Multistability analysis of delayed quaternion-valued neural networks with nonmonotonic piecewise nonlinear activation functions

Applied Mathematics and Computation 341 (2019) 229–255

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Multistability analysis of delayed quaternion-valued neural networks with nonmonotonic piecewise nonlinear activation functions Manchun Tan∗, Yunfeng Liu, Desheng Xu College of Information Science and Technology, Jinan University, Guangzhou 510632, China

a r t i c l e

i n f o

Keywords: Quaternion-valued neural networks Multistability μ-stability Nonmonotonic piecewise nonlinear activation functions Unbounded time delays

a b s t r a c t This paper deals with the multistability problem of the quaternion-valued neural networks (QVNNs) with nonmonotonic piecewise nonlinear activation functions and unbounded time-varying delays. By virtue of the non-commutativity of quaternion multiplication resulting from Hamilton rules, the QVNNs can be separated into four real-valued systems. By using the fixed point theorem and other analytical tools, some novel algebraic criteria are established to guarantee that the QVNNs can have 54n equilibrium points, 34n of which are locally μ-stable. Some criteria that guarantee the multiple exponential stability, multiple power stability, multiple log-stability, multiple log–log-stability are also derived as special cases. The obtained results reveal that the introduced QVNNs in this paper can have larger storage capacity than the complex-valued ones. Finally, one numerical example is presented to clarify the validity of the theoretical results. © 2018 Elsevier Inc. All rights reserved.

1. Introduction In 1843, based on the long-term study on the complex numbers, the Irish mathematician W. R. Hamilton proposed the concept of quaternion. However, the quaternion didn’t cause much attention for quite a long time, because quaternion multiplication does not satisfy the commutative law. Quaternion has got a revival since the 1990s, primarily due to its utility in describing spatial rotations [1–5]. The quaternion, which is a significant achievement in algebra, has recently received increasing attention for its effective applications in adaptive filtering [6], computer graphics [7], robotics [8], array processing [9], color image processing [10], etc. There is no doubt that the quaternion gradually shows its advantages in practical applications. Over the past few decades, the investigation on dynamic behaviors of neural networks has gained in popularity and plentiful results have been obtained for their useful applications in associative memory, pattern recognition, image processing, and so on (see [11–14] and references therein). As is well-known, the two-dimensional data can be well processed by complex-valued neural networks (CVNNs) or many real-valued neural networks (RVNNs). Whereas if the data is three or four dimensional, such as color images, body images, 4-D signals, the quaternion-valued neural networks (QVNNs), which directly encode in terms of quaternions [15], can be more efficient than CVNNs or RVNNs. Moreover, in practical applications

∗

Corresponding author. E-mail address: [email protected] (M. Tan).

https://doi.org/10.1016/j.amc.2018.08.033 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

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M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

such as spatial rotation, color night vision, color image compression, and so on [16–19], QVNNs can have unique advance in efficient information processing. Therefore, it is necessary and important to investigate the QVNNs. As an extension of CVNNs, QVNNs with quaternion-valued states, connections weights and activation functions have gradually aroused researchers’ interests, and many significative results have been obtained in recent years [20–25]. The authors studied the discrete-time QVNNs with linear threshold activation functions in [20]. By virtue of characteristic equation, Lyapunov functional and M-matrix, some sufficient conditions to guarantee the boundedness and global exponential periodicity of the QVNNs were derived. In [21,22], by separating the proposed QVNNs into four real-valued systems, the global exponential stability of the delayed QVNNs was investigated. In [23,24], by separating QVNNs into two complex-valued neural networks, the global μ-stability of the delayed QVNNs was studied by constructing Lyapunov-Krasovskii functional. In [25], by virtue of Homeomorphic mapping theorem, Lyapunov theorem and inequality technique, the robust stability of QVNNs with time delays and parameter uncertainties was studied. As we all know, the number of equilibrium points comes to play a crucial role in the practical applications of neural networks. For example, in optimization application, the networks are required to have only one monostable equilibrium point (see [26–30] and references therein). Nevertheless, some other applications require neural networks to have multistable equilibrium points, such as pattern recognition, digital selection, decision making and associative memory (see [11,31,32] and references therein). Therefore, it is of prime importance to investigate the multistability problem of the neural networks. For RVNNs or CVNNs, numerous results on the multistability have been investigated in [33–47]. In [33], by constructing parameter conditions and using inequality technique, it was verified that the delayed RVNNs with two classes of activation functions can have 3n equilibria and 2n of them are locally exponentially stable. In [35], it was proved that the n-dimensional RVNNs with nondecreasing piecewise linear activation functions with 2r corner points can have (2r + 1 )n equilibria and (r + 1 )n of them are locally exponentially stable. In [39], the multistability problem of CVNNs with real-imaginary-type activation functions was studied. It was found that the n-dimensional CVNNs can have [(2α + 1 )(2β + 1 )]n (α , β ≥ 1 ) equilibria and [(α + 1 )(β + 1 )]n equilibria of them are locally exponentially stable. In [40], for a n-neuron delayed CVNNs with nondecreasing activation functions, sufficient criteria were achieved to assure the existence of 9n equilibria and 4n of them were locally exponentially stable. It is obvious that the number of the equilibria of CVNNs is much more than the ones of RVNNs under the same dimensions. This means that the storage capacity of the CVNNs is larger, which is valuable when the neural networks are applied in associative memory applications. So far, the multistability of the QVNNs hardly has gained attention, which drives us to develop the present research. It is commonly known that the multistability analysis of neural networks heavily depends on the activation functions. Therefore, choosing the proper activation functions is particularly important for the QVNNs. In many previous literature, the activation functions of neural networks are nondecreasing and continuous [33–40,48]. Besides, some studies have obtained for the neural networks with discontinuous activations [42–46]. Heretofore, the multistability for QVNNs with nonmonotonic piecewise nonlinear activation functions has not been considered in the literature, which drives us to study the present research from another motivation. In practical applications, time delays are inevitable factors in neural networks because of the finite switching speed of amplifiers and the interneurons conduction distances. The existence of time delays may become a source of instability or oscillation for the systems. Thus, the dynamics of the neural networks with time delays should be taken into account [49–53]. Besides, many studies about the chaos which can have an impact on inducing instability of systems have been investigated in [54–56] and references therein. In most previous literatures, the multistability analysis of the neural networks were developed with the bounded delays. As pointed in [57,58], time delays may be unbounded in some practical applications, which means that the entire history may affect the present. Therefore, it is significant to study the dynamics of neural networks with unbounded delays. Recently, much attention has been devoted into multiple μ-stability analysis of neural networks with unbounded time-varying delays [59–61], which is a generalization of multiple exponential stability, multiple power stability, multiple log-stability and multiple log–log-stability. To best of the authors’ knowledge, there are few results proposed previously on the multiple μ-stability problem of the QVNNs. Motivated by the above discussions, we address the problem of the multiple μ-stability analysis for the QVNNs with nonmonotonic piecewise nonlinear activation functions and unbounded time-varying delays in this paper. Some novel algebraic criteria are established to guarantee that the QVNNs can have 54n equilibrium points, and 34n of them are locally μ-stability. The novelties and contributions of the paper can be summarized as follows. (1) This paper extends the multiple μ-stability results of RVNNs and CVNNs to QVNNs, which can obtain even more equilibria compared with the previous studies and can achieve larger storage capacity in practical applications. (2) The obtained results in this paper are generalizations of the multiple exponential stability, multiple power stability, multiple log-stability, multiple log–log-stability. (3) The activation functions used in this paper are an extension to the ones used in [59]. The remaining of this paper is organized as follows. In Section 2, model description of the QVNNs and preliminaries are presented. In Section 3, the existence of stability of multiple equilibria is investigated for the QVNNs with unbounded time-varying delays and nonmonotonic piecewise nonlinear activation functions. In Section 4, one numerical simulation is given to clarify the theory and distinct dynamical behaviors for different activation functions. Finally, conclusions are drawn in Section 5. Notations: The notations are quite standard. Throughout this paper, Rm×n , Cm×n , Qm×n denote, respectively, the set of m × n real-valued, complex-valued and quaternion-valued matrices. Function h = hR (t ) + ihI (t ) + jhJ (t ) + khK (t ) denotes quaternion-valued function, where hR (t), hI (t), hJ (t), hK (t) are all real-valued functions and i, j, k obey Hamilton rules:

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

231

ij = −ji = k, jk = −kj = i, ki = −ik = j, i2 = j2 = k2 = ijk = −1. C ((−∞, 0], Rn ) represents the Banach space of continuous vector-valued functions which map the internal (−∞, 0] into Rn with  the topology of uniform convergence. For a vector  z = (z1 , z2 , . . . , zn )T ∈ Rn , ||z||ξ denotes the norm of z with ||z||ξ = max p ξ p |z p | , where ξ = (ξ1 , ξ2 , . . . , ξn )T with ξ p > 0 for p = 1, 2, . . . , n. ‘D− ’ denotes the upper left Dini derivative operator. 2. Problem description and preliminaries Consider the following QVNNs with unbounded time-varying delays n 

h˙ p (t ) = −d p h p (t ) +

a pq f q (hq (t ) ) +

q=1

n 

b pq fq (hq (t − τ pq (t ) ) ) + u p ,

(1)

q=1

where p = 1, 2, . . . , n, h p (t ) ∈ Q is the state of the pth neuron at time t. D = diag(d1 , d2 , . . . , dn ) ∈ Rn×n with dp > 0 is the self-feedback connection weight matrix, A = (a pq )n×n ∈ Qn×n and B = (b pq )n×n ∈ Qn×n are, respectively, the connection weight matrices and the delayed connection weight matrix, where p, q = 1, 2, . . . , n. f p (· ) ∈ Q denotes the activation function. τ pq (t) ≥ 0 represents the time-varying transmission delay. u = (u1 , u2 , . . . , un ) ∈ Qn×1 is the external input vector. The initial value can be expressed as

h p (θ ) =

φ p (θ ) ∈ Q, θ ∈ (−∞, 0], φ p (θ ) = φ pR (θ ) + iφ pI (θ ) + jφ pJ (θ ) + kφ pK (θ ), where An

φ pR (θ ), φ pI (θ ), φ pJ (θ ), φ pK (θ ) ∈ C ((−∞, 0], Rn ), p = 1, 2, . . . , n. equilibrium point h∗ = (h∗1 , h∗2 , . . . , h∗n )T of system (1) satisfies −d p h p ∗ +

n 

∗

a pq f q (hq ) +

q=1

Assumption 1. Let h =

n 

(2) that for p = 1, 2, . . . , n,

∗

b pq fq (hq ) + u p = 0.

q=1

hR

+

ihI

+ jhJ + khK , hR , hI , hJ , hK ∈ R. fp (h) can be described by

f p (h ) = f pR (hR (t )) + i f pI (hI (t )) + j f pJ (hJ (t )) + k f pK (hK (t )), where f pl (· ) : R → R are continuous functions with p = 1, 2, . . . , n and l = R, I, J, K, which are defined as follows:

⎧ l f p,1 (hl ), ⎪ ⎪ ⎪ ⎨ f p,l 2 (hl ), l l l l f p (h ) = f p, 3 (h ), ⎪ l l ⎪ ⎪ ⎩ f p,l 4 (hl ), f p,5 (h ), lim f pl (hl ) = mlp ,

hl →−∞

−∞ < hl < r lp , r lp ≤ hl ≤ slp , slp < hl < t pl , t pl ≤ hl ≤ vlp , vlp < hl < +∞, lim f pl (hl ) = Mlp ,

hl →+∞

l+ l ( vl ) > where mlp < Mlp are constants; r lp , slp , t pl , vlp are constants with −∞ < r lp < slp < t pl < vlp < +∞, f p, (vl ) = f p, 4 p 5 p l+ l (sl ), f l− (r l ) = f l (r l ) = f l− (t l ) = f l (t l ); f l− (· ) and f l+ (· ) denote the left and the right limits. f p, (sl ) = f p, p p 2 p p,2 p p,4 p 3 p p,1 p p,3 p l (· ), f l (· ), f l (· ) and f l (· ) are continuous and monotonically increasing functions, and f l (· ) are Assumption 2. f p, 1 p,2 p,4 p,5 p,3

continuous and monotonically decreasing functions, namely, there exist constants μlp(1 ) , μlp(2 ) , μlp(3 ) , μlp(4 ) , μlp(5 ) , ηlp(1 ) , ηlp(2 ) ,

ηlp(3) , ηlp(4) , ηlp(5) such that

l l l  f p, 1 (h ) − f p,1 (h ) l

0 ≤ μlp(1) ≤

hl − h

l

l l l  f p, 2 (h ) − f p,2 (h )

≤ ηlp(1) ,

∀hl , hl ∈ (−∞, rlp ),

≤ ηlp(2) ,

∀hl , hl ∈ [rlp , slp],

≥ ηlp(3) ,

∀hl , hl ∈ (slp , t pl ),

≤ ηlp(4) ,

∀hl , hl ∈ [t pl , vlp],

≤ ηlp(5) ,

∀hl , hl ∈ (vlp , +∞),

l

0 < μlp(2) ≤

hl

l − h

l l l  f p, 3 (h ) − f p,3 (h ) l

0 > μlp(3) ≥

hl − h

l

l l l  f p, 4 (h ) − f p,4 (h ) l

0 < μlp(4) ≤

hl

l − h

l l l  f p, 5 (h ) − f p,5 (h ) l

0 ≤ μlp(5) ≤

hl − h

l

where l = R, I, J, K and p = 1, 2, . . . , n.

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M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

Remark 1. As is known to all, the type of activation functions plays an important role in the designing of neural networks. Because of the complexity of real world, too strong restriction on the activation functions will limit the ability of the artificial neural networks to model the engineering problem. In most literature, the activation functions are restricted to be either nondecreasing or piecewise linear. For example, the activation functions in [33–39,61] are all monotonically increasing. Recently, some results on the multistability analysis of neural networks with nonmonotonic activation functions were obtained [41,42,59]. According to the Assumptions 1 and 2, the nonmonotonic piecewise nonlinear activation functions employed in this paper are more general than the ones used in [42,59], which are nonlinear in any open interval and nonmonotonic in (−∞, +∞ ). Definition 1. Let h∗ = (h∗1 , h∗2 , . . . , h∗n ) be the equilibrium point of neural networks (1) and h(t ) = (h1 (t ),h2 (t ), . . . , hn (t )) be an arbitrary solution of (1). Suppose that μ(t) is a positive continuous function and satisfies μ(t ) → +∞ as t → +∞. The equilibrium point h∗ is said to be μ-stable, if there exist scalars M > 0 and T > 0 such that

|h p (t ) − h∗p | <

M

μ(t )

,

t ≥ T , p = 1, 2, . . . , n.

By specifying the function μ(t), we can derive the following definitions from Definition 1. Definition 2. The equilibrium point h∗ of neural networks (1) is said to be power stable if there exist scalars M > 0, ε > 0 and T ≥ 0 such that

|h p (t ) − h∗p | <

M , tε

t ≥ T.

Definition 3. The equilibrium point h∗ of neural networks (1) is said to be exponentially stable if there exist scalars M > 0, ε > 0 and T ≥ 0 such that

|h p (t ) − h∗p | <

M , eε t

t ≥ T.

Based on Assumption 1, it is obtained from (1) that

h˙ p (t ) = h˙ Rp (t ) + ih˙ Ip (t ) + jh˙ Jp (t ) + kh˙ Kp (t ) = −d p h p (t ) +

n 

a pq f q (hq (t ) ) +

q=1

n 

b pq fq (hq (t − τ pq (t ) ) ) + u p

q=1

= −d p (hRp (t ) + ihIp (t ) + jhJp (t ) + khKp (t )) +

n 

aRpq + iaIpq + jaJpq + kaKpq



fqR (hRq (t )) + i fqI (hIq (t )) + j fqJ (hJq (t )) + k fqK (hKq (t ))



q=1

+

n 

bRpq + ibIpq + jbJpq + kbKpq



q=1





× fqR (hRq (t − τ pq (t ) ) ) + i fqI (hIq (t − τ pq (t ) ) ) + j fqJ (hJq (t − τ pq (t ) ) ) + k fqK (hKq (t − τ pq (t ) ) + (uRp + iuIp + juJp + kuKp ), where p, q = 1, 2, . . . , n, and

a pq = aRpq + iaIpq + jaJpq + kaKpq , b pq = bRpq + ibIpq + jbJpq + kbKpq , u p = uRp + iuIp + juJp + kuKp . By virtue of the non-commutativity of quaternion multiplication resulting from Hamilton rules: ij = −ji = k, jk = −kj = i, ki = −ik = j, i2 = j2 = k2 = ijk = −1, we can derive from above equation that

h˙ p (t ) = h˙ Rp (t ) + ih˙ Ip (t ) + jh˙ Jp (t ) + kh˙ Kp (t )



= −d p hRp (t ) + ihIp (t ) + jhJp (t ) + khKp (t ) +

n  q=1

aRpq + iaIpq + jaJpq + kaKpq





fqR (hRq (t )) + ifq (hIq (t )) + j fqJ (hJq (t )) + k fqK (hKq (t )) I



M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

+

n 

bRpq + ibIpq + jbJpq + kbKpq



fqR (hRq (t − τ pq (t ) ) ) + i fqI (hIq (t − τ pq (t ) ) )

q=1





+ j fqJ (hJq (t − τ pq (t ) ) ) + k fqK (hKq (t − τ pq (t ) ) + uRp + iuIp + juJp + kuKp

− d p hRp (t ) +

=

n 

233



aRpq fqR (hRq (t )) − aIpq fqI (hIq (t )) − aJpq fqJ (hJq (t )) − aKpq fqK (hKq (t ))



q=1

+

n 

bRpq fqR (hRq (t − τ pq (t ))) − bIpq fqI (hIq (t − τ pq (t )))

q=1





− bJpq fqJ (hJq (t − τ pq (t ))) − bKpq fqK (hKq (t − τ pq (t ))) + uRp

− d p hIp (t ) +

+i

n 

aRpq fqI (hIq (t )) + aIpq fqR (hRq (t )) + aJpq fqK (hKq (t )) − aKpq fqJ (hJq (t ))



q=1

+

n 

bRpq fqI (hIq (t − τ pq (t ))) + bIpq fqR (hRq (t − τ pq (t )))

q=1





+ bJpq fqK (hKq (t − τ pq (t ))) − bKpq fqJ (hJq (t − τ pq (t ))) + uIp

+j

− d p hJp (t ) +

n 

aRpq fqJ (hJq (t )) + aJpq fqR (hRq (t )) − aIpq fqK (hKq (t )) + aKpq fqI (hIq (t ))



q=1

+

n 

bRpq fqJ (hJq (t − τ pq (t ))) + bJpq fqR (hRq (t − τ pq (t )))

q=1





− bIpq fqK (hKq (t − τ pq (t ))) + bKpq fqI (hIq (t − τ pq (t ))) + uJp

+k

− d p hKp (t ) +

n 

aRpq fqK (hKq (t )) + aKpq fqR (hRq (t )) + aIpq fqJ (hJq (t )) − aJpq fqI (hIq (t ))



q=1

+

n 



bRpq fqK (hKq (t − τ pq (t ))) + bKpq fqR (hRq (t − τ pq (t )))

q=1





+ bIpq fqJ (hJq (t − τ pq (t ))) − bJpq fqI (hIq (t − τ pq (t ))) + uKp . Therefore, the four equations are established as follows:

h˙ Rp (t ) = −d p hRp (t ) +

n 

aRpq fqR (hRq (t )) − aIpq fqI (hIq (t )) − aJpq fqJ (hJq (t )) − aKpq fqK (hKq (t ))



q=1

+

n 

bRpq fqR (hRq (t − τ pq (t ))) − bIpq fqI (hIq (t − τ pq (t )))

q=1



− bJpq fqJ (hJq (t − τ pq (t ))) − bKpq fqK (hKq (t − τ pq (t ))) + uRp ,

h˙ Ip (t ) = −d p hIp (t ) +

n 

aRpq fqI (hIq (t )) + aIpq fqR (hRq (t )) + aJpq fqK (hKq (t )) − aKpq fqJ (hJq (t ))

(3a)



q=1

+

n  q=1

bRpq fqI (hIq (t − τ pq (t ))) + bIpq fqR (hRq (t − τ pq (t )))



+ bJpq fqK (hKq (t − τ pq (t ))) − bKpq fqJ (hJq (t − τ pq (t ))) + uIp ,

(3b)

234

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

h˙ Jp (t ) = −d p hJp (t ) +

n 

aRpq fqJ (hJq (t )) + aJpq fqR (hRq (t )) − aIpq fqK (hKq (t )) + aKpq fqI (hIq (t ))



q=1

+

n 

bRpq fqJ (hJq (t − τ pq (t ))) + bJpq fqR (hRq (t − τ pq (t )))

q=1



− bIpq fqK (hKq (t − τ pq (t ))) + bKpq fqI (hIq (t − τ pq (t ))) + uJp ,

h˙ Kp (t ) = −d p hKp (t ) +

n 

(3c)

aRpq fqK (hKq (t )) + aKpq fqR (hRq (t )) + aIpq fqJ (hJq (t )) − aJpq fqI (hIq (t ))



q=1

+

n 

bRpq fqK (hKq (t − τ pq (t ))) + bKpq fqR (hRq (t − τ pq (t )))

q=1



+ bIpq fqJ (hJq (t − τ pq (t ))) − bJpq fqI (hIq (t − τ pq (t ))) + uKp .

(3d)

Next, define the following continuous real-valued functions, p = 1, 2, . . . , n: n 

R

F p (η ) = −d p η + aRpp f pR (η ) + uRp +





n 





n 





n 

max mRq aRpq , MqR aRpq +

q=1,q = p

−

n 



n 





n 

q=1

−

n 

min mJq aJpq , MqJ aJpq −

q=1

min mIq bIpq , MqI bIpq −

q=1

n 

min mJq bJpq , MqJ bJpq −

−





min mRq aRpq , MqR aRpq +



n 





n 

q=1

−

n  q=1

I



n 





n 

max mJq aJpq , MqJ aJpq − max mJq bJpq , MqJ bJpq −





max mKq aJpq , MqK aJpq +



max mIq aRpq , MqI aRpq +

+

n 





max mIq bRpq , MqI bRpq +









min mJq aKpq , MqJ aKpq −

q=1

F Ip (η ) = −d p η + aRpp f pI (η ) + uIp +

n 





min mIq aRpq , MqI aRpq +





n 





n 

min mKq aJpq , MqK aJpq +

q=1

+

n  q=1





max mRq aIpq , MqR aIpq

q=1

n 

n 

n 

(4b)





max mRq bIpq , MqR bIpq







min mJq bKpq , MqJ bKpq ,

(5a)



min mRq aIpq , MqR aIpq



q=1









min mIq bRpq , MqI bRpq +

q=1

min mKq bJpq , MqK bJpq −



max mKq bKpq , MqK bKpq ,

q=1

q=1,q = p

+



q=1

n 

q=1

n 

n 

q=1

max mKq bJpq , MqK bJpq −



max mKq aKpq , MqK aKpq

q=1

n 

q=1



q=1

q=1,q = p

+



(4a)

q=1



n 



min mRq bRpq , MqR bRpq



q=1

F p (η ) = −d p η + aRpp f pI (η ) + uIp + n 

n 

q=1

max mIq bIpq , MqI bIpq −



min mKq bKpq , MqK bKpq ,

q=1



max mIq aIpq , MqI aIpq −



q=1

q=1,q = p n 



min mKq aKpq , MqK aKpq

q=1

q=1

F Rp (η ) = −d p η + aRpp f pR (η ) + uRp +



q=1



min mIq aIpq , MqI aIpq −



max mRq bRpq , MqR bRpq

n 









min mRq bIpq , MqR bIpq

q=1

max mJq aKpq , MqJ aKpq −

n  q=1

max mJq bKpq , MqJ bKpq ,

(5b)

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255 n 

J

F p (η ) = −d p η + aRpp f pJ (η ) + uJp +





n 

max mJq aRpq , MqJ aRpq +

q=1,q = p

+

n 



n 





n 

q=1

+

n 









max mJq bRpq , MqJ bRpq +

q=1

max mIq bKpq , MqI bKpq −

q=1

n 

+



n 





n 

q=1

+

n  q=1









n 

n 

K

min mJq bRpq , MqJ bRpq +

+



+









max mKq aRpq , MqK aRpq +



n 

n 

max mKq bRpq , MqK bRpq +

q=1

n 



min mIq aJpq , MqI aJpq −

+





n 





n 

min mJq aIpq , MqJ aIpq +

q=1

+

n  q=1











max mRq aKpq , MqR aKpq

n 

n 









n 





n 

min mKq aRpq , MqK aRpq + min mKq bRpq , MqK bRpq +

max mRq bKpq , MqR bKpq

















min mIq bJpq , MqI bJpq ,

n 

min mRq aKpq , MqR aKpq

min mRq bKpq , MqR bKpq

max mIq aJpq , MqI aJpq −

max mIq bJpq , MqI bJpq ,

q=1

where η ∈ R. In terms of Assumption 1, we obtain that R

lim F (η ) = +∞, η→−∞ p

lim F (η ) = −∞; η →+∞ p

J

lim F (η ) = +∞, η→−∞ p

lim F (η ) = −∞. η →+∞ p

lim F (η ) = +∞, η→−∞ p

lim F (η ) = −∞; η →+∞ p

lim F (η ) = +∞, η→−∞ p

lim F (η ) = −∞; η →+∞ p

J

I

K

I

K

Finally, for the convenience of description, we denote that



α = (hR1 , hR2 , . . . , hRn , hI1 , hI2 , . . . , hIn , hJ1 , hJ2 , . . . , hJn , hK1 , hK2 , . . . , hKn ) ∈ R4n | hRp ∈ Rp(α p ) , I (αn+ p )

hIp ∈ p

J (α2n+ p )

, hJp ∈ p

(7a)

q=1

q=1

R

(6b)

q=1

q=1

min mJq bIpq , MqJ bIpq −



max mKq bIpq , MqK bIpq ,

q=1

q=1,q = p n 



q=1



q=1

F Kp (η ) = −d p η + aRpp f pK (η ) + uKp +



q=1



q=1



q=1

n 

max mJq bIpq , MqJ bIpq −



min mRq bJpq , MqR bJpq

n 

max mKq aIpq , MqK aIpq −



q=1 n 

(6a)

q=1





max mJq aIpq , MqJ aIpq +



min mRq aJpq , MqR aJpq

n 

q=1,q = p n 



min mKq bIpq , MqK bIpq ,

q=1

q=1

F p (η ) = −d p η + aRpp f pK (η ) + uKp +



q=1

q=1

min mIq bKpq , MqI bKpq −



max mRq bJpq , MqR bJpq

n 

min mKq aIpq , MqK aIpq −

min mJq aRpq , MqJ aRpq +



min mIq aKpq , MqI aKpq +



q=1

q=1,q = p n 

n 

q=1

F Jp (η ) = −d p η + aRpp f pJ (η ) + uJp +



max mRq aJpq , MqR aJpq

q=1



max mIq aKpq , MqI aKpq +

235

K (α3n+ p )

, hKp ∈ p

,



for p = 1, 2, . . . , n ,

where α = (α1 , α2 , . . . , α4n ), α p = 1 or 2 or 3 or 4 or 5 for p = 1, 2, . . . , 4n and

     

lp(1) = hl ∈ R | − ∞ < hl < rlp , lp(2) = hl ∈ R | rlp ≤ hl ≤ slp , lp(3) = hl ∈ R | slp < hl < t pl ,    

lp(4) = hl ∈ R, | t pl ≤ hl ≤ vlp , lp(5) = hl ∈ R | vlp < hl < +∞ ,

(7b)

236

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

where l = R, I, J, K; p = 1, 2, . . . , n. Similarly, set

 α = (hR1 , hR2 , . . . , hRn , hI1 , hI2 , . . . , hIn , hJ1 , hJ2 , . . . , hJn , hK1 , hK2 , . . . , hKn ) ∈ R4n | hRp ∈ Rp(1) or Rp(3) or Rp(5) ,

hIp ∈ Ip(1) or Ip(3) or Ip(5) ,

hJp ∈ Jp(1) or Jp(3) or Jp(5) ,

hKp ∈ Kp (1) or Kp (3) or Kp (5)



for p = 1, 2, . . . , n .

It is obvious that in Qn , there are, respectively, 54n and 34n such kind of regions as α and α .

3. Main results In this section, based on the fixed point theory and the geometrical properties of the activation functions, several sufficient conditions are established to guarantee the existence of 54n equilibrium points, 34n of which are locally μ-stable.

3.1. Coexistence analysis of the equilibrium points for QVNNs Firstly, the coexistence of multiple equilibrium points for systems (3a)–(3d) is investigated under the fixed point theory. R

I

K

J

J

For convenience, the functions G p (· ), GRp (· ), G p (· ), GIp (· ), G p (· ), GKp (· ), G p (· ) and G p (· ) are defined below: R

G p (η ) = −d p η + (aRpp + bRpp ) f pR (η ) +

n 





max mRq (aRpq + bRpq ), MqR (aRpq + bRpq )

q=1,q = p

−

n 





min mIq (aIpq + bIpq ), MqI (aIpq + bIpq ) −

n 

q=1

−

n 





min mJq (aJpq + bJpq ), MqI (aJpq + bJpq )

q=1



 min mKq (aKpq + bKpq ), MqK (aKpq + bKpq ) + uRp ,

(8a)

q=1

GRp (η ) = −d p η + (aRpp + bRpp ) f pR (η ) +

n 





min mRq (aRpq + bRpq ), MqR (aRpq + bRpq )

q=1,q = p

−

n 





max mIq (aIpq + bIpq ), MqI (aIpq + bIpq ) −

q=1

−

n 



n 



max mJq (aJpq + bJpq ), MqI (aJpq + bJpq )

q=1





max mKq (aKpq + bKpq ), MqK (aKpq + bKpq ) + uRp ,

(8b)

q=1

I

G p (η ) = −d p η + (aRpp + bRpp ) f pI (η ) +

n 





max mIq (aRpq + bRpq ), MqI (aRpq + bRpq )

q=1,q = p

+

n 





max mRq (aIpq + bIpq ), MqR (aIpq + bIpq ) +

q=1

−

n 

n 





max mKq (aJpq + bJpq ), MqK (aJpq + bJpq )

q=1



 min mJq (aKpq + bKpq ), MqJ (aKpq + bKpq ) + uIp ,

(9a)

q=1

GIp (η ) = −d p η + (aRpp + bRpp ) f pI (η ) +

n 





min mIq (aRpq + bRpq ), MqI (aRpq + bRpq )

q=1,q = p

+

n 





min mRq (aIpq + bIpq ), MqR (aIpq + bIpq ) +

q=1

−

n  q=1

n 





min mKq (aJpq + bJpq ), MqK (aJpq + bJpq )

q=1





max mJq (aKpq + bKpq ), MqJ (aKpq + bKpq ) + uIp ,

(9b)

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255 n 

J

G p (η ) = −d p η + (aRpp + bRpp ) f pJ (η ) +



237



max mJq (aRpq + bRpq ), MqJ (aRpq + bRpq )

q=1,q = p

+



n 



max mRq (aJpq + bJpq ), MqR (aJpq + bJpq ) +

q=1

−



n 



max mIq (aKpq + bKpq ), MqI (aKpq + bKpq )

q=1



 min mKq (aIpq + bIpq ), MqK (aIpq + bIpq ) + uJp ,

n 

(10a)

q=1 n 

GJp (η ) = −d p η + (aRpp + bRpp ) f pJ (η ) +





min mJq (aRpq + bRpq ), MqJ (aRpq + bRpq )

q=1,q = p

+



n 



min mRq (aJpq + bJpq ), MqR (aJpq + bJpq ) +

q=1

−

n 





min mIq (aKpq + bKpq ), MqI (aKpq + bKpq )

q=1



 max mKq (aIpq + bIpq ), MqK (aIpq + bIpq ) + uJp ,

n 

(10b)

q=1



n 

K

G p (η ) = −d p η + (aRpp + bRpp ) f pK (η ) +



max mKq (aRpq + bRpq ), MqK (aRpq + bRpq )

q=1,q = p

+

n 





max mRq (aKpq + bKpq ), MqR (aKpq + bKpq ) +

q=1

−

n 





max mJq (aIpq + bIpq ), MqJ (aIpq + bIpq )

q=1

n 



 min mIq (aJpq + bJpq ), MqI (aJpq + bJpq ) + uKp ,

(11a)

q=1



n 

GKp (η ) = −d p η + (aRpp + bRpp ) f pK (η ) +



min mKq (aRpq + bRpq ), MqK (aRpq + bRpq )

q=1,q = p

+

n 





min mRq (aKpq + bKpq ), MqR (aKpq + bKpq ) +

q=1

−

n 





min mJq (aIpq + bIpq ), MqJ (aIpq + bIpq )

q=1

n 





max mIq (aJpq + bJpq ), MqI (aJpq + bJpq ) + uKp ,

(11b)

q=1

where η ∈ R. Theorem 1. Under Assumption 1, for any given region α , there exists at least one equilibrium point for systems (3a)–(3d) located just in α if R

G p (r Rp ) < 0, J Gp

( ) < 0, r Jp

GRp (sRp ) > 0, GJp

( ) > 0, sJp

GRp (vRp ) > 0; GJp

( v ) > 0; J p

I

G p (r Ip ) < 0,

K Gp

( ) < 0, r Kp

GIp (sIp ) > 0, GKp

GIp (vIp ) > 0;

( ) > 0, sKp

GKp (vKp ) > 0;

(12)

where p = 1, 2, . . . , n. Proof. To facilitate the discussions, the following four more functions are given: n 

GRp (η ) = −d p η + (aRpp + bRpp ) f pR (η ) +

(aRpq + bRpq ) fqR (hRq ) −

q=1,q = p

−

n  q=1

(aJpq + bJpq ) fqJ (hJq ) −

n 

q=1

q=1

(13)

q=1 n 

(aRpq + bRpq ) fqI (hIq ) +

q=1,q = p n 

(aIpq + bIpq ) fqI (hIq )

(aKpq + bKpq ) fqK (hKq ) + uRp ,

GIp (η ) = −d p η + (aRpp + bRpp ) f pI (η ) + +

n 

(aJpq + bJpq ) fqK (hKq ) −

n  q=1

(aKpq + bKpq ) fqJ (hJq ) + uIp ,

n 

(aIpq + bIpq ) fqR (hRq )

q=1

(14)

238

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255 n 

GJp (η ) = −d p η + (aRpp + bRpp ) f pJ (η ) +

(aRpq + bRpq ) fqJ (hJq ) +

q=1,q = p

+

n 

n 

(aKpq + bKpq ) fqI (hIq ) −

q=1

n  q=1

(aIpq + bIpq ) fqK (hKq ) + uJp ,

(15)

q=1 n 

GKp (η ) = −d p η + (aRpp + bRpp ) f pK (η ) +

n 

(aRpq + bRpq ) fqK (hKq ) +

q=1,q = p

+

(aJpq + bJpq ) fqR (hRq )

n 

(aIpq + bIpq ) fqJ (hJq ) −

n 

q=1

(aKpq + bKpq ) fqR (hRq )

q=1

(aJpq + bJpq ) fqI (hIq ) + uKp ,

(16)

q=1

where η, hRq , hIq , hq , hKq ∈ R for q = 1, 2, . . . , n. It follows obviously from Assumption 1 that J

R

R

I

I

J

J

K

K

F Rp (η ) ≤ GRp (η ) ≤ GRp (η ) ≤ G p (η ) ≤ F p (η ),

(17a)

F Ip (η ) ≤ GIp (η ) ≤ GIp (η ) ≤ G p (η ) ≤ F p (η ),

(17b)

F Jp (η ) ≤ GJp (η ) ≤ GJp (η ) ≤ G p (η ) ≤ F p (η ),

(17c)

F Kp (η ) ≤ GKp (η ) ≤ GKp (η ) ≤ G p (η ) ≤ F p (η ).

(17d)

J J J choose (hR ; hI ; hJ ; hK ) = (hR1 , hR2 , . . . , hRn , hI1 , hI2 , . . . , hIn , h1 , h2 , . . . , hn , hK1 , hK2 , . . . , hKn )T ∈ R4n and substitute R p = 1, 2, . . . , n, the function G p (· ) is continuous with respect to η. Furthermore, it follows from f pR (r Rp ) =

Arbitrarily (13)–(16). For and (12) as well as (17a) that R

GRp (t pR ) < GRp (r Rp ) < G p (r Rp ) < 0,

GRp (sRp ) > GRp (sRp ) > 0,

it into f pR (t pR )

GRp (vRp ) > GRp (vRp ) > 0. R

According to the above discussions and Intermediate Value theorem, it is obvious that there exists h p ∈ Rp(1 ) or Rp(2 ) R

or Rp(3 ) or Rp(4 ) or Rp(5 ) satisfying GRp (h p ) = 0 for p = 1, 2, . . . , n. By the similar discussion, it shows that there respecI tively exist h p ∈ Ip(1 ) J J satisfying G p (h p ) = 0

I

J

J (1 )

or Ip(2 ) or Ip(3 ) or Ip(4 ) or Ip(5 ) satisfying GIp (h p ) = 0, h p ∈ p K hp

Kp (1)

Kp (2)

Kp (3)

Kp (4)

Kp (5)

and ∈ or or or or satisfying Then, for any given region α , we define a mapping H: α → α by

J (2 )

or p

K GKp (h p )

J (3 )

or p

J (4 )

or p

J (5 )

or p

= 0 for p = 1, 2, . . . , n.

H (hR1 , hR2 , . . . , hRn , hI1 , hI2 , . . . , hIn , hJ1 , hJ2 , . . . , hJn , hK1 , hK2 , . . . , hKn ) R

R

R

I

I

I

J

J

J

K

K

K

= (h1 , h2 , . . . , hn , h1 , h2 , · · · , hn , h1 , h2 , . . . , hn , h1 , h2 , . . . , hn ). It is easy to see that the mapping H is continuous. By applying Brouwer’s fixed point theorem, there exists one fixed ∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

J∗

J∗

J∗

∗

∗

∗

∗

R

∗

I

J∗

J

point (hR ; hI ; hJ ; hK ) = (hR1 , hR2 , . . . , hRn , hI1 , hI2 , . . . , hIn , h1 , h2 , . . . , hn , hK1 , hK2 ,. . . , hKn )T with hRp = h p , hIp = h p , h p = h p K

∗

and hKp = h p ( p = 1, 2, . . . , n ) located in α , which is also the equilibrium point of system (1) located in α . Due to the arbitrariness of α , systems (3a)–(3d) (or equivalently, system (1)) under Assumption 1 can have 54n equilibrium points located in Qn . The proof is complete. 3.2. Stability analysis of the equilibrium points In this section, several sufficient conditions proposed to ensure that the system (1) (or equivalently, systems (3a)–(3d)) with unbounded time-varying delays can have 54n equilibrium points, among which 34n are locally μ-stable under the Assumptions 1 and 2. Lemma 1. Suppose that the following conditions R

F p (r Rp ) < 0, F Rp (sRp ) > 0, F Rp (vRp ) > 0; K Fp

(rKp ) < 0, F Kp (sKp ) > 0, F Kp (vKp ) > 0;

I

F p (r Ip ) < 0, F Ip (sIp ) > 0, F Ip (vIp ) > 0; J

F p (r Jp ) < 0, F Jp (sJp ) > 0, F Jp (vJp ) > 0

hold for all p = 1, 2, . . . , n. The system (1) has at least 54n equilibrium and the region α is positively invariant.

(18)

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

239

Proof. It is easy to see that condition (18) implies the condition (12) in Theorem 1 holds. Based on Theorem 1, the existence of at least 54n equilibrium points for system (1) can be achieved under the condition (18). Then, we will prove that the set ˜ ⊂ α , any solution h(t) of system (1) with initial condition (2), h(t) will stay α is positively invariant. That is, for each  ˜ in  for all t ≥ 0. Without loss of generality, we suppose that the initial condition for system (1) is given by

h p (θ ) = hRp (θ ) + ihIp (θ ) + jhJp (θ ) + khKp (θ ) =

φ pR (θ ) + iφ pI (θ ) + jφ pJ (θ ) + kφ pK (θ ), θ ∈ (−∞, 0],

(19)

where p = 1, 2, . . . , n; hRp (θ ) = φ pR (θ ) ∈ (−∞, r Rp ), hIp (θ ) = φ pI (θ ) ∈ (sIp , t pI ), h p (θ ) = φ p (θ ) ∈ (s p , t p ) and hKp (θ ) = φ pK (θ ) ∈ (vKp , +∞ ). Then one can derive that for all t > 0, the solution (hR ; hI ; hJ ; hK ) of system (1) would stay in J

−∞ < hRp (t ) < r Rp , sJp < hJp (t ) < t pJ ,

J

J

J

sIp < hIp (t ) < t pI ,

vKp < hKp (t ) < +∞;

p = 1, 2, . . . , n.

(20)

We achieve this conclusion by contradiction. Suppose that (20) does not hold, then there exist an index p1 ∈ {1, 2, . . . , n} and a time point t∗ > 0 such that for all p = 1, 2, . . . , n :

−∞ < hRp (t ) < r Rp , sIp < hIp (t ) < t pI , sJp < hJp (t ) < t pJ , vKp < hKp (t ) < +∞;

(21)

t ∈ [0, t ∗ )

and one of the following cases holds:

hRp1 (t ∗ ) = r Rp1

and

h˙ Rp1 (t ∗ ) ≥ 0,

(22a)

hIp1 (t ∗ ) = sIp1

and h˙ Is1 (t ∗ ) ≤ 0,

(22b)

hIp1 (t ∗ ) = t pI 1

and h˙ tI1 (t ∗ ) ≥ 0,

(22c)

hJp1 (t ∗ ) = sJp1

and h˙ Jp1 (t ∗ ) ≤ 0,

(22d)

hJp1 (t ∗ ) = t pJ 1

and h˙ Jp1 (t ∗ ) ≥ 0,

(22e)

hKp1 (t ∗ ) = vKp1

and h˙ Kp1 (t ∗ ) ≤ 0.

(22f)

If (22a) is true, it follows from (3a), Assumption 1 and (18) that

h˙ Rp1 (t ∗ ) = −d p1 hRp1 (t ∗ ) +

n 

aRp1 q fqR (hRq (t ∗ )) − aIp1 q fqI (hIq (t ∗ )) − aJp1 q fqJ (hJq (t ∗ )) − aKp1 q fqK (hKq (t ∗ ))



q=1

+

n 

bRp1 q fqR (hRq (t ∗ − τ p1 q (t ∗ ))) − bIp1 q fqI (hIq (t ∗ − τ p1 q (t ∗ )))

q=1



− bJp1 q fqJ (hJq (t ∗ − τ p1 q (t ∗ ))) − bKp1 q fqK (hKq (t ∗ − τ p1 q (t ∗ ))) + uRp1 ≤ −d p1 r Rp1 + aRp1 p1 f pR1 (r Rp1 ) + uRp1 +

n 





max mRq aRp1 q , MqR aRp1 q +

q=1,q = p1

−

n 





min mIq aIp1 q , MqI aIp1 q −

q=1

−

n 

=







min mIq bIp1 q , MqI bIp1 q −

(rRp1 ) < 0,

which contradicts h˙ Rp1 (t ∗ ) ≥ 0 shown in (22a).

n  q=1



max mRq bRp1 q , MqR bRp1 q

q=1



min mJq aJp1 q , MqJ aJp1 q −

q=1

q=1 R F p1

n 

n 

n 









min mKq aKp1 q , MqK aKp1 q

q=1





min mJq bJp1 q , MqJ bJp1 q −

n  q=1

min mKq bKp1 q , MqK bKp1 q



240

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

If (22b) is true, it follows from (3b), Assumption 1 and (18) that

h˙ Ip1 (t ∗ ) = − d p1 hIp1 (t ∗ ) +

n 

aRp1 q fqI (hIq (t ∗ )) + aIp1 q fqR (hRq (t )) + aJp1 q fqK (hKq (t ∗ )) − aKp1 q fqJ (hJq (t ∗ ))



q=1

+

n 

bRp1 q fqI (hIq (t ∗ − τ p1 q (t ∗ ))) + bIp1 q fqR (hRq (t ∗ − τ p1 q (t ∗ )))

q=1



+bJp1 q fqK (hKq (t ∗ − τ p1 q (t ∗ ))) − bKp1 q fqJ (hJq (t ∗ − τ p1 q (t ∗ ))) + uIp



n 

≥ − d p1 sIp1 + aRp1 p1 f pI 1 (sIp1 ) + uIp1 +



min mIq aRp1 q , MqI aRp1 q +

q=1,q = p1

+

n 





n 





n 

min mKq aJp1 q , MqK aJp1 q +

q=1

+

n  q=1



min mRq aIp1 q , MqR aIp1 q



q=1









n 

min mIq bRp1 q , MqI bRp1 q +

q=1

min mKq bJp1 q , MqK bJp1 q −

n 









min mRq bIp1 q , MqR bIp1 q

q=1

max mJq aKp1 q , MqJ aKp1 q −

q=1

n 

max mJq bKp1 q , MqJ bKp1 q

q=1

= F Ip1 (sIp1 ) > 0, which contradicts h˙ Ip1 (t ∗ ) ≤ 0 shown in (22b). If (22e) is true, it follows from (3c), Assumption 1 and (18) that

h˙ Jp1 (t ∗ ) = − d p1 hJp1 (t ∗ ) +

n 

aRp1 q fqJ (hJq (t ∗ )) + aJp1 q fqR (hRq (t )) + aKp1 q fqI (hIq (t ∗ )) − aIp1 q fqK (hKq (t ∗ ))



q=1

+

n 

bRp1 q fqJ (hJq (t ∗ − τ p1 q (t ∗ ))) + bJp1 q fqR (hRq (t ∗ − τ p1 q (t ∗ )))

q=1



+bKp1 q fqI (hIq (t ∗ − τ p1 q (t ∗ ))) − bIp1 q fqK (hKq (t ∗ − τ p1 q (t ∗ ))) + uJp



n 

≥ − d p1 t pJ 1 + aRp1 p1 f pJ 1 (t pJ 1 ) + uJp1 +



min mJq aRp1 q , MqJ aRp1 q +

q=1,q = p1

+

n 





min mIq aKp1 q , MqI aKp1 q +

q=1

+

n 





min mIq bKp1 q , MqI bKp1 q −

n 



min mJq bRp1 q , MqJ bRp1 q +

(

)<

J F p1

(

r Jp1

n 



min mRq bJp1 q , MqR bJp1 q







max mKq aIp1 q , MqK aIp1 q −



n 

max mKq bIp1 q , MqK bIp1 q



q=1

) < 0,

J which contradicts h˙ p1 (t ∗ ) ≥ 0 shown in (22e). If (22f) is true, it follows from (3d), Assumption 1 and (18) that

h˙ Kp1 (t ∗ ) = − d p1 hKp1 (t ∗ ) +



q=1

q=1

t pJ 1

min mRq aJp1 q , MqR aJp1 q

q=1



q=1

q=1 J = F p1

n 



n 

n 

aRp1 q fqK (hKq (t ∗ )) + aKp1 q fqR (hRq (t )) + aIp1 q fqJ (hJq (t ∗ )) − aJp1 q fqI (hIq (t ∗ ))



q=1

+

n 

bRp1 q fqK (hKq (t ∗ − τ p1 q (t ∗ ))) + bKp1 q fqR (hRq (t ∗ − τ p1 q (t ∗ )))

q=1



+bIp1 q fqJ (hJq (t ∗ − τ p1 q (t ∗ ))) − bJp1 q fqI (hIq (t ∗ − τ p1 q (t ∗ ))) + uIp



n 

≥ − d p1 vKp1 + aRp1 p1 f pK1 (vKp1 ) + uKp1 +



min mKq aRp1 q , MqK aRp1 q +

q=1,q = p1

+

n 





min mJq aIp1 q , MqJ aIp1 q +

q=1

+

n 







min mJq bIp1 q , MqJ bIp1 q −

( v ) > 0, K p1

n  q=1



min mRq aKp1 q , MqR aKp1 q

q=1



min mKq bRp1 q , MqK bRp1 q +

q=1

q=1

= F kp1

n 

n 

n 









min mRq bKp1 q , MqR bKp1 q

q=1





max mIq aJp1 q , MqI aJp1 q −

n  q=1

max mIq bJp1 q , MqI bJp1 q



M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

241

which contradicts h˙ Kp1 (t ∗ ) ≤ 0 shown in (22f). Similar discussions derive that cases (22c) and (22d) do not hold. Up to now, we have proved that the set



(hR1 , hR2 , . . . , hRn , hI1 , hI2 , . . . , hIn , hJ1 , hJ2 , . . . , hJn , hK1 , hK2 , . . . , hKn ) ∈ R4n | hRp ∈ Rp(1) , hIp ∈ Ip(3) , hJp ∈ Jp(3) , hKp ∈ Kp (5)

for p = 1, 2, . . . , n



is positively invariant. By means of similar analysis, one can conclude that for any set α , it is also positively invariant. The proof is complete. Lemma 2. Assume that (18) holds and the nondecreasing function μ(t) > 0 with

lim

t→+∞

μ(t ) = +∞, 0 ≤ sup t≥T0

μ˙ (t ) μ(t ) ≤ α , sup ≤ 1 + β, μ(t ) t≥T0 μ (t − τ (t ) )

(23)

where α , β , T0 are nonnegative constants. Under the Assumptions 1 and 2, the 34n equilibria located in α for QVNNs (1) with J unbounded time-varying delays are locally μ-stable, if there are also positive constants ξ pR , ξ pI , ξ p , ξ pK ( p = 1, 2, . . . , n ) such that

 −1   R −1  R  R(1)   R −1  R  R(3)   R −1  R  R(5) (−d p + α ) ξ pR + ξq a pq ηq − ξq a pq ηq + ξq a pq ηq +

q∈N1I

−

q∈N1R

q∈N3R

q∈N5R

  −1   I (3 )   −1   I (5 )   J −1  J  J (1 ) −1   ξqI aIpq ηqI(1) − ξqI aIpq ηq + ξqI aIpq ηq + ξq a pq ηq



q∈N3I

q∈N5I

q∈N5J

q∈N1K

q∈N1J

  J −1  J  J (5 )   −1   K (1 )   −1   K (3 ) −1   ξqJ aJpq ηqJ (3) + ξq a pq ηq + ξqK aKpq ηq − ξqK aKpq ηq



q∈N3J

q∈N3K

   −1     −1   K (5 )   −1   R(3 ) + ξqK aKpq ηq + (1 + β ) ξqR bRpq ηqR(1) − ξqR bRpq ηq q∈N5K

+

q∈N5R

+



q∈N3R

q∈N1I

q∈N3I

  J −1  J  J (5 )   J −1  J  J (5 ) −1   ξqJ bJpq ηqJ (1) − ξq b pq ηq + ξq b pq ηq

 q∈N1J

+

q∈N1R

  −1   I (1 )   −1   I (3 )   −1   I (5 ) −1   ξqR bRpq ηqR(5) + ξqI bIpq ηq − ξqI bIpq ηq + ξqI bIpq ηq



q∈N3J

q∈N5I

q∈N3J

 K −1  K  K (1)   K −1  K  K (3)   K −1  K  K (5)  < 0, ξq b pq ηq − ξq b pq ηq + ξq b pq ηq

q∈N1K

q∈N3K

(24a)

q∈N5K

 −1   I −1  R  I(1)   I −1  R  I(3)   I −1  R  I(5) (−d p + α ) ξ pI + ξq a pq ηq − ξq a pq ηq + ξq a pq ηq +

q∈N1R

−

q∈N1I

q∈N3I

q∈N5I

  −1   R(3 )   −1   R(5 )   −1  J  K (1 ) −1   ξqR aIpq ηqR(1) − ξqR aIpq ηq + ξqR aIpq ηq + ξqK a pq ηq

 

q∈N3R

q∈N5R

q∈N1K

 K −1  J  K (3)   K −1  J  K (5)   J −1  K  J (1)   J −1  K  J (3) ξq a pq ηq + ξq a pq ηq + ξq a pq ηq − ξq a pq ηq

q∈N3K

q∈N5K

q∈N1J

q∈N3J

   −1     J −1   J (5 )   −1   I (3 ) + ξq aKpq ηq + (1 + β ) ξqI bRpq ηqI(1) − ξqI bRpq ηq q∈N5J

+

q∈N5I

+



q∈N1K

+

q∈N1I

q∈N1R

q∈N3R

 J −1  J  K (1)   K −1  J  K (3)   K −1  J  K (5) ξq b pq ηq − ξq b pq ηq + ξq b pq ηq q∈N3K

q∈N5R

q∈N5K

  J −1   J (3 )   J −1   J (5 )  −1   < 0, ξqJ bKpq ηqJ (1) − ξq bKpq ηq + ξq bKpq ηq



q∈N1J

q∈N3I

  −1   R(1 )   −1   R(3 )   −1   R(5 ) −1   ξqI bRpq ηqI(5) + ξqR bIpq ηq − ξqR bIpq ηq + ξqR bIpq ηq



q∈N3J

q∈N5J

(24b)

242

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

 −1   J −1  R  J (1)   J −1  R  J (3)   J −1  R  J (5) (−d p + α ) ξ pJ + ξq a pq ηq − ξq a pq ηq + ξq a pq ηq +

 q∈N1R

−



q∈N1J

q∈N3R



q∈N5I

+



q∈N5J

+



q∈N5R

q∈N1K

q∈N5K

q∈N1I

q∈N3I

   −1     J −1   J (3 ) −1   ξqI aKpq ηqI(5) + (1 + β ) ξqJ bRpq ηqJ (1) − ξq bRpq ηq q∈N1J

q∈N3J

  −1  J  R(1 )   −1  J  R(3 )   −1  J  R(5 ) −1   ξqJ bRpq ηqJ (5) + ξqR b pq ηq − ξqR b pq ηq + ξqR b pq ηq q∈N1R

q∈N3R

 K −1  I  K (1)   K −1  I  K (3)   K −1  I  K (5) ξq b pq ηq − ξq b pq ηq + ξq b pq ηq

q∈N1K

+

q∈N5J

 K −1  I  K (3)   K −1  I  K (5)   I −1  K  I(1)   I −1  K  I(3) ξq a pq ηq + ξq a pq ηq + ξq a pq ηq − ξq a pq ηq

q∈N3K

+

q∈N3J

  −1  J  R(3 )   −1  J  R(5 )   −1   K (1 ) −1   ξqR aJpq ηqR(1) − ξqR a pq ηq + ξqR a pq ηq + ξqK aIpq ηq

q∈N3K



ξqI

q∈N5R

q∈N5K

−1  K  I(1)   K −1  I  K (3)   K −1  I  K (5)  b pq ηq − < 0, ξq b pq ηq + ξq b pq ηq

q∈N1I

q∈N3K

(24c)

q∈N5K

 −1   K −1  R  K (1)   K −1  R  K (3)   K −1  R  K (5) (−d p + α ) ξ pK + ξq a pq ηq − ξq a pq ηq + ξq a pq ηq +

 q∈N1R

−



+

q∈N3R

q∈N5R

q∈N1J

q∈N1I

q∈N5J

q∈N3I

   −1     −1   K (3 ) −1   ξqI aJpq ηqI(5) + (1 + β ) ξqK bRpq ηqK (1) − ξqK bRpq ηq q∈N1K

q∈N3K

 K −1  R  K (5)   R −1  K  R(1)   R −1  K  R(3)   R −1  K  R(5) ξq b pq ηq + ξq b pq ηq − ξq b pq ηq + ξq b pq ηq

q∈N5K

q∈N1R

q∈N3R

q∈N5R

  J −1   J (3 )   J −1   J (5 ) −1   ξqJ bIpq ηqJ (1) − ξq bIpq ηq + ξq bIpq ηq

 q∈N1J

+

q∈N5K

  J −1   J (5 )   −1  J  I (1 )   −1  J  I (3 ) −1   ξqJ aIpq ηqJ (3) + ξq aIpq ηq + ξqI a pq ηq − ξqI a pq ηq



q∈N5I

+

q∈N3K



q∈N3J

+

q∈N1K

  −1   R(3 )   −1   R(5 )   J −1   J (1 ) −1   ξqR aKpq ηqR(1) − ξqR aKpq ηq + ξqR aKpq ηq + ξq aIpq ηq



ξqI

q∈N3J

q∈N5J

q∈N3I

q∈N5I

−1  J  I(1)   I −1  J  I(3)   I −1  J  I(5)  b pq ηq − < 0 ξq b pq ηq + ξq b pq ηq

q∈N1I





(24d)



hold for p = 1, 2, . . . , n, where N1l = q | hlq (t ) ∈ lq(1 ) , q = 1, 2, . . . , n , N3l = q | hlq (t ) ∈ lq(3 ) , q = 1, 2, . . . , n



q | hlq (t ) ∈

lq(5) ,





and N5l =

q = 1, 2, . . . , n for l = R, I, J, K.

Proof. For any given α , according to Lemma 1 we know that the system (1) has an equilibrium point h∗ = ∗ ∗ ∗ ∗ (h∗1 , h∗2 , . . . , h∗n )T with h∗p = hRp + ihIp + jhJp + khKp located in α for p = 1, 2, . . . , n. Let (hR (t); hI (t); hJ (t); hK (t))=

(hR1 (t ), . . . , hRn (t ), hI1 (t ), . . . , hIn (t ), hJ1 (t ), . . . , hJn (t ), hK1 (t ), . . . , hKn (t ))T be a solution of systems (3a)–(3d) with initial condition (hR (s ); hI (s ); hJ (s ); hK (s )) ∈ α (s ∈ (−∞, 0] ), it follows Lemma 1 that (hR (t); hI (t); hJ (t); hK (t)) will stay in α for all t ≥ 0. ∗

˜ R (t ) = hR (t ) − hR∗ , h ˜ R (t ) = hI (t ) − hI∗ , h ˜ J (t ) = hJ (t ) − hJ and h ˜ K (t ) = hK (t ) − hK ∗ , then systems (3a)–(3d) can be Let h p p p p p p p p p p p I transformed into:

     n n n  ∗ ∗ ∗ ˜˙ R (t ) = −d p h ˜ R (t ) + h aRpq fqR (hRq (t )) − fqR (hRq ) − aIpq fqI (hIq (t )) − fqI (hIq ) − aJpq fqJ (hJq (t )) − fqJ (hJq ) p p −

n 

n  q=1

q=1 ∗



aKpq fqK (hKq (t )) − fqK (hKq ) +

q=1

−





n 



q=1

bRpq fqR (hRq (t − τ pq (t ))) − fqR (hRq ) −

q=1 ∗

∗





bJpq fqJ (hJq (t − τ pq (t ))) − fqJ (hJq ) −

n  q=1



q=1 n  q=1 ∗



∗

bIpq fqI (hIq (t − τ pq (t ))) − fqI (hIq )





bKpq fqK (hKq (t − τ pq (t ))) − fqK (hKq ) ,

(25a)

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255







  ∗ ∗ ˜˙ I (t ) = −d p h ˜ I (t ) + h aRpq fqI (hIq (t )) − fqI (hIq ) + aIpq fqR (hRq (t )) − fqR (hRq ) p p n

+

n  q=1

+

n  q=1

+

n 



q=1 ∗

n

q=1



aJpq fqK (hKq (t )) − fqK (hKq ) −

n 



∗

aKpq fqJ (hJq (t )) − fqJ (hJq )

q=1



∗



n 

bRpq fqI (hIq (t − τ pq (t ))) − fqI (hIq ) +



∗

bJpq fqK (hKq (t − τ pq (t ))) − fqK (hKq ) −

q=1





∗

n 



q=1

+

n  q=1

−

n 









q=1 ∗



n

q=1

aIpq fqK (hKq (t )) − fqK (hKq ) +

n 



∗

aKpq fqI (hIq (t )) − fqI (hIq )

q=1



∗



bRpq fqJ (hJq (t − τ pq (t ))) − fqJ (hJq ) +



∗

n 



bIpq fqK (hKq (t − τ pq (t ))) − fqK (hKq ) +





∗

n 



q=1

+

q=1

+

n 



∗

(25c)

q=1

n

n 



bKpq fqI (hIq (t − τ pq (t ))) − fqI (hIq ) ,







  ∗ ∗ ˜˙ K (t ) = −d p h ˜ K (t ) + h aRpq fqK (hKq (t )) − fqK (hKq ) + aKpq fqR (hRq (t )) − fqR (hRq ) p p +

(25b)



bJpq fqR (hRq (t − τ pq (t ))) − fqR (hRq )

q=1

q=1

n 



∗

q=1

n

−



bKpq fqJ (hJq (t − τ pq (t ))) − fqJ (hJq ) ,

  J ∗ ∗ ˜˙ Jp (t ) = −d p h ˜ Jp (t ) + h aRpq fqJ (hJq (t )) − fqJ (hJq ) + a pq fqR (hRq (t )) − fqR (hRq ) n 



bIpq fqR (hRq (t − τ pq (t ))) − fqR (hRq )

q=1



243



q=1 ∗



aIpq fqJ (hJq (t )) − fqJ (hJq ) −

n 

q=1



∗

aJpq fqI (hIq (t )) − fqI (hIq )

q=1



n

∗



bRpq fqK (hKq (t − τ pq (t ))) − fqK (hKq ) +



∗



bIpq fqJ (hJq (t − τ pq (t ))) − fqJ (hJq ) −

q=1

n  q=1

n 







∗

bKpq fqR (hRq (t − τ pq (t ))) − fqR (hRq )



∗





bJpq fqI (hIq (t − τ pq (t ))) − fqI (hIq ) .

(25d)

q=1

˜ R (t ), H I (t ) = μ(t )h ˜ I (t ), H J (t ) = μ(t )h ˜ J (t ), H K (t ) = μ(t )h ˜ K (t ) and H (t ) = (H R (t );HI (t); HJ (t); HK (t)) Let H pR (t ) = μ(t )h p p p p p p p J

J

with H R (t ) = (H1R (t ), . . . , HnR (t ))T , H I (t ) = (H1I (t ), . . . , HnI (t ))T , H J (t ) = (H1 (t ), . . . ,Hn (t ))T and H K (t ) = (H1K (t ), . . . , HnK (t ))T . Define

M(t ) = sup H (s ) ξ , s≤t

t≥T

(26)



where H (s ) ξ = max ||H R (s )||ξ R , ||H I (s )||ξ I , ||H J (s )||ξ J , ||H K (s )||ξ K





max ξqI |H pI (s )| , ||H J (s )||ξ J = max

1≤q≤n

1≤q≤n







with ||H R (s )||ξ R = max ξqR |HqR (s )| , ||H I (s )||ξ I =

    ξqJ |HqJ (s )| , ||H K (s )||ξ K = max ξqK |HqK (s )| .

1≤q≤n

1≤q≤n

We claim that M(t ) is bounded. According to the definition of M(t ), it is obvious that ||H (t )||ξ ≤ M(t ) for all t ≥ T. If at some special time point t that ||H (t )||ξ = M(t ), then there are four possible cases: Case 1. There exists an index pR = pR (t ) depending on t such that

||H(t )||ξ = ||H R (t )||ξ R = ξ pRR |HpRR (t )|.

(27)

Case 2. There exists an index pI = pI (t ) depending on t such that

||H(t )||ξ = ||H I (t )||ξ I = ξ pI I |HpI I (t )|.

(28)

244

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

Case 3. There exists an index pJ = pJ (t ) depending on t such that

||H(t )||ξ = ||H J (t )||ξ J = ξ pJ J |HpJ J (t )|.

(29)

Case 4. There exists an index pK = pK (t ) depending on t such that

||H(t )||ξ = ||H K (t )||ξ K = ξ pKK |HpKK (t )|.

(30)

For case 1, from (25a) and Assumption 1 we know that

(ξ pRR )−1 D− ||H (t )||ξ = D− |HpRR (t )|

˜ RR (t )) μ˙ (t ) h ˜ RR (t ) + sign(h ˜ RR (t )) μ(t ) = sign(h p p p +

n 



q=1

−

n 



+



n 



n 

∗

aIpR q fqI (hIq (t )) − fqI (hIq )

q=1





∗

∗



bRpR q fqR (hRq (t − τ pR q (t ))) − fqR (hRq ) −

n 

bJpR q



fqJ

( (t − τ pR q (t ))) − hJq

fqJ

q=1

(



aKpR q fqK (hKq (t )) − fqK (hKq )

q=1



q=1

−

∗

aJpR q fqJ (hJq (t )) − fqJ (hJq ) −

q=1 n 



∗

aRpR q fqR (hRq (t )) − fqR (hRq ) −

˜ R (t ) − d pR h p

∗ hJq



) −

n 



∗

bIpR q fqI (hIq (t − τ pR q (t ))) − fqI (hIq )

q=1 n 



bKpR q



fqK

( (t − τ pR q (t ))) − hKq

q=1

μ˙ (t )  R    R  R(1)  R    R  R(3)  R  H pR (t ) + a pR q ηq Hq (t ) − a pR q ηq Hq (t ) μ(t ) q∈N1R q∈N3R      I  I (1 )  I    I  I (3 )  I   aRpR q ηqR(5) HqR (t ) + a pR q ηq Hq (t ) − a pR q ηq Hq (t ) +

≤ −d pR +

q∈N5R

q∈N1I

q∈N3I

     J  J (1 )  J    J  J (3 )  J   aIpR q ηqI(5) HqI (t ) + a R ηq Hq (t ) − a R ηq Hq (t ) + p q p q q∈N5I

q∈N1J

q∈N5J

q∈N1K

q∈N3J

     K  K (3 )  K    J  J (5 )  J    a R ηq Hq (t ) + aKpR q ηqK (1) HqK (t ) − a pR q ηq Hq (t ) + p q +

 q∈N5K

q∈N3K

 K  K (5 )  K    R  R (1 )  R  a pR q ηq Hq (t ) + b pR q ηq Hq (t − τ pR q (t ) ) q∈N1R

    bRpR q ηqR(3) HqR (t − τ pR q (t ) ) − q∈N3R

+

    bRpR q ηqR(5) HqR (t − τ pR q (t ) )

q∈N5R

+

    bIpR q ηqI(1) HqI (t − τ pR q (t ) )

q∈N1I

−

    bIpR q ηqI(3) HqI (t − τ pR q (t ) )

q∈N3I

+

    bIpR q ηqI(5) HqI (t − τ pR q (t ) )

q∈N5I

+

   J  J (1 )  J b R ηq Hq (t − τ pR q (t ) ) p q

q∈N1J

−

   J  J (3 )  J b R ηq Hq (t − τ pR q (t ) ) p q

q∈N3J

μ(t ) μ(t − τ pR q (t ) ) μ(t ) μ(t − τ pR q (t ) )

μ(t ) μ(t − τ pR q (t ) ) μ(t ) μ(t − τ pR q (t ) ) μ(t ) μ(t − τ pR q (t ) ) μ(t ) μ(t − τ pR q (t ) ) μ(t ) μ(t − τ pR q (t ) )

μ(t ) μ(t − τ pR q (t ) )

fqK

(

∗ hKq



)



M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

   J  J (5 )  J b R ηq Hq (t − τ pR q (t )) p q

+

q∈N5J



+

q∈N1K



−

q∈N3K



+

 K  K (1 )  K  b pR q ηq Hq (t − τ pR q (t ))

μ(t ) μ(t − τ pR q (t ))

 K  K (3 )  K  b pR q ηq Hq (t − τ pR q (t ))

μ(t ) μ(t − τ pR q (t ))

 K  K (5 )  K  b pR q ηq Hq (t − τ pR q (t ))

μ(t ) μ(t − τ pR q (t ))

q∈N5K

≤



μ(t ) μ(t − τ pR q (t ))

μ˙ (t )  R −1   R −1  R  R(1) + a pR q ηq ξq ξ pR μ(t ) q∈N1R       −1    −1  − ξqR aRpR q ηqR(3) + ξqR aRpR q ηqR(5) H R (t ) −d pR +

q∈N3R

+

q∈N5R

ξR

   −1         −1    −1  ξqI aIpR q ηqI(1) − ξqI aIpR q ηqI(3) + ξqI aIpR q ηqI(5) H I (t )ξ I q∈N1I

q∈N3I

q∈N5I

q∈N1J

q∈N3J

q∈N5J

   −1      J −1  J  J (3 )   J −1  J  J (5 )  + ξqJ aJpR q ηqJ (1) − ξq a pR q ηq + ξq a pR q ηq H J (t )ξ J    −1         −1    −1  + ξqK aKpR q ηqK (1) − ξqK aKpR q ηqK (3) + ξqK aKpR q ηqK (5) H K (t )ξ K q∈N1K

+

μ(t ) μ(t − τ (t )) 

+



q∈N3K



ξqR

q∈N5K

−1  R  R(1)   R −1  R  R(3) b pR q ηq − ξq b pR q ηq

q∈N1R

q∈N3R

  −1   ξqR bRpR q ηqR(5) H R (t − τ pR q (t ))

q∈N5R

+

ξR

   −1         −1    −1  ξqI bIpR q ηqI(1) − ξqI bIpR q ηqI(3) + ξqI bIpR q ηqI(5) H I (t − τ pR q (t ))ξ I q∈N1I

q∈N3I

q∈N5I

q∈N1J

q∈N3J

q∈N5J

   −1      J −1  J  J (3 )   J −1  J  J (5 )  + ξqJ bJpR q ηqJ (1) − ξq b pR q ηq + ξq b pR q ηq H J (t − τ pR q (t ))ξ J +

≤

    −1         −1    −1  ξqK bKpR q ηqK (1) − ξqK bKpR q ηqK (3) + ξqK bKpR q ηqK (5) H K (t − τ pR q (t ))ξ K



+

q∈N1K

  q∈N5K

−

q∈N3R

q∈N3I

q∈N1J

q∈N3J

    −1    −1  −1   ξqJ aJpR q ηqJ (5) + ξqK aKpR q ηqK (1) − ξqK aKpR q ηqK (3) q∈N1K

q∈N3K

   −1    K −1  K  K (5) ξq a pR q ηq + (1 + β ) ξqR bRpR q ηqR(1)



q∈N3R

q∈N1I

  J −1  J  J (1 )   J −1  J  J (3 ) −1   ξqI aIpR q ηqI(5) + ξq a pR q ηq − ξq a pR q ηq



q∈N5J

+

q∈N1R

    −1    −1  −1   ξqR aRpR q ηqR(5) + ξqI aIpR q ηqI(1) − ξqI aIpR q ηqI(3)



q∈N5I

+

q∈N5K

−1   R −1  R  R(1)   R −1  R  R(3) −d pR + α ξ pRR + ξq a pR q ηq − ξq a pR q ηq

q∈N5R

+

q∈N3K



q∈N1R

    −1    −1  −1   ξqR bRpR q ηqR(3) + ξqR bRpR q ηqR(5) + ξqI bIpR q ηqI(1) q∈N5R

q∈N1I

245

246

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

−



ξqI

q∈N3I

−

−1  I  I(3)   I −1  I  I(5)   J −1  J  J (1) b pR q ηq + ξq b pR q ηq + ξq b pR q ηq

q∈N3J

−

q∈N5I

q∈N1J

q∈N5J

q∈N1K

   J −1  J  J (5 )   −1  −1   ξqJ bJpR q ηqJ (3) + ξq b pR q ηq + ξqK bKpR q ηqK (1)



  K −1  K  K (3)   K −1  K  K (5)      b pR q ηq + b pR q ηq M(t ) ξq ξq

 q∈N3K

q∈N5K

≤ 0,

(31)

where ‘D− ’ means the upper left Dini derivative operator. With the similar discussion, one can derive that the other cases do not hold either. According to the above arguments, we can conclude that D− ||H (t )||ξ ≤ 0 which implies that ||H (t )||ξ will be nonincreasing when time passes the point t. Based on the above analysis, one can conclude that M(t ) is bounded. Therefore, there exists a scalar λ > 0 such that M(t ) < λ for all t ≥ T, which directly implies that

|h˜ Rp (t )| <

I J K λR ˜ I (t )| < λ , |h ˜ Jp (t )| < λ , |h ˜ K (t )| < λ , , |h p p μ(t ) μ(t ) μ(t ) μ(t )









(32)



where t ≥ T, λR = λ max1≤q≤n (ξqR )−1 , λI = λ max1≤q≤n (ξqI )−1 , λJ = λ max1≤q≤n (ξq )−1 ∗

∗

∗

J



∗





and λK = λ max1≤q≤n (ξqK )−1 .

That is, the equilibrium point (hR , hI , hJ , hK ) is locally μ-stable located in α . Because α is chosen arbitrarily, we conclude that there is a locally μ-stable equilibrium point in any α . Therefore, system (1) has 34n locally μ-stable equilibrium points. This completes the proof. Combining Lemmas 1 and 2, we get that following result. Theorem 2. Assume that (18) holds and the nondecreasing function μ(t) > 0 with (23). Under the Assumptions 1 and 2 the system (1) with unbounded time-varying delays has 54n equilibria, 34n of which located in α are locally μ-stable, if there are J positive constants ξ pR , ξ pI , ξ p , ξ pK ( p = 1, 2, . . . , n ) such that n  n    n    n        −1  −1   −1   −1 −1 (−d p + α ) ξ pR + ξqR aRpq ρqR + ξqI aIpq ρqI + ξqJ aJpq ρqJ + ξqK aKpq ρqK q=1

q=1

q=1

q=1

  n  n    n    n       −1   −1   −1 −1 + ξqR bRpq ρqR + ξqI bIpq ρqI + ξqJ bJpq ρqJ + ξqK bKpq ρqK (1) (1 + β ) < 0, q=1

q=1

q=1

(33a)

q=1

n    n  n  n          −1  −1   −1   −1 −1 (−d p + α ) ξ pI + ξqI aRpq ρqI + ξqR aIpq ρqR + ξqK aJpq ρqK + ξqJ aKpq ρqJ q=1

q=1

q=1

q=1

 n    n  n    n           −1   −1 −1 −1 + ξqI bRpq ρqI + ξqR bIpq ρqR + ξqJ bJpq ρqK + ξqJ bKpq ρqJ (1 + β ) < 0, q=1

q=1

q=1

(33b)

q=1

n    n  n         −1     −1  −1   −1 −1 (−d p + α ) ξ pJ + ξqJ aRpq ρqJ + ξqR aJpq ρqR + ξqK aIpq ρqK + ξqI aKpq ρqI q=1

q=1

q=1

q∈N1K

 n    n  n  n          −1   −1   −1 −1 + ξqJ bRpq ρqJ + ξqR bJpq ρqR + ξqK bIpq ρqK + ξqI bKpq ρqI (1 + β ) < 0, q=1

q=1

q=1

(33c)

q=1

n  n  n    n          −1  −1   −1   −1 −1 (−d p + α ) ξ pK + ξqK aRpq ρqK + ξqR aKpq ρqR + ξqJ aIpq ρqJ + ξqI aJpq ρqI q=1

q=1

q=1

q=1

 n  n  n    n          −1   −1   −1 −1 + ξqK bRpq ρqK + ξqR bKpq ρqR + ξqJ bIpq ρqJ + ξqI bJpq ρqI (1 + β ) < 0 q=1

q=1

q=1

q=1

hold, where ρql = max(ηql (1 ) , −ηql (3 ) , ηql (5 ) ), p, q = 1, 2, . . . , n, l = R, I, J, K. 3.3. Discussions In this section, the μ-stability is specified with respect to some particular time delays and some particular μ(t).

(33d)

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

247

Corollary 1. (Multiple Exponential Stability). Suppose that (18) holds, and 0 ≤ τ pq (t) ≤ τ , where τ is a positive constant, p, q = 1, 2, · · · , n. Under the Assumptions 1 and 2 the system (1) has 54n equilibrium points, 34n of which located in α are locally J exponentially stable, if there are positive constants ξ pR , ξ pI , ξ p , ξ pK ( p = 1, 2, . . . , n ) such that

−d p

n  n    n    n        R −1  −1   −1   −1 −1 + ξp ξqR aRpq ρqR + ξqI aIpq ρqI + ξqJ aJpq ρqJ + ξqK aKpq ρqK q=1

+

q=1

q=1

−d p

q=1

q=1

q=1

q=1

q=1

q=1

(34b)

q=1

n    n  n         −1     J −1  −1   −1 −1 ξp + ξqJ aRpq ρqJ + ξqR aJpq ρqR + ξqK aIpq ρqK + ξqI aKpq ρqI q=1

n 

q=1

 J −1  R  J ξq b pq ρq +

q=1

−d p

(34a)

q=1

n  n    n         −1   −1   −1 −1 ξqI bRpq ρqI + ξqR bIpq ρqR + ξqJ bJpq ρqK + ξqJ bKpq ρqJ < 0,

−d p

n 

 R −1  J  R ξq b pq ρq +

q=1

q=1

q∈N1K n 

 K −1  I  K ξq b pq ρq +

q=1

n 

 I −1  K  I ξq b pq ρq < 0,

(34c)

q=1

n  n  n    n          K −1  −1   −1   −1 −1 + ξp ξqK aRpq ρqK + ξqR aKpq ρqR + ξqJ aIpq ρqJ + ξqI aJpq ρqI q=1

+

q=1

n   q=1

+

q=1

n    n  n  n          I −1  −1   −1   −1 −1 ξp + ξqI aRpq ρqI + ξqR aIpq ρqR + ξqK aJpq ρqK + ξqJ aKpq ρqJ q=1

+

q=1

n    n    n       −1   −1   −1 −1 ξqR bRpq ρqR + ξqI bIpq ρqI + ξqJ bJpq ρqJ + ξqK bKpq ρqK < 0,

n  

n 

q=1

 K −1  R  K ξq b pq ρq +

q=1

n  q=1

 R −1  K  R ξq b pq ρq +

q=1 n 

 J −1  I  J ξq b pq ρq +

q=1

q=1 n 

 I −1  J  I ξq b pq ρq < 0

(34d)

q=1

hold, where ρql = max(ηql (1 ) , −ηql (3 ) , ηql (5 ) ), p, q = 1, 2, . . . , n, l = R, I, J, K. Corollary 2. (Multiple Power-Stability). Suppose that (18) and (34a)–(34d) hold, and 0 ≤ τ pq (t) ≤ δ t, p, q = 1, 2, . . . , n, where δ is a constant with 0 < δ < 1. Then under the Assumptions 1 and 2 the system (1) has 54n equilibrium points, 34n of which located in α are locally power-stable. Corollary 3. (Multiple Log-Stability). Suppose that (18) and (34a)-(34d) hold, and μ(t ) = ln(1 + t ), 0 ≤ τ pq (t ) ≤ t − lnt t , p, q = 1, 2, . . . , n. Then under Assumptions 1 and 2 the system (1) has 54n equilibrium points, 34n of which located in α are locally log-stable. Corollary 4. (Multiple Log–Log-Stability). Suppose that (18) and (34a)–(34d) hold, and μ(t ) = ln ln(3 + t ), 0 ≤ τ pq (t ) ≤ t − t θ (0 < θ < 1 ), p, q = 1, 2, . . . , n. Then under the Assumptions 1 and 2 the QVNNs (1) has 54n equilibrium points, 34n of which located in α are locally log–log-stable. Since the proofs of the above Corollaries are similar to those in [47], they are omitted here. μ˙ t μt Remark 2. As pointed out in [47], the assumption 0 ≤ sup μ((t )) ≤ α and sup μ(t −(τ )(t )) ≤ 1 + β on the function μ(t) in (23) is t≥T0

t≥T0

weaker than the previous multiple μ-stability results on neural networks [59–61], which were studied under the following typical conditions

lim

t→+∞

μ˙ (t ) = α, μ(t )

lim

t→+∞

μ(t ) = 1 + β. μ(t − τ (t ))

Remark 3. Very recently, the monostability of QVNNs with time delays was studied in [20–25]. However, the problem of multiple stability of QVNNs has seldom been considered [48]. In this paper, the problem of the multiple μ-stability of QVNNs with unbounded time-varying delays is investigated, which is much more complicated than that of CVNNs. Remark 4. Since the quaternion can contain four different kinds of information, QVNNs can combine every four pieces of information into one piece, which facilitate the efficiency of information processing. The real-valued NNs and complexvalued NNs can be regarded as special cases of QVNNs. Remark 5. Among the applications of neural networks, multistability analysis has become a popular tendency. Multistability analysis of real-valued or complex-valued neural networks has attracted much attention of many researchers [33–46,59–61]. In [40], by using real-imaginary-type activation functions, the authors investigated the problem of multistability of CVNNs

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M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

with time delays. Sufficient conditions were derived for assuring the existence of 9n equilibrium points in which 4n of them are locally exponentially stable. The multistability issue for QVNNs with time delays was considered in [48], where criteria were obtained to ensure the existence of 81n equilibrium points, 16n of which are locally stable. In this paper, several sufficient conditions for QVNNs (1) are proposed to achieve the existence of 54n equilibrium points and 34n of them are locally μ-stable. It is obvious that the number of the equilibrium points of QVNNs herein is much more than that of CVNNs in [48]. Remark 6. One purpose in designing neural networks is to increase the storage capacity which can be achieved by increasing the number of stable equilibrium points of neural networks. Obviously, the storage capacity of QVNNs is larger compared with the storage capacity of complex-valued neural networks. Consequently, it is necessary both in theory and in practical application to study the problem of multistability analysis of QVNNs. 4. Numerical example In this section, a numerical example is provided to illustrate the effectiveness of our theoretical results. Example 1. Consider the two-dimensional QVNNs with unbounded time-varying delays:

h˙ (t ) = −Dh(t ) + A f (h(t )) + B f (h(t − τ (t ))) + u,

(35)

where h = hR + ihI + jhJ + khK ∈ Q2×1 ,





3 + 0.04i + 0.03j + 0.04k 0.02 + 0.05i + 0.02j + 0.04k

A=



3 0.02

=



0.04 3



+i

0.04 0.05

0.04 + 0.03i + 0.02j − 0.02k 3 − 0.02i + 0.05j + 0.03k



0.03 −0.02



+j

0.03 0.02



0.02 0.05



+k

0.04 0.04



−0.02 0.03

= AR + iAI + jAJ + kAK ,

 B=

 =



0.02 + 0.02i + 0.02j + 0.01k 0.04 + 0.03i + 0.01 + 0.02k 0.02 0.04



0.06 −0.03



+i

0.02 0.03

0.06 − 0.04i + 0.01j + 0.03k −0.03 + 0.02i − 0.02j − 0.04k



−0.04 0.02



+j

0.02 0.01



0.01 −0.02



+k

0.01 0.02



0.03 −0.04

= BR + iBI + jBJ + kBK , and D = diag(2, 2 ), τ11 (t ) = τ21 (t ) = 0.4t, τ12 (t ) = τ22 (t ) = 0.6t, u = (0.2 − i − 0.2j − 0.8k vation functions are defined as

⎧ 5 1 ⎪ ⎪ tanh(η + 1 ) − , ⎪ 2 4 ⎪ ⎪ ⎪ 11 ⎪ ⎪ −2η2 + 2η + , ⎪ ⎪ 4 ⎪ ⎨ 4 27 f pR (η ) = f pJ (η ) = − η + , 7 28 ⎪ ⎪ ⎪ ⎪ ⎪ 10η − 40, ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎩ 1 tanh η − 35 + 15 , 2

8

4

 7 ⎧ 1 5 ⎪ η tanh + − , ⎪ ⎪ 3 4 5 ⎪ ⎪ ⎪ 24 23 ⎪ ⎪ η+ , ⎪ ⎪ 5 ⎨ 5 4 4 f pI (η ) = f pK (η ) = − η+ , 7 7 ⎪ ⎪ ⎪ 3521 98 ⎪ ⎪ η− , ⎪ ⎪ 10 ⎪ 100 13 7 ⎪ ⎪ 1 ⎩ tanh η − + , 3

4

for p = 1, 2; whose images are shown in Fig. 1.

2

−∞ < η < −1, 1 −1 ≤ η ≤ − , 2 1 31 − <η< , 2 8 35 31 ≤η≤ , 8 8 35 < η < +∞, 8 5 −∞ < η < − , 4 5 3 − ≤η≤− , 4 4 3 69 − <η< , 4 20 79 69 ≤η≤ , 20 20 79 < η < +∞, 20

0.2 − 0.3i + 0.1j − k )T . The acti-

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

249

Fig. 1. Graphs of activation functions f pR (t ), f pJ (t ) and f pI (t ), f pK (t ) for p = 1, 2.

R

I

J

K

The upper bound functions F p (η ), F p (η ), F p (η ), F p (η ) and the lower bound function F Rp (η ), F Ip (η ), F p (η ), F Kp (η ) defined in (4a)–(7b) can be computed directly as follows: J

R

F R1 (η ) = −2η + 3 f1R (η ) − 1.1057;

R

F R2 (η ) = −2η + 3 f2R (η ) − 1.2398;

I

F I1 (η ) = −2η + 3 f1I (η ) − 2.0492;

F 1 (η ) = −2η + 3 f1R (η ) + 1.3746, F 2 (η ) = −2η + 3 f2R (η ) + 1.2918, F 1 (η ) = −2η + 3 f1I (η ) + 0.4442, I

F I2 (η ) = −2η + 3 f2I (η ) − 1.4468;

J

F J1 (η ) = −2η + 3 f1J (η ) − 1.1797;

J

F J2 (η ) = −2η + 3 f2J (η ) − 1.0848;

K

F K1 (η ) = −2η + 3 f1K (η ) − 1.8672;

K

F K2 (η ) = −2η + 3 f2K (η ) − 2.1478

F 2 (η ) = −2η + 3 f2I (η ) + 1.1108, F 1 (η ) = −2η + 3 f1J (η ) + 1.3007, F 2 (η ) = −2η + 3 f2J (η ) + 1.4468, F 1 (η ) = −2η + 3 f1K (η ) + 0.6261, F 2 (η ) = −2η + 3 f2K (η ) + 0.4098,

with their graphs being depicted in Figs. 2–5. It is obvious to see that that condition (18) holds for p = 1, 2. Therefore, the quaternion-valued neural networks (1) with parameters given above can have 58 equilibria according to Lemma 1. J (1 ) Furthermore, on the basis of the activation functions given above, the Assumption 2 is satisfied with μRp(1 ) = μ p = 0,

ηRp(1) = ηJp(1) = 12 , μRp(2) = μJp(2) = 4, ηRp(2) = ηJp(2) = 6, μRp(3) = μJp(3) = ηRp(3) = ηJp(3) = − 74 , μRp(4) = μJp(4) = ηRp(4) = ηJp(4) = 10, I (3 ) μRp(5) = μJp(5) = 0, ηRp(5) = ηJp(5) = 12 , μIp(1) = μKp (1) = 0, ηIp(1) = ηKp (1) = 13 , μIp(2) = μKp (2) = ηIp(2) = ηKp (2) = 24 = μKp (3 ) = 5 , μp I (5 ) ηIp(3) = ηKp (3) = − 74 , μIp(4) = μKp (4) = ηIp(4) = ηKp (4) = 98 = μKp (5 ) = 0, ηIp(5 ) = ηKp (5 ) = 13 for p = 1, 2. In addition, it is eas10 , μ p J

J

J

26 23 I I R ily obtained from the above activation functions that mRp = m p = − 74 , MRp = M p = 17 4 , m p = m p = − 15 , M p = M p = 6 for p = 1, 2. Finally, for given activation functions, by direct computations, one can get that the conditions (34a)–(34d) are satisfied J J with ξ1R = ξ2R = ξ1I = ξ2I = ξ1 = ξ2 = ξ1K = ξ2K = 0.8. Therefore, it directly follows from Corollary 2 that 38 of the 58 equilibria of QVNNs (1) are locally power-stable. For the sake of simulation, we take the following 6 positive sets for instance:

 1 31  21 5  79 159 1 = (−5, −1 ) × − , − × − ,− × , 2 8 4 4 20 20  35 75  1 31  3 69  79 159 ×

8

,

8

× − , 2 8

× − , 4 20

×

20

,

20

,

250

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

R

Fig. 2. Graphs of the upper bound and the lower bound functions F p (η ), F Rp (η ) for p = 1, 2.

I

Fig. 3. Graphs of the upper bound and the lower bound functions F p (η ), F Ip (η ) for p = 1, 2.

J

Fig. 4. Graphs of the upper bound and the lower bound functions F p (η ), F Jp (η ) for p = 1, 2.

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

251

K

Fig. 5. Graphs of the upper bound and the lower bound functions F p (η ), F Kp (η ) for p = 1, 2.

 1 31  79 159  3 69 2 = − , − × (−5, −1 ) × , × − , 2 8 20 20 4 20  1 31  79 159  21 5 × − ,− × (−5, −1 ) × , × − ,− , 2

3 =

8

 35 75

20

 35 75



20

4



4





3 69 21 5 × − , × − ,− 8 8 8 8 4 20 4 4  35 75  21 5  3 69 ×(−5, −1 ) × , × − ,− × − , , 8 8 4 4 4 20 ,

×

4 = (−5, −1 ) ×

,

 35 75 8

,

8

×

 1 31

×(−5, −1 ) × − , 2 8

 79 159 ,

 21

× −

,−

5 4



20 20 4  79 159  21 5 × , × − ,− , 20 20 4 4

 1 31  1 31  3 69  3 69 5 = − , − × − , × − , × − , 2 8 2 8 4 20 4 20  35 75  3 69  3 69 × , × (−5, −1 ) × − , × − , , 8

6 =

8

 35 75 8

,

8

4 20

 1 31

× − , 2 8

 21

× (−5, −1 ) × − ×

 35 75 8

,

8

4

,−

 21

× −

4

5 4

4 20



,−

×

 79 159

5 4

20 ×

,

20

 79 159 20

,

20

.

With 180 random initial conditions, the convergence behaviors of the states of QVNNs (1) are depicted in Figs. 6–11, which show the effectiveness of the obtained results. Remark 7. To be specific, in [59], the illustrative two-dimensional neural network example has been given, where the activation functions are nonmonotonic piecewise linear and it has been proven that this neural network has only 9 locally stable equilibria. In [40,60], the illustrative two-dimensional complex-valued neural network examples have been given, in which it has been revealed that this neural network has only 16 locally stable equilibria. It is obvious that the activation functions in this paper are both nonmonotonic and nonlinear. As shown in the example, the system (35) has 38 locally stable equilibria. Thus, the addressed quaternion-valued networks with nonmonotonic piecewise nonlinear activation functions are less conservative in the type of activation functions and are better than the complex-valued ones for the high-capacity associative memory applications.

252

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

Fig. 6. Trajectories of state variable hR1 (t ) and hR2 (t ).

Fig. 7. Trajectories of state variable hI1 (t ) and hI2 (t ).

Fig. 8. Trajectories of state variable hJ1 (t ) and hJ2 (t ).

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

Fig. 9. Trajectories of state variable hK1 (t ) and hK2 (t ).

Fig. 10. Phase plot of state variable (hR1 (t ), hR2 (t ), hI1 (t ))T and (hI1 (t ), hI2 (t ), hJ1 (t ))T .

Fig. 11. Phase plot of state variable (hJ1 (t ), hJ2 (t ), hK1 (t ))T and (hK1 (t ), hK2 (t ), hR1 (t ))T .

253

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5. Conclusions In this paper, the problem of multiple μ-stability analysis of the QVNNs with nonmonotonic piecewise nonlinear activation functions and unbounded time-varying delays has been considered. By means of fixed point theorem and other analytical tools, several sufficient criteria have been achieved for the adressed QVNNS to guarantee the existence of 54n equilibria, 34n of which are locally μ-stable. Moreover, as a direct application, some criteria that ensure the multiple exponential stability, multiple power stability, multiple log-stability, multiple log–log-stability are also obtained. One numerical example with simulations has been provided to demonstrate the validity of the theoretical results. Acknowledgments The research is supported by grants from the National Natural Science Foundation of China (Nos. 61572233 and 11471083), and the Science and Technology Program of Guangzhou, China (No. 201707010404). Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.amc.2018.08.033. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]

L. Chen, Definition of determinant and cramer solutions over the quaternion field, Acta Math. Sin. 7 (2) (1991) 171–180. P. Arena, L. Fortuna, G. Muscato, M.G. Xibilia, Multilayer perceptrons to approximate quaternion valued functions, Neural Netw. 10 (2) (1997) 335–342. A. Sudbery, Quaternionic analysis, Math. Proc. Camb. Philos. Soc. 85 (1979) 199–225. L. Akila, R. Roopkumar, A natural convolution of quaternion valued functions and its applications, Appl. Math. Comput. 242 (2014) 633–642. G.J. Song, Q.W. Wang, S.W. Yu, Cramers rule for a system of quaternion matrix equations with applications, Appl. Math. Comput. 336 (2018) 490–499. B.C. Ujang, C.C. Took, D.P. Mandic, Quaternion-valued nonlinear adaptive filtering, IEEE Trans. Neural Netw 22 (8) (2011) 1193–1206. S.B. Choe, J.J. Faraway, Modeling head and hand orientation during motion using quaternions, SAE Trans. 113 (2004) 186–192. J.C. Chou, Quaternion kinematic and dynamic differential equations, IEEE Trans. Robot. Autom. 8 (1) (1992) 53–64. S. Miron, N.L. Bihan, J.I. Mars, Quaternion-music for vector-sensor array processing, IEEE Trans. Signal Proces 54 (4) (2006) 1218–1229. T.A. Ell, S.J. Sangwine, Hypercomplex fourier transforms of color images, IEEE Trans. Image Proces 16 (1) (2007) 22–35. M. Morita, Associative memory with nonmonotone dynamics, Neural Netw 6 (1) (1993) 115–126. L.O. Chua, L. Yang, Cellular neural networks: applications, IEEE Trans. Circuits Syst 35 (10) (1988) 1273–1290. K. Dai, J. Zhao, F. Cao, A novel algorithm of extended neural networks for image recognition, Eng. Appl. Artif. Intel. 42 (2015) 57–66. M. Tan, Y. Zhang, New sufficient conditions for global asymptotic stability of Cohen–Grossberg neural networks with time-varying delays, Nonlinear Anal. Real. 10 (4) (2009) 2139–2145. N. Matsui, T. Isokawa, H. Kusamichi, F. Peper, H. Nishimura, J. Intell, Quaternion neural network with geometrical operators, Fuzzy Syst. 15 (2004) 149–164. T. Isokawa, T. Kusakabe, N. Matsui, F. Peper, Quaternion neural network and its application, in: Knowledge-Based Intelligent Information and Engineering Systems, Springer, 2003, pp. 318–324. M. Yoshida, Y. Kuroe, T. Mori, Models of hopfield-type quaternion neural networks and their energy functions, Int.J. Neural Syst. 15 (2005) 129–135. T. Isokawa, N. Matsui, H. Nishimura, Quaternionic neural networks: fundamental properties and applications, in: T. Nitta (Ed.), Complex Valued Neural Networks: Utilizing High Dimensional Parameters, IGI Global, New York, 2009, pp. 411–439. X. Chen, Q. Song, Z. Li, Design and analysis of quaternion-valued neural networks for associative memories, IEEE Trans. Syst. Man Cybern. Syst. (2017), doi:10.1109/TSMC.2017.2717866. J. Hu, C. Zeng, J. Tan, Boundedness and periodicity for linear threshold discrete-time quaternion-valued neural network with time-delays, Neurocomputing 267 (2017) 417–425. D. Zhang, K.I. Kou, Y. Liu, J. Cao, Decomposition approach to the stability of recurrent neural networks with asynchronous time delays in quaternion field, Neural Netw. 94 (2017) 55–66. Y. Liu, D. Zhang, J. Lu, Global exponential stability for quaternion-valued recurrent neural networks with time-varying delays, Nonlinear Dyn. 87 (1) (2017) 553–565. H. Shu, Q. Song, Y. Liu, Z. Zhao, F.E. Alsaadi, Global μ- stability of quaternion-valued neural networks with non-differentiable time-varying delays, Neurocomputing 247 (2017) 202–212. Y. Liu, D. Zhang, J. Lu, J. Cao, Global μ-stability criteria for quaternion-valued neural networks with unbounded time-varying delays, Inf. Sci. 360 (2016) 273–288. X. Chen, Z. Li, Q. Song, J. Hu, Y. Tan, Robust stability analysis of quaternion-valued neural networks with time delays and parameter uncertainties, Neural Netw. 91 (2017) 55–65. W.I. Lee, S.Y. Lee, P. Park, Improved stability criteria for recurrent neural networks with interval time-varying delays via new Lyapunov functionals, Neurocomputing 155 (2015) 128–134. Z. Zhang, S. Yu, Global asymptotic stability for a class of complex-valued Cohen–Grossberg neural networks with time delays, Neurocomputing 171 (2016) 1158–1166. P. Muthukumar, K. Subramanian, Stability criteria for Markovian jump neural networks with mode-dependent additive time-varying delays via quadratic convex combination, Neurocomputing 205 (2016) 75–83. D. Xu, M. Tan, Delay-independent stability criteria for complex-valued BAM neutral-type neural networks with time delays, Nonlinear Dyn. 89 (6) (2017) 1–14. R. Zhang, D. Zeng, S. Zhong, Y. Yu, Event-triggered sampling control for stability and stabilization of memristive neural networks with communication delays, Appl. Math. Comput. 310 (2017) 57–74. S.V. Chakravarthy, J. Ghosh, A complex-valued associative memory for storing patterns as oscillatory states, Biol. Cybern. 75 (3) (1996) 229–238. D. Lee, Improvements of complex-valued hopfield associative memory by using generalized projection rules, IEEE Trans. Neural Netw. 17 (5) (2006) 1341–1347. C. Cheng, K. Lin, C. Shih, Multistability and convergence in delayed neural networks, Physica D 225 (1) (2007) 61–74. X. Nie, J. Cao, Multistability of competitive neural networks with time-varying and distributed delays, Nonlinear Anal. Real. 10 (2) (2009) 928–942. L. Wang, W. Lu, T. Chen, Coexistence and local stability of multiple equilibria in neural networks with piecewise linear nondecreasing activation functions, Neural Netw. 23 (2) (2010) 189–200. Z. Zeng, D. Huang, Z. Wang, Memory pattern analysis of cellular neural networks, Phys. Lett. A 342 (1) (2005) 114–128.

M. Tan et al. / Applied Mathematics and Computation 341 (2019) 229–255

255

[37] Z. Zeng, J. Wang, Multiperiodicity and exponential attractivity evoked by periodic external inputs in delayed cellular neural networks, Neural Comput. 18 (4) (2006) 848–870. [38] X. Nie, J. Cao, S. Fei, Multistability and instability of delayed competitive neural networks with nondecreasing piecewise linear activation functions, Neurocomputing 119 (2013) 281–291. [39] Y. Huang, H. Zhang, Z. Wang, Multistability of complex-valued recurrent neural networks with real-imaginary-type activation functions, Appl. Math. Comput. 229 (2014) 187–200. [40] X. Chen, Z. Zhao, Q. Song, J. Hu, Multistability of complex-valued neural networks with time-varying delays, Appl. Math. Comput. 294 (2017) 18–35. [41] P. Liu, Z. Zeng, J. Wang, Multistability of recurrent neural networks with nonmonotonic activation functions and unbounded time-varying delays, IEEE Trans. Neural Netw. Learn. Syst. (2017), doi:10.1109/TNNLS.2017.2710299. [42] L. Wang, T. Chen, Multistability of neural networks with Mexican-hat-type activation functions, IEEE Trans. Neural Netw. Learn. Syst. 23 (11) (2012) 1816–1826. [43] G. Huang, J. Cao, Multistability of neural networks with discontinuous activation function, Commun. Nonlin. Sci. Numer. Simul. 13 (10) (2008) 2279–2289. [44] J. Liang, W. Gong, T. Huang, Multistability of complex-valued neural networks with discontinuous activation functions, Neural Netw. 84 (2016) 125–142. [45] X. Nie, W.X. Zheng, Multistability and instability of neural networks with discontinuous nonmonotonic piecewise linear activation functions, IEEE Trans. Neural Netw. Learn. Syst. 26 (11) (2015) 2901–2913. [46] Y. Huang, H. Zhang, Z. Wang, Multistability and multiperiodicity of delayed bidirectional associative memory neural networks with discontinuous activation functions, Appl. Math. Comput. 219 (3) (2012) 899–910. [47] M. Tan, D. Xu, Multiple μ-stability analysis for memristor-based complex-valued neural networks with nonmonotonic piecewise nonlinear activation functions and unbounded time-varying delays, Neurocomputing 275 (2018) 2681–2701. [48] Q. Song, X. Chen, Multistability analysis of quaternion-valued neural networks with time delays, IEEE Trans. Neural Netw. Learn. Syst. (2018), doi:10. 1109/TNNLS.2018.2801297. [49] M. Park, O. Kwon, S. Choi, Stability analysis of discrete-time switched systems with time-varying delays via a new summation inequality, Nonlinear Anal. Hybrid Syst. 23 (2017) 76–90. [50] M. Tan, Stabilization of coupled time-delay neural networks with nodes of different dimensions, Neural Process. Lett. 43 (1) (2016) 255–268. [51] W.I. Lee, S.Y. Lee, P. Park, A combined reciprocal convexity approach for stability analysis of static neural networks with interval time-varying delays, Neurocomputing 221 (2017) 168–177. [52] J.J. Oliveira, Global exponential stability of nonautonomous neural network models with unbounded delays, Neural Netw. 96 (2017) 71–79. [53] R. Saravanakumar, M.S. Ali, C.K. Ahn, H.R. Karimi, P. Shi, Stability of Markovian jump generalized neural networks with interval time-varying delays, IEEE Trans. Neural Netw. Learn. Syst. 28 (8) (2017) 1840–1850. [54] D. Zeng, R. Zhang, Y. Liu, S. Zhong, Sampled-data synchronization of chaotic Lur’e systems via input-delay-dependent-free-matrix zero equality approach, Appl. Math. Comput. 315 (2017) 34–46. [55] R. Zhang, X. Liu, D. Zeng, S. Zhong, K. Shi, A novel approach to stability and stabilization of fuzzy sampled-data Markovian chaotic systems, Fuzzy Set. Syst. 344 (2018) 108–128. [56] R. Zhang, D. Zeng, S. Zhong, K. Shi, J. Cui, New approach on designing stochastic sampled-data controller for exponential synchronization of chaotic Lur’e systems, Nonlinear Anal. Hybrid Syst. 29 (2018) 303–321. [57] Y. Zhang, P.A. Heng, K.S. Leung, Convergence analysis of cellular neural networks with unbounded delay, IEEE Trans. Circuits Syst. I 48 (6) (2001) 680–687. [58] L. Zhou, Global asymptotic stability of cellular neural networks with proportional delays, Nonlinear Dyn. 77 (1–2) (2014) 41–47. [59] X. Nie, W. Zheng, J. Cao, Coexistence and local μ-stability of multiple equilibrium points for memristive neural networks with nonmonotonic piecewise linear activation functions and unbounded time-varying delays, Neural Netw. 84 (2016) 172–180. [60] R. Rakkiyappan, G. Velmurugan, J. Cao, Multiple μ-stability analysis of complex-valued neural networks with unbounded time-varying delays, Neurocomputing 149 (2015) 594–607. [61] L. Wang, T. Chen, Multiple μ-stability of neural networks with unbounded time-varying delays, Neural Netw. 53 (2014) 109–118.