Function Projective Synchronization of Two Complex Networks with Unknown Sector Nonlinear Input and Multiple Time-Varying Delay Couplings

Abstract

This paper deals with the function projective synchronization of two complex dynamic networks with unknown sector nonlinear input, multiple time-varying delay couplings, model uncertainty, and external interferences. Based on Lyapunov stability theory and inequality transformation method, the robust adaptive synchronization controller is designed, by which the drive and response systems can achieve synchronization according to the function scaling factor. Different from some existing studies on nonlinear system with sector nonlinear input, this paper studies the synchronization of two complex dynamic networks when the boundary of sector nonlinear input is unknown. The controller does not include the boundary value of the sector nonlinear input and the time delay term, so it is more practical and relatively easy to implement. The corresponding simulation examples demonstrate the effectiveness of the proposed scheme.

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1. Introduction

There are all kinds of complex systems in nature world. These complex systems can be seen as networks, such as Internet, power grid, communication networks, transportation networks, ecological networks, and social networks. The dynamic behavior of complex networks affects almost every aspect of our lives. Among many researches on complex networks, synchronization research is one of the most important branches. So far, many types of synchronization have been investigated, such as complete synchronization [1, 2], antisynchronization [3], exponential synchronization [4–6], quasisynchronization [7], lag synchronization [8, 9], combined synchronization [10], projection synchronization [11], and function projection synchronization [12–14]. Function projection synchronization is a general synchronization form, which means that the driving system and the response system can be synchronized according to a certain function proportional relationship. The complete synchronization, antisynchronization, and projection synchronization are all its exceptional cases. Function projection synchronization has attracted widespread attention because of its implied application in information science and secure communication [15, 16].

It is well known that various time delays are unavoidable in actual engineering applications. The time delay may destroy the dynamic characteristics and decrease the stability of the system, which is extremely detrimental to the control system [17–19]. Multiple time-varying delay couplings mean that multiple different time-varying delays exist in the complex network. The description of multiple time-varying delay couplings is a general description of time delay, and the constant time-delay couplings and single time-varying delay couplings are its special circumstances. The synchronization researches of complex networks with multiple time-varying delay couplings are more realistic and representative [20, 21]. Zhang et al. [22] researched the synchronization of uncertain complex networks with time-varying node delay and multiple time-varying coupling delays via the adaptive control. In [23], the authors researched the synchronization in nonlinear complex networks with multiple time-varying delays. Wang et al. [24] studied the lag synchronization between two coupled complex networks with multiple time-varying delays via the adaptive pinning control. Zhao et al. [25] studied the synchronization issue of uncertain complex networks with multiple time-varying delays. Lu et al. [26] established a robust adaptive synchronization scheme for general complex networks with multiple time-varying coupling delays and uncertainties. Guan et al. [27] studied the synchronization of complex networks with system delay and multiple time-varying coupling delays via impulsive distributed control.

In the actual control system, the backlash, friction, dead zone, and hysteresis will cause the nonlinearity of the control input, which lead to system instability or control performance degeneration [28–34]. Therefore, the synchronization researches of complex networks with input nonlinearity are meaningful. Sector nonlinear input is one type of the nonlinear input, which means that the system input is in a fan-shaped area. Sector nonlinear input represents a large type of input nonlinearity. Many scholars have studied the control of nonlinear systems with sector nonlinear input. Boulkroune and Msaad [35] researched the adaptive variable-structure control of uncertain chaotic MIMO systems with both sector nonlinearities and dead-zones. Fang et al. [36] researched the modified projective synchronization of chaotic systems with sector nonlinearities input. Boubellouta et al. [37] achieved synchronization for a class of fractional-order chaotic systems with sector nonlinearities. Wang and Liu [38] researched the sliding mode control of the master-slave chaotic systems with sector nonlinear input. Yang et al. [39] addressed an adaptive two-stage sliding mode control to realize the synchronization for a class of $n$-dimensional nonlinear systems with sector nonlinearity input. Although the researches on sector nonlinear input have achieved certain results, most existing studies mainly focus on a single system rather than complex networks. Recently, Fang et al. [40] studied the modified function projective synchronization of complex dynamic networks with sector nonlinear input. In the controller design, it is assumed that the range of the sector nonlinear input is known. However, it is difficult to determine the exact boundary value of the sector nonlinear input. Once the restricted boundary of the control input is unknown, the controller designed in [40] is no longer applicable. How to realize function projective synchronization of complex dynamic networks under unknown sector nonlinear input is a challenging research topic.

Based on the results of previous researches, the function projective synchronization for a class of complex dynamic networks with unknown sector nonlinear input, multiple time-varying delay couplings, model uncertainty, and external interferences is studied in this paper. Through the designed adaptive controller, two complex dynamic networks can realize synchronization according to the corresponding function scaling factor. Compared with the existing research results, the contributions of this paper are (a) the complex network model includes the input nonlinearity, multiple time-varying delay couplings, model uncertainty, and external interferences, which is a more general model. (b) Many of the existing studies are concerned with synchronization between complex networks and single systems. This paper studies the synchronization between two complex networks, which is more complex and general. (c) Different from known sector nonlinear inputs in previous studies, this paper investigates the function projective synchronization of complex dynamic networks with unknown sector inputs. The boundary value of the sector nonlinear input and the delay term is not needed in controller design, so it is relatively easy to implement in practical engineering. (d) Function projective synchronization is a more general synchronization form. The controller in this paper can also realize complete synchronization, antisynchronization, and projective synchronization of complex dynamic networks.

2. Model Description

In this article, a type of complex dynamic networks with unknown sector nonlinear input, multiple time-varying delay couplings, model uncertainty, and external interferences is described as the drive system: $$\begin{array}{}\text{(1)}& {\dot{x}}_{i}t=f_i{x}_{i}t+F_i{x}_{i}t{\theta }_i+\sum _{l=0}^{m-1}{c}_{l}t\sum _{j=1}^N{a}_{ij}^l{\Gamma }_l{x}_{j}t-{\tau }_{l}t+d_i^{v}t=f_i{x}_{i}t+F_i{x}_{i}t{\theta }_i+c_{0}t\sum _{j=1}^N{a}_{ij}^0{\Gamma }_0{x}_{j}t-{\tau }_{0}t+c_{1}t\sum _{j=1}^N{a}_{ij}^1{\Gamma }_1{x}_{j}t-{\tau }_{1}t+\cdots +c_{m-1}t\sum _{j=1}^N{a}_{ij}^{m-1}{\Gamma }_{m-1}{x}_{j}t-{\tau }_{m-1}t+d_i^{v}t,\end{array}$$the corresponding response system is $$\begin{array}{}\text{(2)}& {\dot{y}}_{i}t=g_i{y}_{i}t+G_i{y}_{i}t{\eta }_i+\sum _{l=0}^{m-1}{c}_{l}t\sum _{j=1}^N{a}_{ij}^l{\Gamma }_l{y}_{j}t-{\tau }_{l}t+d_i^{s}t+{\varphi }_i{u}_{i}t=g_i{y}_{i}t+G_i{y}_{i}t{\eta }_i+c_{0}t\sum _{j=1}^N{a}_{ij}^0{\Gamma }_0{y}_{j}t-{\tau }_{0}t+c_{1}t\sum _{j=1}^N{a}_{ij}^1{\Gamma }_1{y}_{j}t-{\tau }_{1}t+\cdots +c_{m-1}t\sum _{j=1}^N{a}_{ij}^{m-1}{\Gamma }_{m-1}{y}_{j}t-{\tau }_{m-1}t+d_i^{s}t+{\varphi }_i{u}_{i}t,\end{array}$$where $x_{i}t={x_{i1}t,x_{i2}t,\cdots ,x_{\mathit{\text{in}}}t}^T,i=1,2,\cdots ,N$ is the state vector of the $i$th node in the drive system, $y_{i}t={y_{i1}t,y_{i2}t,\cdots ,y_{\mathit{\text{in}}}t}^T,i=1,2,\cdots ,N$ is the state vector of the $i$th node in the response system. $f_i\times ,g_i\times \widehat{I}{R}^n$ are the continuous nonlinear function vectors, $R^n$ denotes the $n$-dimensional vector space on the real number field $R$, ${\theta }_i,{\eta }_i\in R^w$ are unknown $w$-dimensional constant parameter vector, $F_i\cdot ,G_i\cdot \in R^{n\times w}$ are the continuous nonlinear function matrices, $R^{n\times w}$ denotes the $n\times w$ order matrix on the real number field $R$. $d_i^{v}t$ and $d_i^{s}t$ are the disturbances. The complex network is divided into subnetworks by ${\tau }_{l}t$, ${\tau }_{l}t\ge 0,l=0,1,\cdots ,m-1$ is the different time-varying delays, and especially ${\tau }_{0}t=0$ means that the coupling delay is 0; $c_{l}t$ is the coupling strength; ${\Gamma }_l$ is the inner coupling matrix; $A_l={a_{ij}^l}_{N\times N}$ is weight configuration matrix, representing the topological structure of the network. If nodes $i$ and $jj\ne i$ are connected, then, $a_{ij}^l\ne 0$. If nodes $i$ and $jj\ne i$ have no connection, then. $a_{ij}^l=a_{ji}^l=0.$ The diagonal elements of the matrix $A_l$ are defined as $a_{ii}^l=-{\sum }_{j=1,j\ne i}^N{a}_{ij}^l,i,j=1,2,\cdots ,N.$${\varphi }_i{u}_{i}t={{\varphi }_{i1}{u}_{i1}t,{\varphi }_{i2}{u}_{i2}t,\cdots ,{\varphi }_{in}{u}_{in}t}^{\text{T}}{\varphi }_{i}0=0$ is the control input. ${\varphi }_{ik}{u}_{ik}t$ is in a sector ${pu}_{ik}t,{qu}_{ik}t$, where $p$ and $q$ are two positive numbers and satisfy $p\le {\varphi }_{ik}{u}_{ik}t/u_{ik}t\le q$ when $u_{ik}t\ne 0$. The sector nonlinear input is shown in Figure 1.

Definition 1 (see [15]).

For the complex dynamic networks (1) and (2), if Eq. (3) holds, the complex network (1) and (2) will realize function projective synchronization when $$\begin{array}{}\text{(3)}& \underset{t⟶\infty }{\lim }{e}_{i}t=\underset{t⟶\infty }{\lim }{y}_{i}t-ht{x}_{i}t=0,i=1,2,\cdots ,N.\end{array}$$where $e_{i}t={e_{i1}t,e_{i2}t,\cdots ,e_{\mathit{\text{in}}}t}^{\text{T}},i=1,2,\cdots ,N,\cdot$ denotes the Euclidean norm of a vector. $ht\ne 0$ is function scaling factor, which is a continuously differentiable and bounded function.

Assumption 2.

External disturbances $d_i^{v}t$ and $d_i^{s}t$ are bounded, and there exist positive constants ${\alpha }_i^v,{\alpha }_i^s,$ such that $d_i^{v}t\le {\alpha }_i^v,d_i^{s}t\le {\alpha }_i^s.$

Corollary 3.

Because $ht$ is a continuously differentiable and bounded function, there exists a positive constant $\hslash$ and satisfies $ht\le \hslash$. Under Assumption 2, there exists a positive constant ${\alpha }_i\ge {\alpha }_i^s+\hslash {\alpha }_i^w$, such that $$\begin{array}{}\text{(4)}& d_i^{s}t-ht{d}_i^{v}t\le d_i^{s}t+ht{d}_i^{v}t\le d_i^{s}t+ht{d}_i^{v}t\le {\alpha }_i^s+\hslash {\alpha }_i^v\le {\alpha }_i.\end{array}$$

Assumption 4.

The time-varying coupling strength $c_{l}t$ is bounded, and there exists a positive constant $c$, such that $$\begin{array}{}\text{(5)}& c_{l}t\le c.\end{array}$$

Assumption 5.

The time-varying delay ${\tau }_{l}t,l=0,1,\cdots ,m-1$ is a continuously differentiable function and satisfies $0\le {\dot{\tau }}_{l}t\le \epsilon <1$, so it is easy to get $$\begin{array}{}\text{(6)}& \frac{1-{\dot{\tau }}_{l}t}{21-\epsilon }\ge \frac{1-\epsilon }{21-\epsilon }=\frac{1}{2},\end{array}$$where $0<\epsilon <1$ is positive constant. This assumption is still satisfied if ${\tau }_{l}t$ is zero or some other constants.

Lemma 6 (see [9]).

For any vectors $X,Y\in R^n$ and a positive definite matrix $Q\in R^{n\times n}$ ($R^n$ denotes the $n$-dimensional vector space on the real number field $R$, $R^{n\times n}$ denotes the $n\times n$ order matrix on the real number field $R$), the following matrix inequality holds:$2X^{\text{T}}QY\le X^{\text{T}}{QQ}^{\text{T}}X+Y^{\text{T}}Y$.

Proof.

Let $A={a_{1}t,a_{2}t,\cdots ,a_{n}t}^{\text{T}},B={b_{1}t,b_{2}t,\cdots ,b_{n}t}^{\text{T}}.$

It is easy to get $A^{\text{T}}B,B^{\text{T}}A\in R$ and $A^{\text{T}}B=B^{\text{T}}A,$$A^{\text{T}}A=a_1^2+a_2^2+\cdots +a_n^2,B^{\text{T}}B=b_1^2+b_2^2+\cdots +b_n^2,A^{\text{T}}B=B^{\text{T}}A=a_1\times b_1+a_2\times b_2+\cdots +a_n\times b_n.$

Because $a_1^2+a_2^2+\cdots +a_n^2+b_1^2+b_2^2+\cdots +b_n^2-2\times a_1\times b_1+a_2\times b_2+\cdots +a_n\times b_n={a_1-b_1}^2+{a_2-b_2}^2+\cdots +{a_n-b_n}^2\ge 0,$then, $a_1^2+a_2^2+\cdots +a_n^2+b_1^2+b_2^2+\cdots +b_n^2\ge 2\times a_1\times b_1+a_2\times b_2+\cdots +a_n\times b_n$, i.e., $A^{\text{T}}A+B^{\text{T}}B\ge 2A^{\text{T}}B.$

Let $A=Q^{\text{T}}X,B=Y,$ then, we can get $2X^{\text{T}}QY\le X^{\text{T}}{QQ}^{\text{T}}X+Y^{\text{T}}Y.$

This completes the proof.

3. Controller Design

To realize function projective synchronization, the controller and parameter adaptive laws are designed as follows: $$\begin{array}{}\text{(7)}& u_{ik}t=-\gamma g_{ik}{y}_{i}t-ht{f}_{ik}{x}_{i}t-\dot{h}t{x}_{ik}t+G_{ik}{y}_{i}t{\widehat{\eta }}_i-ht{F}_{ik}{x}_{i}t{\widehat{\theta }}_i+\frac{1}{\gamma }\widehat{\nu }{e}_{ik}t\widehat{\varpi }+{\widehat{\psi }}_{ik}\mathrm{sgn}{e}_{ik}t,i=1,2,\cdots ,Nk=1,2,\cdots ,n.\text{(8)}& {\dot{\widehat{\theta }}}_i=-F_i^{\text{T}}{x}_{i}tht{e}_{i}t,\text{(9)}& {\dot{\widehat{\eta }}}_i=G_i^{\text{T}}{y}_{i}t{e}_{i}t,\text{(10)}& \dot{\widehat{\varpi }}=\sum _{i=1}^N{e}_i^{\text{T}}t{g}_i{y}_{i}t-ht{f}_i{x}_{i}t-\dot{h}t{x}_{i}t+G_i{y}_{i}t{\widehat{\eta }}_i-ht{F}_i{x}_{i}t{\widehat{\theta }}_i+\widehat{\nu }{e}_{i}t,\text{(11)}& {\dot{\widehat{\psi }}}_i=e_{i}t,\text{(12)}& \dot{\widehat{\nu }}=\sum _{i=1}^N{e}_i^{\text{T}}t{e}_{i}t,\end{array}$$where $\gamma ,\nu$ are positive constants and satisfy $\gamma >1,\nu >0.\varpi =1/p>0,{\psi }_i={\alpha }_i/p>0.$$\widehat{\varpi },{\widehat{\psi }}_i,{\widehat{\theta }}_i,{\widehat{\eta }}_i,\widehat{\nu }$ is the estimated parameter for $\varpi ,{\psi }_i,{\theta }_i,{\eta }_i,\nu$, respectively. $F_{ik}\cdot ,G_{ik}\cdot \in R^{1\times w}$ are the $k$th row of the function matrices $F_i\cdot ,G_i\cdot \in R^{n\times w}$.

Remark 7.

Let $u_{ik}t=-{\gamma \mu }_{ik}\mathrm{sgn}{e}_{ik}t$, then, ${\mu }_{ik}=g_{ik}{y}_{i}t-ht{f}_{ik}{x}_{i}t-\dot{h}t{x}_{ik}t+G_{ik}{y}_{i}t{\widehat{\eta }}_i-ht{F}_{ik}{x}_{i}t{\widehat{\theta }}_i+1/\gamma \widehat{\nu }{e}_{ik}t\widehat{\varpi }+{\widehat{\psi }}_{ik}.$

Because $\gamma >1,\nu >0,\varpi =1/p>0,{\psi }_i={\alpha }_i/p>0,$ we can get${\mu }_{ik}>0$.

Lemma 8 (see [40]).

Let $u_{ik}t=-{\gamma \mu }_{ik}\mathrm{sgn}{e}_{ik}t,{\mu }_{ik}>0,$ we can get ${\sum }_{i=1}^N{e}_{ik}t{\varphi }_{ik}{u}_{ik}t\le {\sum }_{i=1}^N-p{\gamma \mu }_{ik}{e}_{ik}t,$ i.e., $e_i^{\text{T}}t{\varphi }_i{u}_{i}t\le -p_i{{\gamma \mu }_i}^{\text{T}}{e}_{i}t.$

Proof.

It can be known from $p\le {\varphi }_{ik}{u}_{ik}t/u_{ik}t\le q$ that ${pu}_{ik}^{2}t\le u_{ik}t{\varphi }_{ik}{u}_{ik}t\le {qu}_{ik}^{2}t.$

When $e_{ik}t=0,$ the equation obviously holds, that is, $e_{ik}t{\varphi }_{ik}{u}_{ik}t=-p{\gamma \mu }_{ik}{e}_{ik}t.$

When $e_{ik}t\ne 0,$ substituting $u_{ik}t=-{\gamma \mu }_{ik}\mathrm{sgn}{e}_{ik}t$ into ${pu}_{ik}^{2}t\le u_{ik}t{\varphi }_{ik}{u}_{ik}t\le {qu}_{ik}^{2}t,$ we can get $p{\gamma }^2{\mu }_{ik}^2{\mathrm{sgn}}^2{e}_{ik}t\le -{\gamma \mu }_{ik}\mathrm{sgn}{e}_{ik}t{\varphi }_{ik}{u}_{ik}t.$

Using $e_{ik}t/e_{ik}t$ instead of $\mathrm{sgn}{e}_{ik}t$, we can get $p{\gamma }^2{\mu }_{ik}^2{e}_{ik}t{e}_{ik}t/e_{ik}t{e}_{ik}t\le -{\gamma \mu }_{ik}{\varphi }_{ik}{\mu }_{ik}t{e}_{ik}t/e_{ik}t$.

Multiplying both sides of the inequality by ${e_{ik}}^{2}t$, we can get $p{\gamma }^2{\mu }_{ik}^2{e_{ik}t}^2\le -{\gamma \mu }_{ik}{e}_{ik}t{e}_{ik}t{\varphi }_{ik}{u}_{ik}t.$ Dividing both sides by ${\gamma \mu }_{ik}{e}_{ik}t$, we can get $e_{ik}t{\varphi }_{ik}{u}_{ik}t\le -p{\gamma \mu }_{ik}{e}_{ik}t.$

It is easy to get ${\sum }_{i=1}^N{e}_{ik}t{\varphi }_{ik}{u}_{ik}t\le {\sum }_{i=1}^N-p{\gamma \mu }_{ik}{e}_{ik}t$, then, $e_i^{\text{T}}t{\varphi }_i{u}_{i}t\le -p_i{{\gamma \mu }_i}^{\text{T}}{e}_{i}t.$

This completes the proof.

Theorem 9.

If Assumptions 2–5 are satisfied, the drive system (1) and the response system (2) can realize function projective synchronization with the controller (7) and adaptive laws (8)–(12).

Proof.

From Definition 1, we have the error term: $$\begin{array}{}\text{(13)}& e_{i}t=y_{i}t-ht{x}_{i}t.\end{array}$$

The time derivative of $e_{i}t$ is $$\begin{array}{}\text{(14)}& {\dot{e}}_{i}t={\dot{y}}_{i}t-ht{\dot{x}}_{i}t-\dot{h}t{x}_{i}t=g_i{y}_{i}t+G_i{y}_{i}t{\eta }_i+\sum _{l=0}^{m-1}{c}_{l}t\sum _{j=1}^N{a}_{ij}^l{\Gamma }_l{y}_{j}t-{\tau }_{l}t+d_i^{s}t+{\varphi }_i{u}_{i}t-ht{f}_i{x}_{i}t-ht{F}_i{x}_{i}t{\theta }_i-ht\sum _{l=0}^{m-1}{c}_{l}t\sum _{j=1}^N{a}_{ij}^l{\Gamma }_l{x}_{j}t-{\tau }_{l}t-ht{d}_i^{v}t-\dot{h}t{x}_{i}t.\end{array}$$

Choosing Lyapunov function as $$\begin{array}{}\text{(15)}& Vt=\frac{1}{2p}\sum _{i=1}^N{e}_i^{\text{T}}t{e}_{i}t+p{{\stackrel{\wedge }{\psi }}_i-{\psi }_i}^2+\frac{1}{2}{\stackrel{\wedge }{\varpi }-\varpi }^2+\frac{1}{2p}{\stackrel{\wedge }{\nu }-{\nu }^{\ast }}^2+\frac{1}{2p}\sum _{i=1}^N{{\stackrel{\wedge }{\theta }}_i-{\theta }_i}^2+\sum _{i=1}^N{{\stackrel{\wedge }{\eta }}_i-{\eta }_i}^2+\frac{1}{1-\epsilon }{\int }_{t-{\tau }_{l}t}^t\sum _{l=1}^{m-1}\sum _{i=1}^N{e}_i^{\text{T}}\delta e_i\delta d\delta ,\end{array}$$where $\widehat{\varpi },{\widehat{\psi }}_i$ is the estimated parameter for $\varpi ,{\psi }_i$. ${\nu }^{\ast }$ is the positive constant to be designed.

Taking the derivative of the Lyapunov function, we can get $$\begin{array}{}\text{(16)}& \dot{V}t=\sum _{i=1}^N\frac{1}{p}{e}_i^{\text{T}}t{\dot{e}}_{i}t+{{\stackrel{\wedge }{\psi }}_i-{\psi }_i}^{\text{T}}{\dot{\widehat{\psi }}}_i+\widehat{\varpi }-\varpi \dot{\widehat{\varpi }}+\frac{1}{p}\widehat{\nu }-{\nu }^{\ast }\dot{\widehat{\nu }}+\frac{1}{p}\sum _{i=1}^N{{\stackrel{\wedge }{\theta }}_i-{\theta }_i}^{\text{T}}{\dot{\widehat{\theta }}}_i+\frac{1}{p}\sum _{i=1}^N{{\stackrel{\wedge }{\eta }}_i-{\eta }_i}^{\text{T}}{\dot{\widehat{\eta }}}_i+\frac{{\sum }_{l=1}^{m-1}{\sum }_{i=1}^N{e}_i^{\text{T}}t{e}_{i}t}{2p1-\epsilon }-\frac{1-{\dot{\tau }}_{l}t}{2p1-\epsilon }\sum _{l=1}^{m-1}\sum _{i=1}^N{e}_i^{\text{T}}t-{\tau }_{l}t{e}_{i}t-{\tau }_{l}t.\end{array}$$

Substituting (8), (9), and (14) into (16), we can get $$\begin{array}{}\text{(17)}& \dot{V}t=\frac{1}{p}\sum _{i=1}^N{e}_i^{\text{T}}t{g}_i{y}_{i}t+G_i{y}_{i}t{\eta }_i+d_i^{s}t+\sum _{l=0}^{m-1}{c}_{l}t\sum _{j=1}^N{a}_{ij}^l{\Gamma }_l{y}_{j}t-{\tau }_{l}t+{\varphi }_i{u}_{i}t-ht{f}_i{x}_{i}t-ht{F}_i{x}_{i}t{\theta }_i-ht\sum _{l=0}^{m-1}{c}_{l}t\sum _{j=1}^N{a}_{ij}^l{\Gamma }_l{x}_{j}t-{\tau }_{l}t-ht{d}_i^{v}t-\dot{h}t{x}_{i}t+\sum _{i=1}^N{{\stackrel{\wedge }{\psi }}_i-{\psi }_i}^{\text{T}}{\dot{\widehat{\psi }}}_i+\widehat{\varpi }-\varpi \dot{\widehat{\varpi }}+\frac{1}{p}\widehat{\nu }-{\nu }^{\ast }\dot{\widehat{\nu }}-\frac{1}{p}\sum _{i=1}^N{{\stackrel{\wedge }{\theta }}_i-{\theta }_i}^{\text{T}}{F}_i^{\text{T}}{x}_{i}tht{e}_{i}t+\frac{1}{p}\sum _{i=1}^N{{\stackrel{\wedge }{\eta }}_i-{\eta }_i}^{\text{T}}{G}_i^{\text{T}}{y}_{i}t{e}_{i}t+\frac{1}{2p1-\epsilon }\sum _{l=1}^{m-1}\sum _{i=1}^N{e}_i^{\text{T}}t{e}_{i}t-\frac{1-{\dot{\tau }}_{l}t}{2p1-\epsilon }\sum _{l=1}^{m-1}\sum _{i=1}^N{e}_i^{\text{T}}t-{\tau }_{l}t{e}_{i}t-{\tau }_{l}t.\end{array}$$

Because ${\sum }_{i=1}^N{e}_i^{\text{T}}tht{F}_i{x}_{i}t{\theta }_i={\sum }_{i=1}^N{{\theta }_i}^{\text{T}}{F_i}^{\text{T}}{x}_{i}tht{e}_{i}t,$${\sum }_{i=1}^N{e}_i^{\text{T}}t{G}_i{y}_{i}t{\eta }_i={\sum }_{i=1}^N{\eta }^{\text{T}}{G_i}^T{y}_{i}t{e}_{i}t,$ we can get $$\begin{array}{}\text{(18)}& -\frac{1}{p}\sum _{i=1}^N{e}_i^{\text{T}}tht{F}_i{x}_{i}t{\theta }_i-\frac{1}{p}\sum _{i=1}^N{{\stackrel{\wedge }{\theta }}_i-{\theta }_i}^{\text{T}}{F}_i^{\text{T}}{x}_{i}tht{e}_{i}t=-\frac{1}{p}\sum _{i=1}^N{{\stackrel{\wedge }{\theta }}_i}^{\text{T}}{F}_i^{\text{T}}{x}_{i}tht{e}_{i}t,\text{(19)}& \frac{1}{p}\sum _{i=1}^N{e}_i^{\text{T}}t{G}_i{y}_{i}t{\eta }_i+\frac{1}{p}\sum _{i=1}^N{{\stackrel{\wedge }{\eta }}_i-{\eta }_i}^{\text{T}}{G}_i^{\text{T}}{y}_{i}t{e}_{i}t=\frac{1}{p}\sum _{i=1}^N{{\stackrel{\wedge }{\eta }}_i}^{\text{T}}{G}_i^{\text{T}}{y}_{i}t{e}_{i}t,\end{array}$$so $$\begin{array}{}\text{(20)}& \dot{V}t=\frac{1}{p}\sum _{i=1}^N{e}_i^{\text{T}}t{g}_i{y}_{i}t+\sum _{l=0}^{m-1}{c}_{l}t\sum _{j=1}^N{a}_{ij}^l{\Gamma }_l{y}_{j}t-{\tau }_{l}t+{\varphi }_i{u}_{i}t+d_i^{s}t-ht{f}_i{x}_{i}t-ht\sum _{l=0}^{m-1}{c}_{l}t\sum _{j=1}^N{a}_{ij}^l{\Gamma }_l{x}_{j}t-{\tau }_{l}t-ht{d}_i^{w}t-\dot{h}t{x}_{i}t+\sum _{i=1}^N{{\stackrel{\wedge }{\psi }}_i-{\psi }_i}^{\text{T}}{\dot{\widehat{\psi }}}_i+\widehat{\varpi }-\varpi \dot{\widehat{\varpi }}-\frac{1}{p}\sum _{i=1}^N{{\stackrel{\wedge }{\theta }}_i}^{\text{T}}{F}_i^{\text{T}}{x}_{i}tht{e}_{i}t+\frac{1}{p}\sum _{i=1}^N{{\stackrel{\wedge }{\eta }}_i}^{\text{T}}{G}_i^{\text{T}}{y}_{i}t{e}_{i}t+\frac{1}{p}\widehat{\nu }-{\nu }^{\ast }\dot{\widehat{\nu }}+\frac{1}{2p1-\epsilon }\sum _{l=1}^{m-1}\sum _{i=1}^N{e}_i^{\text{T}}t{e}_{i}t-\frac{1-{\dot{\tau }}_{l}t}{2p1-\epsilon }\sum _{l=1}^{m-1}\sum _{i=1}^N{e}_i^{\text{T}}t-{\tau }_{l}t{e}_{i}t-{\tau }_{l}t.\end{array}$$

Substituting Lemma 8 and Corollary 3 into (20), we can get $$\begin{array}{}\text{(21)}& \dot{V}t\le \frac{1}{p}\sum _{i=1}^N{e}_i^{\text{T}}t{g}_i{y}_{i}t-ht{f}_i{x}_{i}t-\dot{h}t{x}_{i}t+{\alpha }_i-{{\gamma \mu }_i}^{\text{T}}{e}_{i}t+\sum _{i=1}^N{{\stackrel{\wedge }{\psi }}_i-{\psi }_i}^{\text{T}}{\dot{\widehat{\psi }}}_i+\widehat{\varpi }-\varpi \dot{\widehat{\varpi }}-\frac{1}{p}\sum _{i=1}^N{{\stackrel{\wedge }{\theta }}_i}^{\text{T}}{F}_i^{\text{T}}{x}_{i}tht{e}_{i}t+\frac{1}{p}\sum _{i=1}^N{{\stackrel{\wedge }{\eta }}_i}^{\text{T}}{G}_i^{\text{T}}{y}_{i}t{e}_{i}t+\frac{1}{p}\widehat{\nu }-{\nu }^{\ast }\dot{\widehat{\nu }}+\frac{1}{p}\sum _{i=1}^N{e}_i^{\text{T}}t{c}_l\sum _{j=1}^N{a}_{ij}^0{\Gamma }_0{e}_{j}t+c_l\sum _{l=1}^{m-1}\sum _{j=1}^N{a}_{ij}^l{\Gamma }_l{e}_{j}t-{\tau }_{l}t+\frac{1}{2p1-\epsilon }\sum _{l=1}^{m-1}\sum _{i=1}^N{e}_i^{\text{T}}t{e}_{i}t-\frac{1-{\dot{\tau }}_{l}t}{2p1-\epsilon }\sum _{l=1}^{m-1}\sum _{i=1}^N{e}_i^{\text{T}}t-{\tau }_{l}t{e}_{i}t-{\tau }_{l}t.\end{array}$$

Substituting ${\psi }_i={\alpha }_i/p$ and $\varpi =1/p$ into (21), because ${{\stackrel{\wedge }{\eta }}_i}^{\text{T}}{G_i}^{\text{T}}{y}_i{e}_i-{{\stackrel{\wedge }{\theta }}_i}^{\text{T}}{F_i}^{\text{T}}{x}_i{he}_i={{\stackrel{\wedge }{\eta }}_i}^{\text{T}}{G_i}^{\text{T}}{y}_i-{{\stackrel{\wedge }{\theta }}_i}^{\text{T}}{F_i}^{\text{T}}{x}_{i}h{e}_i\le {{\stackrel{\wedge }{\eta }}_i}^{\text{T}}{G_i}^{\text{T}}{y}_i-{{\stackrel{\wedge }{\theta }}_i}^{\text{T}}{F_i}^{\text{T}}{x}_{i}h{e}_i={{\stackrel{\wedge }{\eta }}_i}^{\text{T}}{G_i}^{\text{T}}{y}_i{e}_i-{{\stackrel{\wedge }{\theta }}_i}^{\text{T}}{F_i}^{\text{T}}{x}_{i}h{e}_i,$where $e_i={e_{i1},e_{i2},\cdots ,e_{\mathit{\text{in}}}}^{\text{T}},$ we can get $$\begin{array}{}\text{(22)}& \dot{V}t\le \sum _{i=1}^N\varpi e_i^{\text{T}}t{g}_i{y}_{i}t-ht{f}_i{x}_{i}t-\dot{h}t{x}_{i}t+e_i^{\text{T}}t{\psi }_i-{{\gamma \mu }_i}^{\text{T}}{e}_{i}t+{{\stackrel{\wedge }{\psi }}_i-{\psi }_i}^{\text{T}}{e}_{i}t+\widehat{\varpi }-\varpi \dot{\widehat{\varpi }}+{\varpi }_i\sum _{i=1}^N{{\stackrel{\wedge }{\eta }}_i}^{\text{T}}{G_i}^{\text{T}}{y}_i{e}_i-{{\stackrel{\wedge }{\theta }}_i}^{\text{T}}{F_i}^{\text{T}}{x}_{i}h{e}_i+\varpi \widehat{\nu }-{\nu }^{\ast }\dot{\widehat{\nu }}+\sum _{i=1}^N{\varpi }_i{e}_i^{\text{T}}t{c}_l\sum _{j=1}^N{a}_{ij}^0{\Gamma }_0{e}_{j}t+c_l\sum _{l=1}^{m-1}\sum _{j=1}^N{a}_{ij}^l{\Gamma }_l{e}_{j}t-{\tau }_{l}t+\frac{{\varpi }_i}{21-\epsilon }\sum _{l=1}^{m-1}\sum _{i=1}^N{e}_i^{\text{T}}t{e}_{i}t-\frac{1-{\dot{\tau }}_{l}t}{21-\epsilon }{\varpi }_i\sum _{l=1}^{m-1}\sum _{i=1}^N{e}_i^{\text{T}}t-{\tau }_{l}t{e}_{i}t-{\tau }_{l}t.\end{array}$$

Substituting (10) and (11) into (22), because ${\sum }_{i=1}^N{e^{\text{T}}}_{i}t{hF}_{i}x{\widehat{\theta }}_i={\stackrel{\wedge }{\theta }}^{\text{T}}{F}^{\text{T}}xhe,$${\sum }_{i=1}^N{e^{\text{T}}}_{i}t{G}_{i}y{\widehat{\eta }}_i={\stackrel{\wedge }{\eta }}^{\text{T}}{G}^{\text{T}}ye,$ then, the above formula can be simplified as $$\begin{array}{}\text{(23)}& \dot{V}t\le \sum _{i=1}^N{e}_i^{\text{T}}t{\psi }_i-\gamma e_i^{\text{T}}t{g}_i{y}_{i}t-ht{f}_i{x}_{i}t-\dot{h}t{x}_{i}t+G_{i}y{\widehat{\eta }}_i-ht{F}_{i}x{\widehat{\theta }}_i+\frac{1}{\gamma }\widehat{\nu }{e}_{i}t\widehat{\varpi }+{\widehat{\psi }}_i+{{\stackrel{\wedge }{\psi }}_i-{\psi }_i}^{\text{T}}{e}_{i}t+\widehat{\varpi }\sum _{i=1}^N{e}_i^{\text{T}}t{g}_i{y}_{i}t-ht{f}_i{x}_{i}t-\dot{h}t{x}_{i}t+G_{i}y{\widehat{\eta }}_i-ht{F}_{i}x{\widehat{\theta }}_i+\sum _{i=1}^N\varpi e_i^{\text{T}}t{c}_l\sum _{j=1}^N{a}_{ij}^0{\Gamma }_0{e}_{j}t+c_l\sum _{l=1}^{m-1}\sum _{j=1}^N{a}_{ij}^l{\Gamma }_l{e}_{j}t-{\tau }_{l}t-{\varpi \nu }^{\ast }\sum _{i=1}^N{e}_i^{\text{T}}t{e}_{i}t+\frac{\varpi }{21-\epsilon }\sum _{l=1}^{m-1}\sum _{i=1}^N{e}_i^{\text{T}}t{e}_{i}t-\frac{1-{\dot{\tau }}_{l}t}{21-\epsilon }\varpi \sum _{l=1}^{m-1}\sum _{i=1}^N{e}_i^{\text{T}}t-{\tau }_{l}t{e}_{i}t-{\tau }_{l}t.\end{array}$$

In order to simplify the proof process, $\dot{V}t$ is decomposed into two parts ${\dot{V}}_{1}t$ and ${\dot{V}}_{2}t$: $$\begin{array}{}\text{(24)}& {\dot{V}}_{1}t=\sum _{i=1}^N\frac{1}{p}{e}_i^{\text{T}}t{c}_l\sum _{j=1}^N{a}_{ij}^0{\Gamma }_0{e}_{j}t+c_l\sum _{l=1}^{m-1}\sum _{j=1}^N{a}_{ij}^l{\Gamma }_l{e}_{j}t-{\tau }_{l}t+\frac{1}{2p1-\epsilon }\sum _{l=1}^{m-1}\sum _{i=1}^N{e}_i^{\text{T}}t{e}_{i}t-\frac{1-{\dot{\tau }}_{l}t}{2p1-\epsilon }\sum _{l=1}^{m-1}\sum _{i=1}^N{e}_i^{\text{T}}t-{\tau }_{l}t{e}_{i}t-{\tau }_{l}t-\frac{{\nu }^{\ast }}{p}\sum _{i=1}^N{e}_i^{\text{T}}t{e}_{i}t,\text{(25)}& {\dot{V}}_{2}t=\sum _{i=1}^N{e}_i^{\text{T}}t{\psi }_i-\gamma e_i^{\text{T}}t{g}_i{y}_{i}t-ht{f}_i{x}_{i}t-\dot{h}t{x}_{i}t+G_{i}y{\widehat{\eta }}_i-ht{F}_{i}x{\widehat{\theta }}_i\widehat{\varpi }+{\widehat{\psi }}_i+{{\stackrel{\wedge }{\psi }}_i}^{\text{T}}{e}_{i}t+\widehat{\varpi }\sum _{i=1}^N{e}_i^{\text{T}}t{g}_i{y}_{i}t-ht{f}_i{x}_{i}t-\dot{h}t{x}_{i}t+G_{i}y{\widehat{\eta }}_i-ht{F}_{i}x{\widehat{\theta }}_i.\end{array}$$

Let $et={e_1^{T}t,e_2^{T}t,\cdots ,e_N^{T}t}^T\in R^{N\times 1},B_0=A_0\otimes {\Gamma }_0,B_1=A_1\otimes {\Gamma }_1,\cdots ,B_l=A_l\otimes {\Gamma }_l,$ where $\otimes$ represents the Kronecker product, then, we can get $$\begin{array}{}\text{(26)}& {\dot{V}}_{1}t\le \frac{1}{p}{c}_l{e}^{\text{T}}t{B}_{0}et+c_l\sum _{l=1}^{m-1}{e}^{\text{T}}t{B}_{l}et-{\tau }_{l}t+\frac{1}{21-\epsilon }\sum _{l=1}^{m-1}{e}^{\text{T}}tet-\frac{1}{2}\sum _{l=1}^{m-1}{e}^{\text{T}}t-{\tau }_{l}tet-{\tau }_{l}t-{\nu }^{\ast }{e}^{\text{T}}tet.\end{array}$$

Based on Lemma 6, it is $$\begin{array}{}\text{(27)}& c_l{e}^{\text{T}}t{B}_{l}et-{\tau }_{l}t\le \frac{1}{2}{c}_l^2{e}^{\text{T}}t{B}_l{B}_l^{\text{T}}et+\frac{1}{2}{e}^{\text{T}}t-{\tau }_{l}tet-{\tau }_{l}t,\end{array}$$so we can get $$\begin{array}{}\text{(28)}& {\dot{V}}_{1}t\le \frac{1}{p}{e}^{\text{T}}t{c}_l{B}_0+\frac{1}{2}{c}_l^2\sum _{l=1}^{m-1}{B}_l{B}_l^{\text{T}}et+\sum _{l=1}^{m-1}\frac{1}{21-\epsilon }{e}^{\text{T}}tet-{\nu }^{\ast }{e}^{\text{T}}tet\le \frac{1}{p}{\lambda }_{\max }{c}_l{B}_0+\frac{1}{2}{c}_l^2\sum _{l=1}^{m-1}{B}_l{B}_l^{\text{T}}+\frac{m-1}{21-\epsilon }-{\nu }^{\ast }{e}^{\text{T}}tet.\end{array}$$

Because $p>0$, if ${\nu }^{\ast }\ge {\lambda }_{\max }{c}_l{B}_0+1/2c_l^2{\sum }_{l=1}^{m-1}{B}_l{B}_l^{\text{T}}+m-1/21-\epsilon$, we can get ${\dot{V}}_{1}t\le 0$, where ${\lambda }_{\max }Q$ is the maximum eigenvalue of the matrix $Q$.

Making a simple equation transformation to ${\dot{V}}_{2}t$, we can get $$\begin{array}{}\text{(29)}& {\dot{V}}_{2}t\le \sum _{i=1}^{N}1-\gamma e_i^{\text{T}}t{g}_i{y}_{i}t-ht{f}_i{x}_{i}t-\dot{h}t{x}_{i}t+G_{i}y{\widehat{\eta }}_i-ht{F}_{i}x{\widehat{\theta }}_i\widehat{\varpi }+{\widehat{\psi }}_i\le \sum _{i=1}^{N}1-\gamma e_i^{\text{T}}t{\mu }_i.\end{array}$$

Because $\gamma >1,\varpi =1/p>0,{\psi }_i={\alpha }_i/p>0$, then $$\begin{array}{}\text{(30)}& {\dot{V}}_{2}t\le \sum _{i=1}^{N}1-\gamma e_i^{\text{T}}t{\mu }_i\le 0.\end{array}$$

Based on the above analysis, we can get that $\dot{V}t\le {\dot{V}}_{1}t+{\dot{V}}_{2}t\le 0$ if ${\nu }^{\ast }\ge {\lambda }_{\max }{c}_l{B}_0+1/2c_l^2{\sum }_{l=1}^{m-1}{B}_l{B}_l^{\text{T}}+m-1/21-\epsilon .$ According to Lyapunov stability theory, we can obtain $e_{i}t⟶0$ as $t⟶\infty$, which means that the function projective synchronization between the drive system (1) and the response system (2) is achieved. This completes the proof.

Remark 10.

In the proof of Theorem 9, based on Lyapunov stability theory and inequality transformation method, by introducing Lemma 6 and 8 and some reasonable Assumptions, the controller is designed flexibly without the boundary value ($p$ and $q$) of the sector nonlinear input and the delay term ${\tau }_{l}t$.

Remark 11.

When ${\tau }_{l}t,l=0,1,2,\cdots ,m-1$ is constant or ${\tau }_{0}t={\tau }_{1}t=\cdots ={\tau }_{m-1}t=\tau t$, the multiple time-varying delay couplings are transformed into constant time-delay coupling or single time-varying delay coupling. When ${\tau }_{l}t,l=0,1,\cdots ,m-1$ is constant or ${\tau }_{0}t={\tau }_{1}t=\cdots ={\tau }_{m-1}t=\tau t,$ Assumption 5 is also satisfied, and the control method in this article is also applicable to constant time-delay coupling or single time-varying delay coupling.

Remark 12.

If $ht$ is a constant, the function projection synchronization is transformed into the projection synchronization. In particular, when $ht=1$ or $ht=-1$, the function projection synchronization turns into complete synchronization or antisynchronization.

4. Numerical Simulation

In order to verify the correctness of the theoretical analysis, we select communication network with chaotic nodes as simulation examples. $$\begin{array}{}\text{(31)}& \begin{array}{c}{\dot{x}}_{i1}t\\ {\dot{x}}_{i2}t\\ {\dot{x}}_{i3}t\end{array}=\begin{array}{c}0\\ -x_{i1}t{x}_{i3}t-x_{i2}t\\ x_{i1}t{x}_{i2}t\end{array}+\begin{array}{ccc}-x_{i1}t+x_{i2}t& 0& 0\\ 0& x_{i1}t& 0\\ 0& 0& -x_{i3}t\end{array}\times \begin{array}{c}{\theta }_{i1}\\ {\theta }_{i2}\\ {\theta }_{i3}\end{array}+\begin{array}{c}{d}_{i1}^{v}t\\ d_{i2}^{v}t\\ d_{i3}^{v}t\end{array}+c_{0}t\sum _{j=1}^4{a}_{ij}^0{\Gamma }_0{x}_{j}t+c_{1}t\sum _{j=1}^4{a}_{ij}^1{\Gamma }_1{x}_{j}t-{\tau }_{1}t+c_{2}t\sum _{j=1}^4{a}_{ij}^2{\Gamma }_2{x}_{j}t-{\tau }_{2}t.\hspace{1em}i=1,2,3,4.\end{array}$$

Example 1.

Considering a communication network with $N=4,n=3,$ the drive system is composed of four Lorenz chaotic systems with two different time-varying delay couplings.

The response system is composed of four Chen chaotic systems with two different time-varying delay couplings. $$\begin{array}{}\text{(32)}& \begin{array}{c}{\dot{y}}_{i1}t\\ {\dot{y}}_{i2}t\\ {\dot{y}}_{i3}t\end{array}=\begin{array}{c}0\\ -y_{i1}t{y}_{i3}t\\ y_{i1}t{y}_{i2}t\end{array}+\begin{array}{ccc}{y}_{i2}t-y_{i1}t& 0& 0\\ -y_{i1}t& y_{i2}t+y_{i1}t& 0\\ 0& 0& -y_{i3}t\end{array}\times \begin{array}{c}{\eta }_{i1}\\ {\eta }_{i2}\\ {\eta }_{i3}\end{array}+\begin{array}{c}{d}_{i1}^{s}t\\ d_{i2}^{s}t\\ d_{i3}^{s}t\end{array}+\begin{array}{c}{\varphi }_{i1}{u}_{i1}t\\ {\varphi }_{i2}{u}_{i2}t\\ {\varphi }_{i3}{u}_{i3}t\end{array}+c_{0}t\sum _{j=1}^4{a}_{ij}^0{\Gamma }_0{y}_{j}t+c_{1}t\sum _{j=1}^4{a}_{ij}^1{\Gamma }_1{y}_{j}t-{\tau }_{1}t+c_{2}t\sum _{j=1}^4{a}_{ij}^2{\Gamma }_2{y}_{j}t-{\tau }_{2}t.i=1,2,3,4.\end{array}$$

In MATLAB numerical simulation, set $c_0=c_1=c_2=0.2,{\tau }_{1}t=t/3+t,{\tau }_{2}t=e^t/3+e^t,$$d_{i}t=0.2\cos t,{\Gamma }_0={\Gamma }_1={\Gamma }_2=I^{3\times 3},ht=2\cos 2t.$ The nonlinear input is ${\varphi }_i{u}_{i}t={\varphi }_{i1}{u}_{i1}t,{\varphi }_{i2}{u}_{i2}t{,{\varphi }_{i3}{u}_{i3}t}^T=0.3+0.2\cos u_{i1}t{u}_{i1}t,0.6+0.2\sin u_{i2}t{u}_{i2}t,$${0.4-0.2\cos u_{i3}t{u}_{i3}t}^T.$ The topological structure matrices $A_0,A_1,A_2$ are as follows: $$\begin{array}{}\text{(33)}& A_0=\begin{array}{cccc}-2& 1& 0& 1\\ 0& -2& 1& 1\\ 1& 1& -2& 0\\ 1& 0& 1& -2\end{array},A_1=\begin{array}{cccc}-1& 0& 0& 1\\ 0& -1& 1& 0\\ 0& 0& -1& 1\\ 0& 1& 0& -1\end{array},A_2=\begin{array}{cccc}-1& 0& 1& 0\\ 0& -2& 1& 1\\ 1& 1& -2& 0\\ 0& 1& 0& -1\end{array}.\end{array}$$

And the topology of the driver network and response network is shown in Figure 2.

[figure(s) omitted; refer to PDF]

The MATLAB simulation results are shown in Figures 3–6. It displays that the error signal between the drive system and the response system can stably approach to zero with the designed adaptive controller, that is, the function projection synchronization of the complex dynamic networks is realized. $$\begin{array}{}\text{(34)}& \begin{array}{c}{\dot{x}}_{i1}t\\ {\dot{x}}_{i2}t\\ {\dot{x}}_{i3}t\\ {\dot{x}}_{i4}t\end{array}=\begin{array}{c}0\\ -x_{i1}t{x}_{i3}t-x_{i2}t\\ x_{i1}t{x}_{i2}t\\ -x_{i1}t\end{array}+\begin{array}{cccc}-x_{i1}t+x_{i2}t& 0& x_{i4}t& 0\\ 0& x_{i1}t& 0& 0\\ 0& 0& 0& -x_{i3}t\\ -x_{i4}t& 0& 0& 0\end{array}\times \begin{array}{c}{\theta }_{i1}\\ {\theta }_{i2}\\ {\theta }_{i3}\\ {\theta }_{i4}\end{array}+\begin{array}{c}{d}_{i1}^{v}t\\ d_{i2}^{v}t\\ d_{i3}^{v}t\\ d_{i4}^{v}t\end{array}+c_{0}t\sum _{j=1}^8{a}_{ij}^0{\Gamma }_0{x}_{j}t+c_{1}t\sum _{j=1}^8{a}_{ij}^1{\Gamma }_1{x}_{j}t-{\tau }_{1}t+c_{2}t\sum _{j=1}^8{a}_{ij}^2{\Gamma }_2{x}_{j}t-{\tau }_{2}t+c_{3}t\sum _{j=1}^8{a}_{ij}^3{\Gamma }_3{x}_{j}t-{\tau }_{3}t.i=1,2,\cdots ,8.\end{array}$$

Example 2.

Considering a communication network with $N=8,n=4$, the drive system is composed of eight LS hyperchaotic systems with three different time-varying delay couplings.

The response system is composed of four hyperchaotic systems with three different time-varying delay couplings. $$\begin{array}{}\text{(35)}& \begin{array}{c}{\dot{y}}_{i1}t\\ {\dot{y}}_{i2}t\\ {\dot{y}}_{i3}t\\ {\dot{y}}_{i4}t\end{array}=\begin{array}{c}0\\ -y_{i1}t{y}_{i3}t+y_{i4}t\\ y_{i1}t{y}_{i2}t\\ 0\end{array}+\begin{array}{cccc}{y}_{i2}t-y_{i1}t& 0& 0& 0\\ 0& y_{i1}t& 0& 0\\ 0& 0& -y_{i3}t& 0\\ 0& 0& 0& -y_{i1}t\end{array}\times \begin{array}{c}{\eta }_{i1}\\ {\eta }_{i2}\\ {\eta }_{i3}\\ {\eta }_{i4}\end{array}+\begin{array}{c}{d}_{i1}^{s}t\\ d_{i2}^{s}t\\ d_{i3}^{s}t\\ d_{i4}^{s}t\end{array}+\begin{array}{c}{\varphi }_{i1}{u}_{i1}t\\ {\varphi }_{i2}{u}_{i2}t\\ {\varphi }_{i3}{u}_{i3}t\\ {\varphi }_{i4}{u}_{i4}t\end{array}+c_{0}t\sum _{j=1}^8{a}_{ij}^0{\Gamma }_0{y}_{j}t+c_{1}t\sum _{j=1}^8{a}_{ij}^1{\Gamma }_1{y}_{j}t-{\tau }_{1}t+c_{2}t\sum _{j=1}^8{a}_{ij}^2{\Gamma }_2{y}_{j}t-{\tau }_{2}t+c_{3}t\sum _{j=1}^8{a}_{ij}^3{\Gamma }_3{y}_{j}t-{\tau }_{3}ti=1,2,\cdots ,8.\end{array}$$

To simplify numerical simulation, set $c_0=c_1=c_2=c_3=0.1,{\tau }_{1}t=t/3+t,{\tau }_{2}t=e^t/3+e^t,{\tau }_{3}t=e^t/6+e^t,d_{i}t=0.2\sin t,{\Gamma }_0={\Gamma }_1={\Gamma }_2=I^{4\times 4},ht=2\sin 2t.$ The nonlinear input is ${\varphi }_i{u}_{i}t={\varphi }_{i1}{u}_{i1}t,{\varphi }_{i2}{u}_{i2}t,{\varphi }_{i3}{u}_{i3}t,{{\varphi }_{i4}{u}_{i4}t}^T=0.3+0.2\cos u_{i1}t{u}_{i1}t,0.6+0.2\sin u_{i2}t{u}_{i2}t,0.4-0.2\cos u_{i3}t{u}_{i3}t,{0.4+0.2\sin u_{i4}t{u}_{i4}t}^T.$ The topological structure matrices $A_0,A_1,A_2,A_3$ are as follows: $$\begin{array}{}\text{(36)}& A_0=\begin{array}{cccccccc}-1& 0& 0& 0& 0& 0& 1& 0\\ 0& -1& 0& 0& 0& 1& 0& 0\\ 1& 0& -1& 0& 0& 0& 0& 0\\ 1& 0& 0& -2& 1& 0& 0& 0\\ 0& 1& 0& 0& -2& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 1& 0& 0& 0& -2\end{array},A_1=\begin{array}{cccccccc}-1& 1& 0& 0& 0& 0& 0& 0\\ 0& -1& 1& 0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& -1& 0& 0& 0\\ 1& 1& 0& 0& 0& -2& 0& 0\\ 0& 0& 1& 0& 0& 1& -2& 0\\ 0& 0& 0& 1& 0& 0& 0& -1\end{array},A_2=\begin{array}{cccccccc}-2& 0& 0& 0& 1& 0& 0& 1\\ 0& -1& 1& 0& 0& 0& 0& 0\\ 1& 0& -1& 0& 0& 0& 0& 0\\ 0& 0& 1& -2& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 1& 1& 1& 0& 0& 0& 0& -3\end{array},A_3=\begin{array}{cccccccc}-1& 0& 0& 1& 0& 0& 0& 0\\ 0& -1& 0& 0& 1& 0& 0& 0\\ 0& 0& -1& 1& 0& 0& 0& 0\\ 1& 0& 1& -3& 1& 0& 0& 0\\ 0& 0& 1& 0& -1& 0& 0& 0\\ 1& 0& 0& 0& 0& -2& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 1& 0& 0& 0& 0& -2\end{array}.\end{array}$$

And the topology of the driver network and response network is shown in Figure 7.

The MATLAB simulation results are shown in Figures 8–15. The function projection synchronization can still be achieved when the number of nodes and the system dimension of the complex network are increased, which further verifies the correctness of the theoretical analysis.

5. Conclusion

In this paper, the function projective synchronization of complex dynamic networks with unknown sector nonlinear input, multiple time-varying delay couplings, model uncertainty, and external interferences is studied. Based on Lyapunov stability theory, adaptive control theory, and inequality theory, the robust adaptive controller is formulated to make the drive and response systems synchronize by the function scaling factor. The controller designed in this paper can effectively overcome the effects of unknown sector input and multiple time-varying delays, so it is more general and easier to implement. Our future research work will focus on how to realize the complex network synchronization with other forms of input constraints and how to apply the research results of this paper to the fields of information security.

Acknowledgments

The work was supported in part by the National Natural Science Foundation of China (Grant no. 61775198) and Henan Province Science and Technology research project (Grant nos. 222102210266 and 222102210059).

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