# Multistability in a Multidirectional Associative Memory Neural Network with Delays

Authors: Min Wang; Tiejun Zhou
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

This paper focuses on the multidirectional associative memory (MAM) neural networks withm fields which is more advanced to realize associative memory. Based on the Brouwer fixed point theorem and Dini upper right derivative, it is confirmed that the multidirectional associative memory neural network can have 3l equilibria and 2l equilibria of them are stable, where l is a parameter associated with the number of neurons. Furthermore, an example is given to illustrate the effectiveness of the results.


---

## Body

## 1. Introduction

The Hopfield neural network was introduced by Hopfield in 1982, which introduced the climax of the research on the neural networks. This network was extended to bidirectional associative memory (BAM) neural network by Kosko in 1987 [1] and to multidirectional associative memory (MAM) neural network by Hagiwara in 1990 [2]. They all can realize associative memory. But by using the MAM neural networks, one can achieve the many-to-many association which is a very advanced function of human brain. The many-to-many association has found wide applications in image denoising, speech recognition, pattern recognition, and intelligent information processing [2–6]. For example, it was shown that today most Indians are derived from the two-ancestor group gene by DNA analyzing [7]. If we need to distinguish which category an Indian belongs to, then this is a many-to-many associative problem.In the designing of the associative memory neural networks to achieve associative memory, it is necessary to ensure their stability. The stability of Hopfield neural networks and BAM neural networks is discussed in a lot of recently published literature works [8–12], but the researchers about MAM neural networks are mainly focused on learning algorithms, fault tolerance, and retrieval efficiency of MAM neural networks [3–6]. To the best of our knowledge, the research on the theory of MAM neural networks was reported only in a few papers [13–18]. Chen et al. proved the stability of some specific types of MAM neural networks in [13, 14]. We studied the existence and global exponential stability of equilibrium for MAM neural networks with constant delays or time-varying delays in [15, 16]. We also obtained a sufficient condition for the global exponential stability of the discrete-time multidirectional associative memory neural network with variable delays in [17].The multistability of a neural network describes coexistence of multiple stable patterns such as equilibria or periodic orbits. In recent years, the multistability issue of neural networks is discussed in some papers [18–32]. In [18], we discussed the existence and the exponential stability of multiple periodic solutions for an MAM neural network. The neural networks with a class of nondecreasing piecewise linear activation functions with 2r corner points were investigated in [19]. It was proposed that the n-neuron dynamical systems can have and only have (2r+1)n equilibria under some conditions, of which (r+1)n are locally exponentially stable and others are unstable. A class of neural networks with Mexican-hat-type activation functions was discussed in [20]. A set of new sufficient conditions were presented to ensure the multistability of the neural networks. The cellular neural networks with time-varying unbounded delays (DCNNs) were discussed in [21]. Under some conditions, it showed that the DCNNs can exhibit 3n equilibrium points. In [22], Huang and Cao found that the 2n-dimensional networks can have 3n equilibria and 2n of them are locally exponentially stable. In [24], with perturbation techniques and the Floquet theory, Campbell et al. discussed the multistability and stable asynchronous periodic oscillations for a network of three identical neurons with multiple discrete signal transmission delays. The possible codimension one bifurcations which occur in the system were determined. In [25], Cheng et al. presented the existence of 2n stationary solutions for a general n-dimensional delayed neural network with several classes of activation functions. It was shown that a two-dimensional neural network has n2 isolated equilibria points which are locally stable, where the activation function has n segments. Furthermore, evoked by periodic external input, n2 periodic orbits which are locally exponentially attractive were found [26]. Some similar results were found on n-neuron Cohen-Grossberg neural networks (CGNNs) with time-varying delays and a general class of activation functions [27]. In [29, 30], the multistability and multiperiodicity issues were discussed for competitive neural networks and high-order competitive neural networks. In [31], the stability of multiple equilibria of neural networks with time-varying delays and concave-convex characteristics was studied. Some sufficient conditions were obtained to ensure that an n-neuron neural network with concave-convex characteristics can have a fixed point located in the appointed region. By partitioning the state space, sufficient conditions were established which ensure that n-dimensional Cohen-Grossberg neural networks with k-level discontinuous activation functions can have kn equilibrium points or kn periodic orbits [32].Generally, the existence of a globally stable equilibrium point or periodic solutions is necessary in solving optimization problems, but to achieve many-to-many associative memory by using MAM neural network, the system which has a globally stable equilibrium point or a globally stable periodic solution can only associate less information. So we should study the multistability of MAM neural network in order to make it achieve many-to-many associative memory. It is necessary to explore the existence, stability, and convergence speed of multiple equilibria or periodic solutions of MAM neural network. Motivated by the above, in this paper, we study the multistability issue for a delayed MAM neural network withm fields and nk neurous in the field k as follows:
(1)dxkidt=-akixki(t)+∑p=1,p≠km‍∑j=1np‍wpjkifpj(xpj(t-τpjki))+Iki,
where k=1,2,…,m, i=1,2,…,nk, xki(t) denotes the membrane voltage of the ith neuron in the field k at time t, aki>0 denotes the decay rate of the ith neuron in the field k, fpj(·) is the neuronal activation function of the jth neuron in the field p, wpjki is the connection weight from the jth neuron in the field p to the ith neuron in the field k, Iki is the external input of the ith neuron in the field k, and τpjki is the time delay of the synapse from the j neuron in the field p to the ith neuron in the field k.Setτ=max{τpjki∣where1≤k≤m,1≤p≤m,p≠k,1≤i≤nk,and1≤j≤np}. The initial conditions with (1) are of the forms
(2)xki(t)=ϕki(t),
where k=1,2,…,m, i=1,2,…,nk, and ϕki:[-τ,0]→ℝ are continuous functions.This paper is organized as follows. In the next section, we discuss the existence of multiple equilibria by using the Brouwer fixed point theorem. In Section3, we analyze the exponential stability of every equilibrium. An example is given to illustrate the effectiveness of our results in Section 4. Finally, conclusions are given in Section 5.


## 2. Existence of Multiple Equilibria

In this section, we consider the existence of multiple equilibria by using the Brouwer fixed point theorem. Since the existence and stability of stationary patterns for neural networks certainly depend on the characteristics of activation functions, we assume the activation functionsfki(·) (1≤i≤nk,1≤k≤m) satisfy the following condition:(H1)fki(·) are continuous, increasing and there exist Mki>0 such that |fki(x)|≤Mki for 1≤i≤nk,1≤k≤m.For convenience, we give some notations as follows:(3)M={m,ifmisanevennumber,m-1,ifmisanoddnumber,IM={1,3,5,…,M-1},li=li+1=min{ni,ni+1},foranyi∈IM,lm=0,hhhhhhhhhhhhhhhhhhhhhhiiiiiiiiiiiiiifM≠m,l=∑i∈IM‍li,n=∑i=1m‍ni,col(bki)=(b11,…,b1n1,b21,…,b2n2,…,bm1,…,bmnm).Set(4)δki=1aki[∑p=1,p≠km‍∑j=1np‍|wpjki|Mpj+|Iki|],Ω={x=col(xki)∈ℝn∣xki∈[-δki,δki],iiiiiiiifor1≤k≤m,1≤i≤nk}.For anyk∈IM, 1≤i≤lk, define
(5)H(k+1)i+(xki)=1a(k+1)i(wki(k+1)ifki(xki)1a(k+1)ic-|wki(k+1)i|Mki)+δ(k+1)i,H(k+1)i-(xki)=1a(k+1)i(wki(k+1)ifki(xki)ccccccccci+|wki(k+1)i|Mki)-δ(k+1)i,Gki+(x(k+1)i)=1aki(w(k+1)ikif(k+1)i(x(k+1)i)1akic-|w(k+1)iki|M(k+1)i)+δki,Gki-(x(k+1)i)=1aki(w(k+1)ikif(k+1)i(x(k+1)i)1akic+|w(k+1)iki|M(k+1)i)-δki.We consider the second parameter condition which is used to establish existence of multiple equilibria for systems (1).(H2) For anyk∈IM, the weights w(k+1)iki·wki(k+1)i>0, and there exist two points (uki1,u(k+1)i1), (uki2,u(k+1)i2), where uki1<uki2, such that
(1)
ifwki(k+1)i,w(k+1)iki>0, then H(k+1)i+(uki1)<u(k+1)i1, Gki+(u(k+1)i1)<uki1, H(k+1)i-(uki2)>u(k+1)i2, Gki-(u(k+1)i2)>uki2;(2)
ifwki(k+1)i,w(k+1)iki<0, then H(k+1)i-(uki1)>u(k+1)i1, Gki+(u(k+1)i1)<uki1, H(k+1)i+(uki2)<u(k+1)i2, Gki-(u(k+1)i2)>uki2.The configuration that motivates (H2) is depicted in Figure1. Under the above assumptions, if wki(k+1)i,w(k+1)iki>0, then there are three crossing points (xki1+,x(k+1)i1+), (xki2-,x(k+1)i2-), and (xki3+,x(k+1)i3+) of the curves x(k+1)i=H(k+1)i+(xki), xki=Gki+(x(k+1)i), and there are three crossing points (xki1-,x(k+1)i1-), (xki2+,x(k+1)i2+), and (xki3-,x(k+1)i3-) of the curves x(k+1)i=H(k+1)i-(xki), xki=Gki-(x(k+1)i) (see Figure 1(a)), where
(6)xki1-<xki1+<uki1<xki2-<xki2+<uki2<xki3-<xki3+,x(k+1)i1-<x(k+1)i1+<u(k+1)i1<x(k+1)i2-<x(k+1)i2+<u(k+1)i2<x(k+1)i3-<x(k+1)i3+.The configurations ofGki+(y), Gki-(y) and H(k+1)i-(x), H(k+1)i+(x). (a) wki(k+1)i,w(k+1)iki>0. (b) wki(k+1)i,w(k+1)iki<0.
(a)
(b)Ifwki(k+1)i,w(k+1)iki<0, then there are three crossing points (xki1-,x(k+1)i1+), (xki2+,x(k+1)i2-), and (xki3-,x(k+1)i3+) of the curves x(k+1)i=H(k+1)i+(xki), xki=Gki-(x(k+1)i), and there are three crossing points (xki1+,x(k+1)i1-), (xki2-,x(k+1)i2+), and (xki3+,x(k+1)i3-) of the curves x(k+1)i=H(k+1)i-(xki), xki=Gki+(x(k+1)i) (see Figure 1(b)), where (7)xki1-<xki1+<uki1<xki2-<xki2+<uki2<xki3-<xki3+,x(k+1)i3-<x(k+1)i3+<u(k+1)i2<x(k+1)i2-<x(k+1)i2+<u(k+1)i1<x(k+1)i1-<x(k+1)i1+.Furthermore, it is easy to confirm thatxkij+,xkij-∈[-δki,δki], for any 1≤k≤m, 1≤i≤nk, and j=1,2,3.Theorem 1.
Under conditions (H1) and (H2), there are at least3l equilibria of the multidirectional associative memory neural network (1).Proof.
Set(8)Ωα={x=col(xki)∈Ω∣(xki,x(k+1)i)iiih∈Ωkiαki,fork∈IM,1≤i≤lk},
where α=(α11,…,α1l1,α31,…,α3l3,…,α(M-1)1,…, α(M-1)lM-1),  αki=1,2, or 3, and
(9)Ωki1={(xki,x(k+1)i)∈ℝ2∣(xki,x(k+1)i)∈[xki1-,xki1+]Ωki1×[x(k+1)i1-,x(k+1)i1+]},Ωki2={(xki,x(k+1)i)∈ℝ2∣(xki,x(k+1)i)∈[xki2-,xki2+]Ωki2×[x(k+1)i2-,x(k+1)i2+]},Ωki3={(xki,x(k+1)i)∈ℝ2∣(xki,x(k+1)i)∈[xki3-,xki3+]Ωki3×[x(k+1)i3-,x(k+1)i3+]}.
Obviously, they are 3l disjoint closed regions, and Ωkij⊆[-δki,δki]×[-δ(k+1)i,δ(k+1)i] for any k∈IM, 1≤i≤lk, j=1,2,3.
For any fixedα and a given x~=col(x~ki)∈Ωα, we study this problem on the following two cases.
Case  1 (1≤i≤lk). Consider the following equations:
(10)-akixki+w(k+1)ikif(k+1)i(x(k+1)i)-w(k+1)ikif(k+1)i(x~(k+1)i)+∑p=1,p≠km‍∑j=1np‍wpjkifpj(x~pj)+Iki=0,(11)-a(k+1)ix(k+1)i+wki(k+1)ifki(xki)-wki(k+1)ifki(x~ki)+∑p=1,p≠km‍∑j=1np‍wpj(k+1)ifpj(x~pj)+I(k+1)i=0.
It follows from (10) that
(12)xki≤1aki[∑j=1,j≠ink+1‍|w(k+1)jki|M(k+1)j+|Iki|w(k+1)ikif(k+1)i(x(k+1)i)iiiiiicii+∑j=1,j≠ink+1‍|w(k+1)jki|M(k+1)j+|Iki|]+1aki∑p=1,p≠k,p≠k+1m∑j=1np‍|wpjki|Mpj=1aki[w(k+1)ikif(k+1)i(x(k+1)i)iiiiiiih-|w(k+1)iki|M(k+1)i+|Iki|]+1aki∑p=1,p≠km‍∑j=1np‍|wpjki|Mpj=1aki[w(k+1)ikif(k+1)i(x(k+1)i)iiiiiicii-|w(k+1)iki|M(k+1)i]+δki=Gki+(x(k+1)i).
Similarly, from (10), we also can obtain that
(13)xki≥1aki[w(k+1)ikif(k+1)i(x(k+1)i)hhhiii+|w(k+1)iki|M(k+1)i]-δki=Gki-(x(k+1)i).
So the curve of (10) is between the curves of xki=Gki-(x(k+1)i) and xki=Gki+(x(k+1)i).
Under similar analysis we can obtain that the curve of (11) is between the curves of x(k+1)i=H(k+1)i-(xki) and x(k+1)i=H(k+1)i+(xki). So we can affirm that there are at least three solutions of (10) and (11); each solution lies separately in Ωki1, Ωki2, and Ωki3. Get one that is in Ωkiαki, and denote it by (x-ki,x-(k+1)i).
Case  2 (lk+1≤i≤nk). Let
(14)x-ki=1aki[∑p=1,p≠km‍∑j=1np‍wpjkifpj(x~pj)+Iki].
Then
(15)|x-ki|≤1aki[∑p=1,p≠km‍∑j=1np‍|wpjkifpj(xpj)|+Iki]≤1aki[∑p=1,p≠km‍∑j=1np‍|wpjki|Mpj+|Iki|]=δki.
From the two cases above, we can obtainx-ki∈Ωα for any given x~∈Ωα. Set x-=col(x-ki). It is obvious that x-∈Ωα. Set mapping Fα:Ωα→Ωα as follows:
(16)Fα(x~)=x-.
Because fki(·) are continuous mappings, so Fα is continuous mapping. By the Brouwer fixed point theorem, there is at least one fixed point of Fα, that is, zero point of (1) in Ωα. Therefore, there are at least 3l equilibria of the multidirectional associative memory neural network (1).


## 3. Stability Analysis

In this section, the stability of the equilibria is considered. Set(17)Ψ={ϕ=col(ϕki)∈C([-τ,0],ℝn)∣-δki≤ϕki(s)≤δki,hhifor1≤k≤m,1≤i≤nk},η=(η11,…,η1l1,η31,…,η3l3,…,η(M-1)1,…,η(M-1)lM-1),
with ηki=1 or ηki=3.We consider the following2l subsets of Ψ:
(18)Ψη={(ϕki,ϕ(k+1)i)ϕ=col(ϕki)∈Ψ∣(ϕki,ϕ(k+1)i)∈Ψkiηki,iiiifork∈IM,1≤i≤lk},
where
(19)Ψki1={(ϕki,ϕ(k+1)i)∈C([-τ,0],ℝ2)∣ϕpi(s)∈[xpi1-,xpi1+],iihiforp=k,k+1},Ψki3={(ϕki,ϕ(k+1)i)∈C([-τ,0],ℝ2)∣ϕpi(s)∈[xpi3-,xpi3+],iiihforp=k,k+1}.Lemma 2.
Under assumptions (H1) and (H2), eachΨη is a positive invariant set with respect to the solution flow generated by system (1).Proof.
For any givenη and any initial condition ϕ=col(ϕki)∈Ψη, let x(t;ϕ)=col(xki(t;ϕ)) be the solution of (1) with initial condition ϕ∈Ψη.
For any1≤k≤m,1≤i≤nk, and any t>0, from (1), we obtain that
(20)-akiδki≤dxki(t;ϕ)dt+akixki(t;ϕ)≤akiδki.
Multiplying both sides of (20) by eakit, we have
(21)-akiδkieakit≤dxki(t;ϕ)eakitdt≤akiδkieakit.
Integrating (21) over [0,t], we obtain
(22)-δki(eakit-1)≤xki(t;ϕ)eakit-ϕki(0)≤δki(eakit-1).
Hence it follows that, for t>0,
(23)|xki(t;ϕ)|≤e-akit(|ϕki(0)|-δki)+δki.
Therefore, for any given initial condition ϕ∈Ψη, we have, for t>0,
(24)|xki(t;ϕ)|≤δki.
That is, x(t,ϕ)∈Ψ for t>0.
We claim thatx(t,ϕ) remains in Ψη for any t>0. If it is not true, there exist k∈IM and 1≤i≤lk that (xki(t),x(k+1)i(t)) firstly (or one of the first) escapes from Ψkiηki.
Case  1 (wki(k+1)i,w(k+1)iki>0). If ηki=1, then exists t0>0, such that (xki(t0),x(k+1)i(t0)) is on the edges of Ψkiηki, and for any t<t0, xki1-<xki(t)<xki1+, x(k+1)i1-<x(k+1)i(t)<x(k+1)i1+. If xki(t0)=xki1+, x(k+1)i(t0)∈[x(k+1)i1-,x(k+1)i1+], then
(25)dxki(t)dt|t=t0=-akixki(t0)+∑p=1,p≠km‍∑j=1np‍wpjkifpj×(xpj(t0-τpjki))+Iki≤-akixki1++w(k+1)ikif(k+1)i(x(k+1)i1+)-|w(k+1)iki|M(k+1)i+∑p=1,p≠km‍∑j=1np‍|wpjki|Mpj+|Iki|=-aki{xki1+-1aki[w(k+1)ikif(k+1)i(x(k+1)i1+)iiiiiihihiiiiiiiiiihhc-|w(k+1)iki|M(k+1)i(x(k+1)i1+)]-δkixki1+-1aki}=-aki[xki1+-Gki+(x(k+1)i1+)]=0.
Therefore, (xki(t),x(k+1)i(t)) cannot escape from Ψkiηki through the edge of xki(t)=xki1+, x(k+1)i(t)∈[x(k+1)i1-,x(k+1)i1+]. With similar proof, we can obtain that (xki(t),x(k+1)i(t)) cannot escape from Ψkiηki through the other three edges. Hence (xki(t),x(k+1)i(t)) cannot escape from Ψki1. We can also prove that (xki(t),x(k+1)i(t)) cannot escape from Ψki3.
Case  2
(
w
k
i
(
k
+
1
)
i
,
w
(
k
+
1
)
i
k
i
<
0
). The proof is similar to that of Case 1.
This completes the proof.We give the criterions concerning the stability for the multiple equilibria of system (1).(H3) There exist constantsLki>0(1≤k≤m and  1≤i≤nk) such that
(26)|fki(x)-fki(y)|≤Lki|x-y|,for eachx,y in a subset Rki⊂ℝ, where Rki is defined as follows:
(27)Rki={[xki1-,xki1+]∪[xki3-,xki3+],for1≤k≤M,1≤i≤lk,[δki-,δki+],for1≤k≤m,lk+1≤i≤nk.(H4) For any1≤k≤m,1≤i≤nk, it is satisfied that
(28)aki-∑p=1,p≠km‍∑j=1np‍|wpjki|Lki>0.Theorem 3.
There are2l exponential stability equilibria of the multidirectional associative memory neural network (1), if the conditions (H1)–(H4) hold.Proof.
According to Theorem1, for any α, the multidirectional associative memory neural network (1) has an equilibrium in Ωα; let it be x-. According to Lemma 2, for any η, the solution x(t,ϕ) of the multidirectional associative memory neural network (1) is in Ψη under initial condition ϕ∈Ψη. Let
(29)y(t)=col{yki(t)}=x(t,ϕ)-x-.
System (1) becomes
(30)dyki(t)dt=-akiyki(t)+∑p=1,p≠km‍∑j=1np‍wpjki{fpj[ypj(t-τpjki)+x-pj]hhhhhhhhhhhhhhhh-fpj(x-pj)},
for 1≤k≤m,1≤i≤nk.
According to condition(H4), there exists μ>0 such that
(31)aki-μ-∑p=1,p≠km‍∑j=1np‍|wpjki|Lkieμτpjki>0.
Define that uki(t)=eμt|yki(t)|, for 1≤k≤m and 1≤i≤nk. It is obvious that uki(t)>0. Denote
(32)Q=max1≤k≤m,1≤i≤nk{supθ∈(-τ,0]|xki(θ)-x-ki|}.
Let δ>1 be an arbitrary real number. For any θ∈(-τ,0], 1≤k≤m, 1≤i≤nk, it is easy to obtain that uki(t)<Qδ. We shall prove that
(33)uki(t)<Qδ,
for 1≤k≤m, 1≤i≤nk, and any t>0. Suppose this is not the case; then there exist k=q, i=r, and a time t0 such that uki(t)≤Qδ for t∈(-τ,t0],1≤k≤mand1≤i≤nk,uqr(t0)=Qδ,andD+uqr(t0)≥0. From (30), we derive that
(34)D+|yqr(t0)|≤-aqr|yqr(t0)|+∑p=1,p≠qm‍∑j=1np‍|wpjqr|Lpj|ypj(t0-τpjqr)|.
Hence, from (31) and (34),
(35)D+|uqr(t0)|≤μeμt0|yqr(t0)|+eμt0×[∑p=1,p≠qm‍∑j=1np‍-aqr|yqr(t0)|+iiiiiiiiiih∑p=1,p≠qm‍∑j=1np‍|wpjqr|Lpj|ypj(t0-τpjqr)|]=-(aqr-μ)eμt0|yqr(t0)|+∑p=1,p≠qm‍∑j=1np‍|wpjqr|Lpjeμτpjqr|upj(t0-τpjqr)|≤-(aqr-μ)uqr(t0)+∑p=1,p≠qm‍∑j=1np‍|wpjqr|Lpjeμτpjqr[supθ∈[t0-τ,t0]upj(θ)]≤-(aqr-μ-∑p=1,p≠qm‍∑j=1np‍|wpjqr|Lpjeμτpjqr)Qδ≤0,
which is contradicting D+uqr(t0)≥0. Hence the inequality (33) holds. Since δ>1 is arbitrary, by allowing δ→1+, we have uki(t)≤Q. Therefore, x(t,ϕ) is exponentially convergent to x-.Remark 4.
The dynamic system (1) studied in this paper is different from the system in [18]. First, the coefficients of system (1) are constants, which are T-periodic functions in [18]. Second, the delays of system (1) are constant delays while they are distributed delays in [18]. Above all, our conclusions in this paper are different from those in [18], and the proof methods are different. In [18], we obtained the existence of 2n0[m/2] exponentially stable T-periodic solutions, where n0=min{n1,n2,…,nm} if m is an even number or n0=min{n1,n2,…,nm-1} if m is an odd number. But we obtain the existence of 3l equilibria by Theorem 1 and the exponential stability of 2l equilibria of them by Theorem 3. Because lk=min{nk,nk+1}, so n0=min{lk∣k∈IM}. It follows that l=∑k∈IMlk≥n0[m/2]. Therefore, the number of equilibria obtained in this paper is more than that of [18] if the conclusions of [18] are used to the constant coefficient system (1).


## 4. Numerical Example

In this section, a numerical example is given to illustrate the validity of results. Consider an MAM neural network with three fields as follows:(36)dxkidt=-akixki(t)+∑p=1,p≠k3‍∑j=1np‍wpjkifpj×(xpj(t-τpjki))+Iki,
where k=1,2,3, n1=n2=2,n3=1, the neuronal signal decay rates a11=a12=a21=a22=a31=1, the external input I11=0.1, I12=-0.4, I21=0.2, I22=-0.2, I31=0.4, the connection weights
(37)(00w2111w2211w311100w2112w2212w3112w1121w122100w3121w1122w122200w3122w1131w1231w2131w22310)=(00-20.30.2000.430.1-3-0.5000.10.3200-0.1121-10),
and the delays
(38)(00τ2111τ2211τ311100τ2112τ2212τ3112τ1121τ122100τ3121τ1122τ122200τ3122τ1131τ1231τ2131τ22310)=(00121.2000.411.510.4000.822000.71.21120).The neuronal activation functionsfpj(x)=tanh(x)(1≤p≤3,1≤j≤np) are continuous, increasing, and bounded functions, and there are Mpj=1>0 such that |fpj(x)|≤Mpj. Hence condition (H1) is satisfied.Obviously,n=5, M=2, l1=l2=2, and l=2. Through calculations, we have
(39)H21±(x11)=-3f11(x11)±0.8,G11±(x21)=-2f21(x21)±0.6,H22±(x12)=2f21(x12)±0.6,G12±(x22)=3f22(x22)±0.9.
There exist four points(u111,u211)=(-0.8,1),(u112,u212)=(0.9,-1.2), (u121,u221)=(-1,-0.9), and (u122,u222)=(1.1,0.9), where u111<u112, u121<u122, such that
(40)H21-(u111)=1.1921>1,G11+(u211)=-0.9232<-0.8,H21+(u112)=-1.3489<-1.2,G11-(u212)=0.9673>0.9,H22+(u121)=-0.9232<-0.9,G12+(u221)=-1.2489<-1,H22-(u122)=1.0010>0.9,G12-(u222)=1.2489>1.1.
Hence condition (H2) holds. The curves x21=H21+(x11), x11=G11-(x21) have three crossing points (x111-,x211+)=(-2.5979,3.7669), (x112+,x212-)=(0.5198,0.6327), and (x113-,x213+)=(1.2886,-1.7763), and the curves x21=H21+(x11), x11=G11-(x21) have three crossing points (x111+,x211-)=(-1.2886,1.7763), (x112-,x212+)=(-0.5198,0.6327), and (x113+,x213-)=(2.5979,-3.7669). The curves x22=H22+(x12),x12=G12+(x22) have three crossing points (x121+,x221+)=(-1.6517,-1.2582), (x122-,x222-)=(-0.6830,-0.5869), and (x123+,x223+)=(3.8670,2.5982), and the curves x22=H22-(x12), x12=G12-(x22) have three crossing points (x121-,x221-)=(-3.8670,-2.5982), (x122+,x222+)=(0.6830,0.5869), and (x123-,x223-)=(1.6517,1.2582). From (27), we obtain the subsets
(41)R11=[-2.2579,-1.2886]∪[1.2886,2.2579],R21=[-3.7669,-1.7763]∪[1.7763,3.7669],R12=[-3.8670,-1.6517]∪[1.6571,3.8670],R22=[-2.5982,-1.2582]∪[1.2582,2.5982],R31=[-5.4,5.4].Because the activation functionsfpj(x)=tanh(x) are differentiable, we can let Lpj=max{fpj′(x)∣x∈Rpj}>0; then L11=0.2625, L12=0.1083, L21=0.1368, L22=0.2765, and L31=1. By Lagrange's mean value theorem, condition (H3) holds. Through calculations we have
(42)1-2L21-0.3L22+0.2L31=0.4434>0,1-0.4L21-3L22+0.1L31=0.0158>0,1-3L11-0.5L12+0.1L31=0.0583>0,1-0.3L11+2L12+0.1L31=0.6047>0,1-L11-2L12-L21-L22=0.1076>0.
Hence condition (H4) holds. Then by Theorems 1 and 3, the MAM neural network (36) has 3l=9 equilibria, and 2l=4 of these equilibria are exponentially stable.The dynamics of the MAM neural network system (36) are illustrated in Figures 2 and 3. Evolutions of sixty initial conditions of the MAM neural network system (36) have been tracked in Figure 2, which clearly displays that there exist four stable equilibria of the dynamical system, as confirmed by our theorems. Figure 3 shows the phases of the evolutions from time 5 to time 60 with sixty initial conditions, which shows that each evolution has converged to one of the four stable equilibria at time 60.Figure 2
The exponential stability of 4 equilibria of the MAM neural network (36).Figure 3
The phase of the MAM neural network (36).


## 5. Conclusions

In this paper, the multistability has been studied for MAM neural networks. Sufficient conditions are obtained which ensure the existence of3l equilibria. It is proved that 2l of the equilibria are exponentially stable. In [18], we have discussed the existence and the exponential stability of multiple periodic solutions for an MAM neural network. Furthermore, the coexistence of multiple stable equilibria and periodic solutions of an MAM neural network is an interesting topic. It will be investigated in the near future.



---

*Source: 592056-2013-11-25.xml*