Multistability in a Multidirectional Associative Memory Neural Network with Delays

Abstract

This paper focuses on the multidirectional associative memory (MAM) neural networks with m 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 3^𝑙 equilibria and 2^𝑙 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.

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 3^𝑛 equilibrium points. In [22], Huang and Cao found that the 2𝑛-dimensional networks can have 3^𝑛 equilibria and 2^𝑛 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 2^𝑛 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 𝑘^𝑛 equilibrium points or 𝑘^𝑛 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 with 𝑚 fields and 𝑛_𝑘 neurous in the field 𝑘 as follows: 𝑑𝑥_𝑘𝑖𝑑𝑡=−𝑎_𝑘𝑖𝑥_𝑘𝑖(𝑡)+^𝑚𝑛_{𝑝=1,𝑝≠𝑘}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑘𝑖}𝑓_𝑝𝑗𝑥_𝑝𝑗𝑡−𝜏_{𝑝𝑗𝑘𝑖}+𝐼_𝑘𝑖, (1) where 𝑘=1,2,…,𝑚, 𝑖=1,2,…,𝑛_𝑘, 𝑥_𝑘𝑖(𝑡) denotes the membrane voltage of the 𝑖th neuron in the field 𝑘 at time 𝑡, 𝑎_𝑘𝑖>0 denotes the decay rate of the 𝑖th neuron in the field 𝑘, 𝑓_𝑝𝑗(⋅) is the neuronal activation function of the 𝑗th neuron in the field 𝑝, 𝑤_{𝑝𝑗𝑘𝑖} is the connection weight from the 𝑗th neuron in the field 𝑝 to the 𝑖th neuron in the field 𝑘, 𝐼_𝑘𝑖 is the external input of the 𝑖th neuron in the field 𝑘, and 𝜏_{𝑝𝑗𝑘𝑖} is the time delay of the synapse from the 𝑗 neuron in the field 𝑝 to the 𝑖th neuron in the field 𝑘.

Set 𝜏=max{𝜏_{𝑝𝑗𝑘𝑖}∣where1≤𝑘≤𝑚,1≤𝑝≤𝑚,𝑝≠𝑘,1≤𝑖≤𝑛_𝑘,and1≤𝑗≤𝑛_𝑝}. The initial conditions with (1) are of the forms 𝑥_𝑘𝑖(𝑡)=𝜙_𝑘𝑖(𝑡), (2) where 𝑘=1,2,…,𝑚, 𝑖=1,2,…,𝑛_𝑘, and 𝜙_𝑘𝑖∶[−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 Section 3, 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 functions 𝑓_𝑘𝑖(⋅) (1≤𝑖≤𝑛_𝑘,1≤𝑘≤𝑚) satisfy the following condition: (H1) 𝑓_𝑘𝑖(⋅) are continuous, increasing and there exist 𝑀_𝑘𝑖>0 such that |𝑓_𝑘𝑖(𝑥)|≤𝑀_𝑘𝑖 for 1≤𝑖≤𝑛_𝑘,1≤𝑘≤𝑚.

For convenience, we give some notations as follows: 𝐼𝑀=𝑚,if𝑚isanevennumber,𝑚−1,if𝑚isanoddnumber,_𝑀={},𝑙1,3,5,…,𝑀−1_𝑖=𝑙_𝑖+1=min𝑛_𝑖,𝑛_𝑖+1,forany𝑖∈𝐼_𝑀,𝑙_𝑚=0,if𝑀≠𝑚,𝑙=_{𝑖∈𝐼𝑀}𝑙_𝑖,𝑛=^𝑚_𝑖=1𝑛_𝑖,col𝑏_𝑘𝑖=𝑏₁₁,…,𝑏_1𝑛1,𝑏₂₁,…,𝑏_2𝑛2,…,𝑏_𝑚1,…,𝑏_𝑚𝑛𝑚. (3)

Set 𝛿_𝑘𝑖=1𝑎_𝑘𝑖^𝑚𝑛_{𝑝=1,𝑝≠𝑘}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑘𝑖}𝑀_𝑝𝑗+𝐼_𝑘𝑖,Ω=𝑥=col𝑥_𝑘𝑖∈ℝ^𝑛∣𝑥_𝑘𝑖∈−𝛿_𝑘𝑖,𝛿_𝑘𝑖,for1≤𝑘≤𝑚,1≤𝑖≤𝑛_𝑘. (4)

For any 𝑘∈𝐼_𝑀, 1≤𝑖≤𝑙_𝑘, define 𝐻⁺_()𝑖𝑘+1𝑥_𝑘𝑖=1𝑎_(𝑘+1)𝑖𝑤_{𝑘𝑖(𝑘+1)𝑖}𝑓_𝑘𝑖𝑥_𝑘𝑖−𝑤_{𝑘𝑖(𝑘+1)𝑖}𝑀_𝑘𝑖+𝛿_(𝑘+1)𝑖,𝐻⁻_()𝑖𝑘+1𝑥_𝑘𝑖=1𝑎_(𝑘+1)𝑖𝑤_{𝑘𝑖(𝑘+1)𝑖}𝑓_𝑘𝑖𝑥_𝑘𝑖+𝑤_{𝑘𝑖(𝑘+1)𝑖}𝑀_𝑘𝑖−𝛿_(𝑘+1)𝑖,𝐺⁺_𝑘𝑖𝑥_(𝑘+1)𝑖=1𝑎_𝑘𝑖𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑓_(𝑘+1)𝑖𝑥_(𝑘+1)𝑖−𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑀_(𝑘+1)𝑖+𝛿_𝑘𝑖,𝐺⁻_𝑘𝑖𝑥_(𝑘+1)𝑖=1𝑎_𝑘𝑖𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑓_(𝑘+1)𝑖𝑥_(𝑘+1)𝑖+𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑀_(𝑘+1)𝑖−𝛿_𝑘𝑖. (5)

We consider the second parameter condition which is used to establish existence of multiple equilibria for systems (1). (H2) For any 𝑘∈𝐼_𝑀, the weights 𝑤_{(𝑘+1)𝑖𝑘𝑖}⋅𝑤_{𝑘𝑖(𝑘+1)𝑖}>0, and there exist two points (𝑢¹_𝑘𝑖,𝑢¹_(𝑘+1)𝑖), (𝑢²_𝑘𝑖,𝑢²_(𝑘+1)𝑖), where 𝑢¹_𝑘𝑖<𝑢²_𝑘𝑖, such that(1)if 𝑤_{𝑘𝑖(𝑘+1)𝑖},𝑤_{(𝑘+1)𝑖𝑘𝑖}>0, then 𝐻⁺_(𝑘+1)𝑖(𝑢¹_𝑘𝑖)<𝑢¹_(𝑘+1)𝑖, 𝐺⁺_𝑘𝑖(𝑢¹_(𝑘+1)𝑖)<𝑢¹_𝑘𝑖, 𝐻⁻_(𝑘+1)𝑖(𝑢²_𝑘𝑖)>𝑢²_(𝑘+1)𝑖, 𝐺⁻_𝑘𝑖(𝑢²_(𝑘+1)𝑖)>𝑢²_𝑘𝑖;(2)if 𝑤_{𝑘𝑖(𝑘+1)𝑖},𝑤_{(𝑘+1)𝑖𝑘𝑖}<0, then 𝐻⁻_(𝑘+1)𝑖(𝑢¹_𝑘𝑖)>𝑢¹_(𝑘+1)𝑖, 𝐺⁺_𝑘𝑖(𝑢¹_(𝑘+1)𝑖)<𝑢¹_𝑘𝑖, 𝐻⁺_(𝑘+1)𝑖(𝑢²_𝑘𝑖)<𝑢²_(𝑘+1)𝑖, 𝐺⁻_𝑘𝑖(𝑢²_(𝑘+1)𝑖)>𝑢²_𝑘𝑖.

The configuration that motivates (H2) is depicted in Figure 1. Under the above assumptions, if 𝑤_{𝑘𝑖(𝑘+1)𝑖},𝑤_{(𝑘+1)𝑖𝑘𝑖}>0, then there are three crossing points (𝑥¹⁺_𝑘𝑖,𝑥¹⁺_(𝑘+1)𝑖), (𝑥²⁻_𝑘𝑖,𝑥²⁻_(𝑘+1)𝑖), and (𝑥³⁺_𝑘𝑖,𝑥³⁺_(𝑘+1)𝑖) of the curves 𝑥_(𝑘+1)𝑖=𝐻⁺_(𝑘+1)𝑖(𝑥_𝑘𝑖), 𝑥_𝑘𝑖=𝐺⁺_𝑘𝑖(𝑥_(𝑘+1)𝑖), and there are three crossing points (𝑥¹⁻_𝑘𝑖,𝑥¹⁻_(𝑘+1)𝑖), (𝑥²⁺_𝑘𝑖,𝑥²⁺_(𝑘+1)𝑖), and (𝑥³⁻_𝑘𝑖,𝑥³⁻_(𝑘+1)𝑖) of the curves 𝑥_(𝑘+1)𝑖=𝐻⁻_(𝑘+1)𝑖(𝑥_𝑘𝑖), 𝑥_𝑘𝑖=𝐺⁻_𝑘𝑖(𝑥_(𝑘+1)𝑖) (see Figure 1(a)), where 𝑥¹⁻_𝑘𝑖<𝑥¹⁺_𝑘𝑖<𝑢¹_𝑘𝑖<𝑥²⁻_𝑘𝑖<𝑥²⁺_𝑘𝑖<𝑢²_𝑘𝑖<𝑥³⁻_𝑘𝑖<𝑥³⁺_𝑘𝑖,𝑥_()𝑖¹⁻_𝑘+1<𝑥_()𝑖¹⁺_𝑘+1<𝑢¹_()𝑖𝑘+1<𝑥_()𝑖²⁻_𝑘+1<𝑥_()𝑖²⁺_𝑘+1<𝑢²_()𝑖𝑘+1<𝑥_()𝑖³⁻_𝑘+1<𝑥_()𝑖³⁺_𝑘+1. (6)

(a)

(b)

Figure 1 

The configurations of 𝐺+𝑘𝑖(𝑦), 𝐺−𝑘𝑖(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.

If 𝑤_{𝑘𝑖(𝑘+1)𝑖},𝑤_{(𝑘+1)𝑖𝑘𝑖}<0, then there are three crossing points (𝑥¹⁻_𝑘𝑖,𝑥¹⁺_(𝑘+1)𝑖), (𝑥²⁺_𝑘𝑖,𝑥²⁻_(𝑘+1)𝑖), and (𝑥³⁻_𝑘𝑖,𝑥³⁺_(𝑘+1)𝑖) of the curves 𝑥_(𝑘+1)𝑖=𝐻⁺_(𝑘+1)𝑖(𝑥_𝑘𝑖), 𝑥_𝑘𝑖=𝐺⁻_𝑘𝑖(𝑥_(𝑘+1)𝑖), and there are three crossing points (𝑥¹⁺_𝑘𝑖,𝑥¹⁻_(𝑘+1)𝑖), (𝑥²⁻_𝑘𝑖,𝑥²⁺_(𝑘+1)𝑖), and (𝑥³⁺_𝑘𝑖,𝑥³⁻_(𝑘+1)𝑖) of the curves 𝑥_(𝑘+1)𝑖=𝐻⁻_(𝑘+1)𝑖(𝑥_𝑘𝑖), 𝑥_𝑘𝑖=𝐺⁺_𝑘𝑖(𝑥_(𝑘+1)𝑖) (see Figure 1(b)), where 𝑥¹⁻_𝑘𝑖<𝑥¹⁺_𝑘𝑖<𝑢¹_𝑘𝑖<𝑥²⁻_𝑘𝑖<𝑥²⁺_𝑘𝑖<𝑢²_𝑘𝑖<𝑥³⁻_𝑘𝑖<𝑥³⁺_𝑘𝑖,𝑥_()𝑖³⁻_𝑘+1<𝑥_()𝑖³⁺_𝑘+1<𝑢²_()𝑖𝑘+1<𝑥_()𝑖²⁻_𝑘+1<𝑥_()𝑖²⁺_𝑘+1<𝑢¹_()𝑖𝑘+1<𝑥_()𝑖¹⁻_𝑘+1<𝑥_()𝑖¹⁺_𝑘+1. (7)

Furthermore, it is easy to confirm that 𝑥^𝑗+_𝑘𝑖,𝑥^𝑗−_𝑘𝑖∈[−𝛿_𝑘𝑖,𝛿_𝑘𝑖], for any 1≤𝑘≤𝑚, 1≤𝑖≤𝑛_𝑘, and 𝑗=1,2,3.

Theorem 1. Under conditions (H1) and (H2), there are at least 3^𝑙 equilibria of the multidirectional associative memory neural network (1).

Proof. Set Ω^𝛼=𝑥=col𝑥_𝑘𝑖∈Ω∣𝑥_𝑘𝑖,𝑥_(𝑘+1)𝑖∈Ω^𝛼𝑘𝑖_𝑘𝑖,for𝑘∈𝐼_𝑀,1≤𝑖≤𝑙_𝑘, (8) where 𝛼=(𝛼₁₁,…,𝛼_1𝑙1,𝛼₃₁,…,𝛼_3𝑙3,…,𝛼_(𝑀−1)1,…, 𝛼_{(𝑀−1)𝑙𝑀−1}), 𝛼_𝑘𝑖=1,2, or 3, and Ω¹_𝑘𝑖=𝑥_𝑘𝑖,𝑥_(𝑘+1)𝑖∈ℝ²∣𝑥_𝑘𝑖,𝑥_(𝑘+1)𝑖∈𝑥¹⁻_𝑘𝑖,𝑥¹⁺_𝑘𝑖×𝑥_()𝑖¹⁻_𝑘+1,𝑥_()𝑖¹⁺_𝑘+1,Ω²_𝑘𝑖=𝑥_𝑘𝑖,𝑥_(𝑘+1)𝑖∈ℝ²∣𝑥_𝑘𝑖,𝑥_(𝑘+1)𝑖∈𝑥²⁻_𝑘𝑖,𝑥²⁺_𝑘𝑖×𝑥_()𝑖²⁻_𝑘+1,𝑥_()𝑖²⁺_𝑘+1,Ω³_𝑘𝑖=𝑥_𝑘𝑖,𝑥_(𝑘+1)𝑖∈ℝ²∣𝑥_𝑘𝑖,𝑥_(𝑘+1)𝑖∈𝑥³⁻_𝑘𝑖,𝑥³⁺_𝑘𝑖×𝑥_()𝑖³⁻_𝑘+1,𝑥_()𝑖³⁺_𝑘+1. (9) Obviously, they are 3^𝑙 disjoint closed regions, and Ω^𝑗_𝑘𝑖⊆[−𝛿_𝑘𝑖,𝛿_𝑘𝑖]×[−𝛿_(𝑘+1)𝑖,𝛿_(𝑘+1)𝑖] for any 𝑘∈𝐼_𝑀, 1≤𝑖≤𝑙_𝑘, 𝑗=1,2,3.
For any fixed 𝛼 and a given 𝑥=col(𝑥_𝑘𝑖)∈Ω^𝛼, we study this problem on the following two cases.
Case  1 (1≤𝑖≤𝑙_𝑘). Consider the following equations: −𝑎_𝑘𝑖𝑥_𝑘𝑖+𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑓_(𝑘+1)𝑖𝑥_(𝑘+1)𝑖−𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑓_(𝑘+1)𝑖𝑥_(𝑘+1)𝑖+^𝑚𝑛_{𝑝=1,𝑝≠𝑘}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑘𝑖}𝑓_𝑝𝑗𝑥_𝑝𝑗+𝐼_𝑘𝑖=0,−𝑎_(𝑘+1)𝑖𝑥_(𝑘+1)𝑖+𝑤_{𝑘𝑖(𝑘+1)𝑖}𝑓_𝑘𝑖𝑥_𝑘𝑖−𝑤_{𝑘𝑖(𝑘+1)𝑖}𝑓_𝑘𝑖𝑥_𝑘𝑖+^𝑚𝑛_{𝑝=1,𝑝≠𝑘}^𝑝_𝑗=1𝑤_{𝑝𝑗(𝑘+1)𝑖}𝑓_𝑝𝑗𝑥_𝑝𝑗+𝐼_(𝑘+1)𝑖=0. (10)(11) It follows from (10) that 𝑥_𝑘𝑖≤1𝑎_𝑘𝑖𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑓_(𝑘+1)𝑖𝑥_(𝑘+1)𝑖+^𝑛𝑘+1_{𝑗=1,𝑗≠𝑖}𝑤_{(𝑘+1)𝑗𝑘𝑖}𝑀_(𝑘+1)𝑗+𝐼_𝑘𝑖+1𝑎^𝑚_𝑘𝑖^𝑛_{𝑝=1,𝑝≠𝑘,𝑝≠𝑘+1}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑘𝑖}𝑀_𝑝𝑗=1𝑎_𝑘𝑖𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑓_(𝑘+1)𝑖𝑥_(𝑘+1)𝑖−𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑀_(𝑘+1)𝑖+𝐼_𝑘𝑖+1𝑎^𝑚_𝑘𝑖^𝑛_{𝑝=1,𝑝≠𝑘}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑘𝑖}𝑀_𝑝𝑗=1𝑎_𝑘𝑖𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑓_(𝑘+1)𝑖𝑥_(𝑘+1)𝑖−𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑀_(𝑘+1)𝑖+𝛿_𝑘𝑖=𝐺⁺_𝑘𝑖𝑥_(𝑘+1)𝑖. (12) Similarly, from (10), we also can obtain that 𝑥_𝑘𝑖≥1𝑎_𝑘𝑖𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑓_(𝑘+1)𝑖𝑥_(𝑘+1)𝑖+𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑀_(𝑘+1)𝑖−𝛿_𝑘𝑖=𝐺⁻_𝑘𝑖𝑥_(𝑘+1)𝑖. (13) So the curve of (10) is between the curves of 𝑥_𝑘𝑖=𝐺⁻_𝑘𝑖(𝑥_(𝑘+1)𝑖) and 𝑥_𝑘𝑖=𝐺⁺_𝑘𝑖(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(𝑥_𝑘𝑖) and 𝑥_(𝑘+1)𝑖=𝐻⁺_(𝑘+1)𝑖(𝑥_𝑘𝑖). So we can affirm that there are at least three solutions of (10) and (11); each solution lies separately in Ω¹_𝑘𝑖, Ω²_𝑘𝑖, and Ω³_𝑘𝑖. Get one that is in Ω^𝛼𝑘𝑖_𝑘𝑖, and denote it by (𝑥_𝑘𝑖,𝑥_(𝑘+1)𝑖).
Case  2 (𝑙_𝑘+1≤𝑖≤𝑛_𝑘). Let 𝑥_𝑘𝑖=1𝑎_𝑘𝑖^𝑚𝑛_{𝑝=1,𝑝≠𝑘}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑘𝑖}𝑓_𝑝𝑗𝑥_𝑝𝑗+𝐼_𝑘𝑖. (14) Then 𝑥_𝑘𝑖≤1𝑎_𝑘𝑖^𝑚𝑛_{𝑝=1,𝑝≠𝑘}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑘𝑖}𝑓_𝑝𝑗𝑥_𝑝𝑗+𝐼_𝑘𝑖≤1𝑎_𝑘𝑖^𝑚𝑛_{𝑝=1,𝑝≠𝑘}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑘𝑖}𝑀_𝑝𝑗+𝐼_𝑘𝑖=𝛿_𝑘𝑖. (15)
From the two cases above, we can obtain 𝑥_𝑘𝑖∈Ω^𝛼 for any given 𝑥∈Ω^𝛼. Set 𝑥=col(𝑥_𝑘𝑖). It is obvious that 𝑥∈Ω^𝛼. Set mapping 𝐹^𝛼∶Ω^𝛼→Ω^𝛼 as follows: 𝐹^𝛼(𝑥)=𝑥. (16) Because 𝑓_𝑘𝑖(⋅) are continuous mappings, so 𝐹^𝛼 is continuous mapping. By the Brouwer fixed point theorem, there is at least one fixed point of 𝐹^𝛼, that is, zero point of (1) in Ω^𝛼. Therefore, there are at least 3^𝑙 equilibria of the multidirectional associative memory neural network (1).

3. Stability Analysis

In this section, the stability of the equilibria is considered. Set Ψ=𝜙=col𝜙_𝑘𝑖∈𝐶[−𝜏,0],ℝ^𝑛∣−𝛿_𝑘𝑖≤𝜙_𝑘𝑖(𝑠)≤𝛿_𝑘𝑖,for1≤𝑘≤𝑚,1≤𝑖≤𝑛_𝑘,𝜂=𝜂₁₁,…,𝜂_1𝑙1,𝜂₃₁,…,𝜂_3𝑙3,…,𝜂_(𝑀−1)1,…,𝜂_{(𝑀−1)𝑙𝑀−1}, (17) with 𝜂_𝑘𝑖=1 or 𝜂_𝑘𝑖=3.

We consider the following 2^𝑙 subsets of Ψ: Ψ^𝜂=𝜙=col𝜙_𝑘𝑖∈Ψ∣𝜙_𝑘𝑖,𝜙_(𝑘+1)𝑖∈Ψ^𝜂𝑘𝑖_𝑘𝑖,for𝑘∈𝐼_𝑀,1≤𝑖≤𝑙_𝑘, (18) where Ψ¹_𝑘𝑖=𝜙_𝑘𝑖,𝜙_(𝑘+1)𝑖∈𝐶[−𝜏,0],ℝ²∣𝜙_𝑝𝑖(𝑠)∈𝑥¹⁻_𝑝𝑖,𝑥¹⁺_𝑝𝑖,,Ψfor𝑝=𝑘,𝑘+1_3𝑘𝑖=𝜙_𝑘𝑖,𝜙_(𝑘+1)𝑖∈𝐶[−𝜏,0],ℝ²∣𝜙_𝑝𝑖(𝑠)∈𝑥³⁻_𝑝𝑖,𝑥³⁺_𝑝𝑖,for𝑝=𝑘,𝑘+1. (19)

Lemma 2. Under assumptions (H1) and (H2), each Ψ^𝜂 is a positive invariant set with respect to the solution flow generated by system (1).

Proof. For any given 𝜂 and any initial condition 𝜙=col(𝜙_𝑘𝑖)∈Ψ^𝜂, let 𝑥(𝑡;𝜙)=col(𝑥_𝑘𝑖(𝑡;𝜙)) be the solution of (1) with initial condition 𝜙∈Ψ^𝜂.
For any 1≤𝑘≤𝑚,1≤𝑖≤𝑛_𝑘, and any 𝑡>0, from (1), we obtain that −𝑎_𝑘𝑖𝛿_𝑘𝑖≤𝑑𝑥_𝑘𝑖𝑡;𝜙𝑑𝑡+𝑎_𝑘𝑖𝑥_𝑘𝑖𝑡;𝜙≤𝑎_𝑘𝑖𝛿_𝑘𝑖. (20) Multiplying both sides of (20) by 𝑒^{𝑎𝑘𝑖𝑡}, we have −𝑎_𝑘𝑖𝛿_𝑘𝑖𝑒^{𝑎𝑘𝑖𝑡}≤𝑑𝑥_𝑘𝑖𝑡;𝜙𝑒^{𝑎𝑘𝑖𝑡}𝑑𝑡≤𝑎_𝑘𝑖𝛿_𝑘𝑖𝑒^{𝑎𝑘𝑖𝑡}. (21) Integrating (21) over [0,𝑡], we obtain −𝛿_𝑘𝑖𝑒^{𝑎𝑘𝑖𝑡}−1≤𝑥_𝑘𝑖𝑡;𝜙𝑒^{𝑎𝑘𝑖𝑡}−𝜙_𝑘𝑖(0)≤𝛿_𝑘𝑖𝑒^{𝑎𝑘𝑖𝑡}−1. (22) Hence it follows that, for 𝑡>0, 𝑥_𝑘𝑖𝑡;𝜙≤𝑒^{−𝑎𝑘𝑖𝑡}𝜙_𝑘𝑖(0)−𝛿_𝑘𝑖+𝛿_𝑘𝑖. (23) Therefore, for any given initial condition 𝜙∈Ψ^𝜂, we have, for 𝑡>0, 𝑥_𝑘𝑖𝑡;𝜙≤𝛿_𝑘𝑖. (24) That is, 𝑥(𝑡,𝜙)∈Ψ for 𝑡>0.
We claim that 𝑥(𝑡,𝜙) remains in Ψ^𝜂 for any 𝑡>0. If it is not true, there exist 𝑘∈𝐼_𝑀 and 1≤𝑖≤𝑙_𝑘 that (𝑥_𝑘𝑖(𝑡),𝑥_(𝑘+1)𝑖(𝑡)) firstly (or one of the first) escapes from Ψ^𝜂𝑘𝑖_𝑘𝑖.
Case  1 (𝑤_{𝑘𝑖(𝑘+1)𝑖},𝑤_{(𝑘+1)𝑖𝑘𝑖}>0). If 𝜂_𝑘𝑖=1, then exists 𝑡₀>0, such that (𝑥_𝑘𝑖(𝑡₀),𝑥_(𝑘+1)𝑖(𝑡₀)) is on the edges of Ψ^𝜂𝑘𝑖_𝑘𝑖, and for any 𝑡<𝑡₀, 𝑥¹⁻_𝑘𝑖<𝑥_𝑘𝑖(𝑡)<𝑥¹⁺_𝑘𝑖, 𝑥¹⁻_(𝑘+1)𝑖<𝑥_(𝑘+1)𝑖(𝑡)<𝑥¹⁺_(𝑘+1)𝑖. If 𝑥_𝑘𝑖(𝑡₀)=𝑥¹⁺_𝑘𝑖, 𝑥_(𝑘+1)𝑖(𝑡₀)∈[𝑥¹⁻_(𝑘+1)𝑖,𝑥¹⁺_(𝑘+1)𝑖], then 𝑑𝑥_𝑘𝑖(𝑡)𝑑𝑡_𝑡=𝑡0=−𝑎_𝑘𝑖𝑥_𝑘𝑖𝑡₀+^𝑚𝑛_{𝑝=1,𝑝≠𝑘}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑘𝑖}𝑓_𝑝𝑗×𝑥_𝑝𝑗𝑡₀−𝜏_{𝑝𝑗𝑘𝑖}+𝐼_𝑘𝑖≤−𝑎_𝑘𝑖𝑥¹⁺_𝑘𝑖+𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑓_(𝑘+1)𝑖𝑥_()𝑖¹⁺_𝑘+1−𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑀_(𝑘+1)𝑖+^𝑚𝑛_{𝑝=1,𝑝≠𝑘}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑘𝑖}𝑀_𝑝𝑗+𝐼_𝑘𝑖=−𝑎_𝑘𝑖𝑥¹⁺_𝑘𝑖−1𝑎_𝑘𝑖𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑓_(𝑘+1)𝑖𝑥_()𝑖¹⁺_𝑘+1−𝑤_{(𝑘+1)𝑖𝑘𝑖}𝑀_(𝑘+1)𝑖−𝛿_𝑘𝑖=−𝑎_𝑘𝑖𝑥¹⁺_𝑘𝑖−𝐺⁺_𝑘𝑖𝑥_()𝑖¹⁺_𝑘+1=0. (25) Therefore, (𝑥_𝑘𝑖(𝑡),𝑥_(𝑘+1)𝑖(𝑡)) cannot escape from Ψ^𝜂𝑘𝑖_𝑘𝑖 through the edge of 𝑥_𝑘𝑖(𝑡)=𝑥¹⁺_𝑘𝑖, 𝑥_(𝑘+1)𝑖(𝑡)∈[𝑥¹⁻_(𝑘+1)𝑖,𝑥¹⁺_(𝑘+1)𝑖]. With similar proof, we can obtain that (𝑥_𝑘𝑖(𝑡),𝑥_(𝑘+1)𝑖(𝑡)) cannot escape from Ψ^𝜂𝑘𝑖_𝑘𝑖 through the other three edges. Hence (𝑥_𝑘𝑖(𝑡),𝑥_(𝑘+1)𝑖(𝑡)) cannot escape from Ψ¹_𝑘𝑖. We can also prove that (𝑥_𝑘𝑖(𝑡),𝑥_(𝑘+1)𝑖(𝑡)) cannot escape from Ψ³_𝑘𝑖.
Case  2 (𝑤_{𝑘𝑖(𝑘+1)𝑖},𝑤_{(𝑘+1)𝑖𝑘𝑖}<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 constants 𝐿_𝑘𝑖>0(1≤𝑘≤𝑚 and 1≤𝑖≤𝑛_𝑘) such that 𝑓_𝑘𝑖(𝑥)−𝑓_𝑘𝑖𝑦≤𝐿_𝑘𝑖𝑥−𝑦, (26) for each 𝑥,𝑦 in a subset 𝑅_𝑘𝑖⊂ℝ, where 𝑅_𝑘𝑖 is defined as follows: 𝑅_𝑘𝑖=𝑥¹⁻_𝑘𝑖,𝑥¹⁺_𝑘𝑖∪𝑥³⁻_𝑘𝑖,𝑥³⁺_𝑘𝑖,for1≤𝑘≤𝑀,1≤𝑖≤𝑙_𝑘,𝛿⁻_𝑘𝑖,𝛿⁺_𝑘𝑖,for1≤𝑘≤𝑚,𝑙_𝑘+1≤𝑖≤𝑛_𝑘. (27)  (H4) For any 1≤𝑘≤𝑚,1≤𝑖≤𝑛_𝑘, it is satisfied that 𝑎_𝑘𝑖−^𝑚𝑛_{𝑝=1,𝑝≠𝑘}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑘𝑖}𝐿_𝑘𝑖>0. (28)

Theorem 3. There are 2^𝑙 exponential stability equilibria of the multidirectional associative memory neural network (1), if the conditions (H1)–(H4) hold.

Proof. According to Theorem 1, for any 𝛼, the multidirectional associative memory neural network (1) has an equilibrium in Ω^𝛼; 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 Ψ^𝜂 under initial condition 𝜙∈Ψ^𝜂. Let 𝑦(𝑡)=col𝑦_𝑘𝑖(𝑡)=𝑥𝑡,𝜙−𝑥. (29) System (1) becomes 𝑑𝑦_𝑘𝑖(𝑡)𝑑𝑡=−𝑎_𝑘𝑖𝑦_𝑘𝑖(𝑡)+^𝑚𝑛_{𝑝=1,𝑝≠𝑘}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑘𝑖}𝑓_𝑝𝑗𝑦_𝑝𝑗𝑡−𝜏_{𝑝𝑗𝑘𝑖}+𝑥_𝑝𝑗−𝑓_𝑝𝑗𝑥_𝑝𝑗, (30) for 1≤𝑘≤𝑚,1≤𝑖≤𝑛_𝑘.
According to condition (H4), there exists 𝜇>0 such that 𝑎_𝑘𝑖−𝜇−^𝑚𝑛_{𝑝=1,𝑝≠𝑘}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑘𝑖}𝐿_𝑘𝑖𝑒^{𝜇𝜏𝑝𝑗𝑘𝑖}>0. (31) Define that 𝑢_𝑘𝑖(𝑡)=𝑒^𝜇𝑡|𝑦_𝑘𝑖(𝑡)|, for 1≤𝑘≤𝑚 and 1≤𝑖≤𝑛_𝑘. It is obvious that 𝑢_𝑘𝑖(𝑡)>0. Denote 𝑄=max_{1≤𝑘≤𝑚,1≤𝑖≤𝑛𝑘}sup_{(]𝜃∈−𝜏,0}𝑥_𝑘𝑖(𝜃)−𝑥_𝑘𝑖. (32) Let 𝛿>1 be an arbitrary real number. For any 𝜃∈(−𝜏,0], 1≤𝑘≤𝑚, 1≤𝑖≤𝑛_𝑘, it is easy to obtain that 𝑢_𝑘𝑖(𝑡)<𝑄𝛿. We shall prove that 𝑢_𝑘𝑖(𝑡)<𝑄𝛿, (33) for 1≤𝑘≤𝑚, 1≤𝑖≤𝑛_𝑘, and any 𝑡>0. Suppose this is not the case; then there exist 𝑘=𝑞, 𝑖=𝑟, and a time 𝑡₀ such that 𝑢_𝑘𝑖(𝑡)≤𝑄𝛿 for 𝑡∈(−𝜏,𝑡₀],1≤𝑘≤𝑚and1≤𝑖≤𝑛_𝑘,𝑢_𝑞𝑟(𝑡₀)=𝑄𝛿,and𝐷⁺𝑢_𝑞𝑟(𝑡₀)≥0. From (30), we derive that 𝐷⁺𝑦_𝑞𝑟𝑡₀≤−𝑎_𝑞𝑟𝑦_𝑞𝑟𝑡₀+^𝑚𝑛_{𝑝=1,𝑝≠𝑞}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑞𝑟}𝐿_𝑝𝑗𝑦_𝑝𝑗𝑡₀−𝜏_{𝑝𝑗𝑞𝑟}. (34) Hence, from (31) and (34), 𝐷⁺𝑢_𝑞𝑟𝑡₀≤𝜇𝑒^𝜇𝑡0𝑦_𝑞𝑟𝑡₀+𝑒^𝜇𝑡0×−𝑎_𝑞𝑟𝑦_𝑞𝑟𝑡₀+_{𝑚𝑛𝑝=1,𝑝≠𝑞𝑝𝑗=1}𝑤_{𝑝𝑗𝑞𝑟}𝐿_𝑝𝑗𝑦_𝑝𝑗𝑡₀−𝜏_{𝑝𝑗𝑞𝑟}=−𝑎_𝑞𝑟−𝜇𝑒^𝜇𝑡0𝑦_𝑞𝑟𝑡₀+^𝑚𝑛_{𝑝=1,𝑝≠𝑞}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑞𝑟}𝐿_𝑝𝑗𝑒^{𝜇𝜏𝑝𝑗𝑞𝑟}𝑢_𝑝𝑗𝑡₀−𝜏_{𝑝𝑗𝑞𝑟}≤−𝑎_𝑞𝑟−𝜇𝑢_𝑞𝑟𝑡₀+^𝑚𝑛_{𝑝=1,𝑝≠𝑞}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑞𝑟}𝐿_𝑝𝑗𝑒^{𝜇𝜏𝑝𝑗𝑞𝑟}sup_{𝑡𝜃∈0−𝜏,𝑡0}𝑢_𝑝𝑗(𝜃𝑎)≤−_𝑞𝑟−𝜇−^𝑚𝑛_{𝑝=1,𝑝≠𝑞}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑞𝑟}𝐿_𝑝𝑗𝑒^{𝜇𝜏𝑝𝑗𝑞𝑟}𝑄𝛿≤0, (35) which is contradicting 𝐷⁺𝑢_𝑞𝑟(𝑡₀)≥0. Hence the inequality (33) holds. Since 𝛿>1 is arbitrary, by allowing 𝛿→1+, we have 𝑢_𝑘𝑖(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 2^{𝑛0[𝑚/2]} exponentially stable 𝑇-periodic solutions, where 𝑛₀=min{𝑛₁,𝑛₂,…,𝑛_𝑚} if 𝑚 is an even number or 𝑛₀=min{𝑛₁,𝑛₂,…,𝑛_𝑚−1} if 𝑚 is an odd number. But we obtain the existence of 3^𝑙 equilibria by Theorem 1 and the exponential stability of 2^𝑙 equilibria of them by Theorem 3. Because 𝑙_𝑘=min{𝑛_𝑘,𝑛_𝑘+1}, so 𝑛₀=min{𝑙_𝑘∣𝑘∈𝐼_𝑀}. It follows that 𝑙=∑_{𝑘∈𝐼𝑀}𝑙_𝑘≥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: 𝑑𝑥_𝑘𝑖𝑑𝑡=−𝑎_𝑘𝑖𝑥_𝑘𝑖(𝑡)+^3𝑛_{𝑝=1,𝑝≠𝑘}^𝑝_𝑗=1𝑤_{𝑝𝑗𝑘𝑖}𝑓_𝑝𝑗×𝑥_𝑝𝑗𝑡−𝜏_{𝑝𝑗𝑘𝑖}+𝐼_𝑘𝑖, (36) 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 00𝑤₂₁₁₁𝑤₂₂₁₁𝑤₃₁₁₁00𝑤₂₁₁₂𝑤₂₂₁₂𝑤₃₁₁₂𝑤₁₁₂₁𝑤₁₂₂₁00𝑤₃₁₂₁𝑤₁₁₂₂𝑤₁₂₂₂00𝑤₃₁₂₂𝑤₁₁₃₁𝑤₁₂₃₁𝑤₂₁₃₁𝑤₂₂₃₁0=,00−20.30.2000.430.1−3−0.5000.10.3200−0.1121−10 (37) and the delays 00𝜏₂₁₁₁𝜏₂₂₁₁𝜏₃₁₁₁00𝜏₂₁₁₂𝜏₂₂₁₂𝜏₃₁₁₂𝜏₁₁₂₁𝜏₁₂₂₁00𝜏₃₁₂₁𝜏₁₁₂₂𝜏₁₂₂₂00𝜏₃₁₂₂𝜏₁₁₃₁𝜏₁₂₃₁𝜏₂₁₃₁𝜏₂₂₃₁0=.00121.2000.411.510.4000.822000.71.21120 (38)

The neuronal activation functions 𝑓_𝑝𝑗(𝑥)=tanh(𝑥)(1≤𝑝≤3,1≤𝑗≤𝑛_𝑝) are continuous, increasing, and bounded functions, and there are 𝑀_𝑝𝑗=1>0 such that |𝑓_𝑝𝑗(𝑥)|≤𝑀_𝑝𝑗. Hence condition (H1) is satisfied.

Obviously, 𝑛=5, 𝑀=2, 𝑙₁=𝑙₂=2, and 𝑙=2. Through calculations, we have 𝐻^±₂₁𝑥₁₁=−3𝑓₁₁𝑥₁₁𝐺±0.8,_±11𝑥₂₁=−2𝑓₂₁𝑥₂₁𝐻±0.6,_±22𝑥₁₂=2𝑓₂₁𝑥₁₂𝐺±0.6,_±12𝑥₂₂=3𝑓₂₂𝑥₂₂±0.9. (39) There exist four points(𝑢¹₁₁,𝑢¹₂₁)=(−0.8,1),(𝑢²₁₁,𝑢²₂₁)=(0.9,−1.2), (𝑢¹₁₂,𝑢¹₂₂)=(−1,−0.9), and (𝑢²₁₂,𝑢²₂₂)=(1.1,0.9), where 𝑢¹₁₁<𝑢²₁₁, 𝑢¹₁₂<𝑢²₁₂, such that 𝐻⁻₂₁𝑢¹₁₁=1.1921>1,𝐺⁺₁₁𝑢¹₂₁𝐻=−0.9232<−0.8,₊₂₁𝑢²₁₁=−1.3489<−1.2,𝐺⁻₁₁𝑢²₂₁𝐻=0.9673>0.9,₊₂₂𝑢¹₁₂=−0.9232<−0.9,𝐺⁺₁₂𝑢¹₂₂𝐻=−1.2489<−1,₋₂₂𝑢²₁₂=1.0010>0.9,𝐺⁻₁₂𝑢²₂₂=1.2489>1.1. (40) Hence condition (H2) holds. The curves 𝑥₂₁=𝐻⁺₂₁(𝑥₁₁), 𝑥₁₁=𝐺⁻₁₁(𝑥₂₁) have three crossing points (𝑥¹⁻₁₁,𝑥¹⁺₂₁)=(−2.5979,3.7669), (𝑥²⁺₁₁,𝑥²⁻₂₁)=(0.5198,0.6327), and (𝑥³⁻₁₁,𝑥³⁺₂₁)=(1.2886,−1.7763), and the curves 𝑥₂₁=𝐻⁺₂₁(𝑥₁₁), 𝑥₁₁=𝐺⁻₁₁(𝑥₂₁) have three crossing points (𝑥¹⁺₁₁,𝑥¹⁻₂₁)=(−1.2886,1.7763), (𝑥²⁻₁₁,𝑥²⁺₂₁)=(−0.5198,0.6327), and (𝑥³⁺₁₁,𝑥³⁻₂₁)=(2.5979,−3.7669). The curves 𝑥₂₂=𝐻⁺₂₂(𝑥₁₂),𝑥₁₂=𝐺⁺₁₂(𝑥₂₂) have three crossing points (𝑥¹⁺₁₂,𝑥¹⁺₂₂)=(−1.6517,−1.2582), (𝑥²⁻₁₂,𝑥²⁻₂₂)=(−0.6830,−0.5869), and (𝑥³⁺₁₂,𝑥³⁺₂₂)=(3.8670,2.5982), and the curves 𝑥₂₂=𝐻⁻₂₂(𝑥₁₂), 𝑥₁₂=𝐺⁻₁₂(𝑥₂₂) have three crossing points (𝑥¹⁻₁₂,𝑥¹⁻₂₂)=(−3.8670,−2.5982), (𝑥²⁺₁₂,𝑥²⁺₂₂)=(0.6830,0.5869), and (𝑥³⁻₁₂,𝑥³⁻₂₂)=(1.6517,1.2582). From (27), we obtain the subsets 𝑅₁₁=[]∪[],𝑅−2.2579,−1.28861.2886,2.2579₂₁=[]∪[],𝑅−3.7669,−1.77631.7763,3.7669₁₂=[]∪[],𝑅−3.8670,−1.65171.6571,3.8670₂₂=[]∪[],𝑅−2.5982,−1.25821.2582,2.5982₃₁=[−5.4,5.4]. (41)

Because the activation functions 𝑓_𝑝𝑗(𝑥)=tanh(𝑥) are differentiable, we can let 𝐿_𝑝𝑗=max{𝑓^′_𝑝𝑗(𝑥)∣𝑥∈𝑅_𝑝𝑗}>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 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. (42) Hence condition (H4) holds. Then by Theorems 1 and 3, the MAM neural network (36) has 3^𝑙=9 equilibria, and 2^𝑙=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 of 3^𝑙 equilibria. It is proved that 2^𝑙 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.