# Multistability and Instability of Competitive Neural Networks with Mexican-Hat-Type Activation Functions

Authors: Xiaobing Nie; Jinde Cao; Shumin Fei
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We investigate the existence and dynamical behaviors of multiple equilibria
for competitive neural networks with a class of general Mexican-hat-type activation
functions. The Mexican-hat-type activation functions are not monotonously increasing, and the
structure of neural networks with Mexican-hat-type activation functions is totally different from
those with sigmoidal activation functions or nondecreasing saturated activation functions, which
have been employed extensively in previous multistability papers. By tracking the dynamics of
each state component and applying fixed point theorem and analysis method, some sufficient conditions
are presented to study the multistability and instability, including the total number of
equilibria, their locations, and local stability and instability. The obtained results extend and improve
the very recent works. Two illustrative examples with their simulations are given to verify
the theoretical analysis.


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## Body

## 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 variables, the state variables of the 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 (CNNs) 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 variables: 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. The CNNs can be written in the following form:(1)STM:x˙i(t)=-dixi(t)+∑j=1Naijfj(xj(t))+Bi∑j=1Pmij(t)yj+Ii,LTM:m˙ij(t)=-αimij(t)+yjβifi(xi(t)),
where i=1,2,…,N, j=1,2,…,P,xi(t) is the neuron current activity level, fj(xj(t)) 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, Ii is the constant input, and αi>0 and βi≥0 denote disposable scaling constants.After settingSi(t)=∑j=1Pmij(t)yj=yTmi(t), where y=(y1,y2,…,yP)T and mi(t)=(mi1(t),mi2(t),…,miP(t))T and assuming the input stimulus y to be normalized with unit magnitude |y|2=y12+⋯+yP2=1, then the above networks are simplified as
(2)x˙i(t)=-dixi(t)+∑j=1Naijfj(xj(t))+BiSi(t)+Ii,S˙i(t)=-αiSi(t)+βifi(xi(t)),i=1,2,…,N.The qualitative analysis of neural dynamics plays an important role in the design of practical neural networks. To solve problems of optimization and signal processing, neural networks have to be designed in such a way that, for a given external input, they exhibit only one globally stable state (i.e., monostability). This matter has been treated in [1–7]. On the other hand, if neural networks are used to analyze associative memories, the coexistence of multiple locally stable equilibria or periodic orbits is required (i.e., multistability or multiperiodicity), since the addressable memories or patterns are stored as stable equilibria or stable periodic orbits. In monostability analysis, the objective is to derive conditions that guarantee that each network contains only one steady state, and all the trajectories of the network converge to it, whereas in multistability analysis, the networks are allowed to have multiple equilibria or periodic orbits (stable or unstable). In general, the usual global stability conditions are not adequately applicable to multistable networks.Recently, the multistability or multiperiodicity of neural networks has attracted the attention of many researchers. In [8, 9], based on decomposition of state space, the authors investigated the multistability of delayed Hopfield neural networks and showed that the n-neuron neural networks can have 2n stable orbits located in 2n subsets of ℝn. Cao et al. [10] extended the above method to the Cohen-Grossberg neural networks with nondecreasing saturated activation functions with two corner points. In [11, 12], the multistability of almost-periodic solution in delayed neural networks was studied. Kaslik and Sivasundaram [13, 14] firstly revealed the effect of impulse on the multistability of neural networks. In [15–17], 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 [18–22], the authors indicated that under some conditions, there exist 3n equilibria for the n-neuron neural networks and 2n of which are locally exponentially stable. In [23], the Hopfield neural networks with nondecreasing piecewise linear activation functions with 2r corner points were considered. It was proved that under some conditions, the n-neuron neural networks can have and only have (2r+1)n equilibria, (r+1)n of which are locally exponentially stable and others are unstable. In [24], the multistability of neural networks with k+m step stair activation functions was discussed based on an appropriate partition of the n-dimensional state space. It was shown that the n-neuron neural networks can have (2k+2m-1)n equilibria, (k+m)n of which are locally exponentially stable. In particular, the case of k=m was previously discussed in [25]. For more references, see [26–32] and references therein.It is well known that the type of activation functions plays a very important role in the multistability analysis of neural networks. In the abovementioned and most existing works, the activation functions employed in multistability analysis were mainly focused on sigmoidal activation functions and nondecreasing saturated activation functions, which are all monotonously increasing. In this paper, we will consider a class of continuous Mexican-hat-type activation functions, which are defined as follows (see Figure1):
(3)fi(x)={ui,-∞<x<pi,li,1x+ci,1,pi≤x≤ri,li,2x+ci,2,ri<x≤qi,ui,qi<x<+∞,
where pi, ri, qi, ui, li,1, li,2, ci,1, ci,2 are constants with -∞<pi<ri<qi<+∞, li,1>0, and li,2<0, i=1,2,…,N. In particular, when pi=-1, ri=1, qi=3, ui=-1, li,1=1, li,2=-1, ci,1=0, and ci,2=2  (i=1,2,…,N), the above activation functions fi reduce to the following special activation functions employed in [33]:
(4)fi(x)={-1,-∞<x<-1,x,-1≤x≤1,-x+2,1<x≤3,-1,3<x<+∞.Figure 1
Mexican-hat-type activation functions (3).It is necessary to point out that the Mexican-hat-type activation functions are not monotonously increasing, which are totally different from sigmoidal activation functions and nondecreasing saturated activation functions. Hence, the results and methods mentioned above cannot be applied to neural networks with activation functions (3). Very recently, the multistability and instability of Hopfield neural networks with activation functions (4) were studied in [33]. Inspired by [33], in this paper, we will investigate the multistability and instability of CNNs with activation functions (3). It should be noted that the structure of CNNs differs from and is more complex than that in [33]. Moreover, the activation functions (3) employed in this paper are more general than activation functions (4). More precisely, the contributions of this paper are three-fold as follows.Firstly, we define four index subsets and present sufficient condition under which the CNNs with activation functions (3) have multiple equilibria, by tracking the dynamics of each state component and applying fixed point theorem. The index subsets are defined in terms of maximum and minimum values, which are different from and less restrictive than those given in [33]. Furthermore, we discuss the exact existence of equilibria for CNNs.Secondly, based on some analysis method, we analyze the dynamical behaviors of each equilibrium point for CNNs, including local stability and instability. The dynamical behaviors of such system are much more complex than those of Hopfield neural networks considered in [33], due to the complexity of the networks structure and generality of activation functions.Thirdly, specializing the model and activation functions to those in [33], we show that the obtained results extend and improve the very recent works in [33].Finally, two examples with their simulations are given to verify and illustrate the validity of the obtained results.


## 2. Main Results

Firstly, we define the four index subsets as follows:(5)ℕ1={{αi-1βiuiBi,αi-1βiviBi}∑j≠i,j=1Nmax⁡{aijuj,aijvj}i:Ii<-max⁡{-dipi+aiifi(pi),-diri+aiifi(ri)}kkkkkkk-∑j≠i,j=1Nmax⁡{aijuj,aijvj}kkkkkkk-max⁡{αi-1βiuiBi,αi-1βiviBi}∑j≠i,j=1Nmax⁡{aijuj,aijvj}},ℕ2={∑j≠i,j=1Nmax⁡{aijuj,aijvj}-max⁡{αi-1βiuiBi,αi-1βiviBi}i:diri-aiifi(ri)-∑j≠i,j=1Nmin⁡{aijuj,aijvj}kkkkkkk-min⁡{αi-1βiuiBi,αi-1βiviBi}kkkkk<Ii<dipi-aiifi(pi)-∑j≠i,j=1Nmax⁡{aijuj,aijvj}kkkkkkk-max⁡{αi-1βiuiBi,αi-1βiviBi}∑j≠i,j=1Nmin⁡{aijuj,aijvj}},ℕ3={{αi-1βiuiBi,αi-1βiviBi}∑j≠i,j=1Nmin⁡{aijuj,aijvj}i:-min⁡{-dipi+aiifi(pi),-diri+aiifi(ri)}kkkkkkk-∑j≠i,j=1Nmin⁡{aijuj,aijvj}kkkkkkk-min⁡{αi-1βiuiBi,αi-1βiviBi}<Ii<diqillllllllllllll-aiifi(qi)-∑j≠i,j=1Nmax⁡⁡{aijuj,aijvj}kkkkkkk-max⁡⁡{αi-1βiuiBi,αi-1βiviBi}∑j≠i,j=1Nmin⁡{aijuj,aijvj}},ℕ4={{αi-1βiuiBi,αi-1βiviBi}∑j≠i,j=1Nmin⁡⁡{aijuj,aijvj}i:Ii>-min⁡{-diqi+aiifi(qi),-diri+aiifi(ri)}kkkkkkk-∑j≠i,j=1Nmin⁡⁡{aijuj,aijvj}kkkikkk-min⁡⁡{αi-1βiuiBi,αi-1βiviBi}∑j≠i,j=1Nmin⁡⁡{aijuj,aijvj}},
where vi=fi(ri). It is easy to see that ui≤fi(x)≤vi for x∈ℝ.Remark 1.
In this paper, the index subsets are defined in terms of maximum and minimum values, which are different from those given in [33], where they are defined in terms of absolute values. In general, our conditions are less restrictive, which have been shown in [16].Remark 2.
The inequalitydi-(aii+αi-1βiBi)li,1<0 holds for all i∈ℕ2.Proof.
By the definition of index subsetℕ2, ui=fi(pi) and vi=fi(ri), we obtain
(6)-dipi+(aii+αi-1βiBi)fi(pi)+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+Ii<0,(7)-diri+(aii+αi-1βiBi)fi(ri)+∑j≠i,j=1Nmin⁡⁡{aijuj,aijvj}+Ii>0.
It follows from (6) and (7) that
(8)-dipi+(aii+αi-1βiBi)fi(pi)<-diri+(aii+αi-1βiBi)fi(ri).
Noting that pi<ri and substituting fi(pi)=li,1pi+ci,1 and fi(ri)=li,1ri+ci,1 into (8), we can derive that di-(aii+αi-1βiBi)li,1<0(i∈ℕ2).Remark 3.
The inequalitydi-(aii+αi-1βiBi)li,2>0 holds for all i∈ℕ2∪ℕ3.Proof.
From Remark2, we get aii+αi-1βiBi>0(i∈ℕ2). Thus, inequality di-(aii+αi-1βiBi)li,2>0 holds for all i∈ℕ2, due to li,2<0.
By the definition of index subsetℕ3 and equalities ui=fi(qi), vi=fi(ri), we get
(9)min⁡{-dipi+aiifi(pi),-diri+aiifi(ri)}+αi-1βiviBi>-diqi+aiifi(qi)+αi-1βiuiBi,
which implies that -diri+(aii+αi-1βiBi)fi(ri)>-diqi+(aii+αi-1βiBi)fi(qi). By using equalities fi(ri)=li,2ri+ci,2, fi(qi)=li,2qi+ci,2 and noting that ri<qi, the inequality di-(aii+αi-1βiBi)li,2>0(i∈ℕ3) can be proved easily.It follows from the second equation of system (2) that
(10)-αiSi(t)+βiui≤S˙i(t)≤-αiSi(t)+βivi,
which leads to
(11)αi-1βiui+(Si(0)-αi-1βiui)e-αit≤Si(t)≤αi-1βivi+(Si(0)-αi-1βivi)e-αit.
Therefore, Si(0)∈[αi-1βiui,αi-1βivi] always implies that Si(t)∈[αi-1βiui,αi-1βivi]. That is, if S(0)∈∏i=1N[αi-1βiui,αi-1βivi], then the solution S(t;S(0)) will stay in ∏i=1N[αi-1βiui,αi-1βivi] for all t≥0.Let(xT(t),ST(t))T be a solution of system (2) with initial state (xT(0),ST(0))T∈ℝN×∏i=1N[αi-1βiui,αi-1βivi]. In the following, we will discuss the dynamics of state components xi(t) for i∈ℕi  (i=1,2,3,4), respectively.Lemma 4.
All the state componentsxi(t), i∈ℕ1, will flow to the interval (-∞,pi] when t tends to +∞.Proof.
According to the different location ofxi(0), there are two cases for us to discuss.
Case (i). Consider xi(0)∈(-∞,pi]. In this case, if there exists some t*≥0 such that xi(t*)=pi, xi(t)≤pi for 0≤t≤t*, then it follows from system (2) and the definition of ℕ1 that
(12)x˙i(t*)=-dixi(t*)+aiifi(xi(t*))+∑j≠i,j=1Naijfj(xj(t*))+BiSi(t*)+Ii≤-dipi+aiifi(pi)+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+max⁡{αi-1βiuiBi,αi-1βiviBi}+Ii<0.
Hence, xi(t) would never get out of (-∞,pi]. Similarly, we can also conclude that once xi(T0)∈(-∞,pi] for some T0>0, then xi(t) would never escape from (-∞,pi] for all t≥T0.
Case (ii). Consider xi(0)∈(pi,+∞). In this case, we claim that xi(t) would monotonously decrease until it reaches the interval (-∞,pi] in some finite time.
In fact, whenxi(t)∈(qi,+∞), noting that the definition of ℕ1 and fi(pi)=fi(qi), we obtain
(13)x˙i(t)=-dixi(t)+aiifi(xi(t))+∑j≠i,j=1Naijfj(xj(t))+BiSi(t)+Ii≤-diqi+aiifi(qi)+∑j≠i,j=1Naijfj(xj(t))+BiSi(t)+Ii≤-dipi+aiifi(pi)+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+max⁡{αi-1βiuiBi,αi-1βiviBi}+Ii<0;
when xi(t)∈(ri,qi], by virtue of equalities fi(ri)=li,2ri+ci,2 and fi(qi)=li,2qi+ci,2, we get
(14)x˙i(t)=-dixi(t)+aii(li,2xi(t)+ci,2)+∑j≠i,j=1Naijfj(xj(t))+BiSi(t)+Ii≤max⁡{(-di+aiili,2)ri,(-di+aiili,2)qi}+aiici,2+∑j≠i,j=1Naijfj(xj(t))+BiSi(t)+Ii=max⁡{-diri+aiifi(ri),-diqi+aiifi(qi)}+∑j≠i,j=1Naijfj(xj(t))+BiSi(t)+Ii≤max⁡{-diri+aiifi(ri),-diqi+aiifi(qi)}+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+max⁡{αi-1βiuiBi,αi-1βiviBi}+Ii≤max⁡{-diri+aiifi(ri),-dipi+aiifi(pi)}+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+max⁡{αi-1βiuiBi,αi-1βiviBi}+Ii<0;
when xi(t)∈(pi,ri], it follows from fi(pi)=li,1pi+ci,1,fi(ri)=li,1ri+ci,1 and system (2) that
(15)x˙i(t)=-dixi(t)+aii(li,1xi(t)+ci,1)+∑j≠i,j=1Naijfj(xj(t))+BiSi(t)+Ii≤max⁡{(-di+aiili,1)pi,(-di+aiili,1)ri}+aiici,1+∑j≠i,j=1Naijfj(xj(t))+BiSi(t)+Ii≤max⁡{-dipi+aiifi(pi),-diri+aiifi(ri)}+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+max⁡{αi-1βiuiBi,αi-1βiviBi}+Ii<0.
In summary, wherever the initial statexi(0)  (i∈ℕ1) is located in, xi(t) would flow to and enter the interval (-∞,pi] and stay in this interval forever.Lemma 5.
All the state componentsxi(t), i∈ℕ3, will flow to the interval [ri,qi] when t tends to +∞.Proof.
We prove it in the following three cases due to the different location ofxi(0).
Case (i). Consider xi(0)∈[ri,qi]. In this case, if there exists some t*≥0 such that xi(t*)=ri, ri≤xi(t)≤qi for 0≤t≤t*, then we have
(16)x˙i(t*)=-dixi(t*)+aiifi(xi(t*))+∑j≠i,j=1Naijfj(xj(t*))+BiSi(t*)+Ii≥-diri+aiifi(ri)+∑j≠i,j=1Nmin⁡{aijuj,aijvj}+min⁡{αi-1βiuiBi,αi-1βiviBi}+Ii>0;
similarly, if there exists some t**≥0 such that xi(t**)=qi, ri≤xi(t)≤qi for 0≤t≤t**, then we get
(17)x˙i(t**)≤-diqi+aiifi(qi)+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+max⁡{αi-1βiuiBi,αi-1βiviBi}+Ii<0.
From the above two inequalities, we know that ifxi(0)∈[ri,qi], then xi(t) would never get out of this interval. In the same way, we can also obtain that if there exists some T0>0 such that xi(T0)∈[ri,qi], then xi(t) would stay in it for all t≥T0.
Case (ii). Consider xi(0)∈(-∞,ri). When xi(t)∈(-∞,pi], from the definition of index subset ℕ3, we get
(18)x˙i(t)≥-dipi+aiifi(pi)+∑j≠i,j=1Nmin⁡{aijuj,aijvj}+min⁡{αi-1βiuiBi,αi-1βiviBi}+Ii>0;
when xi(t)∈(pi,ri), it follows from the definition of index subset ℕ3, equalities fi(pi)=li,1pi+ci,1, fi(ri)=li,1ri+ci,1 that
(19)x˙i(t)=-dixi(t)+aii(li,1xi(t)+ci,1)+∑j≠i,j=1Naijfj(xj(t))+BiSi(t)+Ii≥min⁡{(-di+aiili,1)pi,(-di+aiili,1)ri}+aiici,1+∑j≠i,j=1Naijfj(xj(t))+BiSi(t)+Ii=min⁡{-dipi+aiifi(pi),-diri+aiifi(ri)}+∑j≠i,j=1Naijfj(xj(t))+BiSi(t)+Ii≥min⁡{-dipi+aiifi(pi),-diri+aiifi(ri)}+∑j≠i,j=1Nmin⁡{aijuj,aijvj}+min⁡{αi-1βiuiBi,αi-1βiviBi}+Ii>0.
Thus, in this case, xi(t) would monotonously increase until it reaches the interval [ri,qi].
Case (iii). Consider xi(0)∈(qi,+∞). When xi(t)∈(qi,+∞), it follows that
(20)x˙i(t)≤-diqi+aiifi(qi)+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+max⁡{αi-1βiuiBi,αi-1βiviBi}+Ii<0.
Therefore, xi(t) would monotonously decrease until it enters the interval [ri,qi].
In summary, wherever the initial statexi(0)(i∈ℕ3) is located in, xi(t) would flow to and enter the interval [ri,qi] and stay in it finally.Lemma 6.
All the state componentsxi(t),i∈ℕ4, will flow to the interval [qi,+∞) when t tends to +∞.Proof.
Similar to the proof of Lemmas4 and 5, we will prove it in the following two cases.
Case (i). Consider xi(0)∈[qi,+∞). If there exists some t*≥0 such that xi(t*)=qi, xi(t)≥qi for 0≤t≤t*, then
(21)x˙i(t*)≥-diqi+aiifi(qi)+∑j≠i,j=1Nmin⁡{aijuj,aijvj}+min⁡{αi-1βiuiBi,αi-1βiviBi}+Ii>0.
Therefore, xi(t) would never get out of [qi,+∞). By the same method, we also get that once xi(T0)∈[qi,+∞) for some T0>0, then xi(t) would stay in this interval for all t≥T0.
Case (ii). Consider xi(0)∈(-∞,qi). In this case, we claim that xi(t) would monotonously increase until it enters the interval [qi,+∞).
In fact, whenxi(t)∈(-∞,pi), we have
(22)x˙i(t)≥-dipi+aiifi(pi)+∑j≠i,j=1Nmin⁡{aijuj,aijvj}+min⁡{αi-1βiuiBi,αi-1βiviBi}+Ii≥-diqi+aiifi(qi)+∑j≠i,j=1Nmin⁡{aijuj,aijvj}+min⁡{αi-1βiuiBi,αi-1βiviBi}+Ii>0;
when xi(t)∈[pi,ri], we get
(23)x˙i(t)=-dixi(t)+aii(li,1xi(t)+ci,1)+∑j≠i,j=1Naijfj(xj(t))+BiSi(t)+Ii≥min⁡{-dipi+aiifi(pi),-diri+aiifi(ri)}+∑j≠i,j=1Naijfj(xj(t))+BiSi(t)+Ii≥min⁡{-diqi+aiifi(qi),-diri+aiifi(ri)}+∑j≠i,j=1Nmin⁡{aijuj,aijvj}+min⁡{αi-1βiuiBi,αi-1βiviBi}+Ii>0;
when xi(t)∈(ri,qi), then we obtain
(24)x˙i(t)=-dixi(t)+aii(li,2xi(t)+ci,2)+∑j≠i,j=1Naijfj(xj(t))+BiSi(t)+Ii≥min⁡{-diqi+aiifi(qi),-diri+aiifi(ri)}+∑j≠i,j=1Nmin⁡{aijuj,aijvj}+min⁡{αi-1βiuiBi,αi-1βiviBi}+Ii>0.
In summary, wherever the initial statexi(0)  (i∈ℕ4) is located in, xi(t) would flow to [qi,+∞) when t is big enough and stay in it forever.Denote(-∞,pi]=(-∞,pi]1×[pi,ri]0×[ri,+∞)0; [pi,ri]=(-∞,pi]0×[pi,ri]1×[ri,+∞)0; [ri,+∞)=(-∞,pi]0×[pi,ri]0×[ri,+∞)1. For any i∈ℕ2, let δi=(δi(1),δi(2),δi(3))=(1,0,0) or (0,1,0) or (0,0,1) and define
(25)Ωδi=(-∞,pi]δi(1)×[pi,ri]δi(2)×[ri,+∞)δi(3),Ωδ=∏i∈ℕ1(-∞,pi]×∏i∈ℕ2Ωδi×∏i∈ℕ3[ri,qi]×∏i∈ℕ4[qi,+∞).It is easy to see that there exist3♯ℕ2  Ωδ-type regions, where ♯ℕ2 denotes the number of elements in the set ℕ2. Now, we will prove the following theorem on the existence of multiple equilibria for system (2).Theorem 7.
Suppose thatℕ1∪ℕ2∪ℕ3∪ℕ4={1,2,…,N}. Then, system (2) with activation functions (3) has 3♯ℕ2 equilibria.Proof.
Pick a region arbitrarilyΩδ×∏i=1N[αi-1βiui,αi-1βivi]; we will show that there exists an equilibrium point located in each Ωδ×∏i=1N[αi-1βiui,αi-1βivi].
Denoteℕ2,1={i∈ℕ2:δi(1)=1}, ℕ2,2={i∈ℕ2:δi(2)=1}, and ℕ2,3={i∈ℕ2:δi(3)=1}. It is easy to see that ℕ2,1∪ℕ2,2∪ℕ2,3=ℕ2.
Let(xT(t),ST(t))T be any solution of system (2) with initial state (xT(0),ST(0))T∈Ωδ×∏i=1N[αi-1βiui,αi-1βivi]. Then, for i∈ℕ2,1, if there exists some t*≥0 such that xi(t*)=pi, then we get from the definition of index subset ℕ2 that
(26)x˙i(t*)=-dixi(t*)+aiifi(xi(t*))+∑j≠i,j=1Naijfj(xj(t*))+BiSi(t*)+Ii≤-dipi+aiifi(pi)+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+max⁡{αi-1βiuiBi,αi-1βiviBi}+Ii<0.
Similarly, for i∈ℕ2,3, if xi(t)≥qi, note that fi(pi)=fi(qi); then
(27)x˙i(t)≤-diqi+aiifi(qi)+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+max⁡{αi-1βiuiBi,αi-1βiviBi}+Ii≤-dipi+aiifi(pi)+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+max⁡{αi-1βiuiBi,αi-1βiviBi}+Ii<0,
and if there exists some t**≥0 such that xi(t**)=ri, then
(28)x˙i(t**)≥-diri+aiifi(ri)+∑j≠i,j=1Nmin⁡{aijuj,aijvj}+min⁡{αi-1βiuiBi,αi-1βiviBi}+Ii>0.
That is, the trajectory xi(t) with xi(0)∈[ri,+∞),i∈ℕ2,3, would enter and stay in the interval [ri,qi], which implies that there does not exist any equilibria with the corresponding ith state component located in [qi,+∞).
Combining with Lemmas4–6, it can be concluded that xi(t),i∉ℕ2,2, would never escape from the corresponding interval of Ωδ. Furthermore, denote
(29)Ω~δ=∏i∈ℕ1∪ℕ2,1(-∞,pi]×∏i∈ℕ2,2[pi,ri]×∏i∈ℕ2,3∪ℕ3[ri,qi]×∏i∈ℕ4[qi,+∞).
Then, from (26)–(28), we can derive that if Ωδ×∏i=1N[αi-1βiui,αi-1βivi] has an equilibrium point, it must be located in Ω~δ×∏i=1N[αi-1βiui,αi-1βivi].
Note that any equilibrium point of system (2) is a root of the following equations:
(30)-dixi+∑j=1Naijfj(xj)+BiSi+Ii=0,-αiSi+βifi(xi)=0,i=1,2,…,N.
Equivalently, the above equations can be rewritten as
(31)-dixi+(aii+αi-1Biβi)fi(xi)+∑j≠i,j=1Naijfj(xj)+Ii=0,Si=αi-1βifi(xi),i=1,2,…,N.
InΩ~δ×∏i=1N[αi-1βiui,αi-1βivi], any equilibrium point of system (2) satisfies the following equations:
(32)-dixi+(aii+αi-1βiBi)ui+∑j∈ℕ1∪ℕ2,1j≠iaijuj+∑j∈ℕ2,2aij(lj,1xj+cj,1)+∑j∈ℕ2,3∪ℕ3aij(lj,2xj+cj,2)+∑j∈ℕ4aijuj+Ii=0,i∈ℕ1∪ℕ2,1,-dixi+(aii+αi-1βiBi)(li,1xi+ci,1)+∑j∈ℕ1∪ℕ2,1aijuj+∑j∈ℕ2,2j≠iaij(lj,1xj+cj,1)+∑j∈ℕ2,3∪ℕ3aij(lj,2xj+cj,2)+∑j∈ℕ4aijuj+Ii=0,i∈ℕ2,2,-dixi+(aii+αi-1βiBi)(li,2xi+ci,2)+∑j∈ℕ1∪ℕ2,1aijuj+∑j∈ℕ2,2aij(lj,1xj+cj,1)+∑j∈ℕ2,3∪ℕ3j≠iaij(lj,2xj+cj,2)+∑j∈ℕ4aijuj+Ii=0,i∈ℕ2,3∪ℕ3,-dixi+(aii+αi-1βiBi)ui+∑j∈ℕ1∪ℕ2,1aijuj+∑j∈ℕ2,2aij(lj,1xj+cj,1)+∑j∈ℕ2,3∪ℕ3aij(lj,2xj+cj,2)+∑j∈ℕ4j≠iaijuj+Ii=0,i∈ℕ4.
In the subset region∏i∈ℕ2,2[pi,ri]×∏i∈ℕ2,3∪ℕ3[ri,qi], define a map Γ as follows:
(33)Γi(xi)=1di-(aii+αi-1βiBi)li,1×[∑j∈ℕ2,2j≠iaij(lj,1xj+cj,1)(aii+αi-1βiBi)ci,1+∑j∈ℕ1∪ℕ2,1aijujkkkkk+∑j∈ℕ2,2j≠iaij(lj,1xj+cj,1)kkkkk+∑j∈ℕ2,3∪ℕ3aij(lj,2xj+cj,2)kkk+∑j∈ℕ4aijuj+Ii∑j∈ℕ2,2j≠iaij(lj,1xj+cj,1)],i∈ℕ2,2,Γi(xi)=1di-(aii+αi-1βiBi)li,2×[∑j∈ℕ2,3∪ℕ3j≠iaij(lj,2xj+cj,2)(aii+αi-1βiBi)ci,2+∑j∈ℕ1∪ℕ2,1aijujkkkkk+∑j∈ℕ2,2aij(lj,1xj+cj,1)kkkkk+∑j∈ℕ2,3∪ℕ3j≠iaij(lj,2xj+cj,2)kkk+∑j∈ℕ4aijuj+Ii∑j∈ℕ2,3∪ℕ3j≠iaij(lj,2xj+cj,2)],i∈ℕ2,3∪ℕ3.
For i∈ℕ2,2, substituting fi(pi)=li,1pi+ci,1 into (6) and noting that di-(aii+αi-1βiBi)li,1<0 (Remark 2), we get that
(34)Γi(xi)≥1di-(aii+αi-1βiBi)li,1×[(aii+αi-1βiBi)ci,1+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+Ii]>pi.
Similarly, substituting fi(ri)=li,1ri+ci,1 into (7) results in
(35)Γi(xi)≤1di-(aii+αi-1βiBi)li,1×[(aii+αi-1βiBi)ci,1+∑j≠i,j=1Nmin⁡{aijuj,aijvj}+Ii]<ri.
For i∈ℕ2,3, note that -dipi>-diqi, fi(pi)=fi(qi)=li,2qi+ci,2, and fi(ri)=li,2ri+ci,2; it follows from (6)-(7) and di-(aii+αi-1βiBi)li,2>0 (Remark 3) that
(36)Γi(xi)≥1di-(aii+αi-1βiBi)li,2×[(aii+αi-1βiBi)ci,2+∑j≠i,j=1Nmin⁡{aijuj,aijvj}+Ii]>ri,Γi(xi)≤1di-(aii+αi-1βiBi)li,2×[(aii+αi-1βiBi)ci,2+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+Ii]<qi.
For i∈ℕ3, note that vi=fi(ri)=li,2ri+ci,2 and ui=fi(qi)=li,2qi+ci,2; we can get from the definition of index subset ℕ3 that
(37)-diri+(aii+αi-1βiBi)(li,2ri+ci,2)+∑j≠i,j=1Nmin⁡{aijuj,aijvj}+Ii>0,-diqi+(aii+αi-1βiBi)(li,2qi+ci,2)+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+Ii<0.
That is,
(38)[-di+(aii+αi-1βiBi)li,2]ri+(aii+αi-1βiBi)ci,2+∑j≠i,j=1Nmin⁡{aijuj,aijvj}+Ii>0,[-di+(aii+αi-1βiBi)li,2]qi+(aii+αi-1βiBi)ci,2+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+Ii<0.
Then combining inequality di-(aii+αi-1βiBi)li,2>0 and (38) together gives
(39)Γi(xi)≥1di-(aii+αi-1βiBi)li,2×[(aii+αi-1βiBi)ci,2+∑j≠i,j=1Nmin⁡{aijuj,aijvj}+Ii]>ri,Γi(xi)≤1di-(aii+αi-1βiBi)li,2×[(aii+αi-1βiBi)ci,2+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+Ii]<qi.
Therefore, the map Γ maps a bounded and closed set into itself. Applying Brouwer’s fixed point theorem, there exists one (xi1*,xi2*,…,xis*)T∈∏i∈ℕ2,2[pi,ri]×∏i∈ℕ2,3∪ℕ3[ri,qi] such that
(40)Γ(xi1*,xi2*,…,xis*)=(xi1*,xi2*,…,xis*),
where i1,i2,…,is represent the elements of index subset ℕ2,2∪ℕ2,3∪ℕ3.
Then fori∈ℕ1∪ℕ2,1, define
(41)Fi(xi)=-dixi+(aii+αi-1βiBi)ui+∑j∈ℕ1∪ℕ2,1j≠iaijuj+∑j∈ℕ2,2aij(lj,1xj*+cj,1)+∑j∈ℕ2,3∪ℕ3aij(lj,2xj*+cj,2)+∑j∈ℕ4aijuj+Ii.
By virtue of the definition of index subsets ℕ1 and ℕ2, we have lim⁡ξ→-∞⁡Fi(ξ)=+∞ and
(42)Fi(pi)≤-dipi+(aii+αi-1βiBi)ui+∑j≠i,j=1Nmax⁡{aijuj,aijvj}+Ii<0.
Therefore, there exists the unique xi*∈(-∞,pi) such that Fi(xi*)=0.
Similarly, fori∈ℕ4, define
(43)Fi(xi)=-dixi+(aii+αi-1βiBi)ui+∑j∈ℕ1∪ℕ2,1aijuj+∑j∈ℕ2,2aij(lj,1xj*+cj,1)+∑j∈ℕ2,3∪ℕ3aij(lj,2xj*+cj,2)+∑j∈ℕ4j≠iaijuj+Ii,
and we can also derive that there exists the unique xi*∈(qi,+∞) such that Fi(xi*)=0.
Denote(x*T,S*T)T=(x1*,x2*,…,xN*, β1α1-1f1(x1*), β2α2-1f2(x2*),…,βNαN-1fN(xN*))T. It is easy to see that (x*T,S*T)T is the equilibrium point located in subset Ω~δ×∏i=1N[αi-1βiui,αi-1βivi], which is also the equilibrium point located in subset Ωδ×∏i=1N[αi-1βiui,αi-1βivi]. It should be noted that, by the definition of ℕ1–ℕ4, any state component of x* cannot touch the boundary of Ωδ. That is, x* is located in the interior of Ωδ (see Remark 8). Therefore, system (2) has 3♯ℕ2 equilibria.Remark 8.
x
* is located in the interior of Ωδ.Proof.
Note thatx* satisfies the following equations:
(44)-dixi*+(aii+αi-1Biβi)fi(xi*)+∑j≠i,j=1Naijfj(xj*)+Ii=0,i=1,2,…,N.
In the following, we prove that xi*≠pi, i∈ℕ1. Otherwise, from the definition of ℕ1, we have
(45)0=-dipi+(aii+αi-1Biβi)fi(pi)+∑j≠i,j=1Naijfj(xj*)+Ii≤-dipi+aiifi(pi)+∑j≠i,j=1Nmax⁡{aijujaijvj}+max⁡{αi-1βiuiBi,αi-1βiviBi}+Ii<0,
that is a contradiction. Similarly, from the definition of ℕ2–ℕ4, we can also get
(46)xi*≠pi,ri,qi,i∈ℕ2,xi*≠ri,qi,i∈ℕ3,xi*≠qi,i∈ℕ4.
That is, x* is located in the interior of Ωδ.Remark 9.
Suppose thatℕ1∪ℕ2∪ℕ3∪ℕ4={1,2,…,N}. Furthermore, if the following conditions
(47)∑j∈ℕ2,2j≠i|aijlj,1|+∑j∈ℕ2,3∪ℕ3|aijlj,2|<(aii+αi-1Biβi)li,1-di,i∈ℕ2,2,(48)∑j∈ℕ2,2|aijlj,1|+∑j∈ℕ2,3∪ℕ3j≠i|aijlj,2|<di-(aii+αi-1Biβi)li,2,i∈ℕ2,3∪ℕ3,
hold, then system (2) with activation functions (3) can have and only have 3♯ℕ2 equilibria.Proof.
From the proof of Theorem7, we only need to prove that the fixed point of Γ is unique. In fact, suppose that there exists another fixed point (xi1**,xi2**,…,xis**)T∈∏i∈ℕ2,2[pi,ri]×∏i∈ℕ2,3∪ℕ3[ri,qi] such that
(49)Γ(xi1**,xi2**,…,xis**)=(xi1**,xi2**,…,xis**).
Denote the index il such that
(50)|xil*-xil**|=max⁡i∈ℕ2,2∪ℕ2,3∪ℕ3|xi*-xi**|.
If il∈ℕ2,2, from (ailil+αil-1βilBil)lil,1-dil>0 (Remark 2) and (47), we have
(51)|xil*-xil**|=|Γil(xil*)-Γil(xil**)|=1(ailil+αil-1βilBil)lil,1-dil×|∑j∈ℕ2,2j≠iailjlj,1(xj*-xj**)kkkkk+∑j∈ℕ2,3∪ℕ3ailjlj,2(xj*-xj**)∑j∈ℕ2,2j≠iailjlj,1(xj*-xj**)|≤∑j∈ℕ2,2,j≠il|ailjlj,1|+∑j∈ℕ2,3∪ℕ3|ailjlj,2|(ailil+αil-1βilBil)lil,1-dil×|xil*-xil**|<|xil*-xil**|.
If il∈ℕ2,3∪ℕ3, from dil-(ailil+αil-1βilBil)lil,2>0 (Remark 3) and (48), we have
(52)|xil*-xil**|=|Γil(xil*)-Γil(xil**)|=1dil-(ailil+αil-1βilBil)lil,2×|∑j∈ℕ2,3∪ℕ3j≠ilailjlj,2(xj*-xj**)∑j∈ℕ2,2ailjlj,1(xj*-xj**)+∑j∈ℕ2,3∪ℕ3j≠ilailjlj,2(xj*-xj**)|≤∑j∈ℕ2,2|ailjlj,1|+∑j∈ℕ2,3∪ℕ3,j≠il|ailjlj,2|dil-(ailil+αil-1βilBil)lil,2×|xil*-xil**|<|xil*-xil**|.
By the above two inequalities, we can deduce that xi*=xi** for all i∈ℕ2,2∪ℕ2,3∪ℕ3. That is, the fixed point of map Γ is unique.Theorem 10.
Suppose thatℕ1∪ℕ2∪ℕ3∪ℕ4={1,2,…,N}.(i)
If the following conditions(53)di-aiili,2-∑j∈ℕ2,2|aij||lj,1|-∑j∈ℕ2,3∪ℕ3j≠i|aij||lj,2|-|Bi|>0,(54)αi-βi|li,2|>0hold for all i∈ℕ2,3∪ℕ3, then system (2) with activation functions (3) has 2♯ℕ2 locally stable equilibria.(ii)
Furthermore, if the following conditions(55)-di+aiili,1-∑j∈ℕ2,2j≠i|aij||lj,1|-∑j∈ℕ2,3∪ℕ3|aij||lj,2|-|Bi|>0,(56)αi-βi|li,1|>0hold for all i∈ℕ2,2, then the other 3♯ℕ2-2♯ℕ2 equilibria are unstable.Proof.
In the following, we will discuss the dynamical behaviors of3♯ℕ2 equilibria in two cases, respectively.
Case (i). Consider ℕ2,2=∅. In this case, Ωδ and Ω~δ can be rewritten as
(57)Ωδ=∏i∈ℕ1∪ℕ2,1‍(-∞,pi]×∏i∈ℕ2,3‍[ri,+∞)×∏i∈ℕ3‍[ri,qi]×∏i∈ℕ4‍[qi,+∞),Ω~δ=∏i∈ℕ1∪ℕ2,1‍(-∞,pi]×∏i∈ℕ2,3∪ℕ3‍[ri,qi]×∏i∈ℕ4‍[qi,+∞).
From (53)-(54), we can choose a sufficiently small number η>0 such that
(58)di-aiili,2-∑j∈ℕ2,3∪ℕ3j≠i|aij||lj,2|-|Bi|>η>0,(59)αi-βi|li,2|>η>0
hold for all i∈ℕ2,3∪ℕ3.
Let(xT(t),ST(t))T be any a solution of system (2) with initial state (xT(0),ST(0))T∈Ω~δ×∏i=1N[αi-1βiui,αi-1βivi]. From Lemmas 4–6 and the proof of Theorem 7, we get that there exists some T0≥0 such that (xT(T0),ST(T0))T∈Ω~δ×∏i=1N[αi-1βiui,αi-1βivi]. By the positive invariance of Ω~δ×∏i=1N[αi-1βiui,αi-1βivi], we know that the solution (xT(t),ST(t))T∈Ω~δ×∏i=1N[αi-1βiui,αi-1βivi] for all t≥T0, and its dynamics can be described by
(60)x˙i(t)=-dixi(t)+aiiui+∑j∈ℕ1∪ℕ2,1j≠iaijuj+∑j∈ℕ2,3∪ℕ3aij(lj,2xj(t)+cj,2)+∑j∈ℕ4aijuj+BiSi(t)+Ii,i∈ℕ1∪ℕ2,1,x˙i(t)=-dixi(t)+aii(li,2xi(t)+ci,2)+∑j∈ℕ1∪ℕ2,1aijuj+∑j∈ℕ2,3∪ℕ3j≠iaij(lj,2xj(t)+cj,2)+∑j∈ℕ4aijuj+BiSi(t)+Ii,i∈ℕ2,3∪ℕ3,x˙i(t)=-dixi(t)+aiiui+∑j∈ℕ1∪ℕ2,1aijuj+∑j∈ℕ2,3∪ℕ3aij(lj,2xj(t)+cj,2)+∑j∈ℕ4j≠iaijuj+BiSi(t)+Ii,i∈ℕ4,S˙i(t)=-αiSi(t)+βifi(xi(t)),i=1,2,…,N.
We get from model (60) that
(61)d(xi(t)-xi*)dt=-di(xi(t)-xi*)llllllllllllllllllllll+∑j∈ℕ2,3∪ℕ3aijlj,2(xj(t)-xj*)llllllllllllllllllllll+Bi(Si(t)-Si*),i∈ℕ1∪ℕ2,1∪ℕ4,d(xi(t)-xi*)dt=(-di+aiili,2)(xi(t)-xi*)llllllllllllllllllllll+∑j∈ℕ2,3∪ℕ3j≠iaijlj,2(xj(t)-xj*)llllllllllllllllllllll+Bi(Si(t)-Si*),i∈ℕ2,3∪ℕ3,d(Si(t)-Si*)dt=-αi(Si(t)-Si*),i∈ℕ1∪ℕ2,1∪ℕ4,d(Si(t)-Si*)dt=-αi(Si(t)-Si*)llllllllllllllllllllll+βili,2(xi(t)-xi*),i∈ℕ2,3∪ℕ3.
Let(62)φi(t)=eηt|xi(t)-xi*|,ψi(t)=eηt|Si(t)-Si*|,jjjkkkkki∈ℕ2,3∪ℕ3;
then by using (61)-(62), we can derive that
(63)D+φi(t)=ηeηt|xi(t)-xi*|+eηtsgn⁡(xi(t)-xi*)×[∑j∈ℕ2,3∪ℕ3j≠iaijlj,2(xj(t)-xj*)∑j∈ℕ2,3∪ℕ3j≠iaijlj,2(xj(t)-xj*)(-di+aiili,2)(xi(t)-xi*)lllllllllllll+∑j∈ℕ2,3∪ℕ3j≠iaijlj,2(xj(t)-xj*)+Bi(Si(t)-Si*)]≤(η-di+aiili,2)φi(t)+∑j∈ℕ2,3∪ℕ3j≠i‍|aij||lj,2|φj(t)+|Bi|ψi(t).
Similarly, we have
(64)D+ψi(t)=ηeηt|Si(t)-Si*|+eηtsgn⁡(Si(t)-Si*)×[-αi(Si(t)-Si*)+βili,2(xi(t)-xi*)]≤(η-αi)ψi(t)+βi|li,2|φi(t).
Letσ>0 be an arbitrary real number and
(65)l0=(1+σ)eηT0(max⁡1≤i≤N|xi(T0)-xi*|+max⁡1≤i≤N|Si(T0)-Si*|);
then
(66)φi(T0)=eηT0|xi(T0)-xi*|≤eηT0max⁡1≤i≤N|xi(T0)-xi*|<l0,i∈ℕ2,3∪ℕ3,ψi(T0)=eηT0|Si(T0)-Si*|≤eηT0max⁡1≤i≤N|Si(T0)-Si*|<l0,i∈ℕ2,3∪ℕ3.
In the following, we will prove that
(67)φi(t)≤l0,ψi(t)≤l0,t>T0,i∈ℕ2,3∪ℕ3.
If (67) is not true, then there exists some i∈ℕ2,3∪ℕ3 and t*>T0 such that either
(68)φi(t*)=l0,D+φi(t*)>0,φj(t*)≤l0(T0≤t≤t*,j∈ℕ2,3∪ℕ3),ψj(t*)≤l0(T0≤t≤t*,j∈ℕ2,3∪ℕ3)
or
(69)ψi(t*)=l0,D+ψi(t*)>0,ψj(t*)≤l0(T0≤t≤t*,j∈ℕ2,3∪ℕ3),φj(t*)≤l0(T0≤t≤t*,j∈ℕ2,3∪ℕ3).
For the first case, it follows from (58) and (63) that
(70)D+φi(t*)≤(η-di+aiili,2)φi(t*)+∑j∈ℕ2,3∪ℕ3j≠i|aij||lj,2|φj(t*)+|Bi|ψi(t*)≤[η-di+aiili,2+∑j∈ℕ2,3∪ℕ3j≠i|aij||lj,2|+|Bi|]l0<0,
that is a contradiction. For the second case, by using (59) and (64), we get
(71)D+ψi(t*)≤(η-αi)ψi(t*)+βi|li,2|φi(t*)≤[η-αi+βi|li,2|]l0<0,
that is also a contradiction. So (67) holds; that is, |xi(t)-xi*|≤l0e-ηt, |Si(t)-Si*|≤l0e-ηt hold for all t≥T0 and i∈ℕ2,3∪ℕ3.
From (61), we derive that
(72)|Si(t)-Si*|=|Si(T0)-Si*|e-αi(t-T0),kkkkillkkt≥T0,i∈ℕ1∪ℕ2,1∪ℕ4,
which implies that lim⁡t→+∞⁡|Si(t)-Si*|=0  (i∈ℕ1∪ℕ2,1∪ℕ4). Thus, for any ɛ>0, there exists a constant T1≥T0 such that
(73)∑j∈ℕ2,3∪ℕ3|aij||lj,2||xj(t)-xj*|+|Bi||Si(t)-Si*|<ɛ,kkkkkkkkkkkllkkkkkkt≥T1,i∈ℕ1∪ℕ2,1∪ℕ4.
Then we have
(74)D+(|xi(t)-xi*|)≤-di|xi(t)-xi*|+ɛ,kkkkkkkkkkkkkkkki∈ℕ1∪ℕ2,1∪ℕ4,
which implies that
(75)|xi(t)-xi*|≤e-di(t-T1)|xi(T1)-xi*|+ɛdi(1-e-di(t-T1)),kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkki∈ℕ1∪ℕ2,1∪ℕ4.
That is, lim⁡t→+∞⁡|xi(t)-xi*|=0  (i∈ℕ1∪ℕ2,1∪ℕ4).
To sum up, for alli=1,2,…,N, we have lim⁡t→+∞⁡|xi(t)-xi*|=0, lim⁡t→+∞⁡|Si(t)-Si*|=0. That is, the equilibrium point (x*T,S*T)T is locally stable in Ωδ×∏i=1N[αi-1βiui,αi-1βivi].
Case (ii). Consider ℕ2,2≠∅. In this case, let (xT(t),ST(t))T be a solution of system (2) with initial condition (xT(0),ST(0))T nearby (x*T,S*T)T∈Ω~δ×∏i=1N[αi-1βiui,αi-1βivi]. Without loss of generality, we assume that (xT(t),ST(t))T∈Ω~δ×∏i=1N[αi-1βiui,αi-1βivi] for all t≥0. In fact, we can conclude that if the solution (xT(t),ST(t))T gets out of region Ω~δ×∏i=1N[αi-1βiui,αi-1βivi], the equilibrium point (x*T,S*T)T must be unstable according to the definition of instability. Thus, system (2) can be rewritten as
(76)x˙i(t)=-dixi(t)+aiiui+∑j∈ℕ1∪ℕ2,1j≠iaijuj+∑j∈ℕ2,2aij(lj,1xj(t)+cj,1)+∑j∈ℕ2,3∪ℕ3aij(lj,2xj(t)+cj,2)+∑j∈ℕ4aijuj+BiSi(t)+Ii,i∈ℕ1∪ℕ2,1,x˙i(t)=-dixi(t)+aii(li,1xi(t)+ci,1)+∑j∈ℕ1∪ℕ2,1aijuj+∑j∈ℕ2,2j≠iaij(lj,1xj(t)+cj,1)+∑j∈ℕ2,3∪ℕ3aij(lj,2xj(t)+cj,2)+∑j∈ℕ4aijuj+BiSi(t)+Ii,i∈ℕ2,2,x˙i(t)=-dixi(t)+aii(li,2xi(t)+ci,2)+∑j∈ℕ1∪ℕ2,1aijuj+∑j∈ℕ2,2aij(lj,1xj(t)+cj,1)+∑j∈ℕ2,3∪ℕ3j≠iaij(lj,2xj(t)+cj,2)+∑j∈ℕ4aijuj+BiSi(t)+Ii,i∈ℕ2,3∪ℕ3,x˙i(t)=-dixi(t)+aiiui+∑j∈ℕ1∪ℕ2,1aijuj+∑j∈ℕ2,2aij(lj,1xj(t)+cj,1)+∑j∈ℕ2,3∪ℕ3aij(lj,2xj(t)+cj,2)+∑j∈ℕ4j≠iaijuj+BiSi(t)+Ii,i∈ℕ4,S˙i(t)=-αiSi(t)+βifi(xi(t)),i=1,2,…,N.
It follows from model (76) that
(77)d(xi(t)-xi*)dt=-di(xi(t)-xi*)llllllllllllllllllllll+∑j∈ℕ2,2aijlj,1(xj(t)-xj*)llllllllllllllllllllll+∑j∈ℕ2,3∪ℕ3aijlj,2(xj(t)-xj*)llllllllllllllllllllll+Bi(Si(t)-Si*),i∈ℕ1∪ℕ2,1∪ℕ4,d(xi(t)-xi*)dt=(-di+aiili,1)(xi(t)-xi*)llllllllllllllllllllll+∑j∈ℕ2,2j≠iaijlj,1(xj(t)-xj*)llllllllllllllllllllll+∑j∈ℕ2,3∪ℕ3aijlj,2(xj(t)-xj*)llllllllllllllllllllll+Bi(Si(t)-Si*),i∈ℕ2,2,d(xi(t)-xi*)dt=(-di+aiili,2)(xi(t)-xi*)llllllllllllllllllllll+∑j∈ℕ2,2aijlj,1(xj(t)-xj*)llllllllllllllllllllll+∑j∈ℕ2,3∪ℕ3j≠iaijlj,2(xj(t)-xj*)llllllllllllllllllllll+Bi(Si(t)-Si*),i∈ℕ2,3∪ℕ3,d(Si(t)-Si*)dt=-αi(Si(t)-Si*),i∈ℕ1∪ℕ2,1∪ℕ4,d(Si(t)-Si*)dt=-αi(Si(t)-Si*)+βili,1(xi(t)-xi*),kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkki∈ℕ2,2,d(Si(t)-Si*)dt=-αi(Si(t)-Si*)+βili,2(xi(t)-xi*),kkkkkkkkkkkkkkkkkkkkkkikkkkkkkki∈ℕ2,3∪ℕ3.
Pick an initial state(xT(0),ST(0))T of system (2) such that
(78)max⁡1≤i≤N{|xi(0)-xi*|,|Si(0)-Si*|}=max⁡i∈ℕ2,2|xi(0)-xi*|>0.
In the following, we claim that
(79)max⁡i∈ℕ2,2∪ℕ2,3∪ℕ3{|xi(t)-xi*|,|Si(t)-Si*|}=max⁡i∈ℕ2,2|xi(t)-xi*|
holds for all t≥0. In fact, if (79) is not true, then there exists some t0≥0 such that
(80)max⁡{max⁡i∈ℕ2,3∪ℕ3|xi(t0)-xi*|,max⁡i∈ℕ2,2∪ℕ2,3∪ℕ3|Si(t0)-Si*|}=max⁡i∈ℕ2,2|xi(t0)-xi*|;
then there exists some index i0 such that one of the following three cases holds:(a)
i0∈ℕ2,3∪ℕ3 and |xi0(t0)-xi0*|=max⁡i∈ℕ2,2⁡|xi(t0)-xi*|,(b)
i0∈ℕ2,3∪ℕ3 and |Si0(t0)-Si0*|=max⁡i∈ℕ2,2⁡|xi(t0)-xi*|,(c)
i0∈ℕ2,2 and |Si0(t0)-Si0*|=max⁡i∈ℕ2,2⁡|xi(t0)-xi*|.For the case (a), it follows from (53) and (77) that
(81)D+(|xi0(t0)-xi0*|)≤(-di0+ai0i0li0,2)|xi0(t0)-xi0*|+∑j∈ℕ2,2|ai0j|lj,1|xj(t0)-xj*|+∑j∈ℕ2,3∪ℕ3j≠i0|ai0j||lj,2||xj(t0)-xj*|+|Bi0||Si0(t0)-Si0*|≤(∑j∈ℕ2,3∪ℕ3j≠i0|ai0j||lj,2|-di0+ai0i0li0,2+∑j∈ℕ2,2|ai0j|lj,1kllk+∑j∈ℕ2,3∪ℕ3j≠i0|ai0j||lj,2|+|Bi0|)·|xi0(t0)-xi0*|<0.
For the case (b), by using (54) and (77), we get that
(82)D+(|Si0(t0)-Si0*|)≤(-αi0+βi0|li0,2|)|Si0(t0)-Si0*|<0.
Similarly, for the case (c), by means of (56) and (77), we have
(83)D+(|Si0(t0)-Si0*|)≤(-αi0+βi0|li0,1|)|Si0(t0)-Si0*|<0.
Meanwhile, denotei1∈ℕ2,2 such that |xi1(t0)-xi1*|=max⁡i∈ℕ2,2⁡|xi(t0)-xi*|; then we obtain from (55) and (77) that
(84)D+(|xi1(t0)-xi1*|)≥(-di1+ai1i1li1,1)|xi1(t0)-xi1*|-∑j∈ℕ2,2j≠i1|ai1j|lj,1|xj(t0)-xj*|-∑j∈ℕ2,3∪ℕ3|ai1j||lj,2||xj(t0)-xj*|-|Bi1||Si1(t0)-Si1*|≥(-di1+ai1i1li1,1-∑j∈ℕ2,2j≠i1|ai1j|lj,1kllk-∑j∈ℕ2,3∪ℕ3|ai1j||lj,2|-|Bi1|∑j∈ℕ2,2j≠i1|ai1j|lj,1)·|xi1(t0)-xi1*|>0.
Thus, combining inequalities (79) and (84) together gives
(85)max⁡i∈ℕ2,2∪ℕ2,3∪ℕ3{|xi(t)-xi*|,|Si(t)-Si*|}=max⁡i∈ℕ2,2|xi(t)-xi*|≥max⁡i∈ℕ2,2|xi(0)-xi*|>0
which implies that
(86)max⁡i∈ℕ2,2∪ℕ2,3∪ℕ3|xi(t)-xi*|=max⁡i∈ℕ2,2|xi(t)-xi*|≥max⁡i∈ℕ2,2|xi(0)-xi*|
holds for all t≥0.
Therefore, there must exist an indexi2∈ℕ2,2 and an increasing time sequence {tk}k=1∞ with lim⁡t→+∞⁡tk=+∞ such that |xi2(tk)-xi2*|≥max⁡i∈ℕ2,2⁡|xi(t0)-xi*|. That is, xi2(t) would not converge to xi2* when t tends to +∞. In other words, the equilibrium point x* is unstable.In the following, we will consider the dynamical behaviors of system (2) with activation functions (4). Note that max⁡{-aij,aij}=|aij| and min⁡{-aij,aij}=-|aij|; in this case, ℕ1–ℕ4 reduce to the following index subsets:
(87)ℕ¯1={i:Ii<-|-di+aii|-∑j≠i,j=1N|aij|-αi-1βi|Bi|},ℕ¯2={i:|Ii|<-di+aii-∑j≠i,j=1N|aij|-αi-1βi|Bi|},ℕ¯3={i:|-di+aii|+∑j≠i,j=1N|aij|+αi-1βi|Bi|lllllllllllllll<Ii<3di+aii-∑j≠i,j=1N|aij|-αi-1βi|Bi|},ℕ¯4={∑j≠i,j=1N|aij|i:Ii>max⁡{3di+aii,di-aii}lllllllllllllll+∑j≠i,j=1N|aij|+αi-1βi|Bi|}.
Meanwhile, conditions (47)-(48) become
(88)∑j∈ℕ2,2j≠i|aij|+∑j∈ℕ2,3∪ℕ3|aij|<-di+aii+αi-1Biβi,kkkkkkklkkkki∈ℕ2,2,(89)∑j∈ℕ2,2|aij|+∑j∈ℕ2,3∪ℕ3j≠i|aij|<di+aii+αi-1Biβi,kkkkkki∈ℕ2,3∪ℕ3.Remark 11.
Under index subsetsℕ¯2-ℕ¯3, conditions (88)-(89) always hold.Proof.
From the definition of index subsetℕ¯2, we get
(90)∑j≠i,j=1N|aij|<-di+aii+αi-1βi|Bi|<di+aii+αi-1βi|Bi|.
Therefore, inequalities (88) and (89) hold for i∈ℕ2,2 and i∈ℕ2,3, respectively.
Similarly, it follows from the definition ofℕ¯3 that
(91)di-aii+∑j≠i,j=1N|aij|-αi-1βiBi<3di+aii-∑j≠i,j=1N|aij|+αi-1βiBi,
which implies that
(92)∑j≠i,j=1N|aij|<di+aii+αi-1βiBi.
Therefore, condition (89) holds for i∈ℕ3.From Theorem10 and Remark 11, we can obtain Corollary 12 as follows.Corollary 12.
Suppose thatℕ¯1∪ℕ¯2∪ℕ¯3∪ℕ¯4={1,2,…,N}. If the following conditions
(93)-di+aii-∑j∈ℕ¯2,2j≠i|aij|-∑j∈ℕ¯2,3∪ℕ¯3|aij|-|Bi|>0,i∈ℕ¯2,2,(94)di+aii-∑j∈ℕ¯2,2|aij|-∑j∈ℕ¯2,3∪ℕ¯3j≠i|aij|-|Bi|>0,kkkkikkikki∈ℕ¯2,3∪ℕ¯3,(95)αi>βi,i∈ℕ¯2,2∪ℕ¯2,3∪ℕ¯3
hold, then system (2) with activation functions (4) has exactly 3♯ℕ¯2 equilibria, 2♯ℕ¯2 of them are locally stable and others are unstable.Proof.
First of all, according to Theorem7 and Remark 11, the exact existence of 3♯ℕ¯2 equilibria for system (2) with activation functions (4) can be guaranteed under condition ℕ¯1∪ℕ¯2∪ℕ¯3∪ℕ¯4={1,2,…,N}. In addition, it is easy to see that conditions (93)–(95) imply the conditions (53)–(56) with li,1=1, li,2=-1. From Theorem 10, we derive the result of Corollary 12.LetBi=βi=Si(t)=0  (i=1,2,…,N); then system (2) is transformed into the following neural networks which are investigated in [33]:
(96)x˙i(t)=-dixi(t)+∑j=1Naijfj(xj(t))+Ii.
In this case, ℕ¯1–ℕ¯4 turn into the following index subsets defined in [33]:
(97)ℕ~1={i:Ii<-|-di+aii|-∑j≠i,j=1N|aij|},ℕ~2={i:|Ii|<-di+aii-∑j≠i,j=1N|aij|},ℕ~3={i:|-di+aii|+∑j≠i,j=1N|aij|<Iillllllllllllll<3di+aii-∑j≠i,j=1N|aij|},ℕ~4={i:Ii>max⁡{3di+aii,di-aii}+∑j≠i,j=1N|aij|}.
Applying Corollary 12, we can obtain easily Corollary 13 as follows.Corollary 13.
Suppose thatℕ~1∪ℕ~2∪ℕ~3∪ℕ~4={1,2,…,N} holds. Then system (96) with activation functions (4) has exactly 3♯ℕ~2 equilibria, 2♯ℕ~2 of them are locally stable and others are unstable.Proof.
First of all, it follows from the definition of index subsetℕ~2 that (93)-(94) with Bi=0 hold for i∈ℕ~2,2 and i∈ℕ~2,3, respectively. From the definition of ℕ~3, we have
(98)di-aii+∑j≠i,j=1N|aij|<3di+aii-∑j≠i,j=1N|aij|,
which implies that
(99)di+aii-∑j≠i,j=1N|aij|>0;
that is, inequality (94) with Bi=0 holds for i∈ℕ~3. Inequality (95) is obvious, due to αi>0 and βi=0. From Corollary 12, we derive the result of Corollary 13.Remark 14.
In this paper, we study the multistability and instability of CNNs with activation functions (3). The models are different from and more general than those in [33], and the considered activation functions (3) are also more general than those employed in [33]. Moreover, the index subsets defined in this paper are less restrictive than those defined in [33].Remark 15.
Compared with the results reported in [33], it can be seen that Corollary 13 above is consistent with Theorem  1 in [33]. That is, if we specialize the system and activation functions in Theorem 10 to those considered in [33], we can obtain the main result in [33]. Therefore, Theorem 10 extends and improves the main result in [33].


## 3. Two Illustrative Examples

For convenience, we consider the following two-dimensional CNNs:(100)x˙i(t)=-dixi(t)+∑j=12aijfj(xj(t))+BiSi(t)+Ii,S˙i(t)=-αiSi(t)+βifi(xi(t)),i=1,2.Example 1.
For system (100), take d1=d2=1, a11=a22=2, a12=0.5, a21=-0.5, B1=0.4, B2=-0.4, I1=I2=0, α1=α2=2, β1=β2=1, and
(101)fi(x)={-1,-∞<x<-1,x,-1≤x≤1,-x+2,1<x≤3,-1,3<x<+∞.(i=1,2).
It is easy to see that ℕ2={1,2}. In addition, by simple computations, we have
(102)-d1+a11-|a12|-|B1|=0.1>0,-d2+a22-|a21|-|B2|=0.1>0.
Therefore, the conditions in Corollary 12 hold. According to Corollary 12, system (100) has exactly 32=9 equilibria, 22=4 equilibria are locally stable and others are unstable. In fact, by direct computations, we can obtain the nine equilibria (-2.7,-1.3,-0.5,-0.5)T, (-201/80,-5/8,-1/2,-5/16)T, (-541/280,41/28,-1/2,15/56)T, (5⁄12,-241⁄120,5⁄24,  -1⁄2T)(0,0,0,0)T, (-100⁄361,482⁄361, -50⁄361,120⁄361)T, (39⁄32,-701⁄320,25⁄64, -1⁄2)T, (402/281,100/281,80/281,50/281)T, and (1385/921,1102/921,457/1842,185/921)T. From Figures 2, 3, 4, 5, and 6, it can be seen that the four equilibria (-2.7,-1.3,-0.5,-0.5)T, (-541/280,41/28,-1/2,15/56)T, (39/32,-701/320,25/64,-1/2)T, and (1385/921,1102/921,457/1842,185/921)T are locally stable. Figures 7, 8, 9, 10, and 11 confirm that the others are unstable.Figure 2
Transient behavior ofx1 in Example 1.Figure 3
Transient behavior ofx2 in Example 1.Figure 4
Transient behavior ofS1 and S2 in Example 1.Figure 5
Phase plot of state variable(x1,x2,S1)T in Example 1.Figure 6
Phase plot of state variable(x1,x2,S2)T in Example 1.Figure 7
Transient behavior ofx1 and x2 near the equilibrium point (-201/80,-5/8,-1/2,-5/16)T in Example 1.Figure 8
Transient behavior ofx1 and x2 near the equilibrium point (5/12,-241/120,5/24,-1/2)T in Example 1.Figure 9
Transient behavior ofx1 and x2 near the equilibrium point (0,0,0,0)T in Example 1.Figure 10
Transient behavior ofx1 and x2 near the equilibrium point (-100/361,482/361,-50/361,120/361)T in Example 1.Figure 11
Transient behavior ofx1 and x2 near the equilibrium point (402/281,100/281,80/281,50/281)T in Example 1.Example 2.
For system (100), take d1=1, a11=-1, a12=-0.2, B1=0.2, d2=a22=2, a21=1/3, B2=-0.2, I1=9, I2=0, α1=α2=3, β1=β2=1, and
(103)fi(x)={-3,-∞<x<-2,2x+1,-2≤x≤2,-x+7,2<x≤10,-3,10<x<+∞.(i=1,2).
It is easy to see that ℕ2={2}, ℕ4={1}. Herein, the parameters satisfy conditions in Remark 9 and Theorem 10:
(104)|a21|max⁡{|l1,1|,|l1,2|}=23<2815=min⁡{(a22+α2-1B2β2)l2,1-d2,kkkkkkkkkkkkkkkkkkd2-(a22+α2-1B2β2)l2,2},d2-a22l2,2-|a21|max⁡{|l1,1|,|l1,2|}=103>0,-d2+a22l2,1-|a21|max⁡{|l1,1|,|l1,2|}=43>0.
From Remark 9 and Theorem 10, it follows that the system has exactly three equilibria (62/5,-17/5,-1,-1)T, (59/5,-1/2,-1,0)T, and (3256/295,188/59,-1,75/59)T; the first and the third equilibria are locally stable, while the second equilibrium point is unstable. From Figures 12, 13, 14, 15, and 16, it can be seen that the two equilibria (62/5,-17/5,-1,-1)T and (3256/295,188/59,-1,75/59)T are locally stable. Figure 17 confirms that the equilibrium point (59/5,-1/2,-1,0)T is unstable.Figure 12
Transient behavior ofx1 in Example 2.Figure 13
Transient behavior ofx2 in Example 2.Figure 14
Transient behavior ofS1 and S2 in Example 2.Figure 15
Phase plot of state variable(x1,x2,S1)T in Example 2.Figure 16
Phase plot of state variable(x1,x2,S2)T in Example 2.Figure 17
Transient behavior ofx1 and x2 near the equilibrium point (59/5,-1/2,-1,0)T in Example 2.


## 4. Conclusions

In this paper, the multistability and instability issues have been studied for CNNs with Mexican-hat-type activation functions. We showed that under some conditions, the system has3♯ℕ2 equilibria, 2♯ℕ2 of them are locally stable and others are unstable. Two examples with their computer simulations were given to illustrate the effectiveness of the obtained results. Some thorough analyses are needed further. Here, we only treated neural networks without time delay, how about when time delays are presented? This needs to be investigated in the future.



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