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Synchronization of bidirection multiple neural networks with impulsive coupling control
Advances in Difference Equations volume 2020, Article number: 332 (2020)
Abstract
This paper is devoted to the study of the global synchronization of multiple neural networks (MNNs) via impulsive coupling control. Compared to onelayer neural networks, MNNs have more complex dynamics behavior and greater practical significance. In this paper, two interconnected bidirection MNNs are constructed; for this type of neural network system, each node in their network will be affected by the corresponding node in another network, then the synchronization problem of unidirection MNNs is extended to bidirection MNNs. In order to achieve synchronization of the bidirection MNNs, we adopt the method of impulsive coupling control, and together with the appropriate Lyapunov function, we deduce the relationship of the synchronization error system at the adjacent two impulsive moments, through iteration, and we finally obtain sufficient conditions to realize the global synchronization of the two interconnected bidirection MNNs. We illustrate the effectiveness of the results by simulation.
Introduction
As an important subject, neural networks (NNs) have attracted more and more attention of scholars. Because of their superior performance, they have been widely used in associative memory, signal processing, optimization, machine learning and so on. Common dynamic behaviors just like periodicity, synchronization, chaos, multistability, dissipation, stability, selfassociation and other dynamic behaviors have been widely studied, and new results are still emerging (see [1–3]).
As an important dynamic behavior in neural network, synchronization is of great significance [4]. At present, there are many excellent research results in neural network synchronization. Ramirez et al. [5] study master–slave synchronization, and propose potential solutions to the problem that master–slave synchronization control scheme cannot induce synchronization. In Ref. [6], a new impulsive pinning strategy involving pinning ratio has been proposed to study pinning synchronization under impulsive control. In Ref. [7], Fečkan et al. study the synchronization of a nonlinear fractional system, and analyze the time response and chaotic behaviors of this system. Alsaedi et al. [8] have studied the finitetime synchronization of a class of complex dynamical networks with Markovian hopping and timevarying delay. Generalized multisynchronization is studied in Ref. [9]. CruzAncona et al. [10] study the problem of generalized multisynchronization in a master–slave topology in the framework of differential algebra and design a control protocol that can solve the problem of leadership following consensus. A class of samplingdata synchronization is given in Ref. [11], which is through the method of samplingdata synchronization control to make the system with timevarying sampling period achieve synchronization, the sampling period in [11] is timevarying, switching between two values according to random probability, it is a synchronous way which is defined by the characteristics of the controller.
The dynamic behavior of multiple neural networks (MNNs), which is composed of multiple subnetworks, has been more and more concerned in the past 20 years, and the common examples are the robot group [12], the unmanned aerial vehicle group, and the migration of fish stocks. The synchronization behavior of MNNs is embodied as a common behavior for each subnetwork with different initial state. In the multinetwork synchronization scheme, the coupling between each subnetwork has great effects, resulting in synchronization under various coupling schemes, such as linear coupling. Lu and Chen [13] use variational methods to study the local synchronization of linear coupled NNs, which is also an earlier study of coupled networks. Liu and Liao [14] study a class of finitetime drivenresponse synchronization problems for a general delay memory neural network (DMNN) with interval parameters, fuzzy logic system models, and nonlinear coupling, and they present a unified condition for finitetime synchronization of a fuzzy DMNN with nonlinear coupling. As for fixed topology coupling, Wang and Cao [15] study the exponential synchronization of a class of coupled NNs with probabilistic timevarying delay coupling and timevarying impulsive delay based on random memory. Mathiyalagan et al. [16] study the impulsive synchronization of a class of BAM NNs based on timevarying delay. Li and Cao [17] study the synchronization of inertial memristor system with linear coupling, and use a variety of methods, such as differential inclusion method and matrix measure method, the coupling topology is also fixed. In practical applications, some links are broken, or new links are generated; then the coupling topology is not always fixed, for instance, in limited communication channel, links are usually changed as needed to reduce redundant connections and save resources, so a concept of switching topology is proposed to describe the sudden change of the connection topology of NNs. Because the study of switched topology meets the needs of network change in practical applications, it motivates many scholars. In NNs with switching topology, synchronization is one of the dynamics we have to study. In this regard, Yang et al. [18] study a class of multirecursive NNs with impulsive coupling control and give sufficient conditions for global synchronization in which the coupling topology is variable and introduce the concepts of sequential connection and common connection to describe the switching mode of coupling topology.
In many studies, the coupling scheme has always been effective, but we may only need coupling in some times, and the communication bandwidth is limited, so this kind of coupling scheme causes a lot of unnecessary cost waste. Therefore, if the coupling between NNs only occurs at some discrete time (such as impulsive time), then the cost of the application can be effectively reduced, so it is necessary to study the impulsive coupling control scheme. In [18–21], the impulsive coupling control is used to switch the coupling topology at the impulsive time, so as to achieve better control effect and practical significance. In the nonimpulsive time, each node is independent, and information exchange only occurs at the impulsive time, which greatly reduces the waste of communication resources. There are examples of this in real life, for example, a synchronous flash of fireflies, the flash of each firefly in a population as an impulsive coupled signal, which finally achieves a synchronous flash.
The structure of the NNs is complex and diverse. For bidirectional NNs, as early as 1987, a bidirectional associative memory (BAM) NNs with bilayer is first proposed by Kosko. What is special about this type of neural network is that its ulayer is not independent of the vlayer, but is closely linked through the respective neurons. Because of this particularity, this type of NNs has been used in the fields of artificial intelligence, signal processing, combination optimization and image processing. For this network model, in [22], Cheng et al. investigate the MittagLefflfler synchronization and stability of fractionalorder BAM NNs. Similarly, for this characteristic of the BAM NNs, if we apply it on the MNNs, then we can combine the characteristics of the two networks, which is also very meaningful.
In practice, it is inevitable to study the NNs with timedelay, because of the limited switching speed of the amplifier and the limited information processing speed, thus a large amount of complex dynamical behaviors can be generated. And in the neural network model study, if the type or size of the timedelay is different, then the results may also be completely different. The common delays in currently studies are discrete delay, timevarying delay, statedependent state delay, and infinite timedelay [23].
According to the above discussion, in this paper, we study the synchronization problem of bidirectional MNNs with timedelay under impulsive coupling control. Through the impulsive coupling controller, we finally achieve the synchronization of the two MNNs. The coupling controller in this paper makes the two networks to be coupled only at the discrete time, the coupling topology will switch with time, and it is sequential connection. The two MNNs are not only affected by their respective impulsive coupling controllers, but also by the corresponding nodes of another network. Generally speaking, the main work of this paper can be summed up in the following two points:

(1)
The synchronization problem of two interconnected bidirection MNNs with impulsive coupling control is studied. Because impulsive control is adopted, communication resources can be saved. In addition, the design of impulsive coupling controller is more convenient through coupling intensity selection.

(2)
We give the sufficient conditions of MNNs with timedelay to achieve global synchronization under impulsive coupling control.
The remaining part of this paper is organized as follows. Section 2 gives some preliminaries on graph theory, the system of bidirection MNNs and impulsive coupling controller. Section 3 derives sufficient conditions for global synchronization of the continuoustime MNNs with impulsive interactions. Section 4 gives a numerical example to substantiate the theoretical analysis. Finally, the conclusion of this paper is given in Sect. 5.
Preliminaries
Graph theory
A graph \(\mathtt{G = (V, E)}\) is composed of two sets: V is the set of nodes, and \(\mathtt{E}\subseteq \mathtt{V}\times \mathtt{V} \) is the set of edges. Each graph \(\mathtt{G = (V, E)}\) has a unique nonnegative matrix \(A=(a_{ij})_{N\times N}\in R^{N\times N}\) corresponding to it where \(a_{ij}>0 \) represents the connection relationship between node i and node j, \((i,j)\in \mathtt{V}\times \mathtt{V}\). The union of two graphs \(\mathtt{G}_{1} = (\mathtt{V}, \mathtt{E}_{1})\) and \(\mathtt{G}_{2} = (\mathtt{V}, \mathtt{E}_{2})\) is defined by \(\mathtt{G}_{1}\cup \mathtt{G}_{2} = (\mathtt{V}, \mathtt{E}_{1}\cup \mathtt{E}_{2})\).
Given a graph \(\mathtt{G = (V, E)} \) and a nonempty subset \(\mathtt{N \subseteq V} \), the neighbors of N are defined as the set \(\mathtt{M (N, G)} = \{ {j \in \mathtt{V} \backslash \mathtt{N}  \exists i \in \mathtt{N}},\text{ such that }(i, j) \in \mathtt{E} \} \). If the set N is a singleton, then \(\mathtt{M (N, G)} \) represents the neighbor of one node.
If \(\bigcup^{m}_{i=1}\mathtt{G}_{i} \) contains a spanning tree (a detailed introduction is provided in [19]), then the sequence of graphs \((\mathtt{G}_{i})^{m}_{i=1} \) is jointly connected. If the set of \(\mathtt{V}_{k}\) satisfies \(\mathtt{V}_{k} \subseteq \mathtt{V} (1 \leq k \leq m + 1) \) and \(\mathtt{V}_{k+1} \subseteq \mathtt{V}_{k} \cup \mathtt{N}( \mathtt{G}_{k}, \mathtt{V}_{k} ) \) where \(\mathtt{V}_{1} \) is a singleton, \(\mathtt{V}_{m+1} = \mathtt{V} \), then the sequence of graphs \((\mathtt{G}_{i})^{m}_{i=1} \) is sequentially connected. It is also Tsequential connected with the period \(T=m\). For some more properties of jointly connected and sequentially connected cases refer to [18, 19].
Remark 2.1
Note that each graph of \(\mathtt{G}_{i}\) has the common nodes. In addition, we can see that, if the sequence of graphs \((\mathtt{G}_{i})^{m}_{i=1} \) is Tsequentially connected, then the information spread process of nodes is given by the spanning tree of the sequence of graphs \((\mathtt{G}_{i})^{m}_{i=1} \). At the same time, if a sequence of graphs \((\mathtt{G}_{i})^{m}_{i=1} \) is sequentially connected, then it is also jointly connected, but the converse is not true.
Model description
Let \(\mathtt{V}^{x} = \mathtt{V}^{y} = \{ 1, 2,\ldots, N \} \) be the sets of nodes. In this paper, we study two interconnected bidirection MNNs system and consider each MNNs consisting of N NNs. Each of the NNs corresponds to a node, and the state equations of the ith and the jth NN are given by
or in compact forms
where \(i\in \mathtt{V}^{x} \), \(j\in \mathtt{V}^{y} \), \(t\ge t_{0} \), \(x_{i}(t)=(x_{i1}(t),x_{i2}(t),\ldots,x_{in}(t)) \in R^{n} \) and \(y_{j}(t)=(y_{j1}(t),y_{j2}(t),\ldots, y_{jn}(t)) \in R^{n} \) are the state vectors of the ith NN and jth NN of the two MNNs, respectively. \(C=\operatorname{diag} \{ c_{1},c_{2},\ldots,c_{n} \},c_{i}>0\) and \(\mathfrak{C}=\operatorname{diag} \{ \mathfrak{c}_{1},\mathfrak{c}_{2},\ldots, \mathfrak{c}_{n} \},\mathfrak{c}_{i}>0\) are selfinhibitions of the two neurons. \(I(t) \in R^{n} \), \(\mathfrak{I}(t)\in R^{n} \) represent the input or bias, \(u_{i} \), \(v_{i} \) are control inputs of two neurons, respectively. \(\tau _{1}>0 \), \(\tau _{2}>0 \) are the transmission delays and let \(\tau =\max \{ \tau _{1},\tau _{2} \} \). \(f(y_{j}(t)) = (f_{1}(y_{j1}(t)),f_{2}(y_{j2}(t)),\ldots,f_{n}(y_{jn}(t)))^{T} \), \(g(x_{i}(t))=(g_{1}(x_{i1}(t)),g_{2}(x_{i2}(t)), \ldots,g_{n}(x_{in}(t)))^{T} \) and \(f^{\tau }(y_{j}(t\tau _{1}))=(f^{\tau }_{1}(y_{j1}(t\tau _{1})),f^{ \tau }_{2}(y_{j2}(t\tau _{1})),\ldots,f^{\tau }_{n}(y_{jn}(t\tau _{1})))^{T} \), \(g^{\tau }(x_{i}(t\tau _{2}))=(g^{\tau }_{1}x_{i1}(t\tau _{2}),g^{ \tau }_{2}x_{i2}(t\tau _{2}), \ldots,g^{\tau }_{n}x_{in}(t\tau _{2}))^{T} \) are activation functions, and \(A = [a_{ij} ] _{n \times n}\), \(\mathfrak{A} = [\mathfrak{a}_{ij} ] _{n \times n}\), \(B = [b_{ij}]_{n \times n} \), \(\mathfrak{B} = [\mathfrak{b}_{ij}]_{n \times n} \) are the connection weight matrices and delay connection weight matrices of the two NNs.
Given an impulsive instant sequence \(\{ t_{1},t_{2},t_{3},\ldots \} \) satisfying \(0 < t_{k} < t_{k+1}(k \in Z^{+}) \) and \(t_{k} \to \infty \) when \(k \to \infty \). For the distributed impulsive controllers of the ith and jth nodes, we assign
\(\delta (\cdot )\) is the Dirac impulsive function; \(\varGamma (t_{k}) = \operatorname{diag} \{ \gamma _{1}(t_{k}),\gamma _{2}(t_{k}), \ldots,\gamma _{n}(t_{k}) \} \) and \(\varLambda (t_{k}) = \operatorname{diag} \{ \lambda _{1}(t_{k}),\lambda _{2}(t_{k}), \ldots,\lambda _{n}(t_{k}) \} \) represent the coupling gains between two nodes with \(\gamma _{i}(t_{k})>0 \) and \(\lambda _{i}(t_{k})>0 \). \(D(t_{k})=(d_{ij}(t_{k}))_{N \times N} \) and \(\mathfrak{D}(t_{k})=(\mathfrak{d}_{ji}(t_{k}))_{N \times N} \) represent the impulsive coupling matrices with
In algebraic graph theory, \(\mathtt{G}(D(t_{k} ))\) and \(\mathtt{G}(\mathfrak{D}(t_{k} ))\) are the adjacent matrices of the directed weight graphs \(\mathtt{G}_{k}^{x} \) and \(\mathtt{G}_{k}^{y} \).
Under impulsive controllers (2.5) and (2.6) with the conditions (2.7) and (2.8), then controlled systems (2.1) and (2.2) can be described in the following forms:
where \(x_{i}(t^{}_{k})=x_{i}(t_{k})\), \(y_{j}(t^{}_{k})=y_{j}(t_{k})\).
We use \(D( t_{k} ) \) and \(\mathfrak{D}( t_{k} ) \) to describe the coupling topology of NNs (2.9) and (2.10) at impulsive time \(t_{k}\). The graphs \(\mathtt{G}^{x}_{k}\) and \(\mathtt{G}^{y}_{k}\) correspond to matrices \(D( t_{k} )\) and \(\mathfrak{D}( t_{k} )\), respectively. Coupling matrices \(D( t_{k} ) \) and \(\mathfrak{D}( t_{k} ) \) can switch at impulsive time \(t_{k}\) with period. It should be noted that \(\mathtt{G}^{x}_{k} \) and \(\mathtt{G}^{y}_{k} \) may contain selfloops, which means the conditions \(d_{ii}(t_{k})>0\) and \(\mathfrak{d}_{ii}(t_{k} )>0\) exist.
Remark 2.2
Through (2.7) and (2.8) we can see that the couplings of the two MNNs only occur at the impulsive time \(t_{k}\), and each node of the two MNNs is independent at other times. In addition, there is an interaction between the corresponding nodes of the two MNNs, so that there is also an interaction between the two MNNs. Then how to synchronize the two networks through certain conditions will be a problem we will discuss later.
Remark 2.3
Compared with the controllers used in articles [24–26], the controllers in these three articles are all used in singlelayer network, but the system in this paper is a bidirection MNNs system, for which one needs to consider the coupling between nodes at the impulsive time, so this type of controllers may not be suitable. The controllers used in this paper can be regarded as a generalization of this kind of controller in MNNs.
Note that when each NNs in MNNs (2.9) and (2.10) satisfy
then MNNs (2.9) and (2.10) are globally synchronized.
Let \(\gamma _{\max }=\max_{k \ge 1,i \in N} \{ \gamma _{i}(t_{k} ) \} \), \(\lambda _{\max } = \max_{k \ge 1,i \in N} \{ \lambda _{i}(t_{k} ) \} \) and \(\gamma =\ln (\gamma _{\max })\), \(\lambda =\ln (\lambda _{\max })\) are called coupling strength of \(\mathtt{G}^{x}_{k}\) and \(\mathtt{G}^{y}_{k}\).
For the continuous activation functions \(f_{i}(x) \), \(g_{i}(x) \) and \(f^{\tau }_{i}(x) \), \(g^{\tau }_{i}(x) \), Assumption 1 is made.
Assumption 1
There exist \(p_{i}>0 \), \(p^{\tau }_{i}>0 \) and \(l_{j}>0 \), \(l^{\tau }_{j}>0 \) such that
for any x, \(y \in R \) and \(i=1,2,\ldots,n\), \(j=1,2,\ldots,n \). Denote \(P=\operatorname{diag} \{ p_{1},p_{2},\ldots,p_{n} \} \), \(P^{\tau }=\operatorname{diag} \{ p^{\tau }_{1},p^{\tau }_{2},\ldots,p^{\tau }_{n} \} \) and \(L=\operatorname{diag} \{ l_{1},l_{2},\ldots,l_{n} \} \), \(L^{\tau }=\operatorname{diag} \{ l^{\tau }_{1},l^{\tau }_{2},\ldots,l^{\tau }_{n} \} \).
Definitions and properties
In this paper, we give some concepts of the convex set [19]. Given a set \(M \subseteq R^{n}\), then the convex hull of M is defined as
We define the diameter of M as
Given two sets \(M_{1}, M_{2} \subseteq R^{n}\), define
It is obvious that \(\operatorname{diam}(M_{1} + M_{2}) \leq \operatorname{diam}(M_{1}) + \operatorname{diam}(M_{2})\).
Let \(e_{ij}(t)=x_{i}(t)x_{j}(t),i,j \in \mathtt{V}^{x} \), \(\mathfrak{e}_{ij}(t)=y_{i}(t)y_{j}(t),i,j \in \mathtt{V}^{y} \), The state error between NNs of (2.9) and (2.10) are defined as \(\Vert e_{ij}(t) \Vert \) and \(\Vert \mathfrak{e}_{ij}(t) \Vert \), \(\Vert \cdot \Vert \) represent the 1norm. Besides, let \(\Vert \tilde{e}_{ij}(t) \Vert =\sup_{\theta _{1}\in [ \tau,0]} \Vert e_{ij}(t+\theta _{1}) \Vert \), \(\Vert \tilde{\mathfrak{e}}_{ij}(t) \Vert =\sup_{\theta _{2} \in [\tau,0]} \Vert \mathfrak{e}_{ij}(t+\theta _{2}) \Vert \) and \(h_{k}=t_{k+1}t_{k} \), \(h_{\inf }=\inf_{k\in z^{+}} \{ h_{k} \} \), \(h_{\sup }=\sup_{k\in z^{+}} \{ h_{k} \} \), \(0\leq \tau \leq h_{\mathrm{inf}}\).
Lemma 2.1
For any\(i,j\in \{ 1, 2,\ldots, N \} \)and\(t\in ( t_{k},t_{k+1} ]\), \(k\in Z^{+} \), we have
where\(r \ne 0 \)and satisfies
in which\(c_{\min }=\min_{1\leq i \leq n} \{ c_{i} \} \), \(\mathfrak{c}_{\min }=\min_{1\leq i \leq n} \{ \mathfrak{c}_{i} \} \).
Proof
Let
and \(\mathbb{W}(t)=\max \{ \mathbb{V}(t),\mathfrak{V}(t) \} \).
According to (2.9) and (2.10), when \(t\in (t_{k},t_{k+1})\), we have
Then
Let
Then one has
From (2.13) and (2.15) we have
Let \(\tilde{\mathbb{V}}(t)=\sup_{\theta _{1} \in [\tau,0]} \mathbb{V}(t+\theta _{1})\), from (2.16) and using the Lyapunov–Razumikhin method, we can get
then we can get
hence
Similarly we can get
and
which mean (2.11) and (2.12) are true. □
Remark 2.4
The result of the Lemma 2.1 is important in the subsequent proof, it establishes the relationship of \(\Vert e_{ij}(t) \Vert \) and \(\Vert \mathfrak{e}_{ij}(t) \Vert \) in two adjacent impulsive interval, so we can go through the iteration to derive them up to time \(t=t_{1}\).
Main results
Globally synchronization
Theorem 3.1
Assume that Assumption 1holds and the model satisfies the following conditions:

(1)
There exist two integers\(T\ge 1\)and\(\mathfrak{T}\ge 1\), such that each sequence of graphs\(\{ \mathtt{G}^{x}_{j} \} ^{(m+1)T}_{j=mT+1} \)and\(\{ \mathtt{G}^{y}_{j} \} ^{(m+1)\mathfrak{T}}_{j=m \mathfrak{T}+1} \)are sequentially connected for\(\forall m\in N^{+}\). Furthermore, for any\(i\in \mathtt{V}^{x}_{k+1} \), \(j\in \mathtt{V}^{y}_{k+1} \), there exist\(\alpha _{1}\in (0,1)\)and\(\alpha _{2}\in (0,1)\), satisfying
$$\begin{aligned} &\sum_{j\in \mathtt{V}^{x}_{k}}d_{ij}(t_{k})\ge \alpha _{1} , \end{aligned}$$(3.1)$$\begin{aligned} &\sum_{i\in \mathtt{V}^{y}_{k}}\mathfrak{d}_{ji}(t_{k}) \ge \alpha _{2} . \end{aligned}$$(3.2) 
(2)
\(r\ne 0\)satisfies the inequalities (2.13), (2.14) and
$$\begin{aligned} &\rho ^{T}\exp \{ rq_{m}T\gamma \} \bigl(1(\alpha _{1})^{T} \bigr)< 1 , \end{aligned}$$(3.3)$$\begin{aligned} &\rho ^{\mathfrak{T}}\exp \{ r\mathfrak{q}_{m}\mathfrak{T} \lambda \} \bigl(1(\alpha _{2})^{\mathfrak{T}} \bigr)< 1 , \end{aligned}$$(3.4)where\(q_{m}=t_{(m+1)T+1}t_{mT+1} \), \(\mathfrak{q}_{m}=t_{(m+1)\mathfrak{T}+1}t_{m\mathfrak{T}+1} \)and\(\rho =\max \{ 1,\exp \{ r\tau \} \} \). γandλare the coupling strength of\(\mathtt{G}^{x}\)and\(\mathtt{G}^{y}\), respectively, then MNNs (2.9) and (2.10) are globally synchronized.
Proof
By the condition (1), we can see that the graph sequences \(\{ \mathtt{G}^{x}_{j} \} ^{(m+1)T}_{j=mT+1} \) and \(\{ \mathtt{G}^{y}_{j} \} ^{(m+1)\mathfrak{T}}_{j=m \mathfrak{T}+1} \) are sequentially connected.
First, consider the condition of \(m=0 \), namely, concentrating on the moments of \(t=t_{k}\) with \(k=1,2,\ldots,T \) or \(k=1,2,\ldots,\mathfrak{T} \). Since \(\{ \mathtt{G}^{x}_{k} \} ^{T}_{k=1}\) and \(\{ \mathtt{G}^{y}_{k} \} ^{\mathfrak{T}}_{k=1}\) are sequentially connected, we may suppose that
where \(\mathtt{V}^{x}_{1} \), \(\mathtt{V}^{y}_{1} \) are singletons and \(\mathtt{V}^{x}_{T+1}=\mathtt{V}^{x} \), \(\mathtt{V}^{y}_{\mathfrak{T}+1}=\mathtt{V}^{y} \). For any \(i\in \mathtt{V}^{x}_{k+1} \) and \(j\in \mathtt{V}^{y}_{k+1} \), from (2.9) and (2.10) we have
Let \(\zeta _{i}(t_{k})=\sum_{j\in \mathtt{V}^{x}_{k}}d_{ij}(t_{k})\), \(\eta _{j}(t_{k})=\sum_{i\in \mathtt{V}^{y}_{k}}\mathfrak{d}_{ji}(t_{k})\) and \(\hat{d}_{ij}(t_{k})=\frac{1}{\zeta _{i}(t_{k})}d_{ij}(t_{k})\), \(\hat{\mathfrak{d}}_{ji}(t_{k})=\frac{1}{\eta _{j}(t_{k})} \mathfrak{d}_{ji}(t_{k})\), then we have \(\sum_{j\in \mathtt{V}^{x}_{k}}\hat{d}_{ij}(t_{k})=1\), \(\sum_{i\in \mathtt{V}^{y}_{k}}\hat{\mathfrak{d}}_{ji}(t_{k})=1\) and
Similarly, denote \(\phi _{i}(t_{k})=1\zeta _{i}(t_{k})>0 \), \(\psi _{j}(t_{k})=1\eta _{j}(t_{k})>0 \) and
Obviously
then we have
and
From (3.1), (3.2), we can get \(\zeta _{i}(t_{k})\ge \alpha _{1}\) and \(\eta _{j}(t_{k})\ge \alpha _{2}\), combining (3.5), (3.7), (3.9) and (3.6), (3.8), (3.10) we have
Let
then we get
and we can write (3.5), (3.6) as follows:
Let
and
Then we have
We can also get
Furthermore, let
From (3.11) and (3.12) we can get
According to Lemma 2.1 we have
since \(\max \{ \tau _{1},\tau _{2} \} \leq \tau \leq t_{k+1}t_{k}\). In addition, according to (2.9), (2.10), we have
Combining (3.13), (3.15), (3.17) and by iterations, one obtains
for any \(1\leq v_{1}\leq k\leq T\).
Also combining (3.14), (3.16), (3.18) we can get
for any \(1\leq v_{2}\leq k\leq \mathfrak{T}\).
So, when \(k=T\), \(v_{1}=T1\) and \(k=\mathfrak{T}\), \(v_{2}=\mathfrak{T}1\), from (3.15) and (3.16), we can get
Since \(\mathtt{V}^{x}_{1}\) and \(\mathtt{V}^{y}_{1}\) are singletons, \(\tilde{\Delta }_{1}=0\), \(\tilde{\nabla }_{1}=0\).
Therefore
and
Therefore, (3.19) and (3.21) imply
from (3.20) and (3.22) we can get
It follows from (3.23) and (3.24) that
where
Since \(\mathtt{V}^{x}_{T+1}=\mathtt{V}^{x}\), \(\mathtt{V}^{y}_{\mathfrak{T}+1}=\mathtt{V}^{y}\), we have \(\Delta _{T+1}=E_{T+1}\), \(\nabla _{\mathfrak{T}+1}=R_{\mathfrak{T}+1}\), hence (3.25), (3.26) imply
Now, we get the relationships of \(\tilde{E}_{T+1}\) and \(\tilde{E}_{1}\), \(\tilde{R}_{\mathfrak{T}+1}\) and \(\tilde{R}_{1} \), So when \(m=1\), similar to the analysis of (3.25) and (3.26), we can obtain
By iterations, we can get
Hence, \(\lim_{m\to \infty }\tilde{E}_{mT+1}=0\) and \(\lim_{m\to \infty }\tilde{R}_{mT+1}=0\), which means \(\lim_{t\to \infty }\max \x_{i}(t)x_{j}(t)\=0\), \(i,j\in \mathtt{V}^{x}\) and \(\lim_{t\to \infty }\max \Vert y_{i}(t)y_{j}(t) \Vert =0\), \(i,j\in \mathtt{V}^{y}\) then MNNs (2.9) and (2.10) are in global synchronization. □
Remark 3.1
The NNs models in this paper are slightly distinct from the general NNs model. By observing the models (2.9) and (2.10) we find that the two networks are not only affected by the impulsive coupling control of their own subnetworks, but also each node is affected by the corresponding node of another MNNs. Compared with other network models, this model has more complex dynamic behavior. In addition, impulsive coupling control is used in this paper, which saves communication resources more than continuous coupling.
Remark 3.2
Theorem 3.1 offers a sufficient criterion under which global synchronization of MNNs models (2.9) and (2.10) can be achieved with conditions (1), (2) and Assumption 1. How to make the sufficient conditions more general present challenges for future research.
Remark 3.3
The impulsive sequence in this paper is a set of given random sequences, but in order to achieve global synchronization, not all impulsive instants need to be coupled, so the number of control can be further reduced. In the existing methods, we can use an eventtriggered scheme to further reduce the number of impulsive sequences, but how to design this eventtriggered scheme to work more efficiently is also a direction we will consider in our next work.
Remark 3.4
For proving the synchronization of general networks, we mainly prove the stability of the error system which is composed of a drive system and a response system. However, the synchronization of MNNs requires the trajectory of all nodes to converge to one track. In this paper, we define two convex hulls \(\Delta (t)\) and \(\nabla (t)\), which introduced in Sect. 2.3 (the definition of the convex hull of M), then the diameter of \(\Delta (t)\) and \(\nabla (t)\) means the maximum error between nodes, when the diameter of the two convex hulls tends to zero, the maximum error between the nodes tends to zero, and this means that the two MNNs are synchronized.
Remark 3.5
The main proof ideas of this paper can be summed up as follows: Firstly, we give the relation between the error of impulsive instant \(t^{+}_{k}\) (\(\tilde{e}_{ij}(t^{+}_{k})\)) and the error of nonimpulsive instant \(t\in (t_{k}, t_{k+1} ]\) (\(e_{ij}(t)\)) by Lemma 2.1, then we get the relationship \(\x_{i}(t)x_{j}(t)\\)\(i,j\in \mathtt{V}^{x}\), \(\y_{i}(t)y_{j}(t)\\)\(i,j\in \mathtt{V}^{y}\) between the maximum error at impulsive instant \(t_{k}\) and at \(t\in (t_{k}, t_{k+1} ]\), respectively. In other words, we get the relationships of \(\Delta _{k}\), \(\Delta ^{+}_{k}\) and \(\nabla _{k}\), \(\nabla ^{+}_{k}\). Next, we derive the expansion rates of \(\Delta (t)\) and \(\nabla (t)\) in an impulsive interval \((t_{k}, t_{k+1} ]\), and \(\Delta (t)\), \(\nabla (t)\) will be reduced by impulsive coupling. By conditions (2.15), (2.16), we can see that the decreasing rates of \(\Delta (t)\) and \(\nabla (t)\) are larger than their expansion rates, then the two MNNs are able to achieve synchronization.
Numerical example
In this section, in order to prove the validity of the appeal theorem, we give the following numerical simulation.
Example 4.1
Consider the MNNs (2.9) and (2.10) with \(N=4\)
where \(x_{i}(t)=(x_{i1}(t),x_{i2}(t))^{T}\), \(y_{j}(t)=(y_{j1}(t),y_{j2}(t))^{T}\) are the state vectors. It is assumed that \(f(x)=f^{\tau }(x)=(f_{1}(x_{1}),f_{2}(x_{2}))^{T}\), \(g(y)=g^{\tau }(y)=(g_{1}(y_{1}),g_{2}(y_{2}))^{T}\), and \(f_{i}(x_{i})=(1/2)(x_{i}+1x_{i}1),i=1,2\), \(g_{j}(y_{j})=(1/3)(y_{j}+1y_{j}1),j=1,2\). The delays of \(\tau _{1}\) and \(\tau _{2}\) are set to be 0.1. We give the initial states by \(x_{11}(s)=0.2\), \(x_{12}(s)=0.5\), \(x_{21}(s)=0.1\), \(x_{22}(s)=0.65\), \(x_{31}(s)=0.35\), \(x_{32}(s)=3\), \(x_{41}(s)=0.7\), \(x_{42}(s)=0.6\), \(y_{11}(s)=0.2\), \(y_{12}(s)=0.3\), \(y_{21}(s)=0.4\), \(y_{22}(s)=0.15\), \(y_{31}(s)=0.5\), \(y_{32}(s)=0.8\), \(y_{41}(s)=0.45\), \(y_{42}(s)=0.1\), \(s\in [0.1,0] \).
Taking
from Figs. 1, 2, 3, and 4, we can see that, if we have MNNs (4.1), (4.2) without the impulsive coupling controls, then the synchronizations cannot be obtained.
Next, consider the switching topology shown in Figs. 5 and 6 connected by choosing \(V^{x}_{1}= \{ 1 \} \), \(V^{x}_{2}= \{ 1,2,3 \} \), \(V^{x}_{3}= \{ 1,2,3,4 \} \) and \(V^{y}_{1}= \{ 1 \} \), \(V^{y}_{2}= \{ 1,2,3 \} \), \(V^{x}_{3}= \{ 1,2,3,4 \} \). We can show that \(\{ \mathtt{G}^{x}_{2m+1}, \mathtt{G}^{x}_{2(m+1)} \} \) and \(\{ \mathtt{G}^{y}_{2m+1}, \mathtt{G}^{y}_{2(m+1)} \} \) are sequentially connected (also jointly connected). The matrices corresponding to the graphs are given as
Noting that \(d_{i1}(t_{2m+1})\ge 0.5\), \(\mathfrak{d}_{i1}(t_{2m+1})\ge 0.9\) for any \(i\in \mathtt{V}^{x}_{2}\) and \(i\in \mathtt{V}^{y}_{2}\). Also we have \(\sum_{j\in \mathtt{V}^{x}_{2}}d_{ij}(t_{2(m+1)})=1\), \(\sum_{j\in \mathtt{V}^{y}_{2}}\mathfrak{d}_{ij}(t_{2(m+1)})=1\) for any \(i\in \mathtt{V}^{x}_{3}\), \(i\in \mathtt{V}^{y}_{3}\), then let \(\alpha _{1}=0.5\), \(\alpha _{2}=0.9\). From conditions (2.13) and (2.14) of Lemma 2.1, we solve the following inequality groups by MATLAB:
then we can get \(r<=1.0797 \), choose \(r=6 \), then \(\rho =\mathrm{max} \{ 1,\exp \{ 6\times 0.1 \} \} =1\). Let \(q_{m}=\mathfrak{q}_{m}=0.6\), then the impulsive sequence satisfies \(t_{2(m+1)}t_{2m+1}=0.3\). In order to get the conditions (3.3), (3.4) of Theorem 3.1, then solve the following inequality groups:
we can get \(\gamma >7.7969 \), \(\lambda >10.6544 \), let \(\gamma =7 \), \(\lambda =10 \), then by the definitions \(\gamma _{\max }=\max_{k \ge 1,i \in N} \{ \gamma _{i}(t_{k} ) \} \), \(\lambda _{\max } = \max_{k \ge 1,,i \in N} \{ \lambda _{i}(t_{k} ) \} \) and \(\gamma =\ln (\gamma _{\max })\), \(\lambda =\ln (\lambda _{\max })\), we can get \(\gamma _{\max }=\exp \{ 7 \} \), \(\lambda _{\max }=\exp \{ 10 \} \), then the coupling gains relative to the switching topologies of \(\{ \mathtt{G}^{x} \} \) and \(\{ \mathtt{G}^{y} \} \) are given by
then the conditions (1), (2) of Theorem 3.1 are satisfied, MNNs (4.1) and (4.2) can reach global synchronizations with impulsive coupling controls under switching topologies of Figs. 5 and 6. The state trajectories of all MNNs are illustrated in Figs. 7 and 8. We can see that the two MNNs achieve synchronizations finally. Moreover, the max state errors between four NNs of (4.1) and (4.2) are defined as
Then \(\Delta (t)\to 0\) and \(\nabla (t)\to 0\) as \(t\to \infty \), which are shown in Figs. 9 and 10.
Conclusion
This paper discusses the global synchronizations of the two interconnected bidirection MNNs with timedelay via an impulsive coupling controller in which the coupling topology is variable. After satisfying certain conditions, the two MNNs which affect each other and achieve synchronizations under the action of impulsive coupling. Compared with continuous coupling, impulsive coupling can reduce communication bandwidth. In addition, we give the principle that the model can achieve synchronization under the impulsive coupling controller. Each of the two networks in this paper is not only affected by impulsive coupling, but also affected by the corresponding nodes in another network. The present studies on this kind of problem are few, so the research in this paper has some significance. At the same time, as described in Remark 3.3, the communication data can be further reduced by designing a suitable eventtriggered scheme to determine the impulse sequence, this will be considered in the future.
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Wang, Y., Wu, A. & Li, B. Synchronization of bidirection multiple neural networks with impulsive coupling control. Adv Differ Equ 2020, 332 (2020). https://doi.org/10.1186/s13662020027966
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Keywords
 Multiple neural networks
 Impulsive coupling control
 Synchronization
 Coupling strength
 Variable topology