas S, Hafayed M, 2016. Global Mittag-Lefflerstability of complex valued fractional-order neural net-work with discrete and distributed delays. Rend CircolMatem PalermoSer 2, 65(3):485-505.https://doi.org/10.1007/s12215-016-0248-8

Wang F, Yang YQ, Hu MF, 2015. Asymptotic stability ofdelayed fractional-order neural networks with impulsiveeffects. Neurocomputing, 154:239-244.https://doi.org/10.1016/j.neucom.2014.11.068

Wang H, Yu Y, Wen G, et al., 2015. Global stability analysisof fractional-order Hopfield neural networks with timedelay. Neurocomputing, 154:15-23.https://doi.org/10.1016/j.neucom.2014.12.031

Wang JJ, Jia YF, 2019. Analysis on bifurcation and stabilityof a generalized Gray-Scott chemical reaction model.Phys A, 528:121394.https://doi.org/10.1016/j.physa.2019.121394

Wang LM, Song QK, Liu YR, et al., 2017. Global asymptoticstability of impulsive fractional-order complex-valuedneural networks with time delay. Neurocomputing, 243:49-59. https://doi.org/10.1016/j.neucom.2017.02.086

Wu AL, Zeng ZG, 2017. Global Mittag-Leffler stabilizationof fractional-order memristive neural networks. IEEETrans Neur Netw Learn Syst, 28(1):206-217.https://doi.org/10.1109/TNNLS.2015.2506738

Yang XJ, Li CD, Song QK, et al., 2018. Global Mittag-Lefflerstability and synchronization analysis of fractional-orderquaternion-valued neural networks with linear thresholdneurons. Neur Netw, 105:88-103.https://doi.org/10.1016/j.neunet.2018.04.015

Zeng ZG, Zheng WX, 2012. Multistability of neural networkswith time-varying delays and concave-convex character-istics. IEEE Trans Neur Netw Learn Syst, 23(2):293-305. https://doi.org/10.1109/TNNLS.2011.2179311

Zeng ZG, Huang TW, Zheng WX, 2010. Multistability ofrecurrent neural networks with time-varying delays andthe piecewise linear activation function. IEEE TransNeur Netw, 21(8):1371-1377.https://doi.org/10.1109/TNN.2010.2054106

Zhang FH, Zeng ZG, 2018. Multistability and instabilityanalysis of recurrent neural networks with time-varyingdelays. Neur Netw, 97:116-126.https://doi.org/10.1016/j.neunet.2017.09.013

Zhang XX, Niu PF, Ma YP, et al., 2017. Global Mittag-Leffler stability analysis of fractional-order impulsiveneural networks with one-side Lipschitz condition. NeurNetw, 94:67-75.https://doi.org/10.1016/j.neunet.2017.06.010