1 Introduction

In the past decade, synchronization and control of chaotic complex networks have been extensively investigated. The main reason is that complex networks are an effective tool to solve some complex nonlinear dynamical problems in the real world, such as biological networks, social networks, and neural networks [1–5]. Synchronization is a collective behavior phenomenon which not only exists extensively in nature, but also plays an important role in theory and practice [6]. It is important to study the synchronization problem in coupled networks in engineering applications, such as secure communication and signal generator design [7]. In order to deeply understand synchronization phenomena and make full use of them, researchers have carefully studied about the synchronization problems of chaotic complex dynamical networks and made a great deal of progresses [8–13]. However, most of these studies focus on integer chaotic complex networks.

With the development of fractional calculus, the behavior of many systems can be elegantly described by fractional differential systems, such as viscoelasticity, dielectric polarization, quantum evolution of complex systems and fractional kinetics [14–18]. Compared with the classical integer-order models, fractional-order models provide an excellent instrument for the memory and hereditary properties description of various materials and processes [19–21]. Moreover, it would be more accurate and quite better if practical problems are described by fractional-order dynamical systems rather than integer-order ones. Therefore, it would be more valuable and practical to investigate the synchronization of fractional-order chaotic complex networks.

From the viewpoint of engineering applications and channel characteristics, time delays are inherent in a system due to the finite speeds of the switching and transmitting signals. It is well known that time delays may result in oscillatory behaviors or networks instabilities. Hence, the time delays in coupling dynamical nodes have received considerable attention and the synchronization of complex networks with delays has also been extensively studied. It is worth mentioning that most reported synchronization results about networks are concerning discrete time delays [22–24]. Another kind of time delays, namely continuously distributed delays, existing widely in complex networks due to the presence of an amount of parallel pathways of a variety of node sizes and lengths, has started to get research attention in synchronization, and some progress has been made based on Lyapunov theorem by integer models [25–27].

As far as the stability theorems about fractional system are concerned, there is a small number of achievements, such as the fractional Lyapunov direct theorem and the fractional Lyapunov–Krasovskii stable theorem presented by Li et al. [28] and Baleanu et al. [29], respectively. Both of the two stable theorem present a key issue in designing a positive function V and calculating the fractional derivative of the positive function V. It is usually very difficult to construct a positive function V and calculate the fractional derivatives of the positive function V according to the provided fractional system, especially to the fractional system with delays. By now, there is not an effective approach to design a controller to control a fractional system with time delays according to the fractional direct Lyapunov theorem. Therefore, synchronizing fractional chaotic complex networks with distributed delays is still a challenging research spot and related studies are rarely reported.

Motivated by the above discussion, we study the stability condition of fractional chaotic system with distributed delays by extending Lyapunov–Krasovskii stable theorem to fractional system in this paper and design a controller by constructing a positive Lyapunov function V to synchronize complex networks with distributed delays according to the proposed stability theorem. What need to emphasize is that we take integer derivatives instead of fractional derivatives of the positive function V. Finally, numerical simulations are given to demonstrate the effectiveness of the proposed approach.

This paper is organized as follows: Sect. 2 introduces some definitions, lemmas, and properties of fractional calculus and complex networks. In Sect. 3, Lyapunov stable theorem is extended to fractional chaotic complex networks with distributed delays and the sufficient synchronization criteria are derived; in Sect. 4, synchronizing fractional chaotic complex networks is provided to explain the proposed theorem; finally, a conclusion is drawn in Sect. 5.

2 Preliminaries

In this section, we introduce some preliminaries about fractional derivatives and fractional chaotic complex networks.

2.1 Fractional calculus

We first recall some definitions and properties related to fractional derivatives that will be used in this paper.

There are some definitions for fractional derivatives. The commonly used definitions are Grunwald–Letnikov (GL) [30], Riemann–Liouville (RL) and Caputo (C) definition [31]. The advantage of Caputo approach is that the initial conditions for fractional differential equations with Caputo derivative take on the same form as those for integer-order derivative, which have well-understood physical meanings. In this paper, we adopt the Caputo definition for fractional derivative. The Caputo definition can be expressed as:

$$\begin{aligned} {}_a^CD^\alpha _tf(t)=\frac{1}{\Gamma (n-\alpha )}\times \int _a^t(t-\tau )^{-\alpha +n-1}f^{(n)}(\tau ) \mathrm{d}\tau \end{aligned}$$
(1)

where n is the first integer which is not less than \(\alpha \), i.e., \(n-1\le \alpha \le n\) and \(\Gamma (\cdot )\) is the Gamma function.

$$\begin{aligned} f(t)=f(0)+\frac{1}{\Gamma (\alpha )}\times \int _a^t(t-\tau )^{\alpha -1}\left( _a^CD^\alpha _tf(\tau )\right) \mathrm{d}\tau \end{aligned}$$
(2)

A common fractional nonlinear system can be depicted as:

$$\begin{aligned} _a^CD^\alpha _tx(t)=f(x(t)) \end{aligned}$$
(3)

where \(\alpha \in R\) is fractional order, \(x(t)\in R^n\) is state variable vector, and \(f(\cdot )\) is a nonlinear function.

Lemma 1

For matrices X and Y with appropriate dimensions, the following inequality holds for \(\eta >0\)

$$\begin{aligned} X^{T}Y+Y^{T}X\le \eta X^{T}X+\frac{1}{\eta }Y^{T}Y \end{aligned}$$
(4)

2.2 Fractional complex networks

Consider a complex network consisting of N coupled nodes with distributed delays, in which each node is an n-dimensional fractional chaotic dynamical subsystem expressed as:

$$\begin{aligned} \begin{aligned} _{t_{0}}^CD^\alpha _t{x_{i}}(t)=f(x_{i}(t))+c\sum \limits _{i=1}^kg_{ij}\int _{t-\tau _j}^{t}x_{j}(\xi )\mathrm{d}\xi \end{aligned} \end{aligned}$$
(5)

where \(x_{i}(t)=(x_{i1}(t),x_{i2}(t),\ldots ,x_{in}(t))^T \in R^n\) is the state variable of node i at time t, \(c>0\) is coupling strength, the positive constants \(\tau _1,\tau _2,\ldots ,\tau _k(\tau _i>0,i=1,2,\ldots ,k)\) are the time delays, and function \(f(x_{i}(t))\) is real-valued continuous functions satisfying the Lipschitz condition. The matrix \(G=(g_{ij})\in R^{N\times N}\) is the coupling matrix, which denotes the network topology, and is defined as follows: If there is a connection between node i and node j \((i\ne j)\), then \(g_{ij}=g_{ji}>0\); otherwise, \(g_{ij}=g_{ji}=0\), and the diagonal elements of matrix G are defined by: \(g_{ii}=-\sum _{j=1,j\ne i}^{N}g_{ij}, i=1,2,\ldots , N\).

Next, we introduce some useful definitions and lemmas.

Definition 1

Fractional chaotic dynamical network (5) is said to achieve synchronization if

$$\begin{aligned} \begin{aligned} \lim _{ t\rightarrow \infty }|x_{i}(t)-s(t)|=0 \end{aligned} \end{aligned}$$
(6)

where s(t) is a solution of an isolate node, satisfying \(_{t_{0}}^CD^\alpha _t{s}(t)=f(s(t))\). Without loss of generality, throughout this paper, it is assumed that the trajectory s(t) includes a chaotic attractor. Generally, a fractional chaotic complex network cannot achieve asymptotic synchronization by itself, and a local feedback control strategy is to be designed to achieve synchronization. The state equations of the controlled fractional chaotic complex network are described as:

$$\begin{aligned} \begin{aligned} _{t_{0}}^CD^\alpha _t{x_{i}}(t)=f(x_{i}(t))+c\sum \limits _{i=1}^kg_{ij}\int _{t-\tau _j}^{t}x_{j}(\xi )\mathrm{d}\xi +u_{i} \end{aligned} \end{aligned}$$
(7)

Define the synchronization errors as \(e_{i}(t)=x_{i}(t)-s(t)\) and we can get:

$$\begin{aligned}&_{t_{0}}^CD^\alpha _t{e_{i}}(t)=_{t_{0}}^CD^\alpha _t{x_{i}}(t)-_{t_{0}}^CD^\alpha _t{s}(t)\nonumber \\&\quad =f(x_{i}(t))+c\sum \limits _{i=1}^kg_{ij}\int _{t-\tau _j}^{t}x_{j}(\xi )\mathrm{d}\xi -f(s(t))+u_{i}\nonumber \\&\quad =f(x_{i}(t))+c\sum \limits _{i=1}^kg_{ij}\int _{t-\tau _j}^{t}x_{j}(\xi )\mathrm{d}\xi -f(s(t))\nonumber \\&\qquad -c\sum \limits _{i=1}^kg_{ij}\int _{t-\tau _j}^{t}s(\xi )\mathrm{d}\xi +u_{i}\nonumber \\&\quad =\Psi (e_{i}(t))+c\sum \limits _{i=1}^kg_{ij}\int _{t-\tau _j}^{t}e_{j}(\varepsilon )\mathrm{d}\varepsilon +u_{i} \end{aligned}$$
(8)

where \(u_{i}=ke_{i}\). Here k denotes the feedback strength.

The purpose of this paper is to ascertain the appropriate range of the feedback strength k so that all of the nodes in the fractional chaotic complex network can synchronize with the target node s(t) for any initial states. That is to say that all of the states satisfy:

$$\begin{aligned} \lim _{t\rightarrow \infty }|x_{i}(t)-s(t)|=\lim _{t\rightarrow \infty }|e_{i}(t)|=0, \quad \forall i=1,2,\ldots ,N\nonumber \\ \end{aligned}$$
(9)

3 The main result

Theorem 1

The controlled complex network denoted by (7) synchronizes with s(t) under the controller \(u_{i}\), if the synchronizing errors network given by (8) satisfy:

$$\begin{aligned} \zeta + \frac{\tau }{2}\sum _{j=1}^{n} c( |g_{ij}|+|g_{ji}| )\le 0, \quad \forall j=1,2,\ldots ,N\nonumber \\ \end{aligned}$$
(10)

where \(\zeta =\mathrm{max}(\frac{x^{T}(t)f(x(t))}{x^{T}(t)x(t)})\).

Proof

Construct a positive definite Lyapunov function \(V=V_{1}+V_{2}\), where

$$\begin{aligned} V_{1}=\frac{1}{2}\sum \limits _{i=1}^{N}e_{i}^{2} \end{aligned}$$
(11)

and

$$\begin{aligned} V_{2}= & {} \frac{1}{2}\left[ \sum \limits _{i=1}^{N} \frac{1}{\Gamma (\alpha )}\times \int _{0}^{\tau }\int _{t-u}^t(t-\xi )^{\alpha -1}\right. \nonumber \\&\times \left. \left( c|g_{i1}|e_{1}^{2}(\xi ) + c|g_{i2}|e_{2}^{2}(\xi ) + c|g_{i3}|e_{3}^{2}(\xi ) \right. \right. \nonumber \\&\left. \left. + \cdots + c|g_{in}|e_{n}^{2}\right) (\xi ) \mathrm{d}\xi \mathrm{d}u \right] \end{aligned}$$
(12)

Calculate the time derivative of the function \(V_{1}\):

$$\begin{aligned} \begin{aligned} \dot{V}_{1}&=\sum \limits _{i=1}^{N}e_{i}\dot{e}_{i}=\frac{1}{\Gamma (\alpha )}\sum \limits _{i=1}^{N}e_{i}\frac{\mathrm{d}}{\mathrm{d}t}\int _a^t(t-\xi )^{\alpha -1}\left( _a^CD^\alpha _te_{i}(\xi )\right) \mathrm{d}\xi \\&=\lim _{\delta t\rightarrow 0}\frac{1}{\Gamma (\alpha )}\sum \limits _{i=1}^{N}e_{i}\frac{\int _{t-\delta t}^t(t-\xi )^{\alpha -1}\left( _a^CD^\alpha _te_{i}(\xi )\right) \mathrm{d}\xi }{\mathrm{d}t}\\&=\lim _{\delta t\rightarrow 0}\frac{1}{\Gamma (\alpha )}\sum \limits _{i=1}^{N}\frac{\int _{t-\delta t}^t(t-\xi )^{\alpha -1} e_{i}(t)(\Psi (e_{i}(\xi ))+c\sum \nolimits _{i=1}^kg_{ij} \int _{\xi -\tau }^{\xi }(e_{j}(\varepsilon )\mathrm{d}\varepsilon )\mathrm{d}\xi }{\mathrm{d}t}\\&\le \lim _{\delta t\rightarrow 0}\frac{1}{\Gamma (\alpha )}\sum \limits _{i=1}^{N}\frac{\int _{t-\delta t}^t(t-\xi )^{\alpha -1} e_{i}(t)\Psi (e_{i}(\xi ))+\frac{1}{2}c\sum \nolimits _{i=1}^k|g_{ij}|\left( \tau e_{i}^{2}(t)+\frac{1}{\tau }\int _{\xi -\tau }^{\xi }e_{j}(\varepsilon ) \mathrm{d}\varepsilon \int _{\xi -\tau }^{\xi }e_{j}(\varepsilon ) \mathrm{d}\varepsilon \right) \mathrm{d}\xi }{\mathrm{d}t}\\&\le \lim _{\delta t\rightarrow 0}\frac{1}{\Gamma (\alpha )}\sum \limits _{i=1}^{N}\frac{\int _{t-\delta t}^t(t-\xi )^{\alpha -1} e_{i}(t)\Psi (e_{i}(\xi ))+\frac{1}{2}c\sum \limits _{i=1}^k|g_{ij}|\left( \tau e_{i}^{2}(t)+\int _{\xi -\tau }^{\xi }e_{j}^{2}(\varepsilon )\right) \mathrm{d}\varepsilon \mathrm{d}\xi }{\mathrm{d}t}\\ \end{aligned} \end{aligned}$$
(13)

and the time derivative of the function \(V_{2}\):

$$\begin{aligned} \begin{aligned} \dot{V}_{2}&=\frac{{\mathrm{d}V}_{2}}{\mathrm{d}t}=\lim _{\delta t\rightarrow 0}\frac{1}{2} \frac{\sum \nolimits _{i=1}^{N}\frac{1}{\Gamma (\alpha )} \int _{t-\delta t}^t(t-\xi )^{\alpha -1}((c|g_{i1}|\tau e_{1}^{2}(\xi )+ c|g_{i2}|\tau e_{2}^{2}(\xi ) +\cdots +c|g_{in}|\tau e_{n}^{2}(\xi )\mathrm{d}\xi ))}{\mathrm{d}t}\\&\quad -\lim _{\delta t\rightarrow 0}\frac{1}{2} \frac{\sum \nolimits _{i=1}^{N}\frac{1}{\Gamma (\alpha )} \int _{t-\delta t}^t(t-\xi )^{\alpha -1}\int _{t-\tau }^{t} (c|g_{i1}|e_{1}^{2}(\varepsilon )+ c|g_{i2}|e_{1}^{2}(\varepsilon ) +\cdots +c|g_{in}| e_{n}^{2}(\varepsilon ))\mathrm{d}\varepsilon \mathrm{d}\xi }{\mathrm{d}t}\\ \end{aligned} \end{aligned}$$
(14)

We can get:

$$\begin{aligned} \begin{aligned} \dot{V}&=\dot{V}_{1}+\dot{V}_{2}\le \lim _{\delta t\rightarrow 0}\frac{1}{\Gamma (\alpha )}\sum \limits _{i=1}^{N}\frac{\int _{t-\delta t}^t(t-\xi )^{\alpha -1} e_{i}(t)\Psi (e_{i}(\xi ))+\frac{1}{2}c\sum \nolimits _{i=1}^k|g_{ij}|(\tau e_{i}^{2}(t)+\int _{\xi -\tau }^{\xi }e_{j}^{2}(\varepsilon )) \mathrm{d}\varepsilon \mathrm{d}\xi }{\mathrm{d}t}\\&\quad +\lim _{\delta t\rightarrow 0}\frac{1}{2} \frac{\sum \nolimits _{i=1}^{N}\frac{1}{\Gamma (\alpha )} \int _{t-\delta t}^t(t-\xi )^{\alpha -1}((c|g_{i1}|\tau e_{1}^{2}(\xi )+ c|g_{i2}|\tau e_{2}^{2}(\xi ) +\cdots +c|g_{in}|\tau e_{n}^{2}(\xi )\mathrm{d}\xi }{\mathrm{d}t}\\&\quad -\lim _{\delta t\rightarrow 0}\frac{1}{2} \frac{\sum \nolimits _{i=1}^{N}\frac{1}{\Gamma (\alpha )} \int _{t-\delta t}^t(t-\xi )^{\alpha -1}\int _{t-\tau }^{t} (c|g_{i1}|e_{1}^{2}(\varepsilon )+ c|g_{i2}|e_{1}^{2}(\varepsilon ) +\cdots +c|g_{in}| e_{n}^{2}(\varepsilon ))\mathrm{d}\varepsilon \mathrm{d}\xi }{\mathrm{d}t}\\&=\lim _{\delta t\rightarrow 0}\frac{1}{\Gamma (\alpha )}\sum \limits _{i=1}^{N}\frac{\int _{t-\delta t}^t(t-\xi )^{\alpha -1} e_{i}(t)\Psi (e_{i}(\xi ))+\frac{1}{2}\tau c\sum \nolimits _{i=1}^k(|g_{ij}|+|g_{ij}|) e_{i}^{2}(\xi ) \mathrm{d}\xi }{\mathrm{d}t}\\ \end{aligned} \end{aligned}$$
(15)

Because \(e_{i}(t)\Psi (e_{i}(t))+\frac{1}{2}\tau c\sum \nolimits _{i=1}^k(|g_{ij}|+|g_{ij}|) e_{i}^{2}(t)\le (\zeta + \frac{\tau }{2}\sum _{j=1}^{n} c( |g_{ij}|+|g_{ji}| ))e_{i}^{2}(t)\le 0\) for any time t and the function \(\Psi (e_{i}(t))\) satisfies Lipschitz condition, \(e_{i}(\xi )\Psi (e_{i}(t))+\frac{1}{2}\tau c\sum \nolimits _{i=1}^k(|g_{ij}|+|g_{ij}|) e_{i}^{2}(t)\le 0\) and \((t-\xi )^{\alpha -1}>0\) when \(\delta t\longrightarrow 0\) , \(\xi \subseteq [t-\delta t,t]\) and \(0<\alpha \le 1\). Then, it can be obtained:

$$\begin{aligned} \dot{V}= & {} \dot{V}_{1}+\dot{V}_{2}\le \lim _{\delta t\rightarrow 0}\frac{1}{\Gamma (\alpha )}\sum \limits _{i=1}^{N}\frac{\int _{t-\delta t}^t(t-\xi )^{\alpha -1} e_{i}(t)\Psi (e_{i}(\xi ))+\frac{1}{2}\tau c\sum \nolimits _{i=1}^k(|g_{ij}|+|g_{ij}|) e_{i}^{2}(\xi ) \mathrm{d}\xi }{\mathrm{d}t}\nonumber \\\le & {} \lim _{\delta t\rightarrow 0}\frac{1}{\Gamma (\alpha )}\sum \limits _{i=1}^{N}\frac{\int _{t-\delta t}^t(t-\xi )^{\alpha -1} \left( \zeta +\frac{1}{2}\tau c\sum \nolimits _{i=1}^k(|g_{ij}|+|g_{ij}|)\right) e_{i}^{2}(\xi ) \mathrm{d}\xi }{\mathrm{d}t}\le 0 \end{aligned}$$
(16)

According to Lyapunov–Krasovskii stability theorem, we can draw the conclusion that \(|e_{i}(t)|=0, \quad \forall i=1,2,\ldots ,N\) as \({t\rightarrow \infty }\). The proof of theorem 1 is completed. \(\square \)

4 Numerical examples

Fractional Lorenz hyperchaotic system can be described as [32]:

$$\begin{aligned} \begin{aligned}&^C_aD^\alpha _t x_1(t)=10x_2(t)-10x_1(t)+x_4(t) \\&^C_aD^\alpha _t x_2(t)=28x_1(t)-x_2(t)-x_1(t)x_3(t) \\&^C_aD^\alpha _t x_3(t)=x_1(t)x_2(t)-\frac{8}{3}x_3(t) \\&^C_aD^\alpha _t x_4(t)=x_2(t)x_3(t)-x_4(t) \end{aligned} \end{aligned}$$
(17)

The chaotic attractor of system (17) is shown in Fig. 1 when fractional order \(\alpha =0.99\).

Fig. 1
figure 1

The chaotic attractor \(x_{1}--x_{2}\) of fractional Lorenz hyperchaotic system in system (17)

Full size image

In this section, two numerical examples are provided to verify the effectiveness of the derived results.

Example 1

Consider the drive network consisting of six nodes, which can be described as:

$$\begin{aligned} ^C_aD^\alpha _t x_i(t)=f(x_i(t))+\sum _{j=1}^{6}g_{ij}\int _{t-\tau }^{t}x_{j}(\xi )\mathrm{d}\xi +k_{i}e_{i} \end{aligned}$$
(18)

and we can get the synchronizing error network

$$\begin{aligned} \begin{aligned}&^C_aD^\alpha _t e_i(t)=\Psi (e_i(t))+\sum _{j=1}^{6}g_{ij}\int _{t-\tau }^{t}y_{j}(\xi )\mathrm{d}\xi \end{aligned} \end{aligned}$$
(19)

and \(\Psi (e_i(t))\) satisfying:

$$\begin{aligned} ^C_aD^\alpha _t e_{i1}(t)= & {} 10e_{i2}(t)-10e_{i1}(t)+e_{i4}(t)+k_{i}e_{i1} \nonumber \\ ^C_aD^\alpha _t e_{i2}(t)= & {} 28e_{i1}(t)-e_{i2}(t)-e_{i1}(t)x_{i3}(t)\nonumber \\&\quad -x_{i1}(t)e_{i3}(t)+k_{i}e_{i2} \\ ^C_aD^\alpha _t e_{i3}(t)= & {} e_{i1}(t)x_{i2}(t)+x_{i1}(t)e_{i2}(t)-\frac{8}{3}e_{i3}(t)+k_{i}e_{i3} \nonumber \\ ^C_aD^\alpha _t e_{i4}(t)= & {} e_{i2}(t)x_{i3}(t)+x_{i2}(t)e_{i3}(t)-e_{i4}(t)+k_{i}e_{i4}\nonumber \end{aligned}$$
(20)

We can get:

$$\begin{aligned} {e_i^{T}(t)\Psi (e_i(t))}= & {} 38e_{i2}(t)e_{i1}(t)+(k_{i}-10)e_{i1}^{2}(t)\nonumber \\&+\,\,e_{i4}(t)e_{i1}(t)+(k_{i}-1)e_{i2}^{2}(t)\nonumber \\&-\,\,e_{i1}(t)x_{i3}(t)e_{i2}(t)e_{i1}(t)x_{i2}(t)e_{i3}(t)\nonumber \\&+\,\,\left( k_{i}-\frac{8}{3}\right) e_{i3}^{2}(t) +\,e_{i2}(t)x_{i3}(t)e_{i4}(t)\nonumber \\&+\,\,x_{i2}(t)e_{i3}(t)e_{i4}(t)\nonumber \\&+\,\,(k_{i}-1)e_{i4}^{2}(t) \le (9.5+|0.5x_{i3}(t)|\nonumber \\&+\,\,0.5|x_{i2}(t)|+k_{i})e_{i1}^{2}(t)\nonumber \\&+\,\,(18+0.5|x_{i3}(t)|+0.5|x_{i3}(t)|\nonumber \\&+\,\,k_{i})e_{i2}^{2}(t)\nonumber \\&+\,\,(0.5|x_{i2}(t)|-\frac{8}{3}+0.5|x_{i2}(t)|\nonumber \\&+\,\,k_{i})e_{i3}^{2}(t)+(0.5|x_{i3}(t)|\nonumber \\&+\,\,0.5|x_{i2}(t)|-0.5+k_{i})e_{i4}^{2}(t) \end{aligned}$$
(21)

By numerical simulation, we can get: \(\hbox {max}(|x_{1}|) =37.1566\), \(\hbox {max}(|x_{2}|) =19.6114\), \(\hbox {max}(|x_{3}|) =45.7240\) and \(\hbox {max}(|x_{4}|) =232.3502\).

Then, we can get:

$$\begin{aligned}&\frac{{e_i^{T}(t)\Psi (e_i(t))}}{e_i^{T}(t)e_i(t)}\le ((9. 5+\hbox {max}(|0.5x_{i3}(t)|)\nonumber \\&\qquad +\,\,0.5\hbox {max}(|x_{i2}(t)|)+k_{i})e_{i1}^{2}(t)\nonumber \\&\qquad +\,\,(18+\hbox {max}(|x_{i3}(t)|)+k_{i})e_{i2}^{2}(t)\nonumber \\&\qquad +\,\,\left( -\frac{8}{3}+\hbox {max}(|x_{i2}(t)|)+k_{i}\right) e_{i3}^{2}(t)\nonumber \\&\qquad +\,\,(0.5\hbox {max}(|x_{i3}(t)|)+0.5\hbox {max}(|x_{i2}(t)|\nonumber \\&\qquad -\,\,0.5)+k_{i})e_{i4}^{2}(t))\nonumber \\&\qquad /(e_{i1}^{2}(t)+e_{i2}^{2}(t)+e_{i3}^{2}(t)+e_{i4}^{2}(t))\nonumber \\&\quad \le \hbox {max}(9. 5+0.5\hbox {max}(|x_{i3}(t)|))\nonumber \\&\qquad +\, 0.5\hbox {max}(|x_{i2}(t)|),18+\hbox {max}(|x_{i3}(t)|),\nonumber \\&\qquad -\,\,\frac{8}{3}+\hbox {max}(|x_{i2}(t))|,0.5\hbox {max}(|x_{i3}(t)|)\nonumber \\&\qquad +\,\,0.5\hbox {max}(|x_{i2}(t)|-0.5)\nonumber \\&\quad = \hbox {max}(42.1677+k_{i},63.724+k_{i},16.945\nonumber \\&\qquad +\,\,k_{i},32.1677+k_{i})=63.724+k_{i} \end{aligned}$$
(22)

The synchronizing error network can be written as:

$$\begin{aligned}&_{t_{0}}^CD^\alpha _t{e_{i}}(t)=\Psi (e_{i}(t))+c\sum \limits _{i=1}^kg_{ij}\int _{t-\tau }^{t}e_{j}(\varepsilon )\mathrm{d}\varepsilon \nonumber \\&\quad +\,\,ke_{i}(t) \end{aligned}$$
(23)

We choose \(\tau =2\) and the coupling matrix G is assumed as:

$$\begin{aligned} \begin{aligned} G= \left[ \begin{array}{cccccc} -4 &{} 1 &{} 0 &{} 1 &{} 0 &{} 2 \\ 1 &{} -6 &{} 2 &{} 1 &{} 1 &{} 1 \\ 0 &{} 2 &{} -5&{} 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 1 &{} -4 &{} 1 &{} 1 \\ 0 &{} 1 &{} 1 &{} 1 &{} -4 &{} 1 \\ 2 &{} 1 &{} 1 &{} 1 &{} 1 &{} -6\\ \end{array} \right] \end{aligned} \end{aligned}$$
(24)

According to Theorem 1, the error system is gradually stable to zero according to Theorem 1 when \(k_{i}\le -87.724\).

Fig. 2
figure 2

The synchronizing errors \(e_{i1},e_{i2},e_{i3}, { \mathrm and}~\,e_{i4}\) for Example 1 with time t

Full size image

In numerical simulations, the feedback strength is selected as \(k_{i}=-88(i=1,2,\ldots ,6)\) and the delay time is taken as \(\tau =2\). The synchronizing errors \(e_{i}(t)(1\le i \le 6)\) of the network with distributed delays are shown in Fig. 2. The simulation results show the correctness of Theorem 1.

Example 2

Consider a complex network with distributed delays consisting of N identical nodes in Lorenz hyperchaotic system as shown in system (17) and the coupling matrix G is taken as:

$$\begin{aligned} \begin{aligned} G=\left[ \begin{array}{cccccc} -5 &{} 1 &{} 0 &{} 1 &{} 0 &{} 3 \\ 1 &{} -8 &{} 2 &{} 1 &{} 2 &{} 2 \\ 0 &{} 2 &{} -5&{} 0 &{} 1 &{} 2 \\ 1 &{} 1 &{} 0 &{} -5 &{} 2 &{} 1 \\ 0 &{} 2 &{} 1 &{} 2 &{} -6 &{} 1 \\ 3 &{} 2 &{} 2 &{} 1 &{} 1 &{} -9\\ \end{array} \right] \end{aligned} \end{aligned}$$
(25)

According to Theorem 1, the error system is gradually stable to zero when \(k_{i}\le -99.724\). In numerical simulations, the feedback strength is selected as \(k_{i}=-100(i=1,2,\ldots ,6)\) and the delayed time is selected as \(\tau =2\). The synchronizing errors \(e_{i}(t)(1\le i \le 6)\) of the network with distributed delays are shown in Fig. 3. The numerical simulation results show the correctness of Theorem 1.

Fig. 3
figure 3

The synchronizing errors \(e_{i1},e_{i2},e_{i3},{ \mathrm and }~e_{i4}\) for Example 2 with time t

Full size image

5 Conclusion

In this paper, we extend Lyapunov–Krasovskii stable theorem to fractional system with distributed delays. The sufficient condition of synchronizing fractional complex networks with distributed delays is derived based on the proposed approach. Two numerical examples have been given to show the effectiveness of the proposed approach. The proposed approach provides a new approach to analysis and control fractional system with distributed delays.