1 Introduction

Nowadays, complex networks have become a subject in many different fields. With the continuous progress of the times, the theoretical achievements of complex networks are remarkable.[1,2,3,4] Its earliest theoretical research can be traced back to the “Seven Bridges Problem” in the 18th century, that is, the land is abstracted as a point, and the bridge of continuous land is abstracted as an edge to form a network.[5] In recent years, complex networks with high-dimensional or dynamic nodes of high-dimensional systems have attracted more and more researchers’ interest due to their potential applications in physics, chemistry, aerospace and other fields.[6,7,8] As a result, exploring complex networks has become an important subject today.

Under random initial conditions, due to the existence of coupling, the nodes in the network may evolve into a global synchronization state, or may eventually stabilize in a partial synchronization state.[9] As a common dynamic behavior of complex networks, synchronization has been studied by many scholars, such as lag synchronization,[10] cluster synchronization,[11] projective synchronization,[12] exponential synchronization,[13]. The common methods of network synchronization control include adaptive control, traction control, intermittent control,[14] etc. Projective synchronization is one of the important styles, which refers to a state in which the drive network and the response network can maintain synchronization under a scale factor difference, that is, when the scale factor is 1, the synchronization between such two networks is called complete synchronization.[15] Its accuracy lies in the mixture of projective synchronization and scale factor, which deepens the complexity of synchronization and plays an important role in secure transmission.[16] In recent years, as an important synchronization way, projective synchronization has been widely used for complex networks. For example, in [17] an adaptive synchronization controller is designed to achieve modified function projective synchronization between presented complex networks with known or unknown parameters, which consists of a vector function with dynamic behavior and nodes with an internal coupling matrix. In [18], taking the projective synchronization of Lorenz chaotic system and Rössler chaotic system as an example, the pole placement method is used to design the controller which has the advantages of simplicity and faster synchronization speed, the projective synchronization of two chaotic systems is realized. Asymptotic time synchronization refers to the behavior that the states of all nodes in the network tend to be consistent with time. Based on its limitations, finite-time synchronization is proposed.[19] Finite-time synchronization represents a concept of time limits. In [20], for the stochastic complex networks with time delays, an adaptive control method is used to achieve finite-time synchronization between networks with the same structures and different structures. In [21], for complex networks with time-varying nodal delays and mixed coupling, a simple discontinuous state feedback controller is used to achieve finite-time synchronization between generalized complex networks. In [22], the finite-time synchronization problem of MNNs with mixed delays is proposed. In this paper, a new method is used for the first time to study the finite-time synchronization problem of this network model, and a very simple sign function controller is applied to solve this problem.

Both networks and systems are affected by human control or environmental factors, thus interfering with the control effect of network or system synchronization. In real life, unknown parameter interference is a common phenomenon. In [23], the complete synchronization of two complex systems with uncertain parameters, coupled structure and interference are considered and the synchronization condition is pointed out. In nature, the future development trend of the systems may be jointly determined by the past state and the current state, [24] so time-delay is one of the inevitable phenomena in the networks. It can be divided into discrete time delay, distributed time delay, and mixed time delay.[25] The application of time-delay systems is also widespread in real life. In [26], a sliding mode predictive control for the fast tracking problem of pure time-delay system was explored. In [27], a vehicle-following complex networks model with optimal speed and time-varying delay under random disturbances was considered and congestion of road vehicle is alleviated by the stability control of this model. In [28], a stochastic competitive system with a time-varying delay was discussed by applying appropriate functionals and the theory of neutral equilibrium differential equations, the probability of the stability of this type of model in an equilibrium state was solved. These references fully demonstrate the importance of time-varying delays.

As mentioned above, dynamical system uncertainty and time delay are two essential factors to be considered in the synchronization of complex networks. Motivated by these points, a more general complex network model combining the uncertainty of dynamic parameters with the common mixed delays such as nodal delay, coupling delay and distributed delay is given and discussed, which makes the network model more realistic. Compared with the reference [17], our network does not need to meet the Lipschitz condition, and by constructing a mixed nonlinear control strategy with adaptive law, it can achieve projective synchronization. So our results will be more general. Compared with the reference [21], the model we consider is more general and we apply the new method in [22] to simply deal with the finite-time synchronization problem. To sum up, in comparison with the relevant results, the main contributions of this article are listed as follows.

\(*\) The models used in this paper are new network models that have never appeared before. We fully consider the influence of nodes, coupling structure and distribution state with or without time delays.

\(*\) Dynamic parameters uncertainty and mixed delays are considered into the time-delay complex network model, which makes our model more versatile.

\(*\) The article constructed model itself is used for numerical simulation to make the stability of the model more easily verified and the model is applied to secure communication to show that the theory we adopt can deal with general practical problems.

The main structure of this paper is as follows. In Sect. 2, some preliminaries and model descriptions are introduced. The main methods and proofs are given in Sect. 3. In Sect. 4, we verify the feasibility of the synchronization under mixed control strategy and controller by both theoretical proof and numerical simulations. Then, Sect. 5 is an application to secure communication. Finally, Sect. 6 is the conclusion of the paper.

2 Preliminaries and Model Descriptions

Throughout this article, let \({\mathbb {I}}=\{1,2,\dots ,N\}\), \({\mathbb {R}}^{q}\) and  \({\mathbb {R}}^{N\times N}\) represent the q-dimensional and \(N\times N\) dimensional Euclidean spaces respectively. The superscript \("-1"\) represents the inverse of a matrix or vector and also the superscript "T" stands for the transpose of a vector or a matrix. \(\Vert \cdot \Vert \) represents the Euclidean vector norm.

According to the theory of network dynamics, the state of each node in the network can be described by a dynamic equation \(\dot{x_{i}}(t)=F(x_{i}(t))\). Consider a linear coupling complex network with nodal delay, coupling delay and distributed delay, and take this network as a drive network. Its state equation is described as

$$\begin{aligned} \dot{x_{i}}(t)= & {} F(x_{i}(t))+f_{1}(x_{i}(t-\tau _{0}(t)))+\sum _{j=1}^{N}b_{ij}H_{1}x_{j}(t)\nonumber \\{} & {} +\!\sum _{j=1}^{N}l_{ij}H_{2}x_{j}(t{-}\tau _{1}(t)){+}\!\sum _{j=1}^{N}d_{ij}\int _{t{-}\tau _{2}(t)}^{t}H_{3}x_{j}(\xi )d\xi \nonumber \\ \end{aligned}$$
(1)

where \(i\in {\mathbb {I}}\); \(x_{i}(t)={(x_{i1}(t),x_{i2}(t),\dots ,x_{iq}(t))}^T\in {\mathbb {R}}^q\) represents the state vector of the ith node at time t. The vector function \(F(\cdot )\): \({\mathbb {R}}^q\rightarrow {\mathbb {R}}^q\) drives the network with unknown parameters, and it is guaranteed to be continuous. \(H_{1}(\cdot ),H_{2}(\cdot ),H_{3}(\cdot ):{\mathbb {R}}^{q\times q}\rightarrow {\mathbb {R}}^{q\times q}\) are the internal coupling matrices in the drive network. The vector function \(f_{1}:{\mathbb {R}}^q\rightarrow {\mathbb {R}}^q\) is a nonlinear smooth function, and it represents the dynamic characteristics of the network.  \(\tau _{0}(t)\ge 0\) is the time-varying delay of the node, \(\tau _{1}(t)\ge 0\) and \(\tau _{2}(t)\ge 0\) are the coupled time-varying delay, and the distributed delay of the system. The non-delay outer-coupling matrix  \(B=(b_{ij})_{N\times {N}}\in {\mathbb {R}}^{N\times {N}}\) and the time-delay outer-coupling matrix  \(L=(l_{ij})_{N\times {N}}\in {\mathbb {R}}^{N\times {N}},~D=(d_{ij})_{N\times {N}}\in {\mathbb {R}}^{N\times {N}}\) describe the topology of the networks and they are required to satisfy the following conditions: if there is non-delay coupling between nodes i and j \((i\ne j)\), then  \(b_{ij}>0\), otherwise \(b_{ij}=0\). Similarly, if the node i and j  \((i\ne {j})\)are coupled with time delay, then \(l_{ij}>0,~d_{ij}>0\), otherwise \(l_{ij}=d_{ij}=0\). In addition, the matrices  B, L and D all satisfy the dissipative coupling condition

$$\begin{aligned} b_{ii}= & {} -\sum \limits _{j=1,j\ne {i}}^{N}b_{ij} =-\sum \limits _{j=1,j\ne {i}}^{N}b_{ji},l_{ii}=-\sum \limits _{j=1,j\ne {i}}^{N}l_{ij}\nonumber \\= & {} -\sum \limits _{j=1,j\ne {i}}^{N}l_{ji},\nonumber \\ d_{ii}= & {} -\sum \limits _{j=1,j\ne {i}}^{N}d_{ij}=-\sum \limits _{j=1,j\ne {i}}^{N}d_{ji} \end{aligned}$$
(2)

where\(i\in {\mathbb {I}}\). Since \(F(\cdot )\) is a dynamic function with unknown parameters, we define \(F(x_{i}(t))=A(x_{i}(t))\alpha +f(x_{i}(t))\), where \(A:{\mathbb {R}}^q\rightarrow {\mathbb {R}}^{q\times m}\) and \(f:{\mathbb {R}}^q\rightarrow {\mathbb {R}}^q\) are two functions of the node state vector,\(\alpha =(\alpha _{1},\alpha _{2},\dots ,\alpha _{m})^{T}\in {\mathbb {R}}^m\) are unknown parameter vectors, and m is the number of unknown parameters. In spite that there is often a certain coupling strength between networks, we default the coupling strength of the time-delay and non-delay parts to be 1 for convenience.

Based on the drive-response concept of synchronization, we give the corresponding response network model

$$\begin{aligned} \dot{y_{i}}(t)= & {} G(y_{i}(t))+f_{1}(y_{i}(t-\tau _{0}(t))) +\sum _{j=1}^{N}b_{ij}H_{1}y_{j}(t)\nonumber \\{} & {} +\sum _{j=1}^{N}l_{ij}H_{2}y_{j}(t-\tau _{1}(t)) \nonumber \\{} & {} +\sum _{j=1}^{N}d_{ij}\int _{t-\tau _{2}(t)}^{t}H_{3}y_{j}(\xi )d\xi +u_{i}(t) \end{aligned}$$
(3)

where \(i\in {\mathbb {I}}\);\(y_{i}(t)={(y_{i1}(t),y_{i2}(t),\dots ,y_{iq}(t))}^T\in {\mathbb {R}}^q\) represents the state vector of the ith node at time t. \(u_{i}(t)={(u_{i1}(t),u_{i2}(t),\dots ,u_{iq}(t))}^T\in {\mathbb {R}}^q\) is a control vector function to be designed. According to the state error vector of the nodes between the drive-response network systems, the states of the nodes in the response network are continuously adjusted. The rest of the variables are represented in the same way as the drive network.

The assumptions and related definition and lemmas needed are given below.

Assumption 1

\(\tau _{i}(t)\) is continuously differentiable, satisfying

$$\begin{aligned} 0<\tau _{i}(t)<\tau _{i},0<\dot{\tau _{i}}(t)<{\mu }_{K}<{\mu }<1,~~i\in {\mathbb {I}} \end{aligned}$$

.

Assumption 2

Assume that the network topology matrix D is satisfied the following bounded conditions

$$\begin{aligned} \sum _{i=1}^{N}(d_{ij})^{2}<d,~~~i\in {\mathbb {I}} \end{aligned}$$

Remark 1

The above two assumptions, that is, the time-delay and the conditions that the matrix satisfies, are assumed to ensure the existence of solutions Eqs. (1) and (3) in the corresponding initial conditions. In addition, many systems with time-delay have this condition, and the authors in [22, 23] make a similar assumption. Therefore, the assumptions in this paper are reasonable and necessary.

Definition 1

[29]  For any continuously differentiable scale function matrix  M(t) , if

$$\begin{aligned} \lim _{t\rightarrow \infty }\ \Vert e_{i}(t)\Vert =\lim _{t\rightarrow \infty }\Vert y_{i}(t)-M(t)x_{i}(t)\Vert =0~~~~~ \end{aligned}$$
(4)

then it is called projective synchronization between the n-dimensional drive network and the m-dimensional response network. Here each line element of the function matrix  \(M(t)=(m_{ij}(t))\in {\mathbb {R}}^{m\times n}\) cannot be 0 at the same time, \(e_{i}(t)\) is the error between the state variables of two network nodes.

Lemma 1

[30]  For any positive definite symmetric matrix \(Q\in R^{n\times n}\) and \(x,y\in R^{n}\) , then

$$\begin{aligned} \pm 2x^{T}y\le \ x^{T}Qx+y^{T}Q^{-1}y. \end{aligned}$$

Lemma 2

[31]  For the vector function \(\gamma (t):[a,b]\rightarrow {\mathbb {R}}^n\), it has

$$\begin{aligned} (\int _{a}^{b}{\gamma (t)dt})^{T}W\int _{a}^{b}{\gamma (t)dt}\le (b-a){\int _{a}^{b}\gamma ^{T}(t)W\gamma (t)dt} \end{aligned}$$

for any symmetric positive definite matrix W.

3 Main Results

3.1 Two networks, same structures

In this section, based on the Lyapunov method, a mixed control strategy is presented and the corresponding theoretical criterion is given.

The errors between the state variables of drive-response network nodes are defined as follows

$$\begin{aligned} e_{i}(t)=[e_{i1}(t),e_{i2}(t),\dots ,e_{iq}(t)]^{T}{=}y_{i}(t){-}M(t)x_{i}(t) \end{aligned}$$
(5)

where \(i\in {\mathbb {I}}\), \(M(t)\in {\mathbb {R}}^{q\times q}\) and the derivative of this error can be obtained by

$$\begin{aligned} \dot{e_{i}}(t)=\dot{y_{i}}(t)-{\dot{M}}(t)x_{i}(t)-M(t)\dot{x_{i}}(t) \end{aligned}$$
(6)

Tidy up to get

$$\begin{aligned} \dot{e_{i}}(t)= & {} G(y_{i}(t))+f_{1}(y_{i}(t-\tau _{0}(t))) +\sum _{j=1}^{N}b_{ij}H_{1}y_{j}(t)\nonumber \\{} & {} +\sum _{j=1}^{N}l_{ij}H_{2}y_{j}(t-\tau _{1}(t))\nonumber \\{} & {} +\sum _{j=1}^{N}d_{ij}\int _{t-\tau _{2}(t)}^{t}H_{3}y_{j} (\xi )d\xi -{\dot{M}}(t)x_{i}(t)\nonumber \\{} & {} -M(t)[A(x_{i}(t))\alpha +f(x_{i}(t))\nonumber \\{} & {} +f_{1}(x_{i}(t-\tau _{0}(t)))+\sum _{j=1}^{N}b_{ij}H_{1}x_{j}(t)\nonumber \\{} & {} +\sum _{j=1}^{N}l_{ij}H_{2}x_{j}(t-\tau _{1}(t))\nonumber \\{} & {} +\sum _{j=1}^{N}d_{ij}\int _{t-\tau _{2}(t)}^{t}H_{3}x_{j}(\xi )d\xi ]+u_{i}(t) \end{aligned}$$
(7)

The mixed control strategy made up of nonlinear and adaptive law is given as

$$\begin{aligned} u_{i}(t)=u_{i1}(t)+u_{i2}(t)+u_{i3}(t) \end{aligned}$$
(8)

where

$$\begin{aligned} \left\{ \begin{aligned} u_{i1}(t)&=-G(y_{i}(t))+M(t)f(x_{i}(t))\\&+{\dot{M}}(t)x_{i}(t)-f_{1}(y_{i}(t-\tau _{0}(t))) \\ {}&+M(t)f_{1}(x_{i}(t-\tau _{0}(t)))+M(t)A(x_{i}(t)){\hat{\alpha }}(t)\\ u_{i2}(t)&=\sum _{j=1}^{N}\widetilde{\upsilon _{ij}^{1}}H_{1}y_{j}(t) +\sum _{j=1}^{N}\widetilde{\upsilon _{ij}^{2}}H_{2}y_{j}(t-\tau _{1}(t))\\&+\sum _{j=1}^{N}\widetilde{\upsilon _{ij}^{3}}\int _{t-\tau _{2}(t)}^{t}H_{3}y_{j}(\xi )d\xi \\ u_{i3}(t)&=-E_{i}(t)e_{i}(t) \end{aligned} \right. \end{aligned}$$
(9)

which satisfy

$$\begin{aligned} \left\{ \begin{aligned}&\dot{{\hat{\alpha }}}(t)=\sum _{j=1}^{N}-M(t)A^{T}(x_{i}(t))e_{i}(t)\\&\dot{E_{i}}(t)=\delta _{i}e_{i}^{T}(t)e_{i}(t)\\&\dot{\widetilde{\upsilon _{ij}^{1}}}=-e_{i}^{T}(t)PH_{1}y_{j}(t)\\&\dot{\widetilde{\upsilon _{ij}^{2}}}=-e_{i}^{T}(t)PH_{2}y_{j}(t-\tau _{1}(t))\\&\dot{\widetilde{\upsilon _{ij}^{3}}}=-e_{i}^{T}(t)P\int _{t-\tau _{2}(t)}^{t}H_{3}y_{j}(\xi )d\xi \end{aligned} \right. \end{aligned}$$
(10)

Remark 2

Equation (10) in order to satisfy the conditions needed in the application of Lyapunov function in the following.

Here \(\delta _{i}>0\) is arbitrary constant and \(P=diag(p_{1},p_{2}, \)\( \dots ,p_{q})\) is a positive definite diagonal matrix. Then the following theorem is obtained.

Theorem 1

For drive system Eq. (1) and response system Eq. (3) under Assumptions 1–2. Assume there is a symmetric positive definite matrix \(Q\in {\mathbb {R}}^{q\times q}\) such that

$$\begin{aligned}{} & {} \lambda _{max}(B\otimes H_{1})+d^{2}\varepsilon \lambda _{max}(I)\\{} & {} +\lambda _{max}(Q)-\lambda _{max}(\Upsilon )+\lambda _{max}(L\otimes H_{2})<0\\{} & {} \varepsilon ^{-1}N-1<0 \end{aligned}$$

where \(\lambda _{max}(\cdot )\) denote the maximum eigenvalues, \(\bar{d_{i}}>0\) is an undetermined sufficiently large positive number and \(\varepsilon \) also a positive number, \(I\in {\mathbb {R}}^{q\times q}\) is an identity matrix and \(\Upsilon =diag(\bar{d_{1}},\bar{d_{2}},\dots ,\bar{d_{q}})\), then the drive-response systems achieve projective synchronization under the mixed control strategy Eq. (9) with adaptive law Eq. (10).

Proof

Choose the following Lyapunov function

$$\begin{aligned} V(t)&=\frac{1}{2}\sum _{i=1}^{N}e_{i}^{T}(t)Pe_{i}(t) +\frac{1}{2}\sum _{i=1}^{N}\tilde{\alpha _{i}}^{T}(t)P{\tilde{\alpha }}_{i}(t)\nonumber \\&+\sum _{i=1}^{N}p_{i}\int _{t-\tau _{1}(t)}^{t}\tilde{e_{i}}^{T}(\theta )Q{\tilde{e}}_{i}(\theta )d\theta \nonumber \\&+\frac{1}{2}\sum _{i=1}^{N}\sum _{j=1}^{N}(2b_{ij} +{\widetilde{\upsilon _{ij}^{1}}})^{2} +\frac{1}{2}\sum _{i=1}^{N}\sum _{j=1}^{N}(2l_{ij}+{\widetilde{\upsilon _{ij}^{2}}})^{2}\nonumber \\&+\frac{1}{2}\sum _{i=1}^{N}\sum _{j=1}^{N}(3d_{ij}+{\widetilde{\upsilon _{ij}^{3}}})^{2}\nonumber \\&+\tau _{2}\sum _{i=1}^{N}\int _{-\tau _{2}}^{t} d\theta \int _{\theta +t}^{\theta }[e^{T}_{i}(\xi )H_{3}^{T}H_{3}e_{i}(\xi )]d\xi \nonumber \\&+\frac{1}{2}\sum _{i=1}^{N}\frac{1}{\delta _{i}}(E_{i}(t)-\bar{d_{i}})^{2} \end{aligned}$$
(11)

where:

$$\begin{aligned}{} & {} e_{i}(t)=(e_{i1}(t),e_{i2}(t),\dots ,e_{iq}(t))^{T}\nonumber \\{} & {} \tilde{e_{i}}(t)=(e_{1i}(t),e_{2i}(t),\dots ,e_{qi}(t))^{T}\nonumber \\{} & {} \tilde{\alpha _{i}}(t)=(\alpha _{i1}(t),\alpha _{i2}(t),\dots ,\alpha _{im}(t))^{T}\nonumber \\{} & {} \tilde{\alpha _{i}}(t)=\hat{\alpha _{i}}(t)-\alpha _{i}(t) \end{aligned}$$
(12)

the derivation along the error system can be calculated as follows

$$\begin{aligned} {\dot{V}}(t)= & {} \!\sum _{i=1}^{N}e_{i}^{T}(t)P\dot{e_{i}}(t){+}\!\sum _{i=1}^{N}{\tilde{\alpha }}_{i}^{T}(t)\dot{\tilde{\alpha _{i}}}(t) {+}\!\sum _{i=1}^{N}p_{i}{\tilde{e}}_{i}^{T}(t)Q{{\tilde{e}}_{i}}(t)\nonumber \\{} & {} -\sum _{i=1}^{N}{p_{i}}({1-{\mu }})\tilde{e_{i}}^{T}(t-\tau _{1}(t))Q\tilde{e_{i}}(t-\tau _{1}(t))\nonumber \\{} & {} +\sum _{i=1}^{N}\sum _{j=1}^{N}(2b_{ij}+{\widetilde{\upsilon _{ij}^{1}}})\cdot \dot{{\widetilde{\upsilon _{ij}^{1}}}}\nonumber \\{} & {} +\sum _{i=1}^{N}\sum _{j=1}^{N}(2l_{ij}{+}{\widetilde{\upsilon _{ij}^{2}}})\cdot \dot{{\widetilde{\upsilon _{ij}^{2}}}} {+}\sum _{i=1}^{N}\sum _{j=1}^{N}(3d_{ij}{+}{\widetilde{\upsilon _{ij}^{3}}})\cdot \dot{{\widetilde{\upsilon _{ij}^{3}}}}\nonumber \\{} & {} -\tau _{2}\sum _{i=1}^{N}\int _{t-\tau _{2}(t)}^{t}e^{T}_{i} (\xi )H_{3}^{T}H_{3}e_{i}(\xi )d\xi \nonumber \\{} & {} +\sum _{i=1}^{N}(E_{i}(t)-\bar{d_{i}})\cdot \dot{E_{i}}(t) \end{aligned}$$
(13)

Here

$$\begin{aligned} \sum _{i=1}^{N}e_{i}^{T}(t)P\dot{e_{i}}(t)= & {} \sum _{i=1}^{N}e_{i}^{T}(t)P[G(y_{i}(t)) +f_{1}(y_{i}(t-\tau _{0}(t)))\nonumber \\ {}{} & {} + \sum _{j=1}^{N}b_{ij}H_{1}y_{j}(t) {+}\sum _{j=1}^{N}l_{ij}H_{2}y_{j}(t{-}\tau _{1}(t)) \nonumber \\{} & {} +\sum _{j=1}^{N}d_{ij}\int _{t-\tau _{2}(t)}^{t}H_{3}y_{j}(\xi )d\xi \nonumber \\{} & {} -{\dot{M}}(t)x_{i}(t){-} M(t)(A(x_{i}(t))\alpha {+}f(x_{i}(t))\nonumber \\{} & {} +f_{1}(x_{i}(t-\tau _{0}(t)))\nonumber \\{} & {} +\sum _{j=1}^{N}b_{ij}H_{1}x_{j}(t)+\sum _{j=1}^{N}l_{ij}H_{2}x_{j}(t-\tau _{1}(t)) \nonumber \\ {}{} & {} +\sum _{j=1}^{N}d_{ij}\int _{t{-}\tau _{2}(t)}^{t}H_{3}x_{j}(\xi )d\xi ){+}u_{i}(t)] \nonumber \\ \end{aligned}$$
(14)

Combine together under assumption 1–2 to get

$$\begin{aligned} {\dot{V}}(t)= & {} \sum _{i=1}^{N}e_{i}^{T}(t)P[G(y_{i}(t))+f_{1}(y_{i}(t-\tau _{0}(t)))\nonumber \\{} & {} +\sum _{j=1}^{N}b_{ij}H_{1}y_{j}(t)+\sum _{j=1}^{N}l_{ij}H_{2}y_{j}(t-\tau _{1}(t))\nonumber \\{} & {} +\sum _{j=1}^{N}d_{ij}\int _{t-\tau _{2}(t)}^{t}H_{3}y_{j}(\xi )d\xi -{\dot{M}}(t)x_{i}(t)\nonumber \\- & {} M(t)(A(x_{i}(t))\alpha +f(x_{i}(t)) +f_{1}(x_{i}(t-\tau _{0}(t)))\nonumber \\{} & {} +\sum _{j=1}^{N}b_{ij}H_{1}x_{j}(t)+\sum _{j=1}^{N}l_{ij}H_{2}x_{j}(t-\tau _{1}(t)) \nonumber \\{} & {} +\sum _{j=1}^{N}d_{ij}\int _{t-\tau _{2}(t)}^{t}H_{3}x_{j}(\xi )d\xi )+u_{i}(t)]\nonumber \\{} & {} +\sum _{i=1}^{N}\tilde{\alpha _{i}}^{T}P(-\sum _{i=1}^{N}M(t)A^{T}(x_{i}(t))e_{i}(t))\nonumber \\{} & {} +\sum _{i=1}^{N}p_{i}\tilde{e_{i}}(t)Q\tilde{e_{i}}(t)\nonumber \\{} & {} -\sum _{i=1}^{N}{p_{i}}({1-{\mu }})\tilde{e_{i}}^{T}(t-\tau _{1}(t))Q\tilde{e_{i}}(t-\tau _{1}(t))\nonumber \\{} & {} +\sum _{i=1}^{N}\sum _{j=1}^{N}(2b_{ij}+{\widetilde{\upsilon _{ij}^{1}}})\cdot [-e_{i}^{T}(t)PH_{1}y_{j}(t)]\nonumber \\{} & {} +\sum _{i=1}^{N}\sum _{j=1}^{N}(2l_{ij}+{\widetilde{\upsilon _{ij}^{2}}})\cdot [-e_{i}^{T}(t)PH_{2}y_{j}(t-\tau _{1}(t))]\nonumber \\{} & {} +\sum _{i=1}^{N}\sum _{j=1}^{N}(3d_{ij}+{\widetilde{\upsilon _{ij}^{3}}})\cdot [-e_{i}^{T}(t)P\int _{t-\tau _{2}(t)}^{t}H_{3}y_{j}(\xi )d\xi ]\nonumber \\{} & {} -\tau _{2}\sum _{i=1}^{N}\int _{t-\tau _{2}(t)}^{t}e^{T}_{i}(\xi )H_{3}^{T}H_{3}e_{i}(\xi )d\xi \nonumber \\{} & {} +\sum _{i=1}^{N}(E_{i}(t)-\bar{d_{i}})\cdot {e_{i}(t)}^{T}e_{i}(t) \end{aligned}$$
(15)

According to Lemmas 1 and 2 we have

$$\begin{aligned} {\dot{V}}(t)\le & {} \lambda _{max}(B\otimes H_{1})\sum _{i=1}^{N}p_{i}\tilde{e_{i}}^{T}(t)\tilde{e_{i}}(t)\nonumber \\{} & {} +\lambda _{max}(L\otimes H_{2}) \sum _{i=1}^{N}p_{i}\tilde{e_{i}}^{T}(t)\tilde{e_{i}}(t-\tau _{1}(t))\nonumber \\ {}+ & {} 2\sum _{i=1}^{N}p_{i}{e_{i}}^{T}(t)\sum _{j=1}^{N}d_{ij}\int _{t-\tau _{2}(t)}^{t}H_{3}e_{i}(\xi )d\xi \nonumber \\ {}{} & {} +\sum _{i=1}^{N}p_{i}\tilde{e_{i}}^{T}(t)Q\tilde{e_{i}}(t)\nonumber \\- & {} \sum _{i=1}^{N}(1-{\mu })p_{i}\tilde{e_{i}}^{T}(t-\tau _{1}(t))Q\tilde{e_{i}}(t-\tau _{1}(t)) \nonumber \\ {}- & {} \tau _{2}\sum _{i=1}^{N}\int _{t-\tau _{2}(t)}^{t}e^{T}_{i}(\xi )H_{3}^{T}H_{3}e_{i}(\xi )d\xi \nonumber \\- & {} \sum _{i=1}^{N}{\bar{d}}_{i}{\tilde{{e_{i}}}}^{T}(t)P{\tilde{e_{i}}}(t) \end{aligned}$$
(16)

where the \(\otimes \) is the Kronecker product of the matrices and

$$\begin{aligned} 2\sum _{i=1}^{N}\!\!{} & {} \!\!p_{i}\tilde{e_{i}}^{T}(t)\sum _{j=1}^{N}d_{ij}\int _{t-\tau _{2}(t)}^{t}H_{3}e_{i}(\xi )d\xi \nonumber \\\le & {} 2\sum _{i=1}^{N}\sum _{j=1}^{N}\tilde{e_{i}}^{T}(t)p_{i}d_{ij}\int _{t-\tau _{2}(t)}^{t}H_{3}e_{i}(\xi )d\xi \nonumber \\ {}\le & {} \varepsilon \sum _{i=1}^{N}\sum _{j=1}^{N}\tilde{e_{i}}^{T}(t)p_{i}(d_{ij})^2p_{i}{\tilde{e_{i}}}(t)\nonumber \\{} & {} +\varepsilon ^{-1}\sum _{i=1}^{N}\sum _{j=1}^{N}(\int _{t-\tau _{2}(t)}^{t}H_{3}e_{i} (\xi )d\xi )^{T}\int _{t-\tau _{2}(t)}^{t}H_{3}e_{i}(\xi )d\xi \nonumber \\ {}\le & {} \varepsilon \sum _{i=1}^{N}\sum _{j=1}^{N}\tilde{e_{i}}^{T}(t)p_{i}(d_{ij})^2p_{i}{\tilde{e_{i}}}(t)\nonumber \\{} & {} +\varepsilon ^{-1}\sum _{i=1}^{N}\sum _{j=1}^{N}\tau _{2}\int _{t-\tau _{2}(t)}^{t}e^{T}_{i}(\xi )H_{3}^{T}H_{3}e_{i}(\xi )d\xi \nonumber \\ {}\le & {} d^{2}\varepsilon I\sum _{i=1}^{N}p_{i}{e_{i}}^{T}(t){\tilde{e_{i}}}^{T}(t) +\varepsilon ^{-1}N\tau _{2}\nonumber \\{} & {} \sum _{i=1}^{N}\int _{t-\tau _{2}(t)}^{t}e^{T}_{i}(\xi )H_{3}^{T}H_{3}e_{i}(\xi )d\xi \end{aligned}$$
(17)

sorted it to get

$$\begin{aligned} {\dot{V}}(t)\le & {} \sum _{i=1}^{N}p_{i}\tilde{e_{i}}^{T}(t)[\lambda _{max}(B\otimes H_{1}) +d^{2}\varepsilon \lambda _{max}(I)\nonumber \\{} & {} +\lambda _{max}(Q)-\lambda _{max}(\Upsilon )]\tilde{e_{i}}(t)\nonumber \\ {}{} & {} +\sum _{i=1}^{N}p_{i}\tilde{e_{i}}^{T}(t(\lambda _{max}(L\otimes H_{2})\tilde{e_{i}}(t-\tau _{1}(t))\nonumber \\ {}{} & {} +\sum _{i=1}^{N}p_{i}\tilde{e_{i}}^{T}(t{-}\tau _{1}(t))[{-}\lambda _{max}(Q) ({1{-}\mu })]\tilde{e_{i}}(t{-}\tau _{1}(t))\nonumber \\{} & {} +\tau _{2}(\varepsilon ^{-1}N-1)\sum _{i=1}^{N}\int _{t-\tau _{2}(t)}^{t}e^{T}_{i}(\xi )H_{3}^{T}H_{3}e_{i}(\xi )d\xi \nonumber \\ \end{aligned}$$
(18)

\(\square \)

Here obviously \(-\lambda _{max}(Q)(1-\mu )<0\). Thus when \(\bar{d_{i}}>0\) is large enough, there is \(\lambda _{max}(B\otimes H_{1})+d^{2}\varepsilon \lambda _{max}(I)+\lambda _{max}(Q)-\lambda _{max}(\Upsilon )+\lambda _{max}(L\otimes H_{2})<0\). Therefore \({\dot{V}}(t)<0\) if \(\varepsilon ^{-1}N-1<0\). According to the Lyapunov stability theory and Barbalat’s lemma, when \(t\rightarrow \infty \), the value of the state error vector \(e_{i}(t)=(e_{i1}(t),e_{i2}(t),\dots ,e_{iq}(t))^{T}\rightarrow 0\). It can be concluded that there is a maximum invariant set \({\mathbb {A}}=\{e_{i}=0\,i\in {\mathbb {I}}\}\) in the set \({\mathbb {Z}}=\{e_{i}=0,\Upsilon _{i}(t)=\bar{d_{i}},i\in {\mathbb {I}}\}\) and the trajectory of the error dynamical system running from any initial value will eventually converge globally to set \({\mathbb {Z}}\),at the same time that

$$\begin{aligned} {\lim _{t\rightarrow \infty }}\Vert y_{i}(t)-M(t)x_{i}(t)\Vert =0 \end{aligned}$$

It means that the drive system Eq.(1) and the response system Eq.(3) realize the outer projective synchronization.

3.2 Two networks, different structures

In this section, we change the model a little. Consider a complex network for the drive network with different structures which is described as

$$\begin{aligned} \dot{x_{i}}(t)= & {} F(x_{i}(t))+f_{1}(x_{i}(t-\tau _{0}(t))) +\sum _{j=1}^{N}b_{ij}^{(1)}h_{j}(x_{j}(t))\nonumber \\{} & {} +\sum _{j=1}^{N}l_{ij}^{(1)}h_{j}(x_{j}(t-\tau _{1}(t)))\nonumber \\{} & {} +\sum _{j=1}^{N}d_{ij}^{(1)}\int _{t-\tau _{2}(t)}^{t}h_{j}(x_{j}(s))ds \end{aligned}$$
(19)

and the corresponding node dynamic equation for the response network model

$$\begin{aligned} \dot{y_{i}}(t)= & {} G(y_{i}(t))+f_{1}(y_{i}(t-\tau _{0}(t))) +\sum _{j=1}^{N}b_{ij}^{(2)}h_{j}(y_{j}(t))\nonumber \\{} & {} +\sum _{j=1}^{N}l_{ij}^{(2)}h_{j}(y_{j}(t-\tau _{1}(t)))\nonumber \\{} & {} +\sum _{j=1}^{N}d_{ij}^{(2)}\int _{t-\tau _{2}(t)}^{t}h_{j}(y_{j}(s))ds+u_{i}(t) \end{aligned}$$
(20)

At this point, the networks have different structures. \(h_{j}(\cdot )\in {\mathbb {R}}^{q}\) is a smooth nonlinear function. The initial value of network Eq. (19) is described as \(\phi (s)=(\phi _{1}(s),\phi _{2}(s),\dots \)\( ,\phi _{q}(s))^{T}\in C([-\tau ,0],{\mathbb {R}}^{q})\), where \(\tau =max\{\tau _{0},\tau _{1},\tau _{2}\}\). The same that the initial value of network Eq. (20) such \(\varphi (s)=(\varphi _{1}(s),\varphi _{2}(s),\dots ,\varphi _{q}(s)^{T}\in C([-\tau ,0],{\mathbb {R}}^{q})\). The other representations are the same as the above models. Here we need to add the following assumption and definition.

Assumption 3

For \(\forall x,y\in R\) there exist constants \(\zeta _{i}>0,i\in {\mathbb {I}}\), such that

$$\begin{aligned} \mid h_{i}(y)-h_{i}(x)\mid \le \zeta _{i}\mid y-x\mid \end{aligned}$$

Remark 3

The same under the assumption 3, there exist constants \(\xi _{i},\omega >0\), such that \(\mid h_{i}(x)\mid \le \xi _{i},\mid f_{1}(x)-f_{1}(y)\mid \le \omega \mid x-y\mid \). That is to say that ensure the nonlinear function is continuous and bounded. In this way, the existence of the solution satisfies the networks.

Definition 2

The drive system Eq. (19) is be synchronized with the response system Eq. (20) in finite-time, if there exist constant \(t^{*}>0\), such that

$$\begin{aligned} \lim _{t\rightarrow t^{*}}\ \Vert e_{i}(t)\Vert =\lim _{t\rightarrow t^{*}}\Vert y_{i}(t)-x_{i}(t)\Vert =0,~~t>t^{*} \end{aligned}$$

The errors between the state variables of drive-response network nodes are defined as follows

$$\begin{aligned} e_{i}(t)=y_{i}(t)-x_{i}(t),~~i\in {\mathbb {I}} \end{aligned}$$
(21)

and the derivative of this error can be obtained by

$$\begin{aligned} \dot{e_{i}}(t)=\dot{y_{i}}(t)-\dot{x_{i}}(t) \end{aligned}$$
(22)

Tidy up to get

$$\begin{aligned} \dot{e_{i}}(t)= & {} G(y_{i}(t))-A(x_{i}(t))\alpha (t)-f(x_{i}(t))\nonumber \\{} & {} +f_{1}(e_{i}(t-\tau _{0}(t))) +\sum _{j=1}^{N}b_{ij}^{(1)}g_{j}(e_{j}(t))\nonumber \\ {}{} & {} +\sum _{j=1}^{N}l_{ij}^{(1)}g_{j}(e_{j}(t-\tau _{1}(t)))\nonumber \\{} & {} +\sum _{j=1}^{N}d_{ij}^{(1)}\int _{t-\tau _{2}(t)}^{t}g_{j}(e_{j}(s))ds\nonumber \\{} & {} +\sum _{j=1}^{N}(b_{ij}^{(2)}-b_{ij}^{(1)})h_{j}(y_{j}(t))\nonumber \\ {}{} & {} + \sum _{j=1}^{N}(l_{ij}^{(2)}-l_{ij}^{(1)})h_{j}(y_{j}(t-\tau _{1}(t)))\nonumber \\ {}{} & {} +\sum _{j=1}^{N}(d_{ij}^{(2)}-d_{ij}^{(1)})\int _{t-\tau _{2}(t)}^{t}h_{j}(y_{j}(s))ds+u_{i}(t)\nonumber \\ \end{aligned}$$
(23)

where \(g_{j}(e_{j}(\cdot ))=h_{j}(y_{j}(\cdot ))-h_{j}(x_{j}(\cdot ))\), \(j\in {\mathbb {I}}\). Select the feedback controller that with the delay-independent as follows

$$\begin{aligned} u_{i}(t)= & {} A(x_{i}(t)){\hat{\alpha }}(t)+f(x_{i}(t))-G(y_{i}(t))\nonumber \\{} & {} -\eta e_{i}(t)-\varrho sign(e_{i}(t)) \end{aligned}$$
(24)

Here \(sign(e_{i}(t))=(sign(e_{1}(t)),sign(e_{2}(t)),\dots ,sign\)\((e_{q}(t)))^{T},{\tilde{\alpha }}(t) ={\hat{\alpha }}(t)-{\alpha }(t)({\tilde{\alpha }}(t)=0\),if \(t=0),\dot{{\hat{\alpha }}}(t)=\sum _{i=1}^{N}-A^{T}(x_{i}(t)){\tilde{\alpha }}_{i}(t)\), the normal numbers \(\eta ,\varrho >0\) as the control gains.

Theorem 2

For drive system Eq. (19) and response system Eq. (20) under Assumption 3, if the control gains \(\eta \) and \(\varrho \) satisfy

$$\begin{aligned}{} & {} \eta \ge {\bar{\zeta }}(\Vert {\bar{B}}\Vert _{1}+\frac{\Vert {\bar{L}}\Vert _{1}}{1-\mu }+\tau _{2}\Vert {\bar{D}}\Vert _{1})+\frac{\omega }{1-\mu } \\{} & {} \varrho >{\bar{\xi }}(\Vert B^{(2)}-B^{(1)}\Vert _{\infty }+\Vert L^{(2)}-L^{(1)}\Vert _{\infty }\\{} & {} +\tau _{2}\Vert D^{(2)}-D^{(1)}\Vert _{\infty }) \end{aligned}$$

where \({\bar{B}}=max\{B^{(1)},B^{(2)}\}\), \({\bar{L}}=max\{L^{(1)},L^{(2)}\}\), \({\bar{D}}=max\{D^{(1)},D^{(2)}\}\), \({\bar{\zeta }}=max\{\zeta _{1},\zeta _{2},\dots ,\zeta _{N}\}\), \({\bar{\xi }}=max\{\xi _{1},\xi _{2},\dots ,\xi _{N}\}\). Therefore the drive system Eq. (19) and response system Eq. (20) in finite time synchronization under the controller Eq. (24) and \(t^{*}\) satisfy

$$\begin{aligned} t^{*}< & {} \frac{1}{\beta }[\Vert e(0)\Vert _{1}+\frac{\omega }{1-\mu }\int _{-\tau _{0}(0)}^{0}\Vert e(s)\Vert _{1}ds\\{} & {} +\frac{{\bar{\zeta }}}{1-\mu }\Vert {\bar{L}}\Vert _{1}\int _{-\tau _{1}(0)}^{0}\Vert e(s)\Vert _{1}ds\nonumber \\{} & {} +\frac{{\bar{\zeta }}}{1-\mu }\Vert {\bar{D}}\Vert _{1}\int _{-\tau _{2}(0)}^{0}\int _{s}^{0}\Vert e(w)\Vert _{1}dwds] \end{aligned}$$

where \(\beta =\varrho -{\bar{\xi }}(\Vert B^{(2)}-B^{(1)}\Vert _{\infty } +\Vert L^{(2)}-L^{(1)}\Vert _{\infty }+\tau _{2}\Vert D^{(2)}-D^{(1)}\Vert _{\infty })\)

Proof

Choose the following Lyapunov function

$$\begin{aligned} V(t)= & {} \sum _{i=1}^{N}sign^{T}(e_{i}(t))e_{i}(t)+sign^{T}({\tilde{\alpha }}(t)){\tilde{\alpha }}(t)\nonumber \\{} & {} +\sum _{i=1}^{N}\frac{\omega }{1-\mu }\int _{t-\tau _{0}(t)}^{t}\Vert e_{i}(s)\Vert _{1}ds\nonumber \\{} & {} +\sum _{i=1}^{N}\frac{{\bar{\zeta }}}{1-\mu }\Vert {\bar{L}}\Vert _{1}\int _{t-\tau _{1}(t)}^{t}\Vert e_{i}(s)\Vert _{1}ds\nonumber \\{} & {} +\sum _{i=1}^{N}{\bar{\zeta }}\Vert {\bar{D}}\Vert _{1}\int _{-\tau _{2}(t)}^{0}\int _{t+s}^{t}\Vert e_{i}(w)\Vert _{1}dwds \end{aligned}$$
(25)

the derivation along the error system under the assumption 1 can be calculated as follows

$$\begin{aligned} {\dot{V}}(t)\le & {} \sum _{i=1}^{N}sign^{T}(e_{i}(t))[G(y_{i}(t))-A(x_{i}(t)){\tilde{\alpha }}_{i}(t)\nonumber \\{} & {} -f(x_{i}(t))+f_{1}(e_{i}(t-\tau _{0}(t))) \nonumber \\{} & {} +\sum _{j=1}^{N}{\bar{b}}_{ij}g_{j}(e_{j}(t))+\sum _{j=1}^{N}{\bar{l}}_{ij}g_{j}(e_{j}(t-\tau _{1}(t)))\nonumber \\{} & {} +\sum _{j=1}^{N}{\bar{d}}_{ij}\int _{t-\tau _{2}(t)}^{t}g_{j}(e_{j}(s))ds \nonumber \\{} & {} +\sum _{j=1}^{N}(b_{ij}^{(2)}-b_{ij}^{(1)})h_{j}(y_{j}(t))\nonumber \\{} & {} + \sum _{j=1}^{N}(l_{ij}^{(2)}-l_{ij}^{(1)})h_{j}(y_{j}(t-\tau _{1}(t)))\nonumber \\{} & {} +\sum _{j=1}^{N}(d_{ij}^{(2)}-d_{ij}^{(1)})\int _{t-\tau _{2}(t)}^{t}h_{j}(y_{j}(s))ds+u_{i}(t)]\nonumber \\ {}{} & {} -\sum _{i=1}^{N}sign^{T}({\tilde{\alpha }}_{i}(t))A^{T}(x_{i}(t)){\tilde{\alpha }}_{i}(t)\nonumber \\ {}{} & {} +\sum _{i=1}^{N}\frac{\omega }{1{-}\mu }[\Vert e_{i}(t)\Vert _{1}{-}(1{-}\dot{\tau _{0}}(t))\Vert e_{i}(t{-}\tau _{0}(t))\Vert _{1}]\nonumber \\{} & {} +\!\sum _{i=1}^{N}\Vert {\bar{L}}\Vert _{1}\frac{{\bar{\zeta }}}{1{-}\mu }[\Vert e_{i}(t)\Vert _{1}{-}(1{-}\dot{\tau _{1}}(t))\Vert e_{i}(t{-}\tau _{1}(t))\Vert _{1}]\nonumber \\{} & {} +{\bar{\zeta }}\Vert {\bar{D}}\Vert _{1}\int _{-\tau _{2}}^{0}\Vert e_{i}(t)\Vert _{1}ds-\bar{\zeta }\Vert D^{(1)}\Vert _{1}\nonumber \\{} & {} \times \int _{-\tau _{2}}^{0}\Vert e_{i}(t+s)\Vert _{1}ds \nonumber \\ \end{aligned}$$
(26)

According to assumption 3 we had sorted it to get

$$\begin{aligned} {\dot{V}}(t)\le & {} \sum _{i=1}^{N}sign^{T}(e_{i}(t))[\omega \mid (e_{i}(t-\tau _{0}))\mid \nonumber \\{} & {} +\!\sum _{k=1}^{N}\mid {\bar{b}}_{ik}\mid \zeta _{k}\mid e_{k}(t)\mid {+}\!\sum _{k=1}^{N}\mid {\bar{l}}_{ik}\mid \zeta _{k}\mid e_{k}(t{-}\tau _{1}(t)))\mid \nonumber \\{} & {} +\sum _{k=1}^{N}\mid {\bar{d}}_{ik}\mid \zeta _{k}\int _{t-\tau _{2}}^{t}\mid e_{k}(s)\mid ds\nonumber \\{} & {} +\sum _{k=1}^{N}\mid b_{ik}^{(2)}-b_{ik}^{(1)}\mid h_{k}(y_{k}(t)) \nonumber \\{} & {} +\sum _{k=1}^{N}\mid l_{ik}^{(2)}-l_{ik}^{(1)}\mid h_{k}(y_{k}(t-\tau _{k})) \nonumber \\ {}{} & {} +\sum _{k=1}^{N}\mid d_{ik}^{(2)}-d_{ik}^{(1)}\mid \int _{t-\tau _{2}}^{t}h_{k}(y_{k}(s))ds\nonumber \\{} & {} -\eta \mid e_{i}(t)\mid -\varrho \mid sign(e_{i}(t))\mid ]\nonumber \\ {}{} & {} +\sum _{i=1}^{N}\frac{\omega }{1-\mu }\Vert e_{i}(t)\Vert _{1} -\sum _{i=1}^{N}\omega \Vert e_{i}(t-\tau _{0})\Vert _{1}\nonumber \\{} & {} +\sum _{i=1}^{N}\Vert {\bar{L}}\Vert _{1}\frac{{\bar{\zeta }}}{1-\mu }\Vert e_{i}(t)\Vert _{1} \nonumber \\ {}{} & {} -\sum _{i=1}^{N}\Vert {\bar{L}}\Vert _{1}{\bar{\zeta }}\Vert e_{i}(t-\tau _{1}(t))\Vert _{1} +\sum _{i=1}^{N}\tau _{2}{\bar{\zeta }}\Vert {\bar{D}}\Vert _{1}\Vert e_{i}(t)\Vert _{1} \nonumber \\ {}{} & {} -\sum _{i=1}^{N}{\bar{\zeta }}\Vert {\bar{D}}\Vert _{1}\int _{t-\tau _{2}}^{t}\Vert e_{i}(s)\Vert _{1}ds \end{aligned}$$
(27)

further collation is obtained

$$\begin{aligned} {\dot{V}}(t)\le & {} \omega \Vert e(t{-}\tau _{0})\Vert _{1}+{\bar{\zeta }}\Vert {\bar{B}}\Vert _{1}\Vert e(t)\Vert _{1} {+}{\bar{\zeta }}\Vert {\bar{L}}\Vert _{1}\Vert e(t{-}\tau _{1})\Vert _{1}\nonumber \\{} & {} +{\bar{\zeta }}\Vert {\bar{D}}\Vert _{1}\int _{t-\tau _{2}}^{t}\Vert e(s)\Vert _{1}ds \nonumber \\{} & {} +{\bar{\xi }}\Vert B^{(2)}{-}B^{(1)}\Vert _{\infty }\sum _{i=1}^{N}\chi _{i} {+}{\bar{\xi }}\Vert L^{(2)}{-}L^{(1)}\Vert _{\infty }\sum _{i=1}^{N}\chi _{i}\nonumber \\ {}{} & {} +\tau _{2}{\bar{\xi }}\Vert D^{(2)}{-}D^{(1)}\Vert _{\infty }\sum _{i=1}^{N}\chi _{i} {-}\eta \Vert e(t)\Vert _{1}{-}\varrho \sum _{i=1}^{N}\chi _{i} \nonumber \\ {}{} & {} +\frac{\omega }{1-\mu }\Vert e(t)\Vert _{1}-\omega \Vert e(t-\tau _{0})\Vert _{1}\nonumber \\{} & {} +\Vert {\bar{L}}\Vert _{1}\frac{{\bar{\zeta }}}{1-\mu }\Vert e(t)\Vert _{1}-\Vert {\bar{L}}\Vert _{1}{\bar{\zeta }}\Vert e(t-\tau _{1})\Vert _{1} \nonumber \\ {}{} & {} +\tau _{2}{\bar{\zeta }}\Vert {\bar{D}}\Vert _{1}\Vert e(t)\Vert _{1} {-}{\bar{\zeta }}\Vert {\bar{D}}\Vert _{1}\int _{t{-}\tau _{2}}^{t}\Vert e(s)\Vert _{1}ds \end{aligned}$$
(28)

here \(\chi _{i}=0\) if \(e_{i}(t)=0,i\in {\mathbb {I}}\), otherwise \(\chi _{i}=1\). Therefore,

$$\begin{aligned} {\dot{V}}(t)\le & {} \Vert e(t)\Vert _{1}[{\bar{\zeta }}(\Vert {\bar{B}}\Vert _{1} +\Vert {\bar{L}}\Vert _{1}\frac{1}{1-\mu }\nonumber \\{} & {} +\tau _{2}\Vert {\bar{D}}\Vert _{1})+\frac{\omega }{1-\mu }-\eta ] \nonumber \\{} & {} -\sum _{i=1}^{N}\chi _{i}(-{\bar{\xi }}\Vert B^{(2)} -B^{(1)}\Vert _{\infty }-{\bar{\xi }}\Vert L^{(2)}\nonumber \\{} & {} -L^{(1)}\Vert _{\infty }- \tau _{2}{\bar{\xi }}\Vert D^{(2)}-D^{(1)}\Vert _{\infty }+\varrho ) \end{aligned}$$
(29)

We can get the following inequality if theorem 2 is satisfied,

$$\begin{aligned} {\dot{V}}(t)\le -\beta \sum _{i=1}^{N}\chi _{i} \end{aligned}$$
(30)

where \(\beta =\varrho -({\bar{\xi }}\Vert B^{(2)}-B^{(1)}\Vert _{\infty }+{\bar{\xi }}\Vert L^{(2)}-L^{(1)}\Vert _{\infty }+ \tau _{2}{\bar{\xi }}\Vert D^{(2)}-D^{(1)}\Vert _{\infty })>0\). According in [22], since \(V(t)\ge 0\) and \({\dot{V}}(t)\le 0\), so there must be a constant \(V(0)\ge 0\) when \(t=0\) as well as \(\lim \limits _{t \rightarrow +\infty }V(t)=V(0)\) if \(V(0)\le V(t)\),\(\forall t\ge 0\). At that time there must be exist \(\epsilon =\beta \triangle t\), \(\triangle t\) is a sufficiently small positive constant such that

$$\begin{aligned}{} & {} \mid V(t)-V(0)\mid =V(t)-V(0)<\epsilon , \nonumber \\{} & {} \mid V(t)-V(t+\triangle t)\mid =V(t)-V(t+\triangle t)<\epsilon , ~\forall t\ge t' \nonumber \\ \end{aligned}$$
(31)

here \(\Vert e(t')\Vert _{1}=0\) and \(\Vert e(t)\Vert _{1}=0\)

Remark 3 Proof by contradiction: At the first \(\Vert e(t_{1})\Vert _{1}>0\) and \(\Vert e(t)\Vert _{1}>0\) if there exist \(t_{1}>t'\) for \(t\in [t_{1},t_{1}+\triangle t]\). Since \({\dot{V}}(t)\le -\beta \), integrating both sides of an inequality at the same time such that

$$\begin{aligned} V(t_{1}+\triangle t)-V(t)\le -\beta \triangle t=\epsilon \end{aligned}$$

This contradicts Eq. (31),that is to say that \(\Vert e(t)\Vert _{1}=0\). Second,\(\Vert e(t')\Vert _{1}>0\) if exist sufficiently small positive constant \(\epsilon '\) such that \(\Vert e(t)\Vert _{1}>0\) for \(t\in [t,t+\epsilon ']\), obviously contradicts the formula.

According to definition 2, drive network Eq. (19) and response network Eq. (20) achieve finite time synchronization. Here \(t^{*}=inf\{t>0\mid \Vert e(s)\Vert _{1}=0,s>t\}\). Due to the effect of time delay, we propose \(T=t^{*}+\tau \). Since \({\dot{V}}(t)\le -\beta \) for \(\forall t\in (0,t^{*})\), integrating this inequality we get

$$\begin{aligned} V(t^{*})-V(0)\le -\beta t^{*}\Rightarrow V(t^{*})\le V(0)+\beta t^{*} \end{aligned}$$
(32)

Since \({\dot{V}}(t)\le 0\) for \(\forall t\in (t^{*},T^{*})\), taking the same approach we get \(V(t^{*})\ge V(T^{*})=0\), according to [22] we get that \(V(0)\ge \beta t^{*}\), therefore \(t^{*}\le \frac{V(0)}{\beta }\). \(\square \)

4 Numerical Example

4.1 Example 1

The corresponding numerical simulations are shown in this section. Consider a complex network model with four nodes, its drive system is as follows

$$\begin{aligned} \dot{x_{i}}(t)= & {} F(x_{i}(t))+f_{1}(x_{i}(t-\tau _{0}(t)))+\sum _{j=1}^{4}b_{ij}H_{1}x_{j}(t)\nonumber \\{} & {} +\sum _{j=1}^{4}l_{ij}H_{2}x_{j}(t-\tau _{1}(t)) \nonumber \\{} & {} +\sum _{j=1}^{4}d_{ij}\int _{t-\tau _{2}(t)}^{t}H_{3}x_{j}(\xi )d\xi \nonumber \\ \end{aligned}$$
(33)

where \(F(\cdot )=A(\cdot ){\hat{\alpha }}+f(\cdot )\), \(A(\cdot )=2\cdot tanh(x_{i}(\cdot ))\), \(f(\cdot )=\tanh (x_{i}(\cdot ))\), and take \({\hat{\alpha }}=(a,b,c,d)^{T}\) as the vector estimates of unknown parameters, where the nominal values are \(a=1.0,b=5/8,c=2.8,d=-0.8\). \(f_{1}(\cdot )=\tanh (x_{i}(\cdot ))\) and

$$\begin{aligned} H_{1}=H_{2}=H_{3}=Q=\begin{pmatrix} 1&{}0&{}0&{}0\\ 0&{}1&{}0&{}0\\ 0&{}0&{}1&{}0\\ 0&{}0&{}0&{}1 \end{pmatrix}\nonumber \\ B=(b_{ij})=\begin{pmatrix} \frac{1}{5}&{}0&{}\frac{2}{5}&{}\frac{2}{5}\\ \frac{3}{5}&{}\frac{2}{5}&{}0&{}\frac{2}{5}\\ \ \frac{3}{5}&{}\frac{2}{5}&{}0&{}\frac{2}{5}\\ \frac{2}{5}&{}\frac{2}{5}&{}0&{}\frac{3}{5} \end{pmatrix} \end{aligned}$$
(34)

Similarly, the matrices L,D are expressed as

$$\begin{aligned} L=(l_{ij})=\begin{pmatrix} \frac{3}{5}&{}0&{}\frac{3}{10}&{}\frac{3}{5}\\ 0&{}\frac{3}{5}&{}\frac{3}{5}&{}0\\ 0&{}\frac{3}{5}&{}\frac{3}{5}&{}0\\ \frac{3}{5}&{}\frac{9}{10}&{}\frac{3}{5}&{}0 \end{pmatrix}\nonumber \\ D=(d_{ij})=\begin{pmatrix} \frac{2}{5}&{}0&{}\frac{7}{10}&{}0\\ 0&{}\frac{1}{2}&{}\frac{3}{5}&{}0\\ \frac{3}{5}&{}0&{}0&{}\frac{3}{5}\\ 0&{}0&{}\frac{4}{5}&{}\frac{1}{2} \end{pmatrix} \end{aligned}$$
(35)

The coupling delay is \(\tau _{0}(t)=\tau _{1}(t)=\tau _{2}(t)=0.01+0.02\sin (t)\) for convenience and \(\mu =0.05\). The response system is

$$\begin{aligned} \dot{y_{i}}(t)= & {} G(y_{i}(t))+f_{1}(y_{i}(t-\tau _{0}(t)))+\sum _{j=1}^{4}b_{ij}H_{1}y_{j}(t)\nonumber \\{} & {} +\sum _{j=1}^{4}l_{ij}H_{2}y_{j}(t-\tau _{1}(t))\nonumber \\{} & {} +\sum _{j=1}^{4}d_{ij}\int _{t-\tau _{2}(t)}^{t}H_{3}y_{j}(\xi )d\xi \end{aligned}$$
(36)

where \(G(\cdot )=y_{i}(\cdot )\),\(f_{1}=\tanh (y_{i}(\cdot ))\). The mixed control strategy of the network can be written as

$$\begin{aligned} u_{i1}(t)= & {} -tanh((y_{i}(t)))+M(t)tanh(x_{i}(t))+{\dot{M}}(t)x_{i}(t)\nonumber \\{} & {} -tanh(y_{i}(t-\tau _{0}(t)))\nonumber \\{} & {} +M(t)tanh(x_{i}(t{-}\tau _{0}(t)))\nonumber \\{} & {} +M(t)\cdot 2\cdot tanh(x_{i}(t)){\hat{\alpha }}(t)\nonumber \\ u_{i2}(t)= & {} \sum _{j=1}^{4}\widetilde{\upsilon _{ij}^{1}}H_{1}y_{j}(t) +\sum _{j=1}^{4}\widetilde{\upsilon _{ij}^{2}}H_{2}y_{j}(t-\tau _{1}(t))\nonumber \\{} & {} +\sum _{j=1}^{4}\widetilde{\upsilon _{ij}^{3}}\int _{t-\tau _{2}(t)}^{t}H_{3}y_{j}(\xi )d\xi \nonumber \\ u_{i3}(t)= & {} -E_{i}(t)e_{i}(t) \end{aligned}$$
(37)

with the adaptive parameter updated law

$$\begin{aligned}{} & {} \dot{{\hat{\alpha }}}(t)=\sum _{j=1}^{4}-M(t)\cdot 2\cdot tanh(x_{i}(t))e_{i}(t)\nonumber \\{} & {} \dot{E_{i}}(t)=\delta _{i}e_{i}^{T}(t)e_{i}(t)\nonumber \\{} & {} \dot{\widetilde{\upsilon _{ij}^{1}}}=-e_{i}^{T}(t)PH_{1}y_{j}(t)\nonumber \\{} & {} \dot{\widetilde{\upsilon _{ij}^{2}}}=-e_{i}^{T}(t)PH_{2}y_{j}(t-\tau _{1}(t))\nonumber \\{} & {} \dot{\widetilde{\upsilon _{ij}^{3}}}=-e_{i}^{T}(t)P\int _{t-\tau _{2}(t)}^{t}H_{3}y_{j}(\xi )d\xi \end{aligned}$$
(38)

The scale function is selected as

Fig. 1
figure 1

The trajectories of drive system Eq. (1)

Full size image
$$\begin{aligned} M(t)=\begin{pmatrix} 4+\sin (4t)&{} 0&{} 0&{} 0\\ 0&{} 2\sin (3t)&{} 0&{} 0\\ 0&{} 0&{}0.5+\sin (4t)&{}0\\ 0&{} 0&{}0&{}3\cos (3t) \end{pmatrix}\nonumber \\ \end{aligned}$$
(39)

and the corresponding derivative matrix is

$$\begin{aligned} {\dot{M}}(t)=\begin{pmatrix} 4\cos (4t)&{} 0&{} 0&{} 0\\ 0&{} 6\cos (3t)&{} 0&{} 0\\ 0&{} 0&{}4\cos (4t)&{}0\\ 0&{} 0&{}0&{}-9\sin (3t) \end{pmatrix} \end{aligned}$$
(40)

In the above formula, \(\delta _{i}=2.0\) for \(i=1,2,\dots ,4\), \(P=diag(1,1,1,1)\). According to [32],we take \(\varepsilon =5\), \(d=0.5\),\(\Upsilon =diag(3,4,5,6)\) and by means of calculation, \(\lambda _{max}(Q)=\lambda _{max}(I)=1\),\(\lambda _{max}(B\otimes H_{1})=1.2745,\lambda _{max}(L\otimes H_{2})=1.2,\). Therefore the conditions required by the Theorem 1 are satisfied

The simulations show that the node dynamic behavior can be affected by the mixed control strategy. Figures 1 and 2 show the motion trajectories of the drive system and the response system. Figure 3 shows the motion trajectory of the error uncontrolled system, and it can be seen that the states are not stable. Figure 4 shows the trajectory of the error system under the mixed control strategy. It is not difficult to see that the 4-dimensional node state error tends to zero and remain steady, which means that the drive system and the response system have achieved the outer projective synchronization of the complex networks after a period of time. So this experiment shows that our theoretical result is valid.

Fig. 2
figure 2

The trajectories of response system Eq. (3) under control strategy Eq. (9)

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Fig. 3
figure 3

The trajectories of error system Eq. (7) without control strategy Eq. (9)

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Fig. 4
figure 4

The trajectories of error system Eq. (7) under control strategy Eq. (9)

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4.2 Example 2

Consider a complex network model with two nodes, its drive system is as follows

$$\begin{aligned} \dot{x_{i}}(t)= & {} F(x_{i}(t))+f_{1}(x_{i}(t-\tau _{0}(t))) +\sum _{j=1}^{2}b_{ij}^{(1)}h_{j}(x_{j}(t))\nonumber \\{} & {} +\sum _{j=1}^{2}l_{ij}^{(1)}h_{j}(x_{j}(t-\tau _{1}(t)))\nonumber \\ {}{} & {} +\sum _{j=1}^{2}d_{ij}^{(1)}\int _{t-\tau _{2}(t)}^{t}h_{j}(x_{j}(s))ds \end{aligned}$$
(41)

where \(F(\cdot )=A(\cdot ){\hat{\alpha }}+f(\cdot )\), \(A(\cdot )=2\cdot tanh(x_{i}(\cdot ))\), \(f(\cdot )=\tanh (x_{i}(\cdot ))\), \(h(\cdot )=3tanh(x_{i}(\cdot ))\) and take \({\hat{\alpha }}=(a,b)^{T}\) as the vector estimates of unknown parameters, where the nominal values are \(a=1.0,b=5/8\). \(f_{1}(\cdot )=\tanh (x_{i}(\cdot ))\) and

$$\begin{aligned} B^{(1)}=(b_{ij}^{(1)})=\begin{pmatrix} \frac{1}{2}&{}0\\ \frac{1}{4}&{}\frac{1}{2} \end{pmatrix}\nonumber \\ L^{(1)}=(l_{ij}^{(1)})=\begin{pmatrix} 0&{}\frac{1}{5}\\ \frac{1}{3}&{}\frac{1}{2} \end{pmatrix}\nonumber \\ D^{(1)}=(d_{ij}^{(1)})=\begin{pmatrix} \frac{1}{5}&{}\frac{1}{2}\\ \frac{1}{5}&{}0 \end{pmatrix} \end{aligned}$$
(42)

The response system is

$$\begin{aligned} \dot{y_{i}}(t)= & {} G(y_{i}(t))+f_{1}(y_{i}(t-\tau _{0}(t)))\nonumber \\{} & {} +\sum _{j=1}^{2}b_{ij}^{(2)}h_{j}(y_{j}(t))+\sum _{j=1}^{2}l_{ij}^{(2)}h_{j}(y_{j}(t-\tau _{1}(t)))\nonumber \\ {}{} & {} +\sum _{j=1}^{2}d_{ij}^{(2)}\int _{t-\tau _{2}(t)}^{t}h_{j}(y_{j}(s))ds \end{aligned}$$
(43)

where \(G(\cdot )=y_{i}(\cdot )\), \(f_{1}(\cdot )=tanh(y_{i}(\cdot ))\), \(h(\cdot )=3tanh(y_{i} \)\( (\cdot ))\),\(\tau _{0}(t)=\tau _{1}(t)=\tau _{2}(t)=0.01+0.02\sin (t)\), \(\tau _{0}=\tau _{1}=\tau _{2}=0.03\), \(\omega ={\bar{\zeta }}={\bar{\xi }}=1\), \(\mu =0.05\), \(\eta =3\), \(\varrho =2\). The initial values are \(x(t)=(0.3,2.5)^{T}\), \(y(t)=(0.3,2.5)^{T}\) and

$$\begin{aligned}{} & {} B^{(2)}=(b_{ij}^{(2)})=\begin{pmatrix} \frac{1}{5}&{}0\\ \frac{1}{2}&{}\frac{1}{2} \end{pmatrix} L^{(2)}=(l_{ij}^{(2)})=\begin{pmatrix} 0&{}\frac{1}{2}\\ \frac{1}{6}&{}\frac{1}{4} \end{pmatrix}\nonumber \\{} & {} D^{(2)}=(d_{ij}^{(2)})=\begin{pmatrix} \frac{1}{2}&{}\frac{1}{5}\\ \frac{1}{2}&{}0 \end{pmatrix} \end{aligned}$$
(44)

By calculation, \(\Vert {\bar{B}}\Vert _{1}=0.75\),\(\Vert {\bar{L}}\Vert _{1}=0.75\), \(\Vert {\bar{D}}\Vert _{1}=1\),\(\Vert B^{(2)}-B^{(1)}\Vert _{\infty }=0.3\), \(\Vert L^{(2)}-L^{(1)}\Vert _{\infty }=0.42\),\(\Vert D^{(2)}-D^{(1)}\Vert _{\infty }=0.3\), \(t^{*}\approx 19.97163\), so the conditions of Theorem 2 are satisfied.

The simulations show that the node dynamic behavior can be affected by the controller. Figure 5 shows the motion trajectory of the error uncontrolled system, and it can be seen that the states are not stable. Figure 6 shows the trajectory of the error system under the controller. It is not difficult to see that the 2-dimensional node state error tends to zero and remain steady within \(t^{*}\), which verifies our theoretical results are effective.

Fig. 5
figure 5

The synchronization errors between Eqs. (19) and (20) without controller Eq. (24)

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Fig. 6
figure 6

The synchronization errors between Eqs. (19) and (20) under controller Eq. (24)

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Fig. 7
figure 7

The presented communication system consisting of a transmitter and a receiver

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5 Application to Secure Communication

There will be a lot of information transmission in the actual complex network, information security is becoming more and more important. Therefore, secure communication based on complex networks has attracted more and more attention.[33,34,35] During the encryption process, if there is no decryption key, the document cannot be understood. Secure communication using synchronization between chaotic systems (referred to as chaotic secure communication) is a new concept of secure communication.[36] Referring to the outer projective synchronization in Sect. 3, it is applied to the secure communication scheme shown in the figure for verification.

The secure communication system shown in Fig. 7 consists of a transmitter system and a receiver system. For easy illustration, the transmitted signal is \(m(t)=x(t)+\delta p(t)\). The signal transmitted in the channel is masked by the chaotic signal emitted by the sender to achieve the effect of encryption. Finally, the transmission information is recovered after the chaotic synchronization of the nodes in the network at the receiving end. The transmitter can be described as follows

$$\begin{aligned} \dot{x_{1}}(t)= & {} F(x_{1}(t))+f_{1}(x_{1}(t-\tau _{0}(t)))+\sum _{j=1}^{2}b_{1j}H_{1}x_{j}(t)\nonumber \\{} & {} +\sum _{j=1}^{2}l_{1j}H_{2}x_{j}(t-\tau _{1}(t))\nonumber \\+ & {} \sum _{j=1}^{2}d_{1j}\int _{t-\tau _{2}(t)}^{t}H_{3}x_{j}(\xi )d\xi +\delta p(t) \end{aligned}$$
(45)
$$\begin{aligned} \dot{x_{2}}(t)= & {} F(x_{2}(t))+f_{1}(x_{2}(t-\tau _{0}(t)))+\sum _{j=1}^{2}b_{2j}H_{1}x_{j}(t)\nonumber \\{} & {} +\sum _{j=1}^{2}l_{2j}H_{2}x_{j}(t-\tau _{1}(t))\nonumber \\+ & {} \sum _{j=1}^{2}d_{2j}\int _{t-\tau _{2}(t)}^{t}H_{3}x_{j}(\xi )d\xi +0 \end{aligned}$$
(46)

where \(i,j=1,2.\) Note that p(t) must consume lower power, that is, be weak in comparison with the chaotic carrier. Therefore we take \(\delta =0.01\).

The receiver can be described as follows

$$\begin{aligned} \dot{y_{1}}(t)= & {} G(y_{1}(t))+f_{1}(y_{1}(t-\tau _{0}(t)))\nonumber \\{} & {} +\sum _{j=1}^{2}b_{1j}H_{1}y_{j}(t) +\sum _{j=1}^{2}l_{1j}H_{2}y_{j}(t-\tau _{1}(t))\nonumber \\{} & {} +u_{1}(t)-M(t)y_{1}(t)\nonumber \\ {}{} & {} +\sum _{j=1}^{2}d_{1j}\int _{t-\tau _{2}(t)}^{t}H_{3}y_{j}(\xi )d\xi +m_{1}(t) \end{aligned}$$
(47)
$$\begin{aligned} \dot{y_{2}}(t)= & {} G(y_{2}(t))+f_{1}(y_{2}(t-\tau _{0}(t)))\nonumber \\{} & {} +\sum _{j=1}^{2}b_{2j}H_{1}y_{j}(t) +\sum _{j=1}^{2}l_{2j}H_{2}y_{j}(t-\tau _{1}(t))\nonumber \\{} & {} +u_{2}(t)-M(t)y_{2}(t)\nonumber \\{} & {} +\sum _{j=1}^{2}d_{2j}\int _{t-\tau _{2}(t)}^{t}H_{3}y_{j}(\xi )d\xi +m_{2}(t) \end{aligned}$$
(48)

where

$$\begin{aligned}{} & {} M(t)=\begin{pmatrix} 4+\sin (4t)&{} 0\\ 0&{} 2\sin (3t) \end{pmatrix} \end{aligned}$$
(49)
$$\begin{aligned}{} & {} H_{1}=H_{2}=H_{3}=\begin{pmatrix} 1&{}0\\ 0&{}1 \end{pmatrix} ~~~B=\begin{pmatrix} \frac{1}{5}&{}0\\ \frac{3}{5}&{}\frac{2}{5} \end{pmatrix} \end{aligned}$$
(50)

Similarly, the matrices L,D are expressed as

$$\begin{aligned} L=\begin{pmatrix} \frac{3}{5}&{}0\\ 0&{}\frac{3}{5} \end{pmatrix} ~~~~~~D=\begin{pmatrix} \frac{2}{5}&{}0\\ 0&{}\frac{1}{2} \end{pmatrix} \end{aligned}$$
(51)

and \(m_{1}(t)=x_{1}(t)+\delta p(t)\), \(m_{2}(t)=x_{2}(t)\). Where the system parameters are the same as the numerical Example 1 and take the information message as \(p(t)=sin(0.001t)\). \(m_{1}(t)\)and \(m_{2}(t)\) are the transmitted signal. Due to the fact that the chaotic signal transmitted in the channel is stronger than the transmitted signal, and the latter will be completely covered by the former, it is difficult to be stolen by others by modulating it in the chaotic signal. Driven by \(s(t)=\delta ^{-1}[m_{1}(t)-M(t)y_{1}(t)]\), the chaotic signal at the transmitter and the chaotic signal at the receiver can be approximately synchronized. At the receiver, the signal is extracted by chaotic synchronization as p(t).It can be seen from the Fig. 8 that the error between the transmitted signal and the recovered signal approaches 0 after a period of time. Therefore, the receiver system recovers the transmitted signal quickly and effectively.

Fig. 8
figure 8

Output of continuous signal in chaotic masking secure communication system

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6 Conclusions

A class of complex network models with more general and complex structures was constructed and its the outer projection synchronization and finite-time synchronization were studied. A mixed control strategy with adaptive law was presented to realize the outer projective synchronization. A new controller is used to simplify it to solve some difficult finite-time synchronization problems. For different synchronization methods, the conditions for synchronization are given. In real life, there are more complex network models that can be represented by integer-order or even fractional-order systems. Therefore, constructing new and more complex network models and their synchronization problems are our future research topics.