Multistability and instability of delayed competitive ... - Semantic Scholar

Neurocomputing 119 (2013) 281–291

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Multistability and instability of delayed competitive neural networks with nondecreasing piecewise linear activation functions Xiaobing Nie a,b,n, Jinde Cao a,b, Shumin Fei b a b

Research Center for Complex Systems and Network Sciences, and Department of Mathematics, Southeast University, Nanjing 210096, China School of Automation, Southeast University, Nanjing 210096, China

art ic l e i nf o

a b s t r a c t

Article history: Received 10 July 2012 Received in revised form 29 November 2012 Accepted 13 March 2013 Communicated by H. Jiang Available online 14 June 2013

In this paper, we investigate the exact existence and dynamical behaviors of multiple equilibrium points for delayed competitive neural networks (DCNNs) with a class of nondecreasing piecewise linear activation functions with 2rðr≥1Þ corner points. It is shown that under some conditions, the N-neuron DCNNs can have and only have ð2r þ 1ÞN equilibrium points, ðr þ 1ÞN of which are locally exponentially stable, based on decomposition of state space, fixed point theorem and matrix theory. In addition, for the activation function with two corner points, the dynamical behaviors of all equilibrium points for 2-neuron delayed Hopfield neural networks(DHNNs) are completely analyzed, and a sufficient criterion derived for ensuring the networks have exactly nine equilibrium points, four of which are stable and others are unstable, by discussing the distribution of roots of the corresponding characteristic equation of the linearized delayed system. Finally, two examples with their simulations are presented to verify the theoretical analysis. & 2013 Elsevier B.V. All rights reserved.

Keywords: Delayed competitive neural networks Multistability Instability Piecewise linear activation functions

1. Introduction In the past decades, some famous neural network models, including Hopfield neural networks, cellular neural networks, Cohen–Grossberg neural networks and bidirectional associative memory neural networks, had been proposed in order to solve some practical problems. It should be mentioned that in the above network models only the neuron activity is taken into consideration. That is, there exists only one type of variable—the state variables of the neural neurons in these models. However, in a dynamical network, the synaptic weights also vary with respect to time due to the learning process, and the variation of connection weights may have influences on the dynamics of neural network. Competitive neural networks constitute an important class of neural networks, which model the dynamics of cortical cognitive maps with unsupervised synaptic modifications. In this model, there are two types of state variable: that of the short-term memory (STM) describing the fast neural activity and that of long-term memory (LTM) describing the slow unsupervised synaptic modifications. Up to now, there have been considerable works on mono-stability or synchronization of competitive neural networks [1–5]. On the other hand, in the applications of associative memory storage, pattern recognition, decision making and digital selection,

n

Corresponding author. Tel.: +86 13914753494. E-mail address: [email protected] (X. Nie).

0925-2312/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2013.03.030

existence of many equilibrium points or periodic solutions is a necessary feature [6–9]. For example, in an associative memory neural networks, the addressable memories are stored as stable equilibrium points or periodic orbits. Hence, it is necessary that there exist multiple stable equilibrium points or periodic orbits, which are usually referred to multistability or multiperiodicity. In recent years, the multistability or multiperiodicity of neural networks has attracted the attention of many researchers [10–35]. In [10,11], based on decomposition of state space, the authors investigated the multistability of delayed Hopfield neural networks (DHNNs), and showed that the n-neuron neural networks can have 2n stable orbits located in 2n subsets of Rn . Cao et al. [12] extended above method to the Cohen–Grossberg neural networks swith nondecreasing saturated activation functions with two corner points. In [13,14], the multistability of almost-periodic solution in delayed neural networks was studied. Kaslik et al. [15,16] firstly revealed the effect of impulse on the multistability of neural networks. In [17–19], high-order synaptic connectivity was introduced into neural networks and the multistability and multiperiodicity were considered respectively for high-order neural networks based on decomposition of state space, Cauchy convergence principle and inequality technique. In [20,21], for two special classes of discontinuous activation functions which are piecewise constants in the state space, the multistability of neural networks with and without time delay were investigated. In [22–26], the authors indicated that under some conditions, there exist 3n equilibrium points for the n-neuron neural networks, and 2n of which are locally exponentially stable. It should be pointed out that

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X. Nie et al. / Neurocomputing 119 (2013) 281–291

all the above mentioned works never discussed the exact number of equilibrium points. In addition, they only gave emphasis on 2n stable equilibrium points, never studied the dynamical behaviors of other 3n –2n equilibrium points. To the best of our knowledge, there are few papers studying the exact existence and dynamics of all equilibrium points of neural networks. In order to increase storage capacity, [36] investigated the Hopfield neural networks with nondecreasing piecewise linear activation functions with 2r corner points. It was proposed in [36] that under some conditions, the n-neuron Hopfield neural networks can have and only have ð2r þ 1Þn equilibrium points, ðr þ 1Þn of which are locally exponentially stable and others are unstable. It should be noted that the considered model in [36] is without delay, which is described by ordinary differential equations. It is well known, for ordinary differential equations, that the roots of the corresponding characteristic equation of the linearized equations can be used to determine the dynamics of the trivial solutions. Similar equivalence holds for delay differential equations. However, the characteristic function corresponding to the linearized delay differential equations is no longer ordinary polynomial, rather, is transcendental polynomial. It is more difficult and challenging to study the distribution of roots of transcendental polynomial equation. As far as we know, no result has been obtained about the exact existence and dynamics of all equilibrium points for delayed neural networks. Based on above motivations, in this paper, we shall study the exact existence and dynamical behaviors of multiple equilibrium points for delayed competitive neural networks (DCNNs) with a class of activation functions, which are nondecreasing piecewise linear activation functions with 2rðr≥1Þ corner points. The system is a very general neural network model, including DHNNs [10,13,22,23,25,26] and Hopfield neural networks without delay [36] as special cases. The main contributions of this paper are three-fold as follows:  Firstly, for nondecreasing piecewise linear activation functions with two corner points, we shall present some sufficient conditions under which the N-neuron DCNNs can have and only have 3N equilibrium points, 2N of which are locally exponentially stable, by means of decomposition of state space, fixed point theorem and matrix theory.  Secondly, we shall extend above method to more general activation functions with 2r corner points, and show that the N-neuron DCNNs can have and only have ð2r þ 1ÞN equilibrium points, (r+1)N of which are locally exponentially stable.  Thirdly, For the 2-neuron DHNNs with activation functions possessing two corner points, we shall precisely figure out all equilibrium points and analyze the dynamical behaviors of each equilibrium point. It shall be proved that under some conditions, the networks has exactly nine equilibrium points, four of which are stable and others are unstable, by discussing the distribution of roots of the corresponding characteristic equation of the linearized delay differential equations.

where i ¼ 1; 2; …; N; j ¼ 1; 2; …; P, 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 the connection weight between the ith neuron and the jth neuron, Bi is the strength of the external stimulus, 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 satisfy 0≤τj ðtÞ≤τj . Denote τ ¼ max1≤j≤N fτj g. After setting Si ðtÞ ¼ ∑Pj¼ 1 mij ðtÞyj ¼ yT mi ðtÞ, where y ¼ ðy1 ; y2 ; …; yP ÞT , mi ðtÞ ¼ ðmi1 ðtÞ; mi2 ðtÞ; …; miP ðtÞÞT , and assuming the input stimulus y to be normalized with unit magnitude jyj2 ¼ y21 þ ⋯ þy2P ¼ 1, then the above networks are simplified as 8 N N > > < 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 þ I i ; j¼1

j¼1

> > : S_ i ðtÞ ¼ −αi Si ðtÞ þ β f ðxi ðtÞÞ; i ¼ 1; 2; …; N: i i ð2Þ In this paper, we consider the following nondecreasing piecewise linear activation functions with 2r corner points, which are employed in [18,36]: 8 > u1i ; −∞ o x o p1i ; > > > > 2 1 > u −ui > > > 1i ðx−p1i Þ þ u1i ; p1i ≤x≤q1i ; > 1 > > > qi −pi > > 2 > > q1i o x o p2i ; > ui ; > > > 3 2 > > ui −ui < ðx−p2i Þ þ u2i ; p2i ≤x≤q2i ; ð3Þ f i ðxÞ ¼ q2i −p2i > > > > u3i ; q2i o x o p3i ; > > > > >⋮ ⋮ > > > > > urþ1 −uri > i r r r r > > > qr −pr ðx−pi Þ þ ui ; pi ≤x≤qi ; > > i i > > > rþ1 : ui ; qri o x o þ ∞; where r≥1, uki ðk ¼ 1; 2; …; r þ 1Þ, pki ; qki ðk ¼ 1; 2; …; rÞ are constants with u1i o u2i o⋯ o urþ1 , −∞ o p1i oq1i o p2i oq2i o ⋯ opri o qri o þ i ∞. When r ¼ 1, above activation functions fi reduce to the following piecewise linear function with two corner points at pi ; qi : 8 ui ; −∞ o x o pi ; > > < vi −ui ðx−pi Þ; pi ≤x≤qi ; ð4Þ f i ðxÞ ¼ ui þ qi −pi > > : vi ; qi o x o þ ∞: Especially, activation function (4) includes the following standard activation function as its special case: f i ðxÞ ¼

jx þ 1j−jx−1j : 2

ð5Þ

_ Notations. xðtÞ denotes the derivative of function x(t). Denote mi ¼ minfβi ui ; βi vi g, M i ¼ maxfβi ui ; βi vi g.

3. Main results Finally, two examples with their simulations shall be given to verify and illustrate the validity of the obtained results.

2. Model formulation and preliminaries In this paper, we consider the following DCNNs:

8 N N P > > < x_ i ðtÞ ¼ −di xi ðtÞ þ ∑ aij f j ðxj ðtÞÞ þ ∑ bij f j ðxj ðt−τj 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

ð1Þ

3.1. Multistability of DCNNs with activation functions (4) Denote ð−∞; pi Þ ¼ ð−∞; pi Þ1  ½pi ; qi 0  ðqi ; þ∞Þ0 ; ½pi ; qi  ¼ ð−∞; pi Þ0 ½pi ; qi 1  ðqi ; þ∞Þ0 ; ðqi ; þ∞Þ ¼ ð−∞; pi Þ0  ½pi ; qi 0  ðqi ; þ∞Þ1 ; R ¼ ð−∞; pi Þ∪½pi ; qi ∪ðqi ; þ∞Þ. So RN can be divided into 3N subsets: ( N

ðiÞ

ðiÞ

ðiÞ

ðiÞ ðiÞ ∏ ð−∞; pi Þδ1  ½pi ; qi δ2  ðqi ; þ∞Þδ3 ; ðδðiÞ 1 ; δ2 ; δ 3 Þ

i¼1

)

¼ ð1; 0; 0Þ or ð0; 1; 0Þ orð0; 0; 1Þ; i ¼ 1; 2; …; N ;

ð6Þ

X. Nie et al. / Neurocomputing 119 (2013) 281–291

and RN can be divided into three subsets: ( N

ðiÞ

i¼1

( Ω2 ¼

ðiÞ

Therefore, the equations F i ðξÞ ¼ 0ði ¼ 1; 2; …; NÞ have a unique root xn ¼ ðxn1 ; xn2 ; …; xnN ÞT in Λ1 . Thus, we also obtain a unique equiliT n −1 n brium point ðxn T ; Sn ÞT ¼ ðxn1 ; xn2 ; …; xnN ; β1 α−1 1 f 1 ðx1 Þ; β 2 α2 f 2 ðx2 Þ; …; −1 n T N −1 −1 βN αN f N ðxN ÞÞ of system (2) in Λ1  ∏i ¼ 1 ½αi mi ; αi M i . (ii) Subset Ω2 : Pick Λ2 ¼ Ω2 ¼ ∏N i ¼ 1 ½pi ; qi . Denote li ¼ ðvi − ui Þ=ðqi −pi Þ, ci ¼ ðui qi −vi pi Þ=ðqi −pi Þ, then f i ðxi Þ ¼ li xi þ ci ; xi ∈½pi ; qi . In this case, conditions (7) and (8) are equivalent to the following:

)

∏ ð−∞; pi Þδ  ðqi ; þ∞Þ1−δ ; δðiÞ ¼ 1 or 0; i ¼ 1; 2; …; N ;

Ω1 ¼

)

N

283

∏ ½pi ; qi  ;

i¼1

Ω3 ¼ RN −Ω1 −Ω2 : Hence, Ω1 is composed of 2N regions, Ω2 is composed of one region, and Ω3 is composed of 3N −2N −1 regions. Now, we will prove the following theorem.

−di pi þ ðaii þ bii þ Bi βi α−1 i Þli pi þ

N

∑

lj maxfðaij þ bij Þpj ; ðaij þ bij Þqj g

j≠i;j ¼ 1

N

þ ∑ ðaij þ bij Þcj þ Bi βi α−1 i ci þ I i o 0;

Theorem 1. If the following conditions:

ð12Þ

j¼1

−di pi þ ðaii þ bii þ Bi βi α−1 i Þf i ðpi Þ þ

N

∑

j≠i;j ¼ 1

maxfðaij þ bij Þuj ; ðaij þ bij Þvj g þ I i o0;

ð7Þ

−di qi þ ðaii þ bii þ Bi βi α−1 i Þli qi þ

N

∑

lj minfðaij þ bij Þpj ; ðaij þ bij Þqj g

j≠i;j ¼ 1

−di qi þ ðaii þ bii þ Bi βi α−1 i Þf i ðqi Þ þ

N

N

∑

j≠i;j ¼ 1

minfðaij þ bij Þuj ; ðaij þ bij Þvj g þ I i 4 0

ð8Þ

hold for all i ¼ 1; 2; …; N, then system (2) with activation function (4) can have and only have3N equilibrium points. Proof. Note that any equilibrium point of system (2) is a root of the following equations: 8 N N > < −di xi þ ∑ ð; xi þ ∑ ðaij þ bij Þf j ðxj Þ þ Bi Si þ I i ¼ 0; ð9Þ j¼1 j¼1 > : −αi Si þ βi f i ðxi Þ ¼ 0; i ¼ 1; 2; …; N: Equivalently, the above equations can be rewritten as 8 N > < −di xi þ ðaii þ bii þ Bi βi α−1 Þf i ðxi Þ þ ∑ ðaij þ bij Þf j ðxj Þ þ I i ¼ 0; i > :

i ¼ 1; 2; …; N:

In the following, we discuss the dynamics for every subset. (i) Subset Ω1 : Pick Λ1 ¼ ∏i∈N1 ð−∞; pi Þ  ∏i∈N2 ðqi ; þ∞Þ⊂Ω1 , where N 1 ; N 2 are subsets of f1; 2; …; Ng, and N1 ⋃N2 ¼ f1; 2; …; Ng; N 1 ⋂N 2 ¼ ∅. In subset Λ1 , any equilibrium point of system (2) satisfies the following equations: 8 −d x þ ðaii þ bii þ Bi βi α−1 ∑ ðaij þ bij Þuj þ I i ¼ 0; > i Þui þ ∑ ðaij þ bij Þvj þ > < i i j∈N 2 j∈N 1 ;j≠i

i∈N 1 ;

> −d x þ ðaii þ bii þ Bi βi α−1 ∑ ðaij þ bij Þvj þ I i ¼ 0; > i Þvi þ ∑ ðaij þ bij Þuj þ : i i j∈N j∈N ;j≠i

i∈N 2 :

1

2

ð11Þ Consider the following equations: F i ðξÞ ¼ −di ξ þ ðaii þ bii þ Bi βi α−1 i Þui þ ∑ ðaij þ bij Þvj j∈N2

j∈N 1 ;j≠i

i∈N 1 ;

F i ðξÞ ¼ −di ξ þ ðaii þ bii þ Bi βi α−1 i Þvi þ ∑ ðaij þ bij Þuj j∈N1

þ ∑ ðaij þ bij Þvj þ I i ¼ 0; j∈N 2 ;j≠i

Note that every equilibrium point in subset Λ2 is a root of the following equations: −di xi þ ðaii þ bii þ Bi βi α−1 i Þli xi þ

N

j¼1

F i ð þ ∞Þ ¼ −∞; F i ðqi Þ 4 0;

i∈N2 :

i ¼ 1; 2; …; N:

ð14Þ

~ −d1 x~ 1 þ ða11 þ b11 þ B1 β1 α−1 1 Þl1 x 1 þ ∑ ða1j þ b1j Þlj xj j¼2

N

þ ∑ ða1j þ b1j Þcj þ B1 β1 α−1 1 c1 þ I 1 ¼ 0; j¼1

since 8 N > > −d1 p1 þ ða11 þ b11 þ B1 β1 α−1 Þl1 p1 þ ∑ ða1j þ b1j Þlj xj > > 1 > > j¼2 > > > > N > > > > þ ∑ ða1j þ b1j Þcj þ B1 β1 α−1 1 c1 þ I 1 o 0; > < j¼1 N > > > −d1 q1 þ ða11 þ b11 þ B1 β1 α−1 > 1 Þl1 q1 þ ∑ ða1j þ b1j Þlj xj > > j¼2 > > > > > N > > > ða1j þ b1j Þcj þ B1 β1 α−1 > 1 c1 þ I 1 4 0: : þj ∑ ¼1

Similarly, fixing x1 ; …; xi−1 ; xiþ1 ; …; xN ði ¼ 2; …; NÞ, we can find a unique x~ i ∈ðpi ; qi Þ such that

i∈N 2 :

Then, by virtue of conditions (7) and (8), we have i∈N1 ;

ðaij þ bij Þlj xj

j≠i;j ¼ 1

þ ∑ ðaij þ bij Þcj þ Bi βi α−1 i ci þ I i ¼ 0;

~ −di x~ i þ ðaii þ bii þ Bi βi α−1 i Þli x i þ

F i ð−∞Þ ¼ þ ∞; F i ðpi Þ o 0;

N

∑

N

ð10Þ

þ ∑ ðaij þ bij Þuj þ I i ¼ 0;

ð13Þ

j¼1

For any x ¼ ðx1 ; x2 ; …; xN ÞT ∈Λ2 , fixing x2 ; …; xN , we can find a unique x~ 1 ∈ðp1 ; q1 Þ such that

j≠i;j ¼ 1

−αi Si þ βi f i ðxi Þ ¼ 0;

þ ∑ ðaij þ bij Þcj þ Bi βi α−1 i ci þ I i 4 0:

N

N

∑

j≠i;j ¼ 1

ðaij þ bij Þlj xj

þ ∑ ðaij þ bij Þcj þ Bi βi α−1 i ci þ I i ¼ 0; j¼1

284

X. Nie et al. / Neurocomputing 119 (2013) 281–291

þ ½ðaimþ1 imþ1 þ bimþ1 imþ1 þ Bimþ1 βimþ1 α−1 imþ1 Þlimþ1 −dimþ1 qimþ1 o 0:

since 8 N > −1 > > > −di pi þ ðaii þ bii þ Bi βi αi Þli pi þ ∑ ðaij þ bij Þlj xj > > j≠i;j ¼ 1 > > > > N > > > > þ ∑ ðaij þ bij Þcj þ Bi βi α−1 i ci þ I i o 0; > < j¼1

ð19Þ Then, by comparing with (13), we derive that m

j¼1

N > > > ∑ ðaij þ bij Þlj xj −di qi þ ðaii þ bii þ Bi βi α−1 > i Þli qi þ > > j≠i;j ¼ 1 > > > > > N > −1 > > ðaij þ bij Þcj þ Bi βi αi ci þ I i 4 0: > : þj ∑ ¼1

B B A~ ¼ B B @

−d1 þ ða11 þ b11 þ B1 β1 α−1 1 Þl1 ða21 þ b21 Þl1

⋯ ⋯

1

ða1N þ b1N ÞlN ða2N þ b2N ÞlN

⋮

⋮

⋮

ðaN1 þ bN1 Þl1

⋯

−dN þ ðaNN þ bNN þ BN βN α−1 N ÞlN

C C C C A

ð15Þ

γ imþ1 ¼ k1 γ i1 þ k2 γ i2 þ ⋯ þ km γ im :

sþ1

Bk βk α−1 k ck þ

∑N h ¼ 1 ðakh þ

where J k ¼ ∑h∈Θ ðakh þ bkh Þlh ph þ bkh Þch þ I k ; k ¼ i1 ; …; im . Multiplying each inequality by the corresponding kj and adding them together, we have " # s

∑

j¼1

m

kj ð−dij þ Bij βij α−1 ij lij Þ þ ∑ kh ðaih ij þ bih ij Þlij pij m

þ ∑

j ¼ sþ1

" kj ð−dij þ

h¼1

Bij βij α−1 ij lij Þþ

#

m

m

sþ2

Multiplying each inequality by the corresponding kj and adding them together, we have " # j¼1

m

kj ð−dij þ Bij βij α−1 ij lij Þ þ ∑ kh ðaih ij þ bih ij Þlij qij m

h¼1

"

þ ∑

kj ð−dij þ

j ¼ sþ1

Bij βij α−1 ij lij Þþ

#

m

∑ kh ðaih ij þ bih ij Þlij pij

h¼1

m

m

h¼1

j¼1

þ ∑ kh ðaih imþ1 þ bih imþ1 Þlimþ1 pimþ1 þ ∑ J ij kj 4 0;

ð16Þ

m

sþ2

sþ1

k

s

Without loss of generality, let k1 ; …; ks 4 0, while ksþ1 ; …; km o 0. Denote Θ as the index set except fi1 ; …; im ; imþ1 g. From conditions (12) and (13), we get that 8 s mþ1 > > ð−dk þ Bk βk α−1 lk Þpk þ ∑ ðakij þ bkij Þlij pij þ ∑ ðakij þ bkij Þlij qij > > k > > j¼1 j ¼ sþ1 > > > < þJ o 0; k ¼ i1 ; i2 ; …; is ; k s mþ1 > > ð−d þ B β α−1 l Þq þ ∑ ða þ b Þl p þ ∑ ða þ b Þl q > > k k k k k k kij kij ij ij kij kij ij ij > > j ¼ 1 j ¼ sþ1 > > > : þJ 4 0; k ¼ i ; i ; …; i ;

ð20Þ

On the other hand, it also follows from (12) and (13) that 8 s mþ1 > > ð−dk þ Bk βk α−1 > > k lk Þqk þ ∑ ðakij þ bkij Þlij qij þ ∑ ðakij þ bkij Þlij pij > > j¼1 j ¼ sþ1 > > > < þJ 4 0; k ¼ i1 ; i2 ; …; is ; k s mþ1 > > ð−d þ B β α−1 l Þp þ ∑ ða þ b Þl q þ ∑ ða þ b Þl p > > k k k k k k kij kij ij ij kij kij ij ij > > j¼1 j ¼ sþ1 > > > : þJ o 0; k ¼ i ; i ; …; i :

∑

is invertible. Otherwise, assume that A~ is not invertible. Denote γ i as its i-th row vector. Then, there must exist nonzero constants k1 ; k2 ; …; km ðm≤N−1Þ and i1 ; …; im ; imþ1 ∈f1; 2; …; Ng such that

k

h¼1

h∈Θ

þ Bimþ1 βimþ1 α−1 imþ1 cimþ1 þ I imþ1 :

Define a map H : Λ2 -Λ2 by Hðx1 ; x2 ; …; xN Þ ¼ ðx~ 1 ; x~ 2 ; …; x~ N Þ. It is clear that the map H is continuous. It follows from Brouwer's fixed point theorem that there exists one fixed point xnn ¼ ðxnn 1 ; nn T ; …; x Þ of H in Λ , which is a solution of (14). Thus, we also xnn 2 N 2 T nn −1 nn obtain an equilibrium point ðxnn T ; Snn ÞT ¼ ðxnn 1 ; …; xN ; β1 α1 f 1 ðx1 Þ; −1 nn T N −1 −1 …; βN αN f N ðxN ÞÞ of system (2) in Λ2  ∏i ¼ 1 ½αi mi ; αi M i . In the following, we claim that the coefficient matrix of (14): 0

N

∑ J ij kj o ∑ ðaimþ1 h þ bimþ1 h Þlh ph þ ∑ ðaimþ1 h þ bimþ1 h Þch

ð21Þ

which is equivalent to s

m

m

j ¼ sþ1

j¼1

∑ ðaimþ1 ij þ bimþ1 ij Þlij qij þ ∑ ðaimþ1 ij þ bimþ1 ij Þlij pij þ ∑ J ij kj

j¼1

þ ½ðaimþ1 imþ1 þ bimþ1 imþ1 þ Bimþ1 βimþ1 α−1 imþ1 Þlimþ1 −dimþ1 pimþ1 4 0: ð22Þ Compared with (12), it follows that m

N

∑ J ij kj 4 ∑ ðaimþ1 h þ bimþ1 h Þlh ph þ ∑ ðaimþ1 h þ bimþ1 h Þch

j¼1

h¼1

h∈Θ

þ Bimþ1 βimþ1 α−1 imþ1 cimþ1 þ I imþ1 ;

ð23Þ

which is a contradiction to (20). Therefore, A~ is invertible. It means that the equilibrium point of system (2) is unique. (iii) Subset Ω3 : Pick Λ3 ¼ ∏i∈N1 ð−∞; pi Þ  ∏i∈N2 ðqi ; þ∞Þ ∏i∈N3 ½pi ; qi ⊂Ω3 , where N 1 ∪N 2 ∪N 3 ¼ f1; 2; …; Ng; N i ∩N j ¼ ∅; i≠j. Every equilibrium point in subset Λ3 satisfies the following equations:

∑ kh ðaih ij þ bih ij Þlij qij

h¼1

m

m

h¼1

j¼1

þ ∑ kh ðaih imþ1 þ bih imþ1 Þlimþ1 qimþ1 þ ∑ J ij kj o 0:

ð17Þ

−di xi þ ∑ ðaij þ bij Þlj xj þ ∑ ðaij þ bij Þuj þ ∑ ðaij þ bij Þvj j∈N 3

j∈N1

j∈N2

þ ∑ ðaij þ bij Þcj þ Bi βi α−1 i ui þ I i ¼ 0;

i∈N 1 ;

ð24Þ

j∈N 3

From condition (16), we obtain that 8 m > > ða þ bimþ1 ij Þlij ¼ kj ð−dij þ Bij βij α−1 > ij lij Þ þ ∑ kh ðaih ij þ bih ij Þlij ; < imþ1 ij h¼1

j ¼ 1; 2; …; m;

−di xi þ ∑ ðaij þ bij Þlj xj þ ∑ ðaij þ bij Þuj þ ∑ ðaij þ bij Þvj j∈N 3

j∈N1

j∈N2

þ ∑ ðaij þ bij Þcj þ Bi βi α−1 i vi þ I i ¼ 0;

> −1 > > : ðaimþ1 imþ1 þ bimþ1 imþ1 þ Bimþ1 βimþ1 αimþ1 Þlimþ1 −dimþ1 ¼ ∑ kh ðaih imþ1 þ bih imþ1 Þlimþ1 : m

ð18Þ

j∈N 1

m

m

j¼1

j ¼ sþ1

j¼1

∑ ðaimþ1 ij þ bimþ1 ij Þlij pij þ ∑ ðaimþ1 ij þ bimþ1 ij Þlij qij þ ∑ J ij kj

ð25Þ

½−di þ ðaii þ bii þ Bi βi α−1 i Þli xi þ ∑ ðaij þ bij Þuj þ ∑ ðaij þ bij Þvj

Substituting (18) into (17) results in s

i∈N 2 ;

j∈N 3

h¼1

þ ∑ ðaij þ bij Þcj þ j∈N 3

j∈N 2

∑ ðaij þ bij Þlj xj þ Bi βi α−1 i ci þ I i ¼ 0; i∈N 3 :

j∈N 3 ;j≠i

ð26Þ

X. Nie et al. / Neurocomputing 119 (2013) 281–291

When i∈N 3 , conditions (7) and (8) reduce to ½−di þ ðaii þ bii þ Bi βi α−1 i Þli pi þ

∑ maxfðaij þ bij Þuj ; ðaij þ bij Þvj g

j∈N1 ∪N 2

þ ∑ lj maxfðaij þ bij Þpj ; ðaij þ bij Þqj g

−1 −1 Therefore, Si ð0Þ∈½α−1 i mi ; αi M i  always implies that Si ðtÞ∈½αi mi ; −1 N −1 −1 αi M i . That is, if Sð0Þ∈∏i ¼ 1 ½αi mi ; αi M i , then the solution −1 −1 Sðt; Sð0ÞÞ will stay in ∏N i ¼ 1 ½αi mi ; αi M i  for t≥0. Pick a region arbitrarily N

~ 1 ¼ ∏ ð−∞; pi Þ  ∏ ðqi ; þ∞Þ  ∏ ½α−1 mi ; α−1 M i ⊂Ω1 Ω i i

j∈N 3 ;j≠i

þ ∑ ðaij þ bij Þcj þ Bi βi α−1 i ci þ I i o0;

ð27Þ

i∈N 1

 ∏

i¼1

∑ minfðaij þ bij Þuj ; ðaij þ bij Þvj g

j∈N 1 ∪N2

þ ∑ lj minfðaij þ bij Þpj ; ðaij þ bij Þqj g j∈N 3 ;j≠i

þ ∑ ðaij þ bij Þcj þ Bi βi α−1 i ci þ I i 40:

ð28Þ

j∈N 3

Similar to the proof of case (ii), define a map H~ : ∏ ½pi ; qi - ∏ ½pi ; qi  : ðxi1 ; xi2 ; …; xis Þ-ðx~ i1 ; x~ i2 ; …; x~ is Þ; i∈N 3

i∈N3

in which N3 is denoted as N 3 ¼ fi1 ; …; is g, and x~ ik ; ðk ¼ 1; 2; …; sÞ are defined in the same way as above x~ i in case (ii). Since the map H~ is continuous, it follows from Brouwer's fixed point theorem that nnn nnn T ~ there exists one fixed point ðxnnn i1 ; xi2 ; …; xis Þ of H in ∏j∈N 3 ½pj ; qj , which is a solution of (26). By similar arguments employed in the proof of case (ii), the coefficient matrix of (26) is invertible, which nnn nnn T implies that ðxnnn i1 ; xi2 ; …; xis Þ is unique. Then, replacing xj ; j∈N 3 , nnn for i∈N 1 ∪N2 . by xj ,and solving (24) and (25), we obtain xnnn i nnn T nnn is the unique solution of Denote xnnn ¼ ðxnnn 1 ; …; xN Þ . Then x (24)–(26) in Λ3 . Thus, we also obtain the unique equilibrium T nnn nnn −1 nnn T point ðxnnn T ; Snnn ÞT ¼ ðxnnn 1 ; …; xN ; β1 α1 f 1 ðx1 Þ; …; β N αN f N ðxN ÞÞ N −1 −1 of system (2) in Λ3  ∏i ¼ 1 ½αi mi ; αi M i . From case ((i)) to case (iii), we know that the system (2) can have and only have 3N equilibrium points. □ Theorem 2. If the following conditions: −di pi þ ðaii þ bii Þf i ðpi Þ þ

N

∑

j≠i;j ¼ 1

maxfðaij þ bij Þuj ; ðaij þ bij Þvj g

−1 þ maxfBi βi α−1 i ui ; Bi βi αi vi g þ I i o 0;

ð29Þ

i¼1

i∈N 2

N

j∈N 3

½−di þ ðaii þ bii þ Bi βi α−1 i Þli qi þ

285

½α−1 i mi ;

α−1 i M i ;

where N 1 ; N 2 are subsets of f1; 2; …; Ng, and N 1 ∪N 2 ¼ f1; 2; …; Ng, ~ 1 is an invariant set of the N1 ∩N2 ¼ ∅. We will show that Ω considered model (2). That is, for any initial condition ~ 1 Þ, we ðϕT ; ST ð0ÞÞT ¼ ðϕ1 ðsÞ; …; ϕN ðsÞ; S1 ð0Þ; …; SN ð0ÞT ∈Cð½−τ; 0; Ω ~ 1 for all t≥0. If this claim that the solution ðxT ðt; ϕÞ; ST ðt; Sð0ÞÞÞT ∈Ω is not true, there are two cases for us to discuss: Case1: There exists a component xi(t) of xðt; ϕÞ which is the first (or one of the first) escaping from ∏i∈N1 ð−∞; pi Þ. Restated, there exist some i∈N 1 and t n 4 0 such that xi ðt n Þ ¼ pi ; x_ i ðt n Þ 4 0; xi ðtÞ≤pi −1 for −τ≤t≤t n and Sj ðtÞ∈½α−1 j mj ; αj M j  for j ¼ 1; 2; …; N. It follows from system (2) that N

N

j¼1

j¼1

x_ i ðt n Þ ¼ −di xi ðt n Þ þ ∑ aij f j ðxj ðt n ÞÞ þ ∑ bij f j ðxj ðt n −τj ðt n ÞÞÞ þ Bi Si ðt n Þ þ I i

¼ −di pi þ ðaii þ bii Þf i ðpi Þ þ

∑ ðaij þ bij Þuj

j∈N 1 ;j≠i

þ ∑ ðaij þ bij Þvj þ Bi Si ðt n Þ þ I i j∈N 2

≤−di pi þ ðaii þ bii Þf i ðpi Þ N

þ

∑

j≠i;j ¼ 1

maxfðaij þ bij Þuj ; ðaij þ bij Þvj g

−1 þmaxfBi βi α−1 i ui ; Bi βi αi vi g þ I i o 0:

This is a contradiction. Case2: There exists a component xi(t) of xðt; ϕÞ which is the first (or one of the first) escaping from ∏i∈N2 ðqi ; þ∞Þ. Restated, there exist some i∈N 2 and t n 4 0 such that xi ðt n Þ ¼ qi ; x_ i ðt n Þ o 0; xi ðtÞ≥qi −1 for −τ≤t≤t n and Sj ðtÞ∈½α−1 j mj ; αj M j  for j ¼ 1; 2; …; N. Then we get x_ i ðt n Þ ¼ −di qi þ ðaii þ bii Þf i ðqi Þ þ ∑ ðaij þ bij Þuj j∈N1

þ ∑ ðaij þ bij Þvj þ Bi Si ðt n Þ þ I i −di qi þ ðaii þ bii Þf i ðqi Þ þ

N

∑

j≠i;j ¼ 1

j∈N 2 ;j≠i

minfðaij þ bij Þuj ; ðaij þ bij Þvj g

−1 þ minfBi βi α−1 i ui ; Bi β i αi vi g þ I i 4 0

≥−di qi þ ðaii þ bii Þf i ðqi Þ þ ð30Þ

hold for all i ¼ 1; 2; …; N, then system (2) with activation function (4) has exactly3N equilibrium points, 2N of which are locally exponentially stable. Proof. First of all, it is easy to see that conditions (29) and (30) imply the conditions (7) and (8). Thus, according to Theorem 1, the exact existence of 3N equilibrium points for system (2) can be guaranteed under conditions (29) and (30). In the following, we shall prove the local exponential stability of 2N equilibrium points in two steps. Step I: It follows from the second equation of system (2) that −αi Si ðtÞ þ mi ≤S_ i ðtÞ≤−αi Si ðtÞ þ M i ; which leads to     mi m M M þ Si ð0Þ− i e−αi t ≤Si ðtÞ≤ i þ Si ð0Þ− i e−αi t : αi αi αi αi

N

∑

j≠i;j ¼ 1

minfðaij þ bij Þuj ; ðaij þ bij Þvj g

−1 þminfBi βi α−1 i ui ; Bi βi αi vi g þ I i 4 0:

This contradicts to x_ i ðt n Þ o 0. From the above two cases, we know that the solution ~ 1 for all t≥0. That is, ðxT ðt; ϕÞ; ST ðt; Sð0ÞÞÞT will never escape from Ω ~ 1 is an invariant set of the model (2). Ω Step II: We will prove that the unique equilibrium point T ~ 1 is locally exponentially stable. By ðxn T ; Sn ÞT of system (2) in Ω the positive invariance property, Eq. (2) can be rewritten as 8 < x_ i ðtÞ ¼ −di xi ðtÞ þ ∑ ðaij þ bij Þuj þ ∑ ðaij þ bij Þvj þ Bi Si ðtÞ þ I i ; j∈N1

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

j∈N 2

i ¼ 1; 2; …; N: ð32Þ

ð31Þ

T

T

T

T

Let ðx ðtÞ; S ðtÞÞ ¼ ðx ðt; ϕÞ; S ðt; Sð0ÞÞÞ be any a solution of system ~ 1 Þ. From (2) with initial condition ðϕT ðsÞ; ST ð0ÞÞT ∈Cð½−τ; 0; Ω T T ~ T Step I, we get that ðx ðtÞ; S ðtÞÞ ∈Ω 1 for all t≥0. Note that T

T

286

X. Nie et al. / Neurocomputing 119 (2013) 281–291

~ 1 Þ, we get f i ðxi ðtÞÞ ¼ f i ðxni Þ≡ui or vi when ðϕT ðsÞ; ST ð0ÞÞT ∈Cð½−τ; 0; Ω from model (32) 8 dðxi ðtÞ−xni Þ > > ¼ −di ðxi ðtÞ−xni Þ þ Bi ðSi ðtÞ−Sni Þ; < dt ð33Þ > dðSi ðtÞ−Sni Þ > : ¼ −αi ðSi ðtÞ−Sni Þ: dt From the second equation of system (33), we derive that jSi ðtÞ−Sni j≤jSi ð0Þ−Sni je−αi t ;

ð34Þ

t≥0:

Substituting (34) into the first equation of system (33) results in Dþ jxi ðtÞ−xni j≤−di jxi ðtÞ−xni j þ jBi jjSi ð0Þ−Sni je−αi t : Thus, we obtain that  n  n     xi ðtÞ−xn ≤ ϕi ð0Þ−xn − jBi jjSi ð0Þ−Si j e−di t þ jBi jjSi ð0Þ−Si j e−αi t : i i di −αi di −αi

ð35Þ

ð36Þ

Inequalities (34) and (36) imply that ðxT ðtÞ; ST ðtÞÞT exponentially T converges to ðxn T ; Sn ÞT as t-∞. □ Remark 1. In Refs. [18,19,24], based on the methods of decomposition of state space and geometrical observation, the authors investigated the multistability of high-order competitive neural networks and DCNNs. It was proved that under some conditions, the networks can have 2N locally exponentially stable equilibrium points. However, the exact number of equilibrium points is not considered in [18,19,24]. Let Bi ¼ βi ¼ Si ðtÞ ¼ 0ði; j ¼ 1; 2; …; NÞ, then system (2) is transformed into the following DHNNs which are investigated in [10]: N

N

j¼1

j¼1

x_ i ðtÞ ¼ −di xi ðtÞ þ ∑ aij f j ðxj ðtÞÞ þ ∑ bij f j ðxj ðt−τj ðtÞÞÞ þ I i :

ð37Þ

N

∑

j≠i;j ¼ 1

N

jaij j− ∑ jbij j

ð43Þ

j¼1

hold, then system (37) with di ¼1 has 2N exponentially stable equilibrium points. Since bii −∑N ∑N j≠i;j ¼ 1 jaij þ bij j≥−jbii j− j≠i;j ¼ 1 ðjaij j þ jbij jÞ, it is easy to see that our condition (42) is weaker than condition (43). Moreover, the number of equilibrium points is not discussed in [10]. Thus, Corollary 2 in this paper improves and generalizes the main result in [10]. 3.2. Multistability of DCNNs with activation functions (3) In the above subsection, we study the exact existence and multistability of DCNNs (2) with activation functions (4). The proposed method is also applicable to activation functions (3). Similar to the proof of Theorems 1 and 2, we have the following theorems. Theorem 3. Suppose that k −di pki þ ðaii þ bii þ Bi βi α−1 i Þf i ðpi Þ

þ

N

∑

j≠i;j ¼ 1

maxfðaij þ bij Þu1j ; ðaij þ bij Þurþ1 g þ I i o 0; j

ð44Þ

k −di qki þ ðaii þ bii þ Bi βi α−1 i Þf i ðqi Þ

þ

N

∑

j≠i;j ¼ 1

minfðaij þ bij Þu1j ; ðaij þ bij Þurþ1 g þ Ii 4 0 j

ð45Þ

hold for all i ¼ 1; 2; …; N; k ¼ 1; 2; …; r. Then, system (2) with activation function (3) have exactly ð2r þ 1ÞN equilibrium points.

−di pki þ ðaii þ bii Þf i ðpki Þ þ

Corollary 1. If the following conditions: N

jI i jo aii −1−

Theorem 4. Suppose that

From Theorem 2, we obtain the following corollaries.

−di pi þ ðaii þ bii Þf i ðpi Þ þ ∑ maxfðaij þ bij Þuj ; ðaij þ bij Þvj g þ I i o 0;

Remark 2. In [10], the authors investigated the multistability of (37) with di ¼ 1 and activation functions (5), and proved that if

ð38Þ

N

∑

j≠i;j ¼ 1

maxfðaij þ bij Þu1j ; ðaij þ bij Þujrþ1 g

1 −1 rþ1 þmaxfBi βi α−1 g þ I i o 0; i ui ; Bi β i α i ui

ð46Þ

j≠i;j ¼ 1

−di qi þ ðaii þ bii Þf i ðqi Þ þ

N

∑

j≠i;j ¼ 1

minfðaij þ bij Þuj ; ðaij þ bij Þvj g þ Ii 4 0

ð39Þ

−di qki þ ðaii þ bii Þf i ðqki Þ þ

N

∑

j≠i;j ¼ 1

minfðaij þ bij Þu1j ; ðaij þ bij Þujrþ1 g

1 −1 rþ1 þminfBi βi α−1 g þ Ii 4 0 i ui ; Bi β i α i ui

hold for all i ¼ 1; 2; …; N, then system (37) with activation function (4) can have and only have 3N equilibrium points, and 2N of which are locally exponentially stable.

ð47Þ

hold for all i ¼ 1; 2; …; N; k ¼ 1; 2; …; r. Then, system (2) with activation function (3) has exactly ð2r þ 1ÞN equilibrium points. Among them, (r+1)N of which are locally exponentially stable.

Corollary 2. If the following conditions: di −ðaii þ bii Þ þ

N

∑

j≠i;j ¼ 1

−di þ ðaii þ bii Þ−

jaij þ bij j þ I i o 0;

N

∑

j≠i;j ¼ 1

jaij þ bij j þ I i 4 0

ð40Þ

ð41Þ

hold for all i ¼ 1; 2; …; N, then system (37) with activation function (5) can have and only have 3N equilibrium points, and 2N of which are locally exponentially stable.

3.3. Dynamical behaviors of 2-neuron DHNNs with activation function (4) In this subsection, we will precisely figure out all equilibrium points of DHNNs and analyze the dynamical behaviors of each equilibrium point. For the sake of simplicity, we only consider DHNNs with two neurons: ( x_ 1 ðtÞ ¼ −d1 x1 ðtÞ þ a11 f 1 ðx1 ðtÞÞ þ b12 f 2 ðx2 ðt−τ2 ÞÞ þ I 1 ; ð48Þ x_ 2 ðtÞ ¼ −d2 x2 ðtÞ þ a22 f 2 ðx2 ðtÞÞ þ b21 f 1 ðx1 ðt−τ1 ÞÞ þ I 2 :

Proof. For activation function (5), we have pi ¼ f i ðpi Þ ¼ −1; qi ¼ f i ðqi Þ ¼ 1; ui ¼ −1; vi ¼ 1ði ¼ 1; 2; …; NÞ. Note that maxf−ðaij þ bij Þ; ðaij þ bij Þg ¼ jaij þ bij j, minf−ðaij þ bij Þ; ðaij þ bij Þg ¼ −jaij þ bij j, from Corollary 1, we derive the result of Corollary 2. □

Applying Theorem 2, we can easily obtain the following corollary:

−d1 p1 þ a11 f 1 ðp1 Þ þ maxfb12 u2 ; b12 v2 g þ I 1 o 0;

ð49Þ

If di ¼ 1ði ¼ 1; 2; …; NÞ, then above conditions (40) and(41) are equivalent to the following inequality:

−d1 q1 þ a11 f 1 ðq1 Þ þ minfb12 u2 ; b12 v2 g þ I 1 40;

ð50Þ

−d2 p2 þ a22 f 2 ðp2 Þ þ maxfb21 u1 ; b21 v1 g þ I 2 o 0;

ð51Þ

−d2 q2 þ a22 f 2 ðq2 Þ þ minfb21 u1 ; b21 v1 g þ I 2 40:

ð52Þ

jI i j o−1 þ ðaii þ bii Þ−

N

∑

j≠i;j ¼ 1

jaij þ bij j:

ð42Þ

Corollary 3. Suppose that

X. Nie et al. / Neurocomputing 119 (2013) 281–291

287

Obviously, iwðw 4 0Þ is a root of Eq. (56) if and only if w satisfies −w2 þ i½ðd1 −a11 l1 Þ þ ðd2 −a22 l2 Þw þ ðd1 −a11 l1 Þðd2 −a22 l2 Þ−b12 b21 l1 l2 ð cos wτ−i sin wτÞ ¼ 0: Separating the real and imaginary parts, we get ( −w2 þ ðd1 −a11 l1 Þðd2 −a22 l2 Þ ¼ b12 b21 l1 l2 cos wτ; w½ðd1 −a11 l1 Þ þ ðd2 −a22 l2 Þ ¼ −b12 b21 l1 l2 sin wτ:

ð57Þ

Taking square on the both sides of Eq. (57) and summing up, we have z2 þ ½ðd1 −a11 l1 Þ2 þ ðd2 −a22 l2 Þ2 z Fig. 1. R2 is composed of 9 subsets.

2

Then, system (48) with activation function (4) has exactly nine equilibrium points ðxS1i ; xS2i ÞT ; ðxΛ1 ; xΛ2 ÞT and ðxΞ1 i ; xΞ2 i ÞT , which locate in Si ; Λ and Ξ i ði ¼ 1; 2; 3; 4Þ, respectively (see Fig. 1). Among them, four equilibrium points ðxS1i ; xS2i ÞT ði ¼ 1; 2; 3; 4Þ are locally exponentially stable. Remark 3. Denote li ¼ ðvi −ui Þ=ðqi −pi Þ, ci ¼ ðui qi −vi pi Þ=ðqi −pi Þ ði ¼ 1; 2Þ. Conditions (49) and (50) imply that −di þ aii li 40ði ¼ 1; 2Þ. Proof. Note that f 1 ðp1 Þ ¼ l1 p1 þ c1 and f 1 ðq1 Þ ¼ l1 q1 þ c1 . It follows from conditions (49) and (50) that ð−d1 þ a11 l1 Þp1 o ð−d1 þ a11 l1 Þq1 . Now inequality −d1 þ a11 l1 4 0 follows immediately from the fact p1 o q1 . Inequality −d2 þ a22 l2 4 0 can be proved similarly. □ Letting y1 ðtÞ ¼ x1 ðt−τ1 Þ; y2 ðtÞ ¼ x2 ðtÞ and τ ¼ τ1 þ τ2 , we can rewrite (48) as the following equivalent system: ( y_ 1 ðtÞ ¼ −d1 y1 ðtÞ þ a11 f 1 ðy1 ðtÞÞ þ b12 f 2 ðy2 ðt−τÞÞ þ I 1 ; ð53Þ y_ 2 ðtÞ ¼ −d2 y2 ðtÞ þ a22 f 2 ðy2 ðtÞÞ þ b21 f 1 ðy1 ðtÞÞ þ I 2 : In the following, we shall discuss the dynamical behaviors of equilibrium points ðxΛ1 ; xΛ2 ÞT and ðxΞ1 i ; xΞ2 i ÞT ði ¼ 1; 2; 3; 4Þ, and show that under certain condition the five equilibrium points are unstable by analyzing the associated characteristic equation of system (53). Let yΛ ðt; ϕΛ Þ be a solution of system (53) with initial condition Λ ðϕ1 ; ϕΛ2 ÞT near ðxΛ1 ; xΛ2 ÞT . Without loss of generality, we assume that yΛ ðt; ϕΛ Þ∈Λ for all t≥0. In fact, we can conclude that, if the solution yΛ ðt; ϕΛ Þ get out of region Λ, the equilibrium point ðxΛ1 ; xΛ2 ÞT must be unstable according to the definition of instability. By the translations uΛ1 ðtÞ ¼ yΛ1 ðtÞ−xΛ1 , uΛ2 ðtÞ ¼ yΛ2 ðtÞ−xΛ2 , system (53) can be rewritten as the following equivalent system: (

u_ Λ1 ðtÞ ¼ −d1 uΛ1 ðtÞ þ a11 ½f 1 ðxΛ1 þ uΛ1 ðtÞÞ−f 1 ðxΛ1 Þ þ b12 ½f 2 ðxΛ2 þ uΛ2 ðt−τÞÞ−f 2 ðxΛ2 Þ; u_ Λ2 ðtÞ ¼ −d2 uΛ2 ðtÞ þ a22 ½f 2 ðxΛ2 þ uΛ2 ðtÞÞ−f 2 ðxΛ2 Þ þ b21 ½f 1 ðxΛ1 þ uΛ1 ðtÞÞ−f 1 ðxΛ1 Þ:

ð54Þ xΛi

uΛi ðtÞ∈½pi ; qi ði ¼

xΛ2

Note that þ 1; 2Þ; þ from model (54) ( Λ u_ 1 ðtÞ ¼ ð−d1 þ a11 l1 ÞuΛ1 ðtÞ þ b12 l2 uΛ2 ðt−τÞ;

uΛ2 ðt−τÞ∈½p2 ;

u_ Λ2 ðtÞ ¼ ð−d2 þ a22 l2 ÞuΛ2 ðtÞ þ b21 l1 uΛ1 ðtÞ;

2

2 2

þ ðd1 −a11 l1 Þ2 ðd2 −a22 l2 Þ2 −b12 b21 l1 l2 ¼ 0;

where z ¼ w2 . Till now, we can employ a result from Ruan and Wei [37] to analyze Eq. (56), which is, for the convenience of the readers, stated as follows: Lemma 1 (Ruan and Wei [37]). Consider the exponential polynomial Pðλ; e−λτ1 ; …; e−λτm Þ

¼ λn þ p1ð0Þ λn−1 þ ⋯ þ pð0Þ n−1 λ ð1Þ n−1 ð1Þ −λτ1 þ pð0Þ þ ⋯ þ pn−1 λ þ pð1Þ n þ ½p1 λ n e n−1 ðmÞ −λτm þ ⋯ þ ½pðmÞ þ ⋯ þ pðmÞ ; 1 λ n−1 λ þ pn e

where τi ≥0ði ¼ 1; 2; …; mÞ and pðiÞ j ði ¼ 0; 1; …; m; j ¼ 1; 2; …; nÞ are constants. As ðτ1 ; τ2 ; …; τm Þ vary, the sum of the zeros of Pðλ; e−λτ1 ; …; e−λτm Þ on the open right half plane can change only if a zero appears on or crosses the imaginary axis. Lemma 2. Suppose (49)–(52) hold. If the following condition: ðd1 −a11 l1 Þðd2 −a22 l2 Þ≥jb12 b21 jl1 l2 is satisfied, then the equilibrium point unstable.

ð59Þ ðxΛ1 ; xΛ2 ÞT

of system (48) is

Proof. When τ ¼ 0, Eq. (56) becomes λ2 þ ½ðd1 −a11 l1 Þ þ ðd2 −a22 l2 Þλ þ ðd1 −a11 l1 Þðd2 −a22 l2 Þ−b12 b21 l1 l2 ¼ 0:

ð60Þ Since ðd1 −a11 l1 Þ þ ðd2 −a22 l2 Þ o 0, Eq. (60) has root with positive real part. Note that ðd1 −a11 l1 Þ2 þ ðd2 −a22 l2 Þ2 4 0, it is easy to derive that if condition (59) holds, then Eq. (58) has no positive root, which implies that Eq. (56) has no root with zero real part for all τ≥0. Applying Lemma 1, we can conclude that Eq. (56) has root with positive real part for all τ≥0. Therefore, the equilibrium point ðxΛ1 ; xΛ2 ÞT of system (48) is unstable. □ Similarly, we have the following lemma:

q2 , we get

ð55Þ

ð58Þ

Lemma 3. Suppose (49)–(52) hold. Then the equilibrium points ðxΞ1 i ; xΞ2 i ÞT ði ¼ 1; 2; 3; 4Þ of system (48) are unstable. Proof. We prove only for equilibrium point ðxΞ1 1 ; xΞ2 1 ÞT , and similar arguments can be employed to prove that other equilibrium points are unstable. Let yΞ1 ðt; ϕΞ 1 Þ be a solution of system (53) with initial condition ðϕΞ1 1 ; ϕΞ2 1 ÞT near ðxΞ1 1 ; xΞ2 1 ÞT . Similar to the above discussion, assume that yΞ 1 ðt; ϕΞ1 Þ∈Ξ 1 for all t≥0. Let uΞ1 1 ðtÞ ¼ yΞ1 1 ðtÞ−xΞ1 1 , uΞ2 1 ðtÞ ¼ yΞ2 1 ðtÞ−xΞ2 1 , then system (53) becomes

whose associated characteristic equation is ! −b12 l2 e−λτ λ þ d1 −a11 l1 det ¼ 0: −b21 l1 λ þ d2 −a22 l2 That is

(

λ2 þ ½ðd1 −a11 l1 Þ þ ðd2 −a22 l2 Þλ þ ðd1 −a11 l1 Þðd2 −a22 l2 Þ−b12 b21 l1 l2 e−λτ ¼ 0:

ð56Þ

u_ Ξ1 1 ðtÞ ¼ −d1 uΞ1 1 ðtÞ þ a11 ½f 1 ðxΞ1 1 þ uΞ1 1 ðtÞÞ−f 1 ðxΞ1 1 Þ þ b12 ½f 2 ðxΞ2 1 þ uΞ2 1 ðt−τÞÞ−f 2 ðxΞ2 1 Þ; u_ Ξ2 1 ðtÞ ¼ −d2 uΞ2 1 ðtÞ þ a22 ½f 2 ðxΞ2 1 þ uΞ2 1 ðtÞÞ−f 2 ðxΞ2 1 Þ þ b21 ½f 1 ðxΞ1 1 þ uΞ1 1 ðtÞÞ−f 1 ðxΞ1 1 Þ:

ð61Þ

288

X. Nie et al. / Neurocomputing 119 (2013) 281–291

Note that xΞ1 1 þ uΞ1 1 ðtÞ∈½p1 ; q1 ; xΞ2 1 þ uΞ2 1 ðt−τÞ∈ðq2 ; þ ∞Þ; xΞ2 1 þ uΞ2 1 ðtÞ∈ ðq2 ; þ∞Þ, it follows from (61) that ( Ξ1 u_ 1 ðtÞ ¼ ð−d1 þ a11 l1 ÞuΞ1 1 ðtÞ; ð62Þ u_ Ξ2 1 ðtÞ ¼ −d2 uΞ2 1 ðtÞ þ b21 l1 uΞ1 1 ðtÞ;

70 60 50 40 30

ð63Þ

10 x1(t)

whose associated characteristic equation is ! λ þ d1 −a11 l1 0 det ¼ 0: λ þ d2 −b21 l1

It is easy to see that Eq. (63) has two different real roots λ1 ¼ −d1 þ a11 l1 40; λ2 ¼ −d2 o 0. Therefore, the equilibrium point ðxΞ1 1 ; xΞ2 1 ÞT of system (48) is unstable. □

−10 −20 −30

Theorem 5. Suppose (49)–(52) hold. Then the following results hold. (i) System (48) with activation function (4) has exactly nine equilibrium points ðxS1i ; xS2i ÞT ; ðxΛ1 ; xΛ2 ÞT and ðxΞ1 i ; xΞ2 i ÞT , which locate in Si ; Λ and Ξ i ði ¼ 1; 2; 3; 4Þ, respectively. (ii) Four equilibrium points ðxS1i ; xS2i ÞT ði ¼ 1; 2; 3; 4Þ are locally exponentially stable, and four equilibrium points ðxΞ1 i ; xΞ2 i ÞT ði ¼ 1; 2; 3; 4Þ are unstable. (iii) Moreover, if condition (59) holds as well, then the equilibrium point ðxΛ1 ; xΛ2 ÞT is unstable.

0

2

4

6

8

10

12

14

16

18

20

t Fig. 2. Transient behavior of x1 in Example 1.

70 60 50 40 30

10 x2(t)

4. Two illustrative examples

10 0

Summarizing the results of Corollary 3 and Lemmas 2 and 3, we can easily obtain the following theorem on the dynamics of nine equilibrium points for system (48).

Remark 4. The method given in this subsection can be applied to DCNNs with more neurons, and complex dynamical behaviors can be analyzed, including local stability, instability and Hopf bifurcation, etc.

20

20 10 0

Example 1. Consider the following DCNNs: 8 2 2 > > < 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 þ I i ; j¼1

> > : S_ i ðtÞ ¼ −αi Si ðtÞ þ β f ðxi ðtÞÞ; i i

−10 −20

j¼1

−30

i ¼ 1; 2;

0

2

4

6

8

ð64Þ

Herein, the parameters satisfy conditions in Theorem 5: −d1 p1 þ a11 f 1 ðp1 Þ þ maxfb12 u2 ; b12 v2 g þ I 1 ¼ −0:5 o 0;

14

16

18

20

Fig. 3. Transient behavior of x2 in Example 1.

2 1 0 −1

One can easily check that the conditions in Theorem 4 hold, hence we conclude that the model (64) has exactly ð2  2 þ 1Þ2 ¼ 25 equilibrium points, and among them, (2+1)2 ¼9 equilibrium points are locally exponentially stable . Simulation results with 120 random initial states are depicted in Figs. 2–6.

−2

0

2

4

6

8

10

12

14

16

18

20

12

14

16

18

20

t 2 1

S2(t)

Example 2. For system (48), take d1 ¼ d2 ¼ 2; a11 ¼ 3; b12 ¼ −0:5; a22 ¼ 3:5; b21 ¼ 0:5; I 1 ¼ I 2 ¼ 0; τ1 ¼ τ2 ¼ 1 and 8 > < −1; −∞ o x o −1; −1≤x≤2; f i ðxÞ ¼ x; ði ¼ 1; 2Þ: > : 2; 2 ox o þ ∞;

12

t

S1(t)

where d1 ¼ d2 ¼ 2, a11 ¼ 2:5; a12 ¼ a21 ¼ b12 ¼ b21 ¼ 0:05, a22 ¼ 3:2, b11 ¼ 0:5, b22 ¼ −0:2, B1 ¼ −0:07; B2 ¼ 0:1; α1 ¼ 2; α2 ¼ 3; βi ¼ 1, I i ¼ 0, τi ðtÞ ¼ e−t and 8 −1; −∞ o x o −1; > > > > > x; −1≤x≤1; > < 1 o x o 2; f i ðxÞ ¼ 1; ði ¼ 1; 2Þ: > > > x−1; 2≤x≤4; > > > : 3; 4 o x o þ ∞;

10

0 −1 −2

0

2

4

6

8

10

t Fig. 4. Transient behavior of S1 and S2 in Example 1.

X. Nie et al. / Neurocomputing 119 (2013) 281–291

289

3

4

2

3

1

2

0

x2(t)

S1(t)

5

−1

1 0

−2 80

−1

60 40

10

20 0

x2 (t

)

−20

0

−20

−40 −40

40

20

80

60

−2

t) 10 x 1(

−3 −4

Fig. 5. Phase plot of state variable ðx1 ; x2 ; S1 ÞT in Example 1.

−3

−2

−1

0

1

2

3

4

x1(t) Fig. 8. Phase plot of state variable ðx1 ; x2 ÞT in Example 2.

−1

30

−1.2 −1.4

x1(t)

S2(t)

20 10

−1.6 −1.8

0

−2 −2.2

−10

0

2

4

6

8

10

12

14

16

18

20

12

14

16

18

20

t

−20 80 60 40 20

2 (t)

0 −20 −40

−40

−20

0

20

40

2 1

x2(t)

10 x

3

80

60

(t) 10 x 1

0 −1

Fig. 6. Phase plot of state variable ðx1 ; x2 ; S2 ÞT in Example 1.

−2 0

2

4

6

8

Fig. 9. Transient behavior of x1 and x2 with 30 random initial states near the 1 T equilibrium point ð− 19 12 ; 3Þ in Example 2.

2

x1(t)

10

t

4

0 3

−2

0

2

4

6

8

10

12

14

16

18

20

1 0 −1 −2 0

2

4

6

8

10

12

14

16

18

20

12

14

16

18

20

t 4.2 4 0

2

4

6

8

10

12

14

16

t Fig. 7. Transient behavior of x1 and x2 in Example 2.

−d1 q1 þ a11 f 1 ðq1 Þ þ minfb12 u2 ; b12 v2 g þ I 1 ¼ 1 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 2:5 4 0; ðd1 −a11 l1 Þðd2 −a22 l2 Þ ¼ 1:5≥0:25 ¼ jb12 b21 jl1 l2 :

18

20

3.8

x2(t)

x2(t)

t 5 4 3 2 1 0 −1 −2 −3

x1(t)

2 −4

3.6 3.4 3.2 3

0

2

4

6

8

10

t Fig. 10. Transient behavior of x1 and x2 with 30 random initial states near the T equilibrium point ð1; 15 4 Þ in Example 2.

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X. Nie et al. / Neurocomputing 119 (2013) 281–291

3

x1(t)

2 1 0 −1 0

2

4

6

8

10

12

14

16

18

20

t −1.2

x2(t)

−1.4 −1.6

5. Conclusions

−1.8 −2 −2.2

0

2

4

6

8

10

12

14

16

18

20

t Fig. 11. Transient behavior of x1 and x2 with 40 random initial states near the T equilibrium point ð− 12 ; − 15 8 Þ in Example 2.

3.2

x1(t)

From Theorem 5, it follows that the system has exactly nine T T 5 equilibrium points ð−2; 13 4 Þ ∈ð−∞; −1Þ  ð2; þ∞Þ, ð−4; −2Þ ∈ð−∞; −1Þ T 5 T 5 ð−∞; −1Þ, ð13 ; − Þ ∈ð2; þ∞Þ  ð−∞; −1Þ, ð ; 4Þ ∈ð2; þ∞Þ  ð2; þ∞Þ, ð−19 4 4 2 12 ; 1 T 15 T 1 15 T ð1; 4 Þ ∈½−1; 2  ð2; þ∞Þ, ð−2 ; − 8 Þ ∈½−1; 2 3Þ ∈ð−∞; −1Þ  ½−1; 2, T 2 T ð−∞; −1Þ, ð19 6 ; −3Þ ∈ð2; þ∞Þ  ½−1; 2, ð0; 0Þ ∈½−1; 2  ½−1; 2. Further13 T 5 T more, the four equilibrium points ð−2; 4 Þ ; ð−54; −2ÞT ; ð13 4 ; −4Þ and ð52; 4ÞT are locally exponentially stable, while the other five equilibrium points are unstable. from Figs. 7 and 8, it can be seen that T T 13 T 5 5 T 5 the four equilibrium points ð−2; 13 4 Þ ; ð−4; −2Þ ; ð 4 ; −4Þ and ð2; 4Þ are locally exponentially stable. Figs. 9–13 confirm that the others are unstable.

3 2.8 2.6 2.4

0

2

4

6

8

10

12

14

16

18

20

t

In this paper, the multistability and instability issues have been studied for DCNNs with nondecreasing piecewise linear activation functions possessing 2r corner points. (1) Based on decomposition of state space, fixed point theorem and matrix theory, under some conditions, we showed that the networks have exactly (2r+1)N equilibrium points, (r+1)N of which are locally exponentially stable. (2) For the 2-neuron DHNNs with activation function possessing two corner points, we analyzed completely the dynamical behaviors of all equilibrium points, by discussing the distribution of roots of the corresponding characteristic equation of the linearized delay differential equations. (3) Two examples with their computer simulations are given to illustrate the effectiveness of the obtained results. (4) The method given in this paper can be applied to the study of the exact existence and dynamical behaviors of multiple periodic solutions or almost-periodic solutions for delayed neural networks.

4

x2(t)

3

Acknowledgments

2 1 0 −1 0

2

4

6

8

10

12

14

16

18

20

t Fig. 12. Transient behavior of x1 and x2 with 40 random initial states near the 2 T equilibrium point ð19 6 ; − 3Þ in Example 2.

4

References

x1(t)

3 2 1 0 −1 −2 0

2

4

6

8

10

12

14

16

18

20

12

14

16

18

20

t 4 3

x2(t)

This work was jointly supported by the National Natural Science Foundation of China under Grants nos. 61203300, 61263020 and 11072059, the Specialized Research Fund for the Doctoral Program of Higher Education under Grants nos. 20120092120029 and 20110092110017, the Natural Science Foundation of Jiangsu Province of China under Grants nos. BK2012319 and BK2012741, the China Postdoctoral Science Foundation funded project under Grant 2012M511177, and the Innovation Foundation of Southeast University under Grant 3207012401.

2 1 0 −1 −2 0

2

4

6

8

10

t Fig. 13. Transient behavior of x1 and x2 with 40 random initial states near the equilibrium point ð0; 0ÞT in Example 2.

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Xiaobing Nie received the B.S. degree in Mathematics from Yantai Normal University, Yantai, China, in 1999, the M.S. degree in Applied Mathematics from East China Normal University, Shanghai, China, in 2002, and the Ph.D. degree in Applied Mathematics from Southeast University, Nanjing, China, in 2010. Since July 2002, he has been with the Department of Mathematics, Southeast University, Nanjing, China. He is a very active reviewer for many international journals and has published more than 10 research papers. His research interests include neural networks, multistability theory, nonsmooth system.

Jinde Cao (M'07–SM'07) received the B.S. degree from Anhui Normal University, Wuhu, China, the M.S. degree from Yunnan University, Kunming, China, and the Ph.D. degree from Sichuan University, Chengdu, China, all in Mathematics/Applied Mathematics, in 1986, 1989, and 1998, respectively. From 1989 to 2000, he was with Yunnan University. In 2000, he joined the Department of Mathematics, Southeast University, Nanjing, China. From 2001 to 2002, he was a Postdoctoral Research Fellow with the Department of Automation and Computer-Aided Engineering, Chinese University of Hong Kong, Shatin, Hong Kong. From 2006 to 2008, he was a Visiting Research Fellow and a Visiting Professor with the School of Information Systems, Computing and Mathematics, Brunel University, Uxbridge, U.K. He is currently a TePin Professor and Doctoral Advisor with the Department of Mathematics, Southeast University, prior to which he was a Professor at Yunnan University from 1996 to 2000. He is the author or coauthor of more than 200 research papers and 5 edited books. His research interests include nonlinear systems, neural networks, complex systems, complex networks, stability theory, and applied mathematics. Dr. Cao was an Associate Editor of the IEEE Transactions on neural networks and Neurocomputing. He is an Associate Editor of the Journal of the Franklin Institute, Mathematics and Computers in Simulation, Abstract and Applied Analysis, Journal of Applied Mathematics, The Open Electrical and Electronic Engineering Journal, International Journal of Differential Equations, and Differential Equations and Dynamical Systems. He is a reviewer of Mathematical Reviews and Zentralblatt-Math.

Shumin Fei received the Ph.D. degree from Beijing University of Aeronautics and Astronautics, China, in 1995. From 1995 to 1997, he was doing postdoctoral research at Research Institute of Automation, Southeast University, China. He is currently a Professor and Doctoral Advisor at Southeast University. He has published more than 120 journal papers and his research interests include nonlinear systems, time-delay system, complex systems, and so on.