Stability Analysis of Discrete Hopfield Neural Networks with the Nonnegative Definite Monotone Increasing Weight Function Matrix

Abstract

The original Hopfield neural networks model is adapted so that the weights of the resulting network are time varying. In this paper, the Discrete Hopfield neural networks with weight function matrix (DHNNWFM) the weight changes with time, are considered, and the stability of DHNNWFM is analyzed. Combined with the Lyapunov function, we obtain some important results that if weight function matrix (WFM) is weakly (or strongly) nonnegative definite function matrix, the DHNNWFM will converge to a stable state in serial (or parallel) model, and if WFM consisted of strongly nonnegative definite function matrix and column (or row) diagonally dominant function matrix, DHNNWFM will converge to a stable state in parallel model.

1. Introduction

Discrete Hopfield neural network (DHNN) [1] is one of the famous neural networks with a wide range of applications. With the development of DHNN in theory and application, the model is more and more complex. It is well known that the nonautonomous phenomena often occur in many realistic systems. Particularly when we consider a long-term dynamical behavior of the system and consider seasonality of the changing environment, the parameters of the system usually will change with time [2, 3]. However, the original DHNN is difficult to adapt this change, because the matrixes of DHNN and DHNN with time or time-varying delay are constant matrixes [1, 4–22] and the parameters of DHNN, which change with time, are seldom considered. In order to implement a desired flow vector field distribution by using conventional matrix encoding scheme, a time-varying Hopfield model (TVHM) is proposed [23]. In many applications, the properties of periodic oscillatory solutions are of great interest. For example, the human brain has been in periodic oscillatory or chaos state, hence it is of prime importance to study periodic oscillatory and chaos phenomenon of neural networks. So, the literature [2, 3] studies the global exponential stability and existence of periodic solutions of the high-order Hopfield-type neural networks. In [23, 24], we consider that the weight function matrix and the threshold function vector, respectively, converge to a constant matrix and a constant vector and the weight function matrix is a symmetric function matrix, and, we analyze the stability of the model. In this paper, with the stability of asymmetric Hopfield Neural Networks [4, 5], we work on the stability analysis of discrete Hopfield neural networks with the nonnegative definite monotone increasing weight function matrix.

This paper has the following organization. In Section 1, we provide the introduction. In Section 2, we introduce some basic concepts. In Section 3, we analyze the stability analysis of discrete Hopfield neural networks with the nonnegative definite monotone increasing weight function matrix. The last section offers the conclusions of this paper.

2. Basic Definitions

In this section, we will introduce basic concepts which will be used in the following to obtain some results.

DHNN with weight function matrix (DHNNWFM) varies with the discrete time factor 𝑡 by step length ℎ (in this paper, ℎ=1). Formally, let 𝑁(𝑡)=(𝑊(𝑡),𝜃(𝑡)) be a DHNN with 𝑛 neurons, which have the discrete time factor 𝑡 with step length 1 and are denoted by {𝑥₁,𝑥₂,…,𝑥_𝑛}. In the pair 𝑁(𝑡)=(𝑊(𝑡),𝜃(𝑡)), 𝑊(𝑡)=(𝑤_𝑖𝑗(𝑡)) is an 𝑛×𝑛 function matrix where 𝑤_𝑖𝑗(𝑡) changes with time 𝑡, representing the connection weight from 𝑥_𝑗 to 𝑥_𝑖, and 𝜃(𝑡)=(𝜃_𝑖(𝑡)) is an 𝑛-dimensional function vector where 𝜃_𝑖(𝑡) changes with time 𝑡, representing the threshold attached to the neuron 𝑥_𝑖. The state of the 𝑖 neuron at time 𝑡 is denoted by 𝑥_𝑖(𝑡). Each neuron is assumed to have two possible states: 1 and −1. The state of the network at time 𝑡 is the vector 𝑋(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝑅^𝑛. In general, the state, 𝑋(𝑡+1), of the network at time 𝑡+1 is a function of {𝑊(𝑡),𝜃(𝑡)} and the state, 𝑋(𝑡), of the network at time 𝑡. The network is, thus, completely determined by the parameters {𝑊(𝑡),𝜃(𝑡)}, the initial value, 𝑋(0), of the states, and the manner in which the neurons are updated (evolved).

If at time step 𝑡, a neuron 𝑥_𝑖 is chosen to be updated, then at the next step

𝑥_𝑖(𝑡+1)=sgn(𝑦_𝑖(𝑡))=1,𝑦_𝑖(𝑡)≥0,−1,𝑦_𝑖(𝑡)<0, (2.1) where 𝑦_𝑖(𝑡)=∑^𝑛_𝑗=1𝑤_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−𝜃_𝑖(𝑡),𝑖=1,2,…,𝑛.

(1) The network is updated asynchronously, that is, only one neuron 𝑥_𝑖 is selected at time 𝑡+1. The updating rule is 𝑥_𝑗(𝑡+1)=sgn(𝑦_𝑗𝑥(𝑡)),𝑗=𝑖,_𝑗(𝑡),𝑗≠𝑖. (2.2)

(2) The network is updated synchronously, that is, every neuron 𝑥_𝑗 is selected at time 𝑡+1. The updating rule is 𝑥_𝑗(𝑡+1)=sgn(𝑦_𝑗(𝑡)),𝑗=1,2,…,𝑛. (2.3)

Let 𝑁(𝑡)=(𝑊(𝑡),𝜃(𝑡)) be a DHNNWFM. If the output of 𝑛 neurons does not change any longer after a limited time interval 𝑡 from its initial state 𝑋(0), that is, 𝑋(𝑡+1)=𝑋(𝑡), then we can say that the network is a stable state, and we call 𝑋(𝑡+1)=𝑋(𝑡) a stable point of 𝑁. In addition, we say that the network converges about the initial state 𝑋(𝑡₀). It is easy to know if 𝑋 is a stable point of DHNNWFM, then 𝑋(𝑡)=sgn(𝑊(𝑡)𝑋(𝑡)−𝜃(𝑡)). Sometimes we call 𝑋 that satisfies the above formula a stable attraction factor of the network 𝑁. If 𝑋 is an attraction factor DHNNWFM 𝑁(𝑡)=(𝑊(𝑡),𝜃(𝑡)), we denote attraction domain of 𝑋 as Γ(𝑋), which represents the set which consists of all of the initial state 𝑋(0) attracted to 𝑋.

Let 𝑁(𝑡)=(𝑊(𝑡),𝜃(𝑡)) be a DHNNWFM. 𝑋(0),𝑋(1),…,𝑋(𝑟−1),(𝑟≥1) are 𝑛-dimensional vectors. If𝑋(𝑡+1)=sgn(𝑊(𝑡)𝑋(𝑡)−𝜃(𝑡)),𝑡=0,1,2,…,𝑟−2,and 𝑋(0)=sgn(𝑊(𝑟−1)𝑋(𝑟−1)−𝜃(𝑟−1)), then we called a limit cycle attraction factor of 𝑁(𝑡), sometimes abbreviated to limit cycle; its length is 𝑟, and is denoted by (𝑋(0),𝑋(1),…,𝑋(𝑟−1)). Similar to the attraction domain of a stable attraction factor, the attraction domain of cycle attraction factor (𝑋(0),𝑋(1),…,𝑋(𝑟−1)), denoted by Γ(𝑋(0),𝑋(1),…,𝑋(𝑟−1)), represents the set that consists of all the possible initial states which are attracted to (𝑋(0),𝑋(1),…,𝑋(𝑟−1)).

Definition 2.1. 𝑊(𝑡) is a column or row-diagonally dominant function matrix, if it satisfies the following conditions: 𝑤_𝑖𝑖1(𝑡)≥2_{1≤𝑗≤𝑛,𝑗≠𝑖}|𝑤_𝑗𝑖(𝑡)|,𝑖=1,2,…,𝑛, (2.4) or 𝑤_𝑖𝑖1(𝑡)≥2_{1≤𝑗≤𝑛,𝑗≠𝑖}|𝑤_𝑖𝑗(𝑡)|,𝑖=1,2,…,𝑛. (2.5)

Definition 2.2. let 𝑊(𝑡) be a column or row-diagonally dominant function matrix. 𝑊(𝑡) is called a column or row-diagonally dominant monotone increasing function matrix, if it satisfies the following conditions: Δ𝑤_𝑖𝑖1(𝑡)≥2_{1≤𝑗≤𝑛,𝑗≠𝑖}|Δ𝑤_𝑗𝑖(𝑡)|,𝑖=1,2,…,𝑛, (2.6) or Δ𝑤_𝑖𝑖1(𝑡)≥2_{1≤𝑗≤𝑛,𝑗≠𝑖}|Δ𝑤_𝑖𝑗(𝑡)|,𝑖=1,2,…,𝑛, (2.7) where Δ𝑤_𝑖𝑗(𝑡)=𝑤_𝑖𝑗(𝑡+1)−𝑤_𝑖𝑗(𝑡).

Definition 2.3. 𝑊(𝑡) is called a nonnegative definite matrix on the set {−2,0,2}, if it satisfies 𝑃^𝑇𝑊(𝑡)𝑃≥0 for each 𝑃∈{−2,0,2}^𝑛.

Definition 2.4. let 𝑊(𝑡) be a nonnegative definite function matrix. 𝑊(𝑡) is called nonnegative definite monotone increasing function matrix, if it satisfies 𝑃^𝑇Δ𝑊(𝑡)𝑃≥0, (2.8) where Δ𝑊(𝑡)=𝑊(𝑡+1)−𝑊(𝑡), for each 𝑃∈{−2,0,2}^𝑛.

Definition 2.5. 𝑊(𝑡) is called a weakly nonnegative definite function matrix, if it satisfies 𝑤_𝑖𝑖1(𝑡)≥2_{1≤𝑗≤𝑛}|𝑤_𝑗𝑖(𝑡)−𝑤_𝑖𝑗(𝑡)|,𝑖=1,2,…,𝑛. (2.9)

Definition 2.6. let 𝑊(𝑡) be a weakly nonnegative definite function matrix. 𝑊(𝑡) is called weakly nonnegative definite monotone increasing function matrix, if it satisfies Δ𝑤_𝑖𝑖1(𝑡)≥2_{1≤𝑗≤𝑛}|Δ𝑤_𝑗𝑖(𝑡)−Δ𝑤_𝑖𝑗(𝑡)|,𝑖=1,2,…,𝑛, (2.10) where Δ𝑤_𝑖𝑗(𝑡)=𝑤_𝑖𝑗(𝑡+1)−𝑤_𝑖𝑗(𝑡).
For an 𝑛×𝑛 matrix 𝑊(𝑡), we denote in this paper the corresponding matrix 𝑊^∗(𝑡)=(𝑤^∗_𝑖𝑗(𝑡)) with 𝑤^∗_𝑖𝑗𝑤(𝑡)=^𝑖𝑖1(𝑡)−2_{1≤𝑘≤𝑛}|𝑤_𝑘𝑖(𝑡)−𝑤_𝑖𝑘𝑤(𝑡)|,𝑖=𝑗,_𝑖𝑗(𝑡),𝑖≠𝑗. (2.11)

Definition 2.7. 𝑊(𝑡) is called a strongly nonnegative definite function matrix, if the corresponding matrix 𝑊^∗(𝑡) is a nonnegative definite function matrix.

Definition 2.8. let 𝑊(𝑡) be a strongly nonnegative definite function matrix. 𝑊(𝑡) is called a strongly nonnegative definite monotone increasing function matrix, if the corresponding matrix 𝑊^∗(𝑡) is nonnegative definite monotone increasing function matrix.

In this paper, the function matrix is the sum of the initial weight matrix and increment matrixes, that is,

𝑊(𝑡)=𝑊(0)+_{0≤𝑠<𝑡}Δ𝑊(𝑠). (2.12)

And the 𝑛-dimensional function vector is the sum of the initial vector and increment vectors, that is,

𝜃(𝑡)=𝜃(0)+_{0≤𝑠<𝑡}Δ𝜃(𝑠). (2.13)

In order to describe, let 𝐿(𝑡)={0,1,…,𝑡−1}∪{0}, 𝐼={1,2,…,𝑛}, and 𝑊(0)=Δ𝑊(0). 0 represents 0 of 𝑊(0)

𝐿₁(𝑡)={𝑠∈𝐿(𝑡)∣Δ𝑊(𝑠)=Δ𝑊^𝑇𝐿(𝑠)},₂(𝑡)={𝑠∈𝐿(𝑡)∣𝑃^𝑇Δ𝑊^∗𝐿(𝑠)𝑃≥0},₃(𝑡)=.𝑠∈𝐿(𝑡)∣Δ𝑊(𝑠)isweaklynonnegativedefinitefunctionmatrix (2.14)

3. Main Results

Theorem 3.1. Let 𝑁(𝑡)=(𝑊(𝑡),𝜃(𝑡)) be a DHNNWFM.
(1) If 𝑊(𝑡) is a weakly nonnegative definite monotone increasing function matrix and lim_𝑡→∞𝜃(𝑡)→𝜃 , then 𝑁(𝑡) will converge to a stable state in serial mode.
(2) If 𝑊(𝑡) is a strongly nonnegative definite monotone increasing function matrix and lim_𝑡→∞𝜃(𝑡)→𝜃, then 𝑁(𝑡) will converge to a stable state in parallel mode.

Proof. Based on (2.12) and (2.13), we have the following.
Let Δ𝜀_𝑖(𝑡)=max{Δ𝛿_𝑖(𝑡)+Δ𝜃_𝑖(𝑡)∣Δ𝛿_𝑖(𝑡)+Δ𝜃_𝑖(𝑡)<0}, where Δ𝛿_𝑖(𝑡)=^𝑛_𝑗=1Δ𝑤_𝑖𝑗(𝑡)𝑥_𝑗(𝑡),𝑥_𝑗(𝑡)∈{1,−1},𝑗∈𝐼. (3.1)
If ∀𝑥_𝑗∈{1,−1},𝑗∈𝐼,Δ𝛿_𝑖(𝑡)≥0, then Δ𝛿_𝑖(𝑡) is assigned an arbitrary negative number. Suppose Δ𝜃(0)=𝜃(0), 𝜃(𝑡)=∑_{𝑠∈𝐿(𝑡)}Δ𝜃(𝑠), where Δ𝜃(𝑠)=(Δ𝜃₁(𝑠),…,Δ𝜃_𝑛(𝑠)) and Δ𝜃_𝑖(𝑠)=Δ𝜃_𝑖(𝑠)−(Δ𝜀_𝑖(𝑠)/2),𝑖∈𝐼. Then we consider energy function (Lyapunov function) of the DHNNWFM as follows: 1𝐸(𝑡)=−2𝑋^𝑇(𝑡)𝑊(𝑡)𝑋(𝑡)−𝑋^𝑇(𝑡)𝜃(𝑡). (3.2)
Combined with (2.12) and (2.13), we have 1𝐸(𝑡)=−2𝑋^𝑇(𝑡)(_{𝑠∈𝐿(𝑡)}Δ𝑊(𝑠))𝑋(𝑡)−𝑋^𝑇(𝑡)(_{𝑠∈𝐿(𝑡)}1Δ𝜃(𝑠)),𝐸(𝑡+1)=−2𝑋^𝑇(𝑡+1)𝑊(𝑡)𝑋(𝑡+1)−𝑋^𝑇(𝑡+1)𝜃(𝑡)+𝜒(𝑡), (3.3) where 1𝜒(𝑡)=−2𝑋^𝑇(𝑡+1)Δ𝑊(𝑡)𝑋(𝑡+1)−𝑋^𝑇(𝑡+1)Δ𝜃(𝑡). (3.4)𝜒(𝑡)that is the increasing energy for the connected weight matrix increases. So, the change of energy is Δ𝐸(𝑡)=𝐸(𝑡+1)−𝐸(𝑡)=_{𝑠∈𝐿(𝑡)}𝛽_𝑠(𝑡)+𝜒(𝑡), (3.5) where 𝛽_𝑠1(𝑡)=−2𝑋^𝑇(𝑡+1)Δ𝑊(𝑠)𝑋(𝑡+1)−𝑋^𝑇(𝑡+1)+1Δ𝜃(𝑠)2𝑋^𝑇(𝑡)Δ𝑊(𝑠)𝑋(𝑡+1)+𝑋^𝑇(𝑡)Δ𝜃(𝑠). (3.6) According to (3.4) and Δ𝜃_𝑖(𝑡)=Δ𝜃_𝑖(𝑡)−(Δ𝜀_𝑖(𝑡)/2), we obtain 1𝜒(𝑡)=−2𝑋^𝑇(𝑡+1)Δ𝑊(𝑡)𝑋(𝑡+1)+𝑋^𝑇(𝑡+1)Δ𝜀(𝑡)2−𝑋^𝑇1(𝑡+1)Δ𝜃(𝑡)=−2𝑋^𝑇1(𝑡+1)Δ𝑊(𝑡)𝑋(𝑡+1)+2𝑋^𝑇(𝑡+1)Δ𝑊(𝑡)𝑋(𝑡)−𝑋^𝑇(𝑡+1)Δ𝜃(𝑡). (3.7) By Δ𝑋(𝑡)=2𝑋(𝑡+1)=−2𝑋(𝑡), 𝜒(𝑡)=−𝑋^𝑇(𝑡+1)Δ𝑊(𝑡)𝑋(𝑡+1)−𝑋^𝑇(𝑡+1)Δ𝜃(𝑡). (3.8)
Because lim_𝑡→∞𝜃(𝑡)→𝜃 (i.e., lim_𝑡→∞Δ𝜃(𝑡)→0) and Δ𝑊(𝑡) is nonnegative definite, we have 𝜒(𝑡)≤0. (3.9)
(1) Here, 𝑊(𝑡) is a weakly nonnegative definite monotone increasing function matrix, so Δ𝑊(𝑠),𝑠∈𝐿₃(𝑡) is weakly nonnegative definite matrixes. Then, based on [4], when 𝑁(𝑡) is operating in serial mode, we obtain 𝛽_𝑠(𝑡)≤0,𝑠∈𝐿₃(𝑡). Then Δ𝐸(𝑡)≤0. Therefore, 𝑁(𝑡) will converge to a stable state in serial mode.
(2) Here, 𝑊(𝑡) is a strongly nonnegative definite monotone increasing function matrix, so Δ𝑊(𝑠),𝑠∈𝐿₂(𝑡) is strongly nonnegative definite matrixes. According to [4], we know that 𝛽_𝑠(𝑡)≤0 in parallel mode. Then Δ𝐸(𝑡)≤0. Therefore, 𝑁(𝑡) will converge to a stable state in parallel mode. The proof is completed.

Theorem 3.2. Let 𝑁(𝑡)=(𝑊(𝑡),𝜃(𝑡)) be a DHNNWFM.
(1) If there exits an integer constant 𝐾 such that Δ𝑊(𝑠),0≤𝑠≤𝐾 is symmetric or weakly nonnegative definite matrix, Δ𝑊(𝑠),𝐾<𝑠 is weakly nonnegative definite matrix and lim_𝑡→∞𝜃(𝑡)→𝜃, then 𝑁(𝑡) will converge to a stable state in serial mode.
(2) If there exits an integer constant 𝐾 such that Δ𝑊(𝑠),0≤𝑠≤𝐾 is symmetric or strongly nonnegative definite matrix Δ𝑊(𝑠),𝐾<𝑠 is strongly nonnegative definite matrix and lim_𝑡→∞𝜃(𝑡)→𝜃, then 𝑁(𝑡) will converge to a limit cycle of length at most 2 or a stable state in parallel mode.

Proof. We consider energy function (Lyapunov function) of the DHNNWFM as follows: 1𝐸(𝑡)=−2𝑋^𝑇(𝑡)𝑊(𝑡)𝑋(𝑡)−𝑋^𝑇(𝑡)𝜃(𝑡). (3.10)
According to the proof of Theorem 3.1, now we have Δ𝐸(𝑡)=∑_{𝑠∈𝐿(𝑡)}𝛽_𝑠(𝑡)+𝜒(𝑡)
(1) Here, we know 𝐿(𝑡)=𝐿₁(𝑡)∪𝐿₃(𝑡)⋀𝐿₁(𝑡)∩𝐿₃(𝑡)=∅. Then Δ𝐸(𝑡)=_{𝑠∈𝐿1(𝑡)}𝛽_𝑠(𝑡)+_{𝑠∈𝐿3(𝑡)}𝛽_𝑠(𝑡)+𝜒(𝑡). (3.11)
Based on [1, 4], when 𝑁(𝑡) is operating in serial mode, we obtain 𝛽_𝑠(𝑡)≤0,𝑠∈𝐿(𝑡) for Δ𝑊(𝑠),𝑠∈𝐿(𝑡) is symmetric or weakly nonnegative definite matrixes and according to (3.9) we know 𝜒(𝑡)≤0. Then Δ𝐸(𝑡)≤0. Therefore, 𝑁(𝑡) will converge to a stable state in serial mode.
(2) Here, we know 𝐿(𝑡)=𝐿₁(𝑡)∪𝐿₂(𝑡)⋀𝐿₁(𝑡)∩𝐿₂(𝑡)=∅. Then Δ𝐸(𝑡)=_{𝑠∈𝐿1(𝑡)}𝛽_𝑠(𝑡)+_{𝑠∈𝐿2(𝑡)}𝛽_𝑠(𝑡)+𝜒(𝑡). (3.12)
If Δ𝑊(𝑠)(𝑠∈𝐿₁(𝑡)), according to [4] we know that 𝛽_𝑠(𝑡)≤0 if and only if 𝑋(𝑡)≠𝑋(𝑡+1)=𝑋(𝑡−1) or 𝑋(𝑡)=𝑋(𝑡+1)=𝑋(𝑡−1) in parallel mode.
If Δ𝑊(𝑠)(𝑠∈𝐿₂(𝑡)), according to [4] we know that 𝛽_𝑠(𝑡)≤0 in parallel mode.
According to (3.9), we know 𝜒(𝑡)≤0.
Based on the above, we obtain that if 𝑋(𝑡)≠𝑋(𝑡+1)=𝑋(𝑡−1) or 𝑋(𝑡)=𝑋(𝑡+1)=𝑋(𝑡−1), then Δ𝐸(𝑡)≤0. So, 𝑁(𝑡) will converge to a limit cycle of length at most 2 or a stable state in parallel mode. The proof is completed.

Combined with [25], we have the following.

Theorem 3.3. Let 𝑁(𝑡)=(𝑊(𝑡)+𝑈(𝑡)+𝑂(𝑡),𝜃(𝑡)) be a DHNNWFM. If 𝑈(𝑡) is column diagonally dominant monotone increasing function matrix, 𝑂(𝑡) is row diagonally dominant monotone increasing function matrix, and 𝑊(𝑡) is strongly nonnegative definite monotone increasing function matrix, then 𝑁(𝑡) will converge to a stable state in parallel mode.

Proof. Let Δ𝜀_𝑖(𝑡)=max{Δ𝛿_𝑖(𝑡)+Δ𝜃_𝑖(𝑡)∣Δ𝛿_𝑖(𝑡)+Δ𝜃_𝑖(𝑡)<0}, where Δ𝛿_𝑖(𝑡)=^𝑛_𝑗=1Δ𝑤_𝑖𝑗(𝑡)𝑥_𝑗(𝑡),𝑥_𝑗(𝑡)∈{1,−1},𝑗∈𝐼. (3.13)
If ∀𝑥_𝑗∈{1,−1},𝑗∈𝐼,Δ𝛿_𝑖(𝑡)≥0, then Δ𝛿_𝑖(𝑡) is assigned an arbitrary negative number. Suppose Δ𝜃(0)=𝜃(0), 𝜃(𝑡)=∑_{𝑠∈𝐿(𝑡)}Δ𝜃(𝑠) where Δ𝜃(𝑠)=(Δ𝜃₁(𝑠),…,Δ𝜃_𝑛(𝑠)) and Δ𝜃_𝑖(𝑠)=Δ𝜃_𝑖(𝑠)−(Δ𝜀_𝑖(𝑠)/2),i∈I. Then we consider energy function (Lyapunov function) of the DHNNWFM as follows: 1𝐸(𝑡)=−2𝑋^𝑇(𝑡)𝑊(𝑡)𝑋(𝑡)−𝑋^𝑇(𝑡)𝑈(𝑡)𝑋(𝑡)−𝑋^𝑇(𝑡)𝑂(𝑡)𝑋(𝑡)−𝑋^𝑇(𝑡)𝜃(𝑡), (3.14) where 𝑊(𝑡)=∑_{𝑠∈𝐿(𝑡)}Δ𝑊(𝑠), 𝑈(𝑡)=∑_{𝑠∈𝐿(𝑡)}Δ𝑈(𝑠), 𝑂(𝑡)=∑_{𝑠∈𝐿(𝑡)}Δ𝑂(𝑠), and 𝜃(𝑡)=∑_{𝑠∈𝐿(𝑡)}Δ𝜃(𝑠).
We have that the change of energy is 1Δ𝐸(𝑡)=𝐸(𝑡+1)−𝐸(𝑡)Δ𝐸(𝑡)=2Δ𝑋^𝑇(𝑡)𝑊(𝑡)𝑋^𝑇1(𝑡)−2𝑋^𝑇(𝑡)𝑊(𝑡)Δ𝑋^𝑇−1(𝑡)2Δ𝑋^𝑇(𝑡)𝑊(𝑡)Δ𝑋^𝑇(𝑡)−Δ𝑋^𝑇(𝑡)((𝑊(𝑡)+𝑈(𝑡)+𝑂(𝑡))𝑋^𝑇(𝑡)+𝜃)−𝑋^𝑇1(𝑡+1)(𝑈(𝑡)+𝑂(𝑡))Δ𝑋(𝑡)−2𝑋^𝑇(𝑡+1)Δ𝑊(𝑡)𝑋(𝑡+1)−𝑋^𝑇(𝑡+1)Δ𝑈(𝑡)𝑋(𝑡+1)−𝑋^𝑇(𝑡+1)Δ𝑂(𝑡)𝑋(𝑡+1)−𝑋^𝑇(𝑡+1)Δ𝜃(𝑡)=−𝜑(𝑡)−𝜂(𝑡)−𝜅(𝑡)−𝛾(𝑡), (3.15)(3.16) where 1𝜑(𝑡)=−2Δ𝑋^𝑇(𝑡)𝑊(𝑡)𝑋^𝑇1(𝑡)+2𝑋^𝑇(𝑡)𝑊(𝑡)Δ𝑋^𝑇1(𝑡)+2Δ𝑋^𝑇(𝑡)𝑊(𝑡)Δ𝑋^𝑇(𝑡),𝜂(𝑡)=Δ𝑋^𝑇(𝑡)((𝑊(𝑡)+𝑈(𝑡)+𝑂(𝑡))𝑋^𝑇(𝑡)+𝜃),𝜅(𝑡)=𝑋^𝑇1(𝑡+1)(𝑈(𝑡)+𝑂(𝑡))Δ𝑋(𝑡),𝛾(𝑡)=2𝑋^𝑇(𝑡+1)Δ𝑊(𝑡)𝑋(𝑡+1)+𝑋^𝑇(𝑡+1)Δ𝑈(𝑡)𝑋(𝑡+1)+𝑋^𝑇(𝑡+1)Δ𝑂(𝑡)𝑋(𝑡+1)+𝑋^𝑇(𝑡+1)Δ𝜃(𝑡). (3.17)
Based on [4, Theorem  2], we obtain 𝜑(t)≥0.
Obviously, when Δ𝑋(𝑡)≠0, Δ𝑋(𝑡)=2𝑋(𝑡+1)=−2𝑋(𝑡), 𝜂(𝑡)≥0.
When it is operating in parallel mode, let 𝐼₁(𝑡)={𝑖∈𝐼∣Δ𝑥_𝑖(𝑡)≠0},𝐼₂(𝑡)=𝐼⧵𝐼₁(𝑡).
According to the property of column (or row) diagonally dominant matrix, we have 𝜅(𝑡)=2𝑋^𝑇(𝑡+1)𝑈(𝑡)𝑋(𝑡+1)+2𝑋^𝑇(𝑡+1)𝑂(𝑡)𝑋(𝑡+1)≥_{𝑖∈𝐼1(𝑡)}(𝑢_𝑖𝑖(𝑡)−_𝑗≠𝑖|𝑢_𝑗𝑖(𝑡)|)+_{𝑖∈𝐼1(𝑡)}(𝑜_𝑖𝑖(𝑡)−_𝑗≠𝑖|𝑜_𝑖𝑗(𝑡)|)≥0. (3.18)
According to Definition 2.2 and (3.9), we know 𝛾(𝑡)≥0.
Based on the above, we have Δ𝐸(𝑡)≤0. So, 𝑁(𝑡) will converge to a stable state in parallel mode. The proof is completed.

4. Examples

Example 4.1. Let 𝑁(𝑡)=(𝑊(𝑡),𝜃(𝑡)) be a DHNNWFM, where 𝑊(𝑡)=[^{2𝑡−𝑡𝑡𝑡3𝑡−2𝑡}_{𝑡2𝑡3𝑡}] and 𝜃(𝑡)=0. 𝑁(𝑡) will converge to a stable state in parallel mode.

Example 4.2. Let N(t)=(W(t),𝜃(t)) be a DHNNWFM, where 𝑊(𝑡)=[^{2𝑡−6𝑡𝑡𝑡𝑡2−2𝑡}_{8𝑡1/22𝑡𝑡2}] and 𝜃(𝑡)=0. 𝑁(𝑡) will converge to a stable state in parallel mode.

Example 4.3. Let 𝑁(𝑡)=(𝑊(𝑡)+𝑈(𝑡)+𝑂(𝑡),𝜃(𝑡)) be a DHNNWFM, where 𝑊(𝑡)=[^{2𝑡−𝑡𝑡𝑡3𝑡−2𝑡}_{𝑡2𝑡3𝑡}], 𝑂(𝑡)=0, 𝑈(𝑡)=[^{3𝑡3𝑡𝑡2𝑡5𝑡−2𝑡}_{−𝑡𝑡3𝑡}] and 𝜃(𝑡)=0. 𝑁(𝑡) will converge to a stable state in parallel mode.

5. Conclusion

In this paper, we firstly introduce the DHNNWFM. Then we mainly discuss stability of DHNNWFM that WFM is a symmetric or nonnegative definite, or column (or row) diagonally dominant function matrix. This work widens the DHNN model. And we obtain some important results, which supply some theoretical principles to the application. DHNNWFM has many interesting phenomena. We will continue for the theoretic and the practical research about DHNNFWM.