Remarks on the Article by X.B. Nie, J.D. Cao and S.M. Fei “Multistability and instability of delayed competitive neural networks with nondecreasing piecewise linear activation functions”

Neurocomputing 177 (2016) 628–635

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Remarks on the Article by X.B. Nie, J.D. Cao and S.M. Fei “Multistability and instability of delayed competitive neural networks with nondecreasing piecewise linear activation functions” Shuang Ding, Xiangyu Gao n, Julong Tan School of Mathematical Science, Heilongjiang University, Harbin 150080, China

art ic l e i nf o

a b s t r a c t

Article history: Received 2 May 2015 Received in revised form 14 August 2015 Accepted 24 November 2015 Communicated by Haijun Jiang Available online 11 December 2015

In this paper, a simple and elementary proof of the equilibrium uniqueness theorem (i.e. Part of Theorem 1 in Nie et al. [2013. Multistability and instability of delayed competitive neural networks with nondecreasing piecewise linear activation functions. Neurocomputing 119, 281–291]) is given, which shows the equilibrium uniqueness of delayed competitive neural networks (DCNNs) with the activation function possessing two corner points in a given subset. In addition, for the piecewise linear activation function with two corner points, the dynamical behaviors of all equilibrium points of n-neuron delayed Hopfield neural networks (DHNNs) are completely analyzed, and a sufficient condition is obtained to guarantee that the n-neuron DHNNs have exactly 3n equilibrium points, where 2n of them are stable, and the others are unstable. Finally, an example is provided to show the theoretical analysis. & 2015 Elsevier B.V. All rights reserved.

Keywords: DCNNs Piecewise linear activation function Equilibrium point Multistability Instability DHNNs

1. Introduction Multistability and instability problems of delayed competitive neural networks with nondecreasing piecewise linear activation functions are investigated in [1]. A sufficient condition of the exact existence of multiple equilibrium points for DCNNs with the piecewise linear activation function possessing two corner points is given in the form of several elementary inequalities (i.e. Theorem 1 in [1]). Under this sufficient condition, DCNNs can have and only have 3n equilibrium points in three different subsets Ω1 , Ω2 and Ω3: (i) DCNNs have 2n equilibrium points in Ω1, (ii) DCNNs have and only have one equilibrium point in Ω2; (iii) DCNNs have 3n
2n
1 equilibrium points in Ω3, where Ω1 , Ω2 and Ω3 will be defined in the next section. However, the equilibrium uniqueness proof of DCNNs in the subset Ω2 is very complex, which appears in many different literature (see [1–5]). This motivates us to reduce its complexity. In this paper, a simple and elementary proof of the uniqueness of equilibrium point for DCNNs in the subset Ω2 is given, which is different from one proposed in [1–5]. In addition, a sufficient condition is provided to completely analyze the n

Corresponding author. Tel.: þ 86 45186608282; fax: þ86 45186608240. E-mail address: [email protected] (X. Gao).

http://dx.doi.org/10.1016/j.neucom.2015.11.067 0925-2312/& 2015 Elsevier B.V. All rights reserved.

dynamical behaviors of all equilibrium points of n-neuron DHNNs, which can be considered as the generalization of Corollary 3 in [1]. This sufficient condition not only ensures that the n-neuron DHNNs have exactly 3n equilibrium points, but also shows that 2n of these equilibrium points are stable, and others are unstable. It is worth mentioning that the sufficient condition is derived by analyzing the distribution of roots of the characteristic equation of the delay differential equations. Since this characteristic equation is not an ordinary polynomial equation but a transcendental polynomial equation, it is very difficult and challenging to study the distribution of roots of transcendental polynomial equation. The remaining part of the paper is organized as follows. In Section 2, the model of DCNNs is introduced. Based on this model, and the corresponding problem is proposed. In Section 3.1, a simple and elementary proof has been given to show the equilibrium uniqueness in Ω2 of delayed competitive neural networks with the activation function possessing two corner points. Furthermore, a sufficient condition has been obtained in Section 3.2 to analyze the dynamical behaviors of all equilibrium points of nneuron DHNNs with the activation function possessing two corner points. In Section 4, a numerical example has been given to demonstrate the effectiveness of the obtained theoretical results. Finally, some conclusions are drawn in Section 5.

S. Ding et al. / Neurocomputing 177 (2016) 628–635

Obviously, Ω1 is composed of 2n regions, Ω2 is composed of one region, and Ω3 is composed of 3n
2n
1 regions.

2. Problem formulation Consider the following DCNNs: 8 p n n X X X > > < x_ i ðtÞ ¼
di xi ðtÞ þ aij f j ðxj ðtÞÞ þ bij f j ðxj ðt
τ j ðtÞÞÞ þ Bi mij ðtÞyj þ I i j¼1

> > :m _ ij ðtÞ ¼
αi mij ðtÞ þ yj βi f i ðxi ðtÞÞ;

j¼1

j¼1

i AN½1; n; j A N½1; p

ð1Þ where xi(t) is the neuron current activity level, f j ðxj ðtÞÞ is the output of neurons, mij(t) is the synaptic efficiency, yj is the constant external stimulus, aij represents connection weights between the ith neuron and the jth neuron, bij represents connection weights of delayed feedback, Ii is the constant input, αi 4 0, βi denote disposable scaling constants, τj ðtÞ corresponds to the transmission delay and satisfies 0 r τj ðtÞ r τj , N½s; t ¼ fs; s þ1; …; tg. P Let Si ðtÞ ¼ pj¼ 1 mij ðtÞyj , where y21 þ y22 þ ⋯ þ y2p ¼ 1, then the system (1) can be rewritten as 8 n n X X > > < x_ i ðtÞ ¼
di xi ðtÞ þ aij f j ðxj ðtÞÞ þ bij f j ðxj ðt
τj ðtÞÞÞ þ Bi Si ðtÞ þI i j¼1

> > : S_ ðtÞ ¼
α S ðtÞ þ β f ðx ðtÞÞ; i i i i i i

, i A N½1; C kn . Similarly, [6], the ith element of Nk is denoted as N ðiÞ k let M n
k be a set, whose element is the form of fj1 ; j2 ; …; jn
k g, js A f0; 1g; s A N½1; n
k; M ðjÞ is the jth element of M n
k , n
k jA N½1; 2n
k .

ðiÞ

N k ¼ N½1; n=NðiÞ k

Let

fik þ 1 ; ik þ 2 ; …; in g, ik þ 1 o ik þ 2 o ⋯ oin , δ

be ðik þ l Þ

represented

as

represents the lth

1 , l A ½1; n
k. Then, Ω3 ¼ ⋃nk
element of M ðjÞ ¼ 1 Ξ k , where n
k n o ðjÞ ðjÞ k n Ξ k ¼ Ξ ðiÞ k  Ξ k j iA N½1; C n ; M n
k A M n
k ; j A N½1; 2  ðjÞ

ðsÞ

ðsÞ

Ξ k ¼ ∏ ð
1; ps Þδ  ðqs ; 1Þ1
δ ; δðsÞ A MðjÞ n
k ðiÞ

sANk ðiÞ k

Ξ ¼ ∏ ½ps ; qs : s A N ðiÞ k

3.1. The simple proof of the equilibrium uniqueness of DCNNs

i A N½1; n:

In addition, we introduce the following nondecreasing piecewise linear activation functions with two corner points proposed in [1]: 8 ui ;
1o x o pi > > < vi
ui ðx
pi Þ; pi r x r qi ð3Þ f i ðxÞ ¼ ui þ qi
pi > > : vi ; qi o x o þ 1: Now, our goal of this paper may be stated formally as follows: (i) a simple and elementary method is provided to show system (2) with activation function (3) has only one equilibrium point in the subset Ω2; (ii) for activation function (3), the dynamical behaviors of all equilibrium points of n-neuron DHNNs are completely analyzed.

Now, we shall show the following theorem by a simple and elementary method. Theorem 1. If the following inequalities: 8 n X >
1 > > maxfðaij þ bij Þuj ; ðaij þ bij Þvj g þ I i o 0 >
di pi þ ðaii þ bii þ Bi β i αi Þf i ðpi Þ þ > < j a i;j ¼ 1 n X > > > minfðaij þ bij Þuj ; ðaij þ bij Þvj g þ I i 4 0
di qi þ ðaii þ bii þ Bi βi αi
1 Þf i ðqi Þ þ > > : j a i;j ¼ 1

ð4Þ hold for any i A N½1; n, then system (2) with activation function (3) has and only has one equilibrium point in the subset Ω2. Proof. From Theorem 1 in [1], it follows that system (2) with activation function (3) has at least one equilibrium point in Ω2 , we shall further show system (2) with activation function (3) have only one equilibrium point in the subset Ω2. Let

3. Main results Let ð
1; pi Þ ¼ ð
1; pi Þ1  ½pi ; qi 0  ðqi ; þ 1Þ0

li ¼

½pi ; qi  ¼ ð
1; pi Þ0  ½pi ; qi 1  ðqi ; þ 1Þ0

ui qi
v i pi ; qi
pi

iA N½1; n;

iA N½1; n:

Thus, every equilibrium point in equations:

R ¼ ð
1; pi Þ [ ½pi ; qi  [ ðqi ; þ 1Þ:


di xi þ ðaii þ bii þ Bi βi αi
1 Þli xi þ

Then, Rn can be divided into the following 3n subsets: ðiÞ

ci ¼

f i ðxi Þ ¼ li xi þ ci ;

and

ðiÞ

vi
ui ; qi
pi

then

ðqi ; þ 1Þ ¼ ð
1; pi Þ0  ½pi ; qi 0  ðqi ; þ 1Þ1

n

Remark 1. For k A N½1; n
1, fi1 ; i2 ; …; ik g are any k numbers taken from N½1; n, which is regarded as an element, all these elements form a set Nk. According to the dictionary-ordering method in

j¼1

ð2Þ

(

629

ðiÞ

∏ ð
1; pi Þδ1  ½pi ; qi δ2  ðqi ; þ 1Þδ3 :ðδ1 ; δ2 ; δ3 Þ ¼ ð1; 0; 0Þ; ð0; 1; 0Þ; ð0; 0; 1Þ ðiÞ

ðiÞ

ðiÞ

i¼1

Ω2 is a root of the following

n X

ða1j þ b1j Þlj xj

j a i;j ¼ 1

o

þ

n X

ðaij þ bij Þcj þ Bi β i αi
1 ci þ I i ¼ 0;

iA N½1; n:

ð5Þ

j¼1

Let

Thus,

A~ ¼ ða~ ij Þnn ;

Rn ¼ Ω1 [ Ω2 [ Ω3 ; where (

Ω1 ¼

n

) δðiÞ

∏ ð
1; pi Þ

i¼1

 ðqi ; þ1Þ

1
δ

ðiÞ

ðiÞ

: δ ¼ 0; 1

where ( a~ ij ¼


di þ ðaij þ bij þ Bi βi αi
1 Þli ;

i¼j

ðaij þ bij Þlj ;

i aj;

Ω2 ¼ ∏ ½pi ; qi 

we can show that the coefficient matrix A~ of Eq. (5) is invertible. From (4), it is clear that

Ω3 ¼ Rn
Ω1
Ω2 :


di qi þ ðaii þbii þBi βi αi
1 Þli qi

n

i¼1

630

S. Ding et al. / Neurocomputing 177 (2016) 628–635 n X

þ

simple and elementary proof is provided in this paper to reduce its complexity.

minfðaij þ bij Þlj pj ; ðaij þ bij Þlj qj g

j a i;j ¼ 1

4
di pi þ ðaii þ bii þ Bi β i αi
1 Þli pi n X

þ

3.2. Dynamical behaviors of n-neuron DHNNs with activation function (3)

maxfðaij þ bij Þlj pj ; ðaij þ bij Þlj qj g

j a i;j ¼ 1

In this section, we will further study all equilibrium points of nneuron DHNNs with activation function (3) and analyze their dynamical behaviors. Consider the following DHNNs with n-neurons: ( x_ 1 ðtÞ ¼
d1 x1 ðtÞ þ a11 f 1 ðx1 ðtÞÞ þ b1n f n ðxn ðt
τn ÞÞ þI 1 x_ i ðtÞ ¼
di xi ðtÞ þ aii f i ðxi ðtÞÞ þ bii
1 f i
1 ðxi
1 ðt
τi
1 ÞÞ þ I i ; i A N½2; n:

i.e., ð
di þ ðaii þ bii þ Bi β i αi
1 Þli Þðqi
pi Þ n X

4

maxfðaij þbij Þlj pj ; ðaij þ bij Þlj qj g

ð8Þ

j a i;j ¼ 1 n X




From (8), it follows minfðaij þ bij Þlj pj ; ðaij þ bij Þlj qj g

ð6Þ

(

j a i;j ¼ 1

In the following, there are two cases to be discussed: lj ðaij þ bij Þ Z 0 and lj ðaij þ bij Þ r 0. (i)

When lj ðaij þ bij Þ Z 0, then

x_ 1 ðt
σ 1 Þ ¼
d1 x1 ðt
σ 1 Þ þ a11 f 1 ðx1 ðt
σ 1 ÞÞ þ b1n f n ðxn ðt
τ n
σ 1 ÞÞ þ I 1 x_ i ðt
σ i Þ ¼
di xi ðt
σ i Þ þ aii f i ðxi ðt
σ i ÞÞþ bii
1 f i
1 ðxi
1 ðt
τ i
1
σ i ÞÞ þ I i ;

ð9Þ where

σ i ¼
τn


maxfðaij þ bij Þlj pj ; ðaij þ bij Þlj qj g ¼ ðaij þ bij Þlj qj

i
1 X j¼1

τj þ

n i X τ; n j¼1 j

i A N½1; n:

and

Let yi ðtÞ ¼ xi ðt
σ i Þ, the system (9) can become

minfðaij þ bij Þlj pj ; ðaij þbij Þlj qj g ¼ ðaij þ bij Þlj pj

(

(ii) When lj ðaij þ bij Þ r 0, then

y_ 1 ðtÞ ¼
d1 y1 ðtÞ þ a11 f 1 ðy1 ðtÞÞ þ b1n f n ðyn ðt
τn
σ 1 þ σ n ÞÞ þ I 1 y_ i ðtÞ ¼
di yi ðtÞ þ aii f i ðyi ðtÞÞ þ bii
1 f i
1 ðyi
1 ðt
τi
1
σ i þ σ i
1 ÞÞ þ Ii ;

i A N½2; n

ð10Þ

maxfðaij þ bij Þlj pj ; ðaij þ bij Þlj qj g ¼ ðaij þ bij Þlj pj Furthermore, (10) can be rewritten as ( y_ 1 ðtÞ ¼
d1 y1 ðtÞ þ a11 f 1 ðy1 ðtÞÞ þ b1n f n ðyn ðt
τÞÞ þ I 1 y_ i ðtÞ ¼
di yi ðtÞ þ aii f i ðyi ðtÞÞ þ bii
1 f i
1 ðyi
1 ðt
τÞÞ þ I i ;

and minfðaij þ bij Þlj pj ; ðaij þbij Þlj qj g ¼ ðaij þ bij Þlj qj

i A N½2; n ð11Þ

Combining (i) with (ii), we obtain ð
di þ ðaii þ bii þ Bi β i αi
1 Þli Þðqi
pi Þ 4

n X

∣lj ðaij þ bij Þ∣ðqj
pj Þ;

iA N½1; n:

where τ ¼ ð7Þ

j a i;j ¼ 1

From Corollary A 2 a~ 11 ðq1
p1 Þ 6~ 6 a 21 ðq1
p1 Þ B~ ¼ 6 6 ⋮ 4 a~ n1 ðq1
p1 Þ

in [7], it follows that 3 ⋯ a~ 1n ðqn
pn Þ 7 ⋯ a~ 2n ðqn
pn Þ 7 7 7 ⋮ ⋮ 5 ⋯ a~ nn ðqn
pn Þ

is invertible. Furthermore, B~ can be rewritten as B~ ¼ A~ diagfq1
p1 ; q1
p1 ; …; qn
pn g: According to qi
pi 4 0, iA N½1; n; it is easy to see that diagf q1
p1 ; q1
p1 ; …; qn
pn g is invertible. Thus, A~ is invertible, which implies that (5) has a unique solution. Therefore, system (2) with activation function (3) has and has only one equilibrium point in subset Ω2. □ Remark 2. From Theorem 1 in [1], it follows that system (2) with activation function (3) has and only has 3n equilibrium points in Rn. In other words, the system (2) with activation function (3) has 2n equilibrium points in Ω1, one equilibrium point in Ω2, and 3n
2n
1 equilibrium points in Ω3. However, the proof process of only one equilibrium point in Ω2 is very complex (see Theorem 1 in [1]), which often appears in other different literature [2–5]. A

1 Pn τ: n j¼1 j

In the following, we shall discuss the dynamical behaviors of equilibrium points of n-neuron DHNNs, and show that under certain conditions n-neuron DHNNs with activation function (3) have exactly 3n equilibrium points, where the 2n equilibrium points in Ω1 are locally exponentially stable, and other 3n
2n equilibrium points in Ω2 and Ω3 are unstable. Theorem 2. Suppose
d1 p1 þ a11 f 1 ðp1 Þ þ maxfb1n un ; b1n vn g þ I 1 o 0

ð12aÞ


d1 q1 þ a11 f 1 ðq1 Þ þ minfb1n un ; b1n vn g þ I 1 40

ð12bÞ


di pi þ aii f i ðpi Þ þmaxfbii
1 ui
1 ; bii
1 vi
1 g þ I i o 0;

i A N½2; n ð12cÞ


di qi þ aii f i ðqi Þ þminfbii
1 ui
1 ; bii
1 vi
1 g þ I i 4 0;

i A N½2; n; ð12dÞ

then system (8) with activation function (3) has exactly 3n equilibrium points in Rn, and 2n equilibrium points in Ω1 are locally exponentially stable. Proof. From the form of the system (8), we can observe that it is a special case of system (2). According to (12a)–(12d), it follows that

S. Ding et al. / Neurocomputing 177 (2016) 628–635

631

the known conditions of Theorem 2 in [1] are satisfied. Thus, the proof is obtained immediately.□

It is clear that λ1 ¼
d1 þ a11 l1 is a real root of (15). From (3), it follows that

Theorem 3. Suppose (12a)–(12d) hold, then the equilibrium points ðx1 ; x2 ; …; xn ÞT A Ω3 of system (8) are unstable.

f 1 ðpÞ ¼ l1 p1 þ c1 ;

Proof. From Remark 1, it follows there k A N½1; n
1; i A ½1; C kn ; j A ðiÞ

ðjÞ

½1; 2n
k  such that ðx1 ; x2 ; …; xn ÞT A Ξ k  Ξ k . Without loss of ðC 2 Þ

ð2n
2 Þ

ðC 2 Þ

ð2n
2 Þ

generality, choose k ¼ 2, i ¼ C 2n , j ¼ 2n
2 , then Ξ 2 n  Ξ 2 ¼ ½p1 ; q1   ½p2 ; q2   ∏ni¼ 3 ðqi ; þ 1Þ. In the following, we shall prove is that the equilibrium point ðx1 ; x2 ; …; xn ÞT in subset Ξ 2 n  Ξ 2 unstable. Let yðt; ϕÞ be a solution of system (11) with initial con-

dition ðϕ1 ; ϕ2 ; …; ϕn Þ near ðx1 ; x2 ; …; xn Þ . If U is any neighborhood T

T

f 1 ðq1 Þ ¼ l1 q1 þc1

ð16Þ

where l1 ¼

v1
u1 ; q1
p1

c1 ¼

u1 q1
v1 p1 q1
p1

According to (12a) and (12b), we can obtain
d1 p1 þ a11 f 1 ðp1 Þ o
d1 q1 þ a11 f 1 ðq1 Þ: This, together with (16), yields λ1 ¼
d1 þ a11 l1 4 0: Therefore, the equilibrium point ðx1 ; x2 ; …; xn ÞT of system (8) is unstable in subset ð2n
2 Þ

2

of the equilibrium point ðx1 ; x2 ; …; xn ÞT , there always exists an

nÞ Ξ ðC  Ξ2 2

initial condition ðϕ1 ; ϕ2 ; …; ϕn ÞT A U such that the solution yðt; ϕÞ

that the equilibrium points in other subsets

with this initial condition ðϕ1 ; ϕ2 ; …; ϕn ÞT gets out of region

Ξ

ðC 2n Þ 2

Ξ

ð2n
2 Þ . 2

Then, the equilibrium point ðx1 ; x2 ; …; xn ÞT must be ðC 2 Þ

ð2n
2 Þ

unstable. Thus, we may assume that yðt; ϕÞ A Ξ 2 n  Ξ 2 t Z 0. Let ui ðtÞ ¼ yi ðtÞ
xi , then system (11) becomes (

for all

ð2n
2 Þ

Ξ 2

Þ

jA N½1; 2

n
k

. Following similar arguments above, we can show

are .

where

k A N½1; n
1; i A N½1; C kn ,

□

vi
ui ; qi
pi

ci ¼

u_ 1 ðtÞ ¼
d1 u1 ðtÞ þa11 ðf 1 ðx1 þ u1 ðtÞÞ
f 1 ðx1 ÞÞ
b1n f n ðxn Þ þ b1n f n ðxn þ un ðt
τÞÞ u_ i ðtÞ ¼
di ui ðtÞ þ aii ðf i ðxi þ ui ðtÞÞ
f i ðxi ÞÞ
bii
1 f i
1 ðxi
1 Þ þ bii
1 f i
1 ðxi
1 þ ui
1 ðt
τÞÞ;

ui qi
v i pi ; qi
pi

i ¼ 1; 2; …; n:

ð13Þ

iA N½2; n

If (12a)–(12d) and

x1 þu1 ðtÞ A ½p1 ; q1 ;

x1 þu1 ðt
τÞ A ½p1 ; q1 

x2 þu2 ðtÞ A ½p2 ; q2 ;

x2 þu2 ðt
τÞ A ½p2 ; q2 

n

∏ ∣di
aii li ∣ Z∣b1n bnn
1 bn
1n
2 ⋯b21 ∣l1 l2 ⋯ln

ð17Þ

i¼1

Table 2 Equilibrium points in Ω2 and Ω3.

and xi þ ui ðtÞ A ðqi ; þ 1Þ;

xi þ ui ðt
τÞ A ðqi ; þ 1Þ;

i A N½3; n;

this, together with (13) yields 8 u_ 1 ðtÞ ¼ ð
d1 þ a11 l1 Þu1 ðtÞ > > > > < u_ 2 ðtÞ ¼ ð
d2 þ a22 l2 Þu2 ðtÞ þ b21 l1 u1 ðt
τÞ u_ 3 ðtÞ ¼
d3 u3 ðtÞ þ b32 l2 u2 ðt
τÞ > > > > : u_ ðtÞ ¼
d u ðtÞ; iA N½4; n; i

Equilibrium point

Value

Region

Region symbol

xe9

ð0; 0; 0ÞT 1  1 14401 T 12;
144;
6480  1 1 14401T
6; 72; 3240  1 19801  2 T 330; 5940 ;
55  1  1 T
660;
19801 11880; 55  19801 1  1 T
11880; 12;
660 19801 1 1 T 5940 ;
6; 330  1649 31 1 T
990 ;
18; 55 1649 31  2 T 495 ; 9 ;
55 1  359 101 T 12;
216;
45

½
1; 2  ½
1; 2Þ  ½
1; 2

Ωð1Þ 2

½
1; 2  ½
1; 2  ð
1;
1Þ

Ωð1Þ 3

½
1; 2  ½
1; 2  ð2; 1Þ

Ωð2Þ 3

½
1; 2  ð2; 1Þ  ½
1; 2

Ωð3Þ 3

½
1; 2  ð
1;
1Þ  ½
1; 2

Ωð4Þ 3

ð
1;
1Þ  ½
1; 2  ½
1; 2

Ωð5Þ 3

ð2; 1Þ  ½
1; 2  ½
1; 2

Ωð6Þ 3

xe10 xe11

ð14Þ

xe12 xe13

i i

xe14

whose characteristic equation is 2 3 λ þ d1
a11 l1 0 0 n 6
b l e
λτ λ þ d
a l 0 7 det4 5 ∏ ðλ þ di Þ ¼ 0 21 1 2 22 2 0
b32 l2 e
λτ λ þ d3 i ¼ 4

xe15 xe16

ð15Þ

xe17 xe18

ð
1;
1Þ  ð
1;
1Þ  ½
1; 2 Ωð7Þ 3 ð2; 1Þ  ð2; 1Þ  ½
1; 2

Ωð8Þ 3

½
1; 2  ð
1;
1Þ  ð
1;
1Þ Ωð9Þ 3

Table 1 Equilibrium points in Ω1. Equilibrium point

Value

xe1



xe2 xe3 xe4 xe5 xe6 xe7 xe8

2 nÞ

Theorem 4. Let li ¼

Note that

unstable,

ðjÞ

ðC Ξij ðΞ ðiÞ k  Ξ k a Ξ2

 101 T

31
31 18;
18;
45 31 31 202T 9 ; 9 ; 45  14 31 199T
9 ;
18; 45 59 31 98T 18; 9 ;
45  14 31 202T
9 ; 9 ; 45  31 31 98T
18; 9 ;
45 31 14 199T 9 ;
18; 45 59 14 101T 18;
8 ;
45

Region

Region symbol

ð
1;
1Þ  ð
1;
1Þ  ð
1;
1Þ

Ωð1Þ 1

ð2; 1Þ  ð2; 1Þ  ð2; 1Þ

Ωð2Þ 1

ð
1;
1Þ  ð
1;
1Þ  ð2; 1Þ

Ωð3Þ 1

ð2; 1Þ  ð2; 1Þ  ð
1;
1Þ

Ωð4Þ 1

ð
1;
1Þ  ð2; 1Þ  ð2; 1Þ

Ωð5Þ 1

ð
1;
1Þ  ð2; 1Þ  ð
1;
1Þ

Ωð6Þ 1

ð2; 1Þ  ð
1;
1Þ  ð2; 1Þ

Ωð7Þ 1

ð2; 1Þ  ð
1;
1Þ  ð
1;
1Þ

Ωð8Þ 1

632

S. Ding et al. / Neurocomputing 177 (2016) 628–635

Theorem 3, we assume that yðt; ϕÞ A Λ for all t Z 0. Let

Table 3 Equilibrium points in Ω3. Equilibrium point

ui ðtÞ ¼ yi ðtÞ
xi ;

Value

Region symbol

½
1; 2  ð2; 1Þ  ð2; 1Þ

Ωð10Þ 3

xe19



xe20

59

ð2; 1Þ  ½
1; 2  ð
1;
1Þ



Ωð11Þ 3

xe21

ð
1;
1Þ  ½
1; 2  ð2; 1Þ

xe22



Ωð12Þ 3 Ωð13Þ 3 Ωð14Þ 3 Ωð15Þ 3 Ωð16Þ 3 Ωð17Þ 3 Ωð18Þ 3

xe23 xe24 xe25 xe26 xe27

202
16; 359 108; 45

T

Region

T
16;
601 270  14 1 2401 T
9 ; 12; 540

18;

 59 2 T
826 495; 18;
55 3301 14 1 T 990 ;
9 ; 55 31 1 1199T 9 ;
6; 270  31 1 T
18; 12;
1199 540  1 181 199T
6;
108; 45  1 721 98T 12; 216;
45

ð
1;
1Þ  ð2; 1Þ  ½
1; 2 ð2; 1Þ  ð
1;
1Þ  ½
1; 2 ð2; 1Þ  ½
1; 2  ð2; 1Þ ð
1;
1Þ  ½
1; 2  ð
1;
1Þ ½
1; 2  ð
1;
1Þ  ð2; 1Þ ½
1; 2  ð2; 1Þ  ð
1;
1Þ

i A N½1; n;

then system (11) is transformed into (13). Note that xi þ ui ðt
τÞ A ½pi ; qi ;

xi þ ui ðtÞ A ½pi ; qi ;

i A N½1; n

According to (3), (13) can be rewritten as ( u_ 1 ðtÞ ¼ ð
d1 þ a11 l1 Þu1 ðtÞ þ b1n ln un ðt
τÞ u_ i ðtÞ ¼ ð
di þ aii li Þui ðtÞ þ bii
1 li
1 ui
1 ðt
τÞ;

ð18Þ

iA N½2; n:

whose characteristic equation is 2

λ þ d1
a11 l1 6 6
b21 l1 e
λτ 6 6 0 6 det6 6 ⋮ 6 6 0 4 0

⋯

0


b1n ln e
λτ

λ þ d2
a22 l2 ⋯

0

0

⋱

0

0

0
b32 l2 e


λτ

⋮ 0

⋱

0

⋯

⋮

⋮

λ þdn
1
an
1n
1 ln
1

0


bnn
1 ln
1 e
λτ

λ þ dn
ann ln

3 7 7 7 7 7 7¼0 7 7 7 5

i.e.,

λn þ

n X

n
1

ðdi
aii li Þλ

þ⋯þ

i¼1

n X

n

∏

j ¼ 1 i ¼ 1;j a i

n

ðdi
aii li Þλ þ ∏ ðdi
aii li Þ i¼1


ðbnn
1 bn
1n
2 ⋯b21 b1n l1 l2 ⋯ln e
λnτ Þ ¼0

ð19Þ

It is clear that iωðω A RÞ is a roof of (19) if and only if

ω satisfies

p0 ðiωÞ þ p1 ðiωÞ þ ⋯ þ pn
1 ðiωÞ þ pn
pn þ 1 ð cos ωnτ
i sin ωnτÞ ¼ 0 n
1

n

ð20Þ

where p0 ¼ 1 n X p1 ¼ ðdi1
ai1 i1 li1 Þ i1 ¼ 1

Fig. 1. Transient behavior of the equilibrium point xe1.

p2 ¼

n
1 X

n X

ðdi1
ai1 i1 li1 Þðdi2
ai2 i2 li2 Þ

i1 ¼ 1 i2 ¼ i1 þ 1

p3 ¼

n
2 X

n
1 X

n X

ðdi1
ai1 i1 li1 Þðdi2
ai2 i2 li2 Þðdi3
ai3 i3 li3 Þ

i1 ¼ 1 i2 ¼ i1 þ 1 i3 ¼ i2 þ 1

⋮ pn
1 ¼

n X

n

∏

ðdi
aii li Þ

j ¼ 1 i ¼ 1;j a i

n

pn ¼ ∏ ðdi
aii li Þ i¼1

pn þ 1 ¼ bnn
1 ⋯b32 b21 b1n l1 l2 ⋯ln :

Fig. 2. Transient behavior of the equilibrium point xe2.

are satisfied, then, the equilibrium point ðx1 ; x2 ; …; xn ÞT in Λ of system (8) is unstable, where Λ ¼ ∏ni¼ 1 ½pi ; qi . Proof. In the following, we shall prove that the equilibrium point ðx1 ; x2 ; …; xn ÞT is unstable in Λ. Let yðt; ϕÞ be a solution of system (11) with initial condition ðϕ1 ; ϕ2 ; …; ϕn ÞT near ðx1 ; x2 ; …; xn ÞT . Similar to the discussion of

In the following, there are two cases to be discussed: n is even and n is odd. (i) When n is even, from (20), it follows 8 n=2 X > > > ð
1Þi pn
2i ω2i ¼ pn þ 1 cos ωnτ > > > i > 2i þ 1 > ð
1Þ pn
2i
1 ω ¼
pn þ 1 sin ωnτ: > : i¼0

(ii) When n is odd, similar to (21a), it is easy to compute 8 ðn
1Þ=2 X > > > ð
1Þi pn
2i ω2i ¼ pn þ 1 cos ωnτ > > < i¼0 ðn X
1Þ=2 > > > > ð
1Þi pn
2i
1 ω2i þ 1 ¼
pn þ 1 sin ωnτ: > : i¼0

ð21bÞ

S. Ding et al. / Neurocomputing 177 (2016) 628–635

Fig. 6. Transient behavior of the equilibrium point xe6.

Fig. 3. Transient behavior of the equilibrium point xe3.

Fig. 7. Transient behavior of the equilibrium point xe7.

Fig. 4. Transient behavior of the equilibrium point xe4.

Fig. 8. Transient behavior of the equilibrium point xe8.

Fig. 5. Transient behavior of the equilibrium point xe5.

Taking square on both sides of (21a) and (21b) and summing up, we get p0 ω

2n

þðp21
2p2 p0 Þ

ω

2ðn
1Þ

þ ðp22
2p3 p1 þ 2p4 p0 Þ

ω

2ðn
2Þ

¼ 0;

where p21
2p2 p0 ¼

þ⋯

n X

ðdi1
ai1 i1 li1 Þ2

i1 ¼ 1

þ ðp2n
2
2pn
1 pn
3 þ2pn pn
4 Þω4 þ ðp2n
1
2pn pn
2 Þω2 þ p2n
p2n þ 1

633

ð22Þ

p22
2p3 p1 þ 2p4 p0 ¼

n
1 X

n X

i1 ¼ 1 i2 ¼ i1 þ 1

ðdi1
ai1 i1 li1 Þ2 ðdi2
ai2 i2 li2 Þ2

634

S. Ding et al. / Neurocomputing 177 (2016) 628–635

Fig. 9. Transient behavior of the equilibrium point xe9. Fig. 11. Transient behavior of the equilibrium point xe11 .

b13 ¼ b21 ¼ 0:05; I 1 ¼ I 2 ¼ I 3 ¼ 0; a33 ¼ 2; b32 ¼ 0:02; τij ðtÞ ¼ j: Choose the activation functions with two corner points 8 > <
1;
1 ox o
1;
1 r x r 2; f i ðxÞ ¼ x; > : 2; 2 ox o þ 1;

ð24Þ

where i A N½1; 3. It is easy to verify that the parameters are satisfied with (12a)– (12d) and (17), i.e.,
d1 p1 þ a11 f 1 ðp1 Þ þ maxfb13 u3 ; b13 v3 g þ I 1 ¼
0:5 o 0
d1 q1 þ a11 f 1 ðq1 Þ þ minfb13 u3 ; b13 v3 g þ I 1 ¼ 1:15 4 0
d2 p2 þ a22 f 2 ðp2 Þ þ maxfb21 u1 ; b21 v1 g þ I 2 ¼
0:5 o 0
d2 q2 þ a22 f 2 ðq2 Þ þ minfb21 u1 ; b21 v1 g þ I 2 ¼ 1:15 4 0
d3 p3 þ a33 f 3 ðp3 Þ þ maxfb32 u2 ; b32 v2 g þ I 3 ¼
1:06 o 0
d3 q3 þ a33 f 3 ðq3 Þ þ minfb32 u2 ; b32 v2 g þ I 3 ¼ 2:18 4 0

Fig. 10. Transient behavior of the equilibrium point xe10 .

⋮ p2n
1
2pn pn
2

and

n X

¼

n

∏

j ¼ 1 i ¼ 1;j a i

ðdi
aii li Þ

2

3

∏ ∣ðdi
aii li Þ∣ ¼ 0:396 Z 0:00005 ¼ ∣b13 b32 b21 ∣l1 l2 l3 :

i¼1

From (17), it follows that p2n
p2n þ 1 4 0, this means that coefficients of (22) are all positive. Thus, (22) has no positive root, which implies that (19) has no root with zero real part for all τ Z0. On the other hand, when τ ¼ 0, (19) becomes

λn þ

n X

ðdi
aii li Þλ

n
1

þ⋯þ

i¼1

n X

n

∏

j ¼ 1 i ¼ 1;j a i

n

points in Ω1 are locally exponentially stable, and 33
23 in Ω2 and Ω3 equilibrium points are unstable, where Ω1 ¼ ⋃8i ¼ 1 ΩðiÞ 1 , Ω2 ¼

ðdi
aii li Þλ

þ ∏ ðdi
aii li Þ
bnn
1 bn
1n
2 ⋯b21 b1n l1 l2 ⋯ln ¼ 0:

From Theorem 2, it follows that the system (8) with the activation functions (24) has exactly 33 equilibrium points. After the simple computation, it is found that the 33 equilibrium points are listed as Tables 1–3. xn3 ¼
14401 6480 
2:22. According to Theorems 2–4, we obtain that 23 equilibrium

ð23Þ

ð3Þ 18 Ωð1Þ 2 and Ω3 ¼ ⋃i ¼ 1 Ω1 . To further illustrate the theoretical result,

From (12a)–(12d), it is easy to observe di
aii li o 0; i A N½1; n. Thus, Pn i ¼ 1 ðdi
aii li Þ o 0. Therefore, (23) has root with positive real part. Applying Lemma 1 in [1], we can conclude that (19) has root with positive real part for all τ Z0. Therefore, the equilibrium point ðx1 ; x2 ; …; xn ÞT in Λ of system (8) is unstable. □

we choose all equilibrium points xe1 , xe2, xe3 , xe4 , xe5, xe6 ; xe7 , xe8 A Ω1 , xe9 A Ω2 , xe10 , xe11 , A Ω3 and 20 random points near them. By using MATLAB, the trajectories of the system (8) with the activation functions (24) are shown in Figs. 1–11, we can observe that equilibrium points xe1 , xe2, xe3 , xe4 , xe5, xe6 , xe7 and xe8 are locally exponentially stable, but other equilibrium points xe9 , xe10 and xe11 are unstable.

4. Numerical example

5. Conclusions

i¼1

Example 1 (Zeng and Wang [8]). Consider the system (8) with the parameters: a11 ¼ a22 ¼ 1:5;

d1 ¼ d2 ¼ d3 ¼ 0:9;

In this paper, a simple and elementary proof of the equilibrium uniqueness of delayed competitive neural networks with the activation function possessing two corner points is given, which can greatly simplify the proof complexity of Theorem 1 in [1]. In

S. Ding et al. / Neurocomputing 177 (2016) 628–635

addition, for n-neuron DHNNs with the activation function possessing two corner points, a sufficient condition is obtained to analyze the dynamical behaviors of all equilibrium points of the nneuron DHNNs. In other words, under this sufficient condition, the n-neuron DHNNs have exactly 3n equilibrium points, 2n of which are stable, and others are unstable. This sufficient condition is a slight generalization of Corollary 3 in [1]. Finally, a numerical example with their computer simulations is given to show the effectiveness of the obtained results.

635 Shuang Ding received B.S. degree from Harbin University in China in 2013. She is studying for the M.S. degree in School of Mathematical Science Heilongjiang University in China. Her research interests include the theory of genetic regulatory networks and neural networks.

Acknowledgment This work was partially supported by the National Natural Science Foundation of China (No. 11371006) and the Natural Science Foundation of Heilongjiang Province (No. A201416).

References [1] Xiaobing Nie, Jinde Cao, Shumin Fei, Multistability and instability of delayed competitive neural networks with nondecreasing piecewise linear activation functions, Neurocomputing 119 (2013) 281–291. [2] Lili Wang, Wenlian Lu, Tianping Chen, Coexistence and local stability of multiple equilibria in neural networks with piecewise linear nondecreasing activation functions, Neural Netw. 23 (2) (2010) 189–200. [3] Jinde Cao, Gang Feng, Yanyan Wang, Multistability and multiperiodicity of delayed Cohen–Grossberg neural networks with a general class of activation functions, Phys. D: Nonlinear Phenom. 237 (13) (2008) 1734–1749. [4] Xiaobing Nie, Zhenkun Huang, Multistability and multiperiodicity of high-order competitive neural networks with a general class of activation functions, Neurocomputing 82 (2012) 1–13. [5] Xiaobing Nie, Jinde Cao, Multistability of second-order competitive neural networks with nondecreasing saturated activation functions, IEEE Trans. Neural Netw. 22 (11) (2011) 1694–1708. [6] E.F. Wang, S.M. Shi, Multivariate polynomial, Advanced Algebra, High Education Press, Beijing, 2003, p. 35 (in Chinese). [7] P.J. Erdelsky, A general theorem on dominant-diagonal matrices, Linear Algebra Appl. 1 (2) (1968) 203–209. [8] Zhigang Zeng, Jun Wang, Multiperiodicity of discrete-time delayed neural networks evoked by periodic external inputs, IEEE Trans. Neural Netw. 17 (5) (2006) 1141–1151.

Xiangyu Gao received B.S. degree from Zhengzhou University, Zhengzhou, China, in 2002. In 2008, he had M.S. degree from Heilongjiang University, Harbin, China. In 2014, he graduated from Harbin Institute of Technology with a Ph.D. degree in Control Theory and Engineering. Since 2002, he has been at Heilongjiang University, where he is currently an associate professor in School of Mathematical Science. His current research interests include neural networks, genetic regulatory networks and optimal control theory and its application. He has authored more than 10 research papers.

Julong Tan received B.S. degree from Daqing Normal University in China in 2013. He is studying for the M.S. degree in School of Mathematical Science Heilongjiang University in China. His research interests include genetic regulatory networks and neural networks.