Dynamical Analysis and Synchronization of Complex Network Dynamic Systems under Continuous-Time

Abstract

In multilayer complex networks, the uncertainty in node states leads to intricate behaviors. It is, therefore, of great importance to be able to estimate the states of target nodes in these systems, both for theoretical advancements and practical applications. This paper introduces a state observer-based approach for the state estimation of such networks, focusing specifically on a class of complex dynamic networks with nodes that correspond one-to-one. Initially, a chaotic system is employed to model the dynamics of each node and highlight the essential state components for analysis and derivation. A network state observer is then constructed using a unique diagonal matrix, which underpins the driver and response-layer networks. By integrating control theory and stability function analysis, the effectiveness of the observer in achieving synchronization between complex dynamic networks and target systems is confirmed. Additionally, the efficacy and precision of the proposed method are validated through simulation.

1. Introduction

In the real world, complex dynamic networks are omnipresent and ubiquitous. Complex networks can be considered as dynamical systems in which connections between nodes represent interactions between nodes. The study of complex network dynamics helps to explain various real-world problems and increase the understanding of dynamical properties such as network stability, consistency, optimization and synchronization. Many natural or man-made systems can be modeled and described by complex dynamic networks, such as the World Wide Web, genetic networks [1,2], power networks [3,4,5], social communication networks [6,7,8], transportation networks [9,10,11,12], and so on. In addition, complex network topologies can be studied to solve problems in real networks, such as large-scale power outages in power networks and virus propagation in networks.

Euler adeptly resolved the “Königsberg Seven Bridges Problem” using foundational graph theory [13]. Erdős and Rényi collaboratively developed random graph theory, laying the groundwork for systematic studies in complex network theory [14]. Watts and Strogatz discovered the small world of networks [15], and Barabási and Albert revealed the scale-free characteristics and small-world properties of complex dynamic networks through the scale-free nature of networks [16]. These mathematical models provided a robust theoretical foundation that inspired increasing scholarly interest in complex dynamic networks. Targui et al. proposed an observer based on the hydrodynamic model to estimate the state of each node in the communication network [17]. Fan studies the topology identification of complex dynamic networks through systems with chaotic nodes [18]. Wu studies the state estimation problem of discrete-time general complex dynamic networks with packet loss in the transmission channel between the original network and the observation network [19]. Alsaadi et al. also discussed the problem of network state estimation on single-layer complex networks [20,21,22,23]. Lü summarizes the link prediction in complex networks [24]. Yu analyzed the complex network of China’s innovation and development [25]. Laumann discusses the complex network of climate sustainable development [26]. These studies highlight the ubiquity and importance of complex networks in nature and human society, not only advancing theoretical progress in mathematics and physics, but also having a profound impact on practical applications in fields such as engineering, economics, social sciences, and environmental science.

In complex dynamic networks, the phenomenon of synchronization is a striking collective behavior. However, since most networks cannot achieve synchronization spontaneously, control is often considered an effective method to realize network synchronization. Consequently, this topic has attracted considerable attention from scholars, encompassing various types such as complete synchronization [27], cluster synchronization [28], exponential synchronization [29,30], finite-time synchronization [31,32,33], and other synchronization schemes [34,35]. Soudbakhsh studies the problem of wide-area power system control using synchronization measurement in the presence of network delay [36]. Yu studies the long-term prediction of continuous-time dynamic graph networks [37]. Wu gives new criteria for the boundedness and asymptotic stability of neural networks under impulsive control on time scales [38]. Li studies the model following adaptive control problem for a class of time-varying complex dynamic networks [39]. Chen focuses on a new observer-based synchronization problem of discrete-time complex networks with Markov switching topology, and develops a non-homogeneous observer with asynchronous topology and gain to estimate the state of nodes [40]. However, these studies typically assume that complex networks are single-layer, neglecting the interactions between multiple networks that occur in real-world scenarios. For example, in social networks, daily communications among individuals occur through various channels, including telephone and email. Transportation networks consist of trains, cars, and airplanes. Given the limitations of single-layer network models, which fail to adequately describe many practical situations, it is essential to explore multilayer complex networks.

The existing synchronization research literature primarily focuses on achieving uniformity across all state components of network nodes [41,42]. However, practical network applications often require synchronization of only certain state components. In networks composed of high-dimensional nodes, attention may only be necessary for the dynamics of specific state components in certain applications. Therefore, studying the synchronization of partial state components is not only more cost effective but also economically feasible, especially in networks with varying dynamics, such as different parameters or structures. For instance, Motter explored the dynamics of nonlinear networks [43], Li studied state estimation in multilayer complex dynamic networks under time-varying delay conditions [44], and Wu et al. used observable state information from certain nodes in multilayer networks to estimate the states of target nodes in other layers [45]. Berner discussed the spectra of multilayer complex dynamic networks and the correlations between the eigenvalues of different layers [46]. Liu examined state changes in each layer of a dual-layer complex network [47]. Currently, theoretical research on the synchronization of partial state components primarily focuses on achieving synchronization over an infinite duration. However, in practical applications, there is a preference for achieving synchronization within a finite time to save time and reduce costs; yet, studies on finite-time partial state component synchronization in complex dynamic networks are still lacking.

This paper primarily examines the continuous-time state component synchronization issue of a specific type of complex dynamic network with one-to-one corresponding nodes. The key contributions of this paper are outlined as follows: In order to resolve the issue of target-layer state estimation in multilayer complex dynamic networks, an adaptive observer is designed, and the observability is studied by utilizing functional analysis. The structure of this paper is as follows: In Section 2, a mathematical model of a complex network dynamic system is established, and the stability of complete synchronization is analyzed. In Section 3, the state estimation problem of a two-layer complex dynamic network with one-to-one corresponding nodes is studied. The original network is regarded as the driving layer to construct the response-layer network. The topology identification of the network is realized by combining the Lyapunov stability principle. In Section 4, the efficacy of the method is validated through numerical simulations. Finally, the conclusions are presented in Section 5.

2. Construction of Mathematical Model of Complex Network Dynamic System

2.1. Problem Description

A network is considered, comprising N chaotic oscillator rings with an M-layer topology and N identical nonlinear systems, which are considered nodes. The state equation of the first node can be expressed as

$${\dot{X}}_i\left(t\right)=f\left(X_i\left(t\right)\right)+c\sum _{j=1}^N{a}_{ij}\mathsf{\Gamma }(X_j\left(t\right)-X_i\left(t\right))$$

(1)

where $X_i\left(t\right)={(x_i^1,x_i^2,\cdots ,x_i^n)}^T\in R^n$ is the state variable of the i-th node; $f\left(X_i\left(t\right)\right):R^n\to R^n$ is a mapping of nonlinear functions, representing the dynamic behavior of each node; the constant c is the coupling strength of the state variable of the i-th node of the network, that is, the coupling strength $c>0$ of the network, and $\mathrm{A}={\left(a_{ij}\right)}_{N\times N}$ represents the topological structure of the network. The topological structure of the network is shown by $\mathrm{A}$, which meets the dissipative coupling condition ${\displaystyle \sum _{j=1}^N}{a}_{ij}=0,i=1,\cdots ,N$, and is called the external coupling matrix $a_{ij}$. It meets the following conditions:

$\mathrm{A}$ satisfies the dissipative coupling condition ${\displaystyle \sum _{j=1}^N}{a}_{ij}=0,i=1,\cdots ,N$, $\mathrm{A}$ is called external coupling $a_{ij}$, which satisfies the following conditions:

when the external coupling $\mathrm{A}$ is a simple topological structure with no right and no direction, if there is a connected edge between node i and node j, then $a_{ij}=a_{ji}=1$, otherwise $a_{ij}=a_{ji}=1$, the diagonal element is $a_{ii}=-{\displaystyle \sum _{j=1,j\ne i}^N}{a}_{ij}=-{\displaystyle \sum _{j=1,j\ne i}^N}{a}_{ji}=-k_i,i=1,2,\cdots ,N$, where $k_i$ is the degree of node i; $\mathsf{\Gamma }$, the coupling matrix, links the internal state variables of each node and is known as the output function of each node.

2.2. Stability Analysis of Complete Synchronization of Complex Dynamic Networks

For the network Equation (1), can express the dynamic equation of the network’s synchronous target trajectory as follows:

$$\dot{S}\left(t\right)=f\left(S\left(t\right)\right)$$

(2)

In the dynamic network Equation (1), the coupling $\mathrm{A}$ has a single zero eigenvalue with multiplicity one. The corresponding eigenvector represents the invariant manifold of the system, which is given by the vector ${(1,1,1,\cdots ,1)}^T$. When $t\to \infty$, $X_1\left(t\right)\to X_2\left(t\right)\to \cdots \to X_N\left(t\right)\to S\left(t\right)$, that is,

$$\underset{t\to \infty }{lim}{X}_i\left(t\right)=S\left(t\right)$$

(3)

Due to the dissipative coupling condition, the synchronized state vector $S\left(t\right)\in R^m$ must be the outcome of a solitary node, satisfying $\dot{S}\left(t\right)=f\left(S\left(t\right)\right)$. $S\left(t\right)$ may signify the equilibrium point of a single node, periodic orbits, or chaotic orbits.

For discrete cases, the discrete-time dissipative coupled dynamic network corresponding to Equation (1) is as follows:

$${\dot{X}}_i^{(n+1)}\left(t\right)=f\left(X_i^{\left(n\right)}\left(t\right)\right)+c\sum _{j=1}^N{a}_{ij}\mathsf{\Gamma }{X}_j^n\left(t\right)$$

(4)

where $X_j^n\left(t\right)$ represents the state of the j-th node at the n-th time.

Similarly, if in the dynamic network model (4) there exists a condition where $n\to \infty$ implies $\underset{n\to \infty }{lim}{X}_i^n\left(t\right)=S\left(t\right)(i=1,2,\cdots ,N)$, then the dynamic network is said to have achieved complete synchronization.

By analyzing the linear stability of Equation (1) of a continuous-time dynamic network with respect to the synchronous state $S\left(t\right)$, and linearizing the state Equation (1) with respect to the synchronous state $S\left(t\right)$, where ${\xi }_i$ is the state component of the i-th node, we can obtain

$${\dot{\xi }}_i=Df\left(S\right){\xi }_i+c\sum _{j=1}^N{a}_{ij}D\mathsf{\Gamma }\left(S\right){\xi }_j$$

(5)

where $Df\left(S\right)$ and $D\mathsf{\Gamma }\left(S\right)$ are the Jacobi matrices of $f\left(S\right)$ and $\mathsf{\Gamma }\left(S\right)$ with respect to S, respectively, and then, $\xi =[{\xi }_1,{\xi }_2,\cdots ,{\xi }_N]$ is given; then, the above Equation (1) can be written in matrix form:

$$\dot{\xi }=Df\left(S\right)\xi +cD\mathsf{\Gamma }\left(S\right)\xi {\mathrm{A}}^T$$

(6)

Let ${\mathrm{A}}^T=P\mathsf{\Lambda }{P}^{-1}$, perform the Jacobi–Jordan factorization of $\mathrm{A}$, assuming that $\mathsf{\Lambda }$ is a diagonal matrix, then $\mathsf{\Lambda }=\mathrm{diag}({\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_N)$, where ${\lambda }_i(i=1,2,\cdots ,N)$ is the eigenvalue of $\mathrm{A}$ and ${\lambda }_1=0$. Then, making $\eta =[{\eta }_1,{\eta }_2,\cdots ,{\eta }_N]=\xi P$, gives

$$\dot{\eta }=Df\left(S\right)\eta +cD\mathsf{\Gamma }\left(S\right)\eta \mathsf{\Lambda }$$

(7)

In other words, the above formula can be equivalent to

$${\dot{\eta }}_k=[Df\left(S\right)+c{\lambda }_{i}D\mathsf{\Gamma }\left(S\right)]{\eta }_i$$

(8)

where $i=2,3,\cdots ,N$.

Secondly, the Lyapunov exponent method judges the stability of a synchronous manifold. In Equation (8), only ${\lambda }_i$ is connected to ${\eta }_i$ and i. If the external coupling matrix $\mathrm{A}$ is not symmetric, its eigenvalue might be complex, which leads to the main stability equation:

$$\dot{Y}=[Df\left(S\right)+c(\alpha +i\beta )D\mathsf{\Gamma }\left(S\right)]Y$$

(9)

The principal stability function of a dynamic network is the maximum Lyapunov exponent ${\gamma }_{max}$, which is a function of the real variables $\alpha$ and $\beta$. By identifying the fixed point $c{\lambda }_i$ on the complex plane where $(\alpha ,\beta )$ is located, we can determine the stability of the network, represented by the positive and negative values of the maximum Lyapunov exponent at this point. The coupling strength c plays a crucial role in this regard.

By identifying the fixed point $c{\lambda }_i$ on the complex plane where $(\alpha ,\beta )$ is located, we can determine the stability of the network, represented by the positive and negative values of the maximum Lyapunov exponent at this point.

When the maximum Lyapunov exponent corresponding to this point is positive, ${\gamma }_{max}>0$, the characteristic state is deemed unstable.

When the maximum Lyapunov exponent corresponding to this point is positive, ${\gamma }_{max}>0$, the characteristic state is deemed unstable.

Conversely, when the maximum Lyapunov exponent corresponding to this point is negative, ${\gamma }_{max}<0$, the characteristic state is considered stable. If all the Lyapunov exponents corresponding to the eigenvalues ${\lambda }_i(i=1,2,\cdots ,N)$ are negative, the entire network’s synchronization manifold becomes asymptotically stable under the coupling strength c.

When the network has no power and no connection, the eigenvalues of the external coupling matrix $\mathrm{A}$ are all real numbers, i.e., $0={\lambda }_1>{\lambda }_2\ge \cdots \ge {\lambda }_N$, and the main stability equation is

$$\dot{Y}=[Df\left(S\right)+c\alpha D\mathsf{\Gamma }\left(S\right)]Y$$

(10)

So, the negative value range G of the main stability function is set by the dynamic function f, the coupling strength c, the outer coupling matrix $\mathrm{A}$, and the inner coupling function $\mathsf{\Gamma }$ on the isolated node. This range is also known as the dynamic network’s synchronization region. If the product of the coupling strength c and each negative eigenvalue of the outer coupling matrix $\mathrm{A}$ is within the synchronization region, that is,

$$c{\lambda }_i\in G$$

(11)

then, the synchronous manifold of the network is asymptotically stable.

3. State Estimation of Dynamic Systems with Multilayer Complex Networks

According to Equation (1), we can obtain the following dynamic equation for a multilayer complex network dynamic model with the same number of nodes in each layer and one-to-one correspondence between layers:

$$\begin{array}{c}{\dot{x}}_i^K=f\left(x_i^K\right)+c_K{\displaystyle \sum _{j=1}^N}{d}_{ij}^K\mathsf{\Gamma }{x}_j^K+{\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{x}_i^R\hfill \\ y_i^K=H{x}_i^K\hfill \end{array}$$

(12)

where $1\le i\le R$; $x_i^K=[x_{i1}^K,x_{i2}^K,\cdots ,x_{in}^K]\in R^N$ represents the state variable of the i-th node in the $K(1\le K\le M)$-th layer in the M-layer network, $y_i^K\in R$ represents the state variable of the i-th node in the K-th layer, and $f:R^n\to R^n$ is the dynamic equation of the node; $c_K$ is the intra-layer coupling strength of the K-layer network; $D^K=\left(d_{ij}^K\right)\in R^{N\times N}$ is the intra-layer coupling matrix of the K-layer network. If there is a connecting edge from node i to node j, then $d_{ij}^K=1$, otherwise $d_{ij}^K=0$, and $d_{ij}^K=-{\displaystyle \sum _{j=1,j\ne i}^N}{d}_{ij}^K$; the coupling strength between the i-th node of the K-th layer network and the i-th node of the R$(1\le R\le M)$-layer network is denoted by $d_i^{KR}$, and $d_i^{KK}=-{\displaystyle \sum _{R=1,R\ne K}^N}{d}_i^{KR}$; $\mathsf{\Gamma }$ is the inline matrix within and between layers of network nodes, which is a unit matrix; $H=[h_1,h_2,\cdots ,h_n]$ is the output matrix of nodes.

When the network layer’s topology is unknown or uncertain, we can identify it by constructing the driving response. In this paper, we use the original network’s output variables as state observers at the driving end, and construct state observers at the response end to approximate the multilayer complex dynamic network. The state observers at the response end are constructed as follows:

$$\begin{array}{c}{\widehat{x}}_i^K=f\left({\widehat{x}}_i^K\right)+c_K{\displaystyle \sum _{j=1}^N}{d}_{ij}^K\mathsf{\Gamma }{\widehat{x}}_j^K+{\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{\widehat{x}}_i^R+{\sigma }_{Ki}(y_i^K-{\widehat{y}}_i^K)\hfill \\ {\widehat{y}}_i^K=H{\widehat{x}}_i^K\hfill \end{array}$$

(13)

In the formula, $1\le i\le R$; ${\widehat{x}}_i^K=[{\widehat{x}}_{i1}^K,{\widehat{x}}_{i2}^K,\cdots ,{\widehat{x}}_{in}^K]\in R^N$ represents the state observation value of the i-th node of the K-th layer; ${\widehat{y}}_i^K\in R$ represents the output observation value of the i-th node of the K-th layer network; and ${\sigma }_{Ki}$ is the adaptive parameter.

Making the state observation error $e_i^K={\widehat{x}}_i^K-x_i^K$, the state error of the system can be obtained from the driving-layer network (Equation (12)) and the response-layer network (Equation (13)) as follows:

$$\begin{array}{c}{\dot{e}}_i^K={\widehat{x}}_i^K-{\dot{x}}_i^K\hfill \\ =f\left({\widehat{x}}_i^K\right)-f\left({\dot{x}}_i^K\right)+c_K{\displaystyle \sum _{j=1}^N}{d}_{ij}^K\mathsf{\Gamma }{\widehat{x}}_j^K-c_K{\displaystyle \sum _{j=1}^N}{d}_{ij}^K\mathsf{\Gamma }{x}_j^K+{\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{\widehat{x}}_i^R-{\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{x}_i^R+{\sigma }_{Ki}(y_i^K-{\widehat{y}}_i^K)\hfill \\ =f\left({\widehat{x}}_i^K\right)-f\left({\dot{x}}_i^K\right)+c_K{\displaystyle \sum _{j=1}^N}{d}_{ij}^K\mathsf{\Gamma }{e}_j^K+{\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{e}_i^R+{\sigma }_{Ki}H{e}_i^K\hfill \end{array}$$

(14)

It can be observed that the state observer requires information solely about nodes and network trajectories, and not about other nodes in the network. Consequently, the continuous-time state component synchronous controller proposed in this paper exhibits the advantages of low dimensionality, low cost, and easy implementation. It is worth noting that the finite-time synchronization controller proposed in the literature [10,22,23,24,25,26] not only requires information about its own node and reference trajectory, but also requires information about other nodes in the network. This differs from the state observation designed in this paper, which does not require information about other nodes.

This paper presents an effective state observer designed to determine the adaptive parameter ${\sigma }_{Ki}$. Upon synchronizing the complex networks of the driving and response layers, the state error of the network converges to 0, allowing the corresponding layer to identify the driving-layer network. The following assumptions and lemmas are provided for the subsequent derivation and proof.

Suppose 1.

$f\left(x\right)$ is Lipschitz continuous, that is, there is a positive constant $\alpha$ so that for any vector $x\in R^N$, $y\in R^N$, $∥f\left(x\right)-f\left(y\right)∥\le \alpha ∥x-y∥$ holds. Where $∥•∥$ is the European norm.

Lemma 1.

For any vector $x\in R^N$, $y\in R^N$, the following inequalities hold:

$$2x^{T}y\le x^{T}x+y^{T}y$$

(15)

Lemma 2.

Schur’s theorem: For a given symmetric $S=\left(\begin{array}{cc}{S}_{11}& S_{12}\\ S_{21}& S_{22}\end{array}\right)$, where $S_{11}=S_{11}^T$, $S_{12}=S_{21}^T$, $S_{22}=S_{22}^T$, then $S<0$ is equivalent to

$$S_{11}<0,S_{22}-S_{21}{S}_{11}^{-1}{S}_{12}<0$$

$$S_{22}<0,S_{11}-S_{12}{S}_{22}^{-1}{S}_{21}<0$$

Theorem 1.

If hypothesis 1 holds, given an arbitrary constant $k_i(k_i>0)$, the following equation holds:

$$\begin{array}{c}{d}_{ij}^K=-c_K{\left(e_i^K\right)}^T\mathsf{\Gamma }{\widehat{x}}_j^K\hfill \\ d_i^{KR}={\left(e_i^K\right)}^T\mathsf{\Gamma }{\widehat{x}}_i^R\hfill \\ {\sigma }_{Ki}=k_i{\left(e_i^K\right)}^{T}H{e}_i^K\hfill \end{array}$$

(16)

When time tends to infinity, the error dynamic network (Equation (14)) will converge to the origin. This is demonstrated by the fact that when $x_j^K$ is approximately equal to ${\widehat{x}}_j^K$, the designed state observer network (Equation (13)) can estimate the state between the practical nodes of the complex network in the response layer.

Theorem 2.

For a network as Equation (12) composed of chaotic nodes, let the maximum Lyapunov exponent of isolated nodes be $h_{max}$. If $H=I_n$ and satisfies

$$\left|c{\lambda }_2\right|>h_{max}$$

(17)

then, the synchronous flow form is exponentially stable.

Proof. 

Build the Lyapunov function:

$$V=\frac{1}{2}\sum _{i=1}^N{\left(e_i^K\right)}^T{e}_i^K$$

(18)

Deriving it leads to

$$\dot{V}=\frac{1}{2}\sum _{i=1}^N{\left(({\dot{e}}_i^K\left(t\right)\right)}^T{e}_i^K\left(t\right)+{\left(e_i^K\left(t\right)\right)}^T{\dot{e}}_i^K\left(t\right))=\sum _{i=1}^N{\left(e_i^K\left(t\right)\right)}^T{\dot{e}}_i^K\left(t\right)$$

(19)

$\dot{V}>0$, $V\ge 0$, by bringing in Equation (14):

$$\begin{array}{c}\dot{V}=\frac{1}{2}{\displaystyle \sum _{i=1}^N}\left({\left({\dot{e}}_i^K\left(t\right)\right)}^T{e}_i^K\left(t\right)+{\left(e_i^K\left(t\right)\right)}^T{\dot{e}}_i^K\left(t\right)\right)\hfill \\ =\frac{1}{2}{\displaystyle \sum _{i=1}^N}\left(\begin{array}{c}{\left(f\left({\widehat{x}}_i^K\right)-f\left({\dot{x}}_i^K\right)\right)}^T{e}_i^K\left(t\right)+{\left(c_K{\displaystyle \sum _{j=1}^N}{d}_{ij}^K\mathsf{\Gamma }{e}_j^K\right)}^T{e}_i^K\left(t\right)\hfill \\ +{\left({\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{e}_i^R\right)}^T{e}_i^K\left(t\right)+{\left({\sigma }_{Ki}H{e}_i^K\right)}^T{e}_i^K\left(t\right)+\left(f\left({\widehat{x}}_i^K\right)-f\left({\dot{x}}_i^K\right)\right){\left(e_i^K\left(t\right)\right)}^T\hfill \\ +{\left(c_K{\displaystyle \sum _{j=1}^N}{d}_{ij}^K\mathsf{\Gamma }{e}_j^K\right)}^T{\left(e_i^K\left(t\right)\right)}^T+{\left({\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{e}_i^R\right)}^T{\left(e_i^K\left(t\right)\right)}^T+{\sigma }_{Ki}H{e}_i^K{\left(e_i^K\left(t\right)\right)}^T\hfill \end{array}\right)\hfill \end{array}$$

(20)

According to Hypothesis 1 and Lemma 1, we can simplify Equation (20) and obtain

$$\begin{array}{c}\dot{V}=\frac{1}{2}{\displaystyle \sum _{i=1}^N}\left(\begin{array}{c}{\left(f\left({\widehat{x}}_i^K\right)-f\left({\dot{x}}_i^K\right)\right)}^T{e}_i^K\left(t\right)+{\left(c_K{\displaystyle \sum _{j=1}^N}{d}_{ij}^K\mathsf{\Gamma }{e}_j^K\right)}^T{e}_i^K\left(t\right)\hfill \\ +{\left({\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{e}_i^R\right)}^T{e}_i^K\left(t\right)+{\left({\sigma }_{Ki}H{e}_i^K\right)}^T{e}_i^K\left(t\right)+\left(f\left({\widehat{x}}_i^K\right)-f\left({\dot{x}}_i^K\right)\right){\left(e_i^K\left(t\right)\right)}^T\hfill \\ +{\left(c_K{\displaystyle \sum _{j=1}^N}{d}_{ij}^K\mathsf{\Gamma }{e}_j^K\right)}^T{\left(e_i^K\left(t\right)\right)}^T+{\left({\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{e}_i^R\right)}^T{\left(e_i^K\left(t\right)\right)}^T+{\sigma }_{Ki}H{e}_i^K{\left(e_i^K\left(t\right)\right)}^T\hfill \end{array}\right)\hfill \\ =\frac{1}{2}{\displaystyle \sum _{i=1}^N}\left(\begin{array}{c}{\left(f\left({\widehat{x}}_i^K\right)-f\left({\dot{x}}_i^K\right)\right)}^T{e}_i^K\left(t\right)+\left(f\left({\widehat{x}}_i^K\right)-f\left({\dot{x}}_i^K\right)\right){\left(e_i^K\left(t\right)\right)}^T\hfill \\ +{\left(c_K{\displaystyle \sum _{j=1}^N}{d}_{ij}^K\mathsf{\Gamma }{e}_j^K\right)}^T{e}_i^K\left(t\right)+{\left(c_K{\displaystyle \sum _{j=1}^N}{d}_{ij}^K\mathsf{\Gamma }{e}_j^K\right)}^T{\left(e_i^K\left(t\right)\right)}^T\hfill \\ +{\left({\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{e}_i^R\right)}^T{e}_i^K\left(t\right)+{\left({\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{e}_i^R\right)}^T{\left(e_i^K\left(t\right)\right)}^T\hfill \\ +{\left({\sigma }_{Ki}H{e}_i^K\right)}^T{e}_i^K\left(t\right)+{\sigma }_{Ki}H{e}_i^K{\left(e_i^K\left(t\right)\right)}^T\hfill \end{array}\right)\hfill \\ =\frac{1}{2}{\displaystyle \sum _{i=1}^N}\left(\begin{array}{c}{\left(e_i^K\left(t\right)\right)}^{T}2\alpha e_i^K\left(t\right)+{\left(c_K{\displaystyle \sum _{j=1}^N}{d}_{ij}^K\mathsf{\Gamma }{e}_j^K\right)}^T{e}_i^K\left(t\right)+{\left(c_K{\displaystyle \sum _{j=1}^N}{d}_{ij}^K\mathsf{\Gamma }{e}_j^K\right)}^T{\left(e_i^K\left(t\right)\right)}^T\hfill \\ +{\left({\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{e}_i^R\right)}^T{e}_i^K\left(t\right)+{\left({\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{e}_i^R\right)}^T{\left(e_i^K\left(t\right)\right)}^T\hfill \\ +{\left({\sigma }_{Ki}H{e}_i^K\right)}^T{e}_i^K\left(t\right)+{\sigma }_{Ki}H{e}_i^K{\left(e_i^K\left(t\right)\right)}^T\hfill \end{array}\right)\hfill \end{array}$$

(21)

$$\begin{array}{c}\dot{V}\le \frac{1}{2}{\displaystyle \sum _{i=1}^N}\left(\begin{array}{c}{\left(e_i^K\left(t\right)\right)}^{T}2\alpha e_i^K\left(t\right)+e_i^K\left(t\right){\left({\sigma }_{Ki}H{e}_i^K\right)}^T+{\left(e_i^K\left(t\right)\right)}^T{\sigma }_{Ki}H{e}_i^K\hfill \\ +{\left(e_i^K\left(t\right)\right)}^T{c}_K{\displaystyle \sum _{j=1}^N}{d}_{ij}^K\mathsf{\Gamma }{e}_j^K+e_i^K\left(t\right){\left({\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{e}_i^R\right)}^T+2{\left(e_i^K\left(t\right)\right)}^T{\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{e}_i^R\hfill \end{array}\right)\hfill \\ \le \frac{1}{2}{\displaystyle \sum _{i=1}^N}\left(\begin{array}{c}\alpha {\left(e_i^K\left(t\right)\right)}^T{e}_i^K\left(t\right)+{{\sigma }_{Ki}}^T{H}^T{\left(e_i^K\left(t\right)\right)}^T{e}_i^K\left(t\right)+{\left(e_i^K\left(t\right)\right)}^T{\sigma }_{Ki}H{e}_i^K\hfill \\ +{\left(e_i^K\left(t\right)\right)}^T{c}_K{\displaystyle \sum _{j=1}^N}{d}_{ij}^K\mathsf{\Gamma }{e}_j^K+e_i^K\left(t\right){\left({\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{e}_i^R\right)}^T+2{\left(e_i^K\left(t\right)\right)}^T{\displaystyle \sum _{R=1}^M}{d}_i^{KR}\mathsf{\Gamma }{e}_i^R\hfill \end{array}\right)\hfill \\ \le \frac{1}{2}\left({\left(E^K\right)}^T\alpha E^K+c_K{\left(E^K\right)}^T{Q}_1{E}^K+{\displaystyle \sum _{R=1}^M}{\left(E^K\right)}^T{Q}_2{E}^R\right)\hfill \\ \le \frac{1}{2}\left({\left(E^K\right)}^T\alpha E^K+c_K{\left(E^K\right)}^T{Q}_1{E}^K+\frac{1}{2}{\displaystyle \sum _{R=1}^M}{\left(E^K\right)}^T{E}^K+\frac{1}{2}{\displaystyle \sum _{R=1}^M}{\left(Q_2{E}^R\right)}^T{Q}_2{E}^R\right)\hfill \\ \le \frac{1}{2}\left({\left(E^K\right)}^T(\alpha +c_K{Q}_1+\frac{M}{2})E^K+\frac{M}{2}{\left(E^R\right)}^T{Q_2}^T{Q}_2{E}^R\right)\hfill \end{array}$$

(22)

Among them, $E^K=\left({\left(e_1^K\right)}^T,{\left(e_2^K\right)}^T,\cdots ,{\left(e_N^K\right)}^T\right)$; $E^R=\left({\left(e_1^R\right)}^T,{\left(e_2^R\right)}^T,\cdots ,{\left(e_N^R\right)}^T\right)$; $Q_1=\left(d_{ij}^K\otimes \mathsf{\Gamma }\right)$; and $Q_2=\left(d_i^{KR}\otimes \mathsf{\Gamma }\right)$. $Q_1$ and $Q_2$ depend on the parameters in the network layer. When $(\alpha +c_K{Q}_1+\frac{M}{2})$ is small enough, the latter term of the above Equation (22) can be obtained. According to the Lasalle invariance principle, the dynamics of the error system is asymptotically stable, and the error Equation (14) will gradually converge to the origin and remain unchanged. That is, the driving-layer network Equation (12) and the response-layer network Equation (13) reach a synchronous state, and the topology of the driving-layer network Equation (12) can be successfully identified. □

4. Simulation and Result Analysis

A two-layer dynamic network represents a specific instance of a multilayer complex dynamic network. This section employs a two-layer complex dynamic network as a case study to assess the viability of the adaptive synchronization scheme. The selection of a chaotic system as the node dynamic characteristic of a complex dynamic network is motivated by its well-documented sensitivity to changes in its initial value. This paper examines a two-layer complex dynamic network with a topological structure of chaotic oscillators and nonlocal interactions. The network’s nodes are Lorenz oscillators. The single Lorenz oscillator’s dynamic equation is as follows:

$$\left\{\begin{array}{c}\dot{x}=-ax+ay\\ \dot{y}=bx-y-xz\\ \dot{z}=xy-cz\end{array}\right.$$

(23)

where $x,y$, and z are the dynamic variables and $a,b$, and c are the control parameters of the oscillator. Parameters $a,b$, and c determine the characteristics of the dynamic state, and their change factors affect the attractor of the phase space of the system. According to Equation (22), three fixed solutions are obtained, $x=y=z=0$, $x=y=\pm {\left[c(b-1)\right]}^{\frac{1}{2}}$, and $z=b-1$, where the last two solutions exist at $b\ge 1$. When $a=10,b=28$, and $c=1$, the system will appear as a chaotic phenomenon and produce strange attractors. The phase diagram corresponding to the system Equation (23) is as follows:

The chaotic attractor in Figure 1 can be made using numerical simulation, and it can be shown that the Lorenz oscillator has rich chaotic behavior. Figure 2 displays the complexity diagram of the Lorenz oscillator. The parameter settings are consistent with the dynamic phenomena studied above. The different colors in the diagram represent the chaotic, periodic, and divergent states of the Lorenz oscillator. The diagram reveals that chaotic areas dominate, and complex chaotic behaviors emerge. The boundary between high and low complexity is obvious. The evolution diagram of the x-, y-, and z-direction values in the Lorenz oscillator with time is shown in Figure 3 below. We find that while their trajectories remain within the Lorenz oscillator’s attractors’ range, they lack synchronization.

Through numerical simulation, the chaotic attractor shown in Figure 1 can be generated, and it can be determined that the Lorenz oscillator has rich chaotic behavior; Figure 2 shows the Lorenz oscillator’s complexity diagram. The parameter settings are consistent with the dynamic phenomena studied above. The diagram depicts the chaotic, periodic, and divergent states of the Lorenz oscillator, with different colors representing these states. It can be observed that the majority of the diagram is occupied by chaotic areas, which exhibit complex chaotic behaviors. The boundary between high and low complexity is clearly delineated. Figure 3 below illustrates the evolution diagram of the x-, y-, and z-direction values in the Lorenz oscillator over time. It is observed that although the trajectories of the two systems were not outside the range of the attractors of the Lorenz oscillator, they are not synchronized with each other.

Through computer simulation, the Lyapunov exponents of isolated nodes of Lorenz oscillator can be obtained as $h_1\approx -0.5839$, $h_2\approx 0.0569$, and $h_3\approx -11.4730$, namely, $h_{max}\approx 0.06$. According to Theorem 2, the synchronization condition of isolated nodes of the Lorenz oscillator can be obtained as $c>0.06/3=0.02$. Figure 4 shows the Lyapunov exponent diagram of the Lorenz oscillator. To sum up, by analyzing the dynamics of the Lorenz oscillator, we can find that the Lorenz oscillator has abundant chaotic characteristics.

For any two state variables $x_i$ and $x_j$ of a Lorenz oscillator, because the chaotic attractor is bounded, there exists a constant $\beta$ such that $∥x_{ik}∥\le \beta$, $∥x_{jk}∥\le \beta$$(k=1,2,3)$ and

$$\begin{array}{c}∥f\left(x_i\right)-f\left(x_j\right)∥\hfill \\ =\sqrt{{\left(-x_{i3}(x_{i1}-x_{j1})-x_{j1}(x_{i3}-x_{j3})\right)}^2+{\left(x_{i2}(x_{i1}-x_{j1})-x_{j1}(x_{i2}-x_{j2})\right)}^2}\hfill \\ =\sqrt{{\left(x_{j1}{x}_{j3}-x_{j1}{x}_{i3}\right)}^2+{\left(x_{i1}{x}_{i2}-x_{j1}{x}_{j2}\right)}^2}\hfill \\ \le 2\beta ∥x_i-x_j∥\hfill \\ =\alpha ∥x_i-x_j∥\hfill \end{array}$$

(24)

Therefore, the Lorenz oscillator satisfies Hypothesis 1. The dynamic model of a two-layer complex network with the same number of nodes in each layer and one-to-one correspondence is selected, and the state equation of the driving layer of the two-layer network is as follows:

$$\begin{array}{c}{\dot{x}}_i^1=f\left(x_i^1\right)+c_1{\displaystyle \sum _{j=1}^N}{d}_{ij}^1\mathsf{\Gamma }{x}_j^1+d_i^{11}\mathsf{\Gamma }{x}_i^1+d_i^{12}\mathsf{\Gamma }{x}_i^2\hfill \\ {\dot{x}}_i^2=f\left(x_i^2\right)+c_2{\displaystyle \sum _{j=1}^N}{d}_{ij}^2\mathsf{\Gamma }{x}_j^2+d_i^{21}\mathsf{\Gamma }{x}_i^1+d_i^{22}\mathsf{\Gamma }{x}_i^2\hfill \end{array}$$

(25)

The topological structure of the two-layer complex network consists of a ring network with 10 nodes, where the nodes of the two-layer network have a one-to-one correspondence. The couplings of the first-layer network ($D^1$), the second layer ($D^2$), and the inter-layer ($D^{11},D^{12},D^{21},D^{22}$) are as follows:

$$D^1=D^2=\left[\begin{array}{cccccccccc}-2\hfill & 1& 0& 0& 0& 0& 0& 0& 0& 1\\ 1\hfill & -2& 1& 0& 0& 0& 0& 0& 0& 0\\ 0\hfill & 1& -2& 1& 0& 0& 0& 0& 0& 0\\ 0\hfill & 0& 1& -2& 1& 0& 0& 0& 0& 0\\ 0\hfill & 0& 0& 1& -2& 1& 0& 0& 0& 0\\ 0\hfill & 0& 0& 0& 1& -2& 1& 0& 0& 0\\ 0\hfill & 0& 0& 0& 0& 1& -2& 1& 0& 0\\ 0\hfill & 0& 0& 0& 0& 0& 1& -2& 1& 0\\ 0\hfill & 0& 0& 0& 0& 0& 0& 1& -2& 1\\ 1\hfill & 0& 0& 0& 0& 0& 0& 0& 1& -2\end{array}\right]$$

(26)

$$D^{11}=D^{22}=\left[\begin{array}{cccccccccc}-1\hfill & 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0\hfill & -1& 1& 0& 0& 0& 0& 0& 0& 0\\ 0\hfill & 0& -1& 1& 0& 0& 0& 0& 0& 0\\ 0\hfill & 0& 0& -1& 1& 0& 0& 0& 0& 0\\ 0\hfill & 0& 0& 0& -1& 1& 0& 0& 0& 0\\ 0\hfill & 0& 0& 0& 0& -1& 1& 0& 0& 0\\ 0\hfill & 0& 0& 0& 0& 0& -1& 1& 0& 0\\ 0\hfill & 0& 0& 0& 0& 0& 0& -1& 1& 0\\ 0\hfill & 0& 0& 0& 0& 0& 0& 0& -1& 1\\ 0\hfill & 0& 0& 0& 0& 0& 0& 0& 1& -1\end{array}\right]$$

(27)

$$D^{12}=D^{21}=\left[\begin{array}{cccccccccc}1\hfill & 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0\hfill & 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0\hfill & 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0\hfill & 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0\hfill & 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 0\hfill & 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0\hfill & 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0\hfill & 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0\hfill & 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0\hfill & 0& 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]$$

(28)

The parameters in the network are set as follows: the coupling strength of each layer of network is $c_1=c_2=1$; inline $\mathsf{\Gamma }=diag(1,1,1)$ within and between layers of network nodes; output $H=[1,0,0]$ of nodes. The complex network for building the response side is

$$\begin{array}{c}{\dot{x}}_i^1=f\left(x_i^1\right)+c_1{\displaystyle \sum _{j=1}^N}{d}_{ij}^1\mathsf{\Gamma }{x}_j^1+d_i^{11}\mathsf{\Gamma }{x}_i^1+d_i^{12}\mathsf{\Gamma }{x}_i^2+{\sigma }_{Ki}(x_i^1-{\widehat{x}}_i^1)\hfill \\ {\dot{x}}_i^2=f\left(x_i^2\right)+c_2{\displaystyle \sum _{j=1}^N}{d}_{ij}^2\mathsf{\Gamma }{x}_j^2+d_i^{21}\mathsf{\Gamma }{x}_i^1+d_i^{22}\mathsf{\Gamma }{x}_i^2+{\sigma }_{Ki}(x_i^2-{\widehat{x}}_i^2)\hfill \end{array}$$

(29)

The dynamic network equation of its error system is ${\dot{e}}_i^1={\widehat{x}}_i^1-{\dot{x}}_i^1$, ${\dot{e}}_i^2={\widehat{x}}_i^2-{\dot{x}}_i^2$.

Coupling the $x,y$, and z variables, the network equation of the Lorenz oscillator can be written as

$$\left\{\begin{array}{c}{\dot{x}}_{i1}=-a{x}_{i1}+a{x}_{i2}+c{\displaystyle \sum _{j=1}^{10}}{d}_{ij}^1\mathsf{\Gamma }{x}_j^1\\ {\dot{x}}_{i2}=b{x}_{i1}-x_{i2}-x_{i1}{x}_{i3}+c{\displaystyle \sum _{j=1}^{10}}{d}_{ij}^1\mathsf{\Gamma }{x}_{j2}^1\\ {\dot{x}}_{i3}=x_{i1}{x}_{i2}-c{x}_{i3}+c{\displaystyle \sum _{j=1}^{10}}{d}_{ij}^1\mathsf{\Gamma }{x}_{j3}^1\end{array}\right.$$

(30)

The evolution of the x, y, and z components of 10 nodes in a complex network dynamic system with a Lorenz oscillator can be obtained by estimating the states of the 10 nodes over time.

Figure 5, Figure 6 and Figure 7 illustrate the temporal evolution of errors between each node of the two-layer complex dynamic network and the target system. From the figure, it can be observed that when the initial values are disparate, the network error of the system is conspicuous in the initial period of time. However, with the implementation of the state observer, the system attains a synchronous state and the error value tends to 0 asymptotically, thereby realizing synchronous tracking between the driving-layer network and the response-layer network and achieving the desired synchronous state.

Due to the initial values and network topology, the driving and response layers initially exhibit significant errors. However, over time, the state observer begins to function, and after a certain period, the network’s total error tends towards zero. As the state error approaches zero, the rate of parameter change in the adaptive synchronization scheme also approaches zero, ultimately leading to a stable state. The above simulation results demonstrate that the discrepancy between the driving-layer network and the response-layer network rapidly diminishes and remains at a negligible level from approximately $t=0.4$, as illustrated in Figure 8 and Figure 9. Furthermore, the node state information within the driving-layer network gradually aligns with that of the response-layer network and persists in maintaining alignment with the node state information of the original network over time. The structure of the multilayered, complex dynamic network can be identified through the adaptive synchronization approach. It has been demonstrated that the adaptive synchronization scheme can precisely assess the state of unfamiliar nodes in multilayered, complex dynamic networks with a one-to-one correlation.

5. Conclusions

This paper investigates the synchronization issue between complex dynamic networks and chaotic systems. It proposes a state estimation scheme for multilayer complex networks, based on a state observer. The proposed scheme targets a multilayer complex network dynamic model, which exhibits the same number of nodes in each layer, and follows one-to-one correspondence. Firstly, the study focuses on estimating the state of a general multilayer complex dynamic network. We construct a corresponding network state observer based on the information of the node dynamics and topology of the original network, and establish the driving- and response-layer networks. Secondly, we analyze the stability of the system through the primary stability function analysis method, resulting in proof of the network state observer’s ability to achieve synchronization between the complex dynamic network and the target system. Finally, we present numerical simulations of a network consisting of 10 Lorenz oscillators and demonstrate the effectiveness of the proposed method through the results obtained. In the future, we will further study the synchronization problem of continuous-time partial state components in directed time-varying time-delay complex networks, and will consider using different control methods such as sliding mode control, pulse control, and intermittent control to achieve synchronization of the network.

6. Use of AI Tools Declaration

The authors declare they have not used artificial intelligence (AI) tools in the creation of this article.