1 Introduction

Synchronous behavior of networked nonlinear dynamic systems attracted a lot of research effort owing to a wide range of applications, such as swarm robotics [1], secure communication [2,3,4] in recent years. The study of synchronous behavior in complex neural networks has attracted the attention of many scholars [5,6,7,8,9]. For instances, in literature [10], reaction–diffusion neural networks were controlled to achieve synchronization which applies to secure communication. In these works, several neural networks were interconnected by single type neurotransmitter. However, in duplex neural networks, neurons transmit signals by more than one neurotransmitter, such as electrical or chemical types. There are many types of neurotransmitters, and different types of neurotransmitters have different effects. So there are different connection modes between nodes in neural networks, which are multi-layer neural networks. On another hand, asymptotic synchronization is a method which is often adopted by scholars [11]. However, due to better convergence rate of exponentially stable, exponentially stable-based synchronization in networks is concerned.

Interlayer synchronization is a special dynamic behavior of multi-layer networks, which is described as synchronization between each node in a given layer and its corresponding node in another layer. Some achievements have been reported on asymptotic stability-based interlayer synchronization [12, 13]. Research on exponentially stability-based synchronization of networks has also been reported, such as exponentially stability-based synchronization in complex-valued neural networks and delay neural networks [14, 15]. However, so far, few scholars have paid attention to exponentially stable-based interlayer synchronization in DANNs, which has attracted our interest.

Time delays are very common in nature [16, 17]. In networks, there are delays in communication between nodes, which is usually represented by internal communication delays, such as internal communication delays for inertial neural networks [17], internal communication delays in two-layer multiplex networks [18], and internal communication delays for neural networks [19], so it is necessary to consider internal communication delays in the study of DANNs.

In recently, more and more cyber attacks occur in networks, which directly affect people’s work and live [20, 21]. Cyber attacks in communication engineering networks have become hot topics [22, 23]. Due to the application value of artificial neural networks in secure communication, artificial neural networks under cyber attacks have been concerned, such as multiplex-neural networks under cyber attacks [22], deception attacks in master–slave neural networks [23]. Stochastic cyber attacks may lead to communication lag or network congestion in networks. So stochastic internal delays in DANNs are considered when study stochastic cyber attacks.

Due to stochastic cyber attacks and stochastic communication delays in networks, numerous uncertain factors will emerge, which have a significant impact on the stability of networks [24]. At this point, because intelligent adaptive controllers can automatically adjust its own parameters based on the uncertain characteristics of neural networks [25, 26], an intelligent adaptive control method is needed to improve the robustness of the networks [27,28,29,30,31]. Quantitative control is an approach which can be applied to characterize communication constraints in DANNs [32, 33]. On another hand, secure communication about signal encryption is concerned by some scholars [34, 35]. However, secure communication based on exponentially interlayer synchronization in DANNs under random cyber attacks is still an open problem. To resist random uncertainties and cyber attacks which is conceled in transmission of the signals, based on the Lyapunov’s second method, a class of intelligent adaptive quantitative controller can be designed to achieve the exponentially interlayer synchronization and secure communication.

Based on the above analysis, we construct DANNs with stochastic internal communication delays under random cyber attacks. To resist random cyber attacks and uncertain factors, an intelligent adaptive quantization controller is designed to realize the exponential interlayer synchronization in DANNs. At the same time, we obtain some sufficient conditions to guarantee the exponential stable-based synchronization. The design method of the controller is derived. Finally, a secure communication framework based on the synchronous policy is derived. The effectiveness of this secure communication scheme has been validated via an example.

Notation. R characterizes the real number set. Matrix \(H^{T}\) is the transpose matrix of matrix H. \(\left| \cdot \right| \) characterizes absolute value. \(\left\| \cdot \right\| \) is Euclidean norm. \(I_n\) is n-dimensional unit matrix. \({\mathscr {P}r}[ \bullet ]\) is probability. e is the natural index. \({E}[ \bullet ]\) is the expectation. \(\left\{ {\begin{array}{*{20}{c}} {sign\left( w \right) = 1,w > 0^{}}\\ {sign\left( w \right) = 0,w = 0}\\ {sign\left( w \right) = - 1,w < 0} \end{array}} \right. .\)

2 Preliminaries

Consider delay networks composed of N artificial neural networks. The first layer (driving) networks in duplex networks are as follows:

$$\begin{aligned} {\dot{x}_i}\left( t \right) =&- C{x_i}(t) + Af({x_i}(t))\nonumber \\&+(1 - \alpha (t))p\sum \limits _{j = 1}^N {{b_{ij}}\Gamma {x_j}\left( t \right) }\nonumber \\&+ \alpha (t)q\sum \limits _{j = 1}^N {{d_{ij}}\Gamma {x_j}\left( {t - \tau \left( t \right) } \right) },\nonumber \\&i = 1,2, \ldots ,N, \end{aligned}$$
(1)

where \({x_i}\left( t \right) {=}{\left( {{x_{i1}}\left( t \right) ,{x_{i2}}\left( t \right) , \ldots ,{x_{in}}\left( t \right) } \right) ^T}\), A is the weight matrix for coupling between neurons; \(- C{x_i}(t) + Af({x_i}(t))\) are neural sub-networks; \(f\left( {{x_i}\left( t \right) } \right) \) is the neuron activation function of the artificial neural networks as a network node; \({{p > 0}}\) and \({{q > 0}}\) stand for coupling strength; \(\Gamma \) stands for inner coupling; \(B = {\left( {{b_{ij}}} \right) _{N \times N}}\), \(D = {\left( {{d_{ij}}} \right) _{N \times N}}\). \({{{b}}_{ij}} > 0\) represents that there exists a interconnection between neural networks j and i, else \(b_{ij}=0\). Similarly, \({d_{ij}} > 0\) characterizes that there is a stochastic occuring signal transmission delay \(\tau \left( t \right) \) between nodes j and i, else \(d_{ij}=0\). \({b_{ii}} = - \sum _{j = 1,j \ne i}^N {{b_{ij}}}\), \({d_{ii}} = - \sum _{j = 1,j \ne i}^N {{d_{ij}}}\), \(0 \le \tau \left( t \right) \le \tau \) and \(0< \dot{\tau }(t) \le {\tau _M} < 1\). Bernoulli random variable is \(\alpha (t)\). \(\alpha (t)\) satisfies \(\alpha \left( t \right) = \left\{ \begin{array}{l} 1\\ 0 \end{array} \right. \), \(\mathscr {P}r(\alpha (t) = 1)=\alpha ,\mathscr {P}r(\alpha (t) = 0)= 1 - {\alpha }\), \(E(\alpha (t) - \alpha )=0,E(\alpha (t) -\alpha )^2= \alpha (1 -\alpha )\).

The second(response) layer networks in duplex networks are as follows:

$$\begin{aligned} {\dot{y}_i}\left( t \right) =&- C{y_i}(t) + Af({y_i}(t)) \nonumber \\&+ (1 - \alpha ( t ))p\sum \limits _{j = 1}^N {{b_{ij}}\Gamma {y_j}\left( t \right) } \nonumber \\&+\alpha (t)q\sum \limits _{j = 1}^N {{d_{ij}}\Gamma {y_j}\left( {t - \tau \left( t \right) } \right) } \nonumber \\&+ {u_i}\left( t \right) , \end{aligned}$$
(2)

where \({y_i}(t)\) is similar to \({x_i}(t)\); \({u_i}\left( t \right) \) is the designed adaptive quantitative controller. The other parameters are the same as these in (1).

The second layer networks under random cyber attacks are

$$\begin{aligned} {\dot{y}_i}\left( t \right) =&- C{y_i}(t) + Af({y_i}(t)) \nonumber \\&+(1 - \alpha (t))p\sum \limits _{j = 1}^N {{b_{ij}}\Gamma {y_j}\left( t \right) }\nonumber \\&+\alpha (t)q\sum \limits _{j = 1}^N {{d_{ij}}\Gamma {y_j}\left( {t - \tau \left( t \right) } \right) } \nonumber \\&+ {U_i}(t). \end{aligned}$$
(3)

\({U_i}(t)\) is the designed adaptive quantitative controller under stochastic cyber attacks.

Fig. 1
figure 1

Duplex networks

Full size image

In Fig. 1, different colors are used to represent the states of nodes, and the states of nodes within the same layer of networks is generally different. From Fig. 1, when \(\alpha (t)\) changes from 0 to 1, cyber attacks occur and signal transmission delays occur in the networks. When the delays are large enough, it may cause communication interruption between some nodes in duplex networks.

Definition 1

([33]) The quantizer \(Q( \cdot ):R \rightarrow \Delta \) is as follows: quantized set is \(\Delta =\left\{ { \pm {{H}_i}:{{H}_i} = {\rho ^i}{{H}_0},i} \right. \left. {= \pm 1, \pm 2, \ldots } \right\} \cup \left\{ { \pm {{H}_0}} \right\} \cup \left\{ 0 \right\} \), where \(0< \rho < 1\); the constant \({{H}_0} > 0\). Sector bound and quantized density are \(\sigma \) and \(\rho \) respectively, where \(\sigma = \frac{{1 - \rho }}{{1 + \rho }}.\)

$$\begin{aligned} Q(\mathcal {u}(t)) = \left\{ \begin{array}{l} {H_i},\begin{array}{*{20}{c}} {} \end{array}if\begin{array}{*{20}{c}} {} \end{array}\frac{1}{{1 + \sigma }}{H_i}< \mathcal {u}(t) \le \frac{1}{{1 - \sigma }}{H_i}\\ 0,\begin{array}{*{20}{c}} {} \end{array}if\begin{array}{*{20}{c}} {} \end{array}\mathcal {u}(t) = 0,\\ - Q\left( { - \mathcal {u}(t)} \right) ,\begin{array}{*{20}{c}} {} \end{array}if\begin{array}{*{20}{c}} {} \end{array}\mathcal {u}(t) < 0. \end{array} \right. \end{aligned}$$

is a quantization function, where \(\mathcal {u}(t) \in R\). According to [33], a Filippov solution \(\vartheta \in [ - \sigma ,\sigma )\) satisfies \(Q(\mathcal {u}(t)) = (1 + \vartheta )\mathcal {u}(t)\).

Definition 2

([36]) Let \({\omega _i}\left( t \right) \mathrm{{ = }}{y_i}\left( t \right) - {x_i}\left( t \right) \). If

$$\begin{aligned} \left\| {{\omega _i}\left( t \right) } \right\| \le K{e^{ - {\eta _1}t}}, \end{aligned}$$

then networks (1) and (2) or (1) and (3) are exponentially asymptotical interlayer synchronization.

Assumption 1

([37]) Nonlinear function \(h\left( \cdot \right) \) satisfies

$$\begin{aligned} \left\| {h\left( {{\varphi _i}\left( t \right) } \right) } \right\| \le {M_i}, \end{aligned}$$

where \({M_i} > 0\).

Assumption 2

([38]) Function \(-C\lambda _i(t)+f(\lambda _i(t))\) and \(-C\theta _i(t)+f(\theta _i(t))\) satisfies

$$\begin{aligned}&{\left( {{\lambda _i}\left( t \right) - {\theta _i}\left( t \right) } \right) ^T}\left\{ -C{\lambda _i}(t)+ A f({{\lambda _i}(t)})\right. \\&\left. \quad -[-C{\theta _i}(t)+ Af({{\theta _i}(t)})] \right\} \\&\quad \le {L_i}{\left( {{\lambda _i}\left( t \right) - {\theta _i}\left( t \right) } \right) ^T}\left( {{\lambda _i}\left( t \right) - {\theta _i}\left( t \right) } \right) , \end{aligned}$$

where \({L_i} > 0\).

3 Main results

3.1 Interlayer synchronization under attacks

The first layer (driving) network is (1). The second layer (response) network is (3). The systems on errors are as follows [37]:

$$\begin{aligned} {{\dot{\omega }}_i}\left( t \right) =&-Cy_i(t)+Cx_i(t)+f\left( {{y_i}\left( t \right) } \right) - f\left( {{x_i}\left( t \right) } \right) \nonumber \\&+(1-\alpha (t))p\sum \limits _{j = 1}^N {{b_{ij}}\Gamma (y_j(t) - x_j(t))} \nonumber \\&+ \alpha (t) q\sum \limits _{j = 1}^N {{d_{ij}}\Gamma \left( {{\omega _j}(t - \tau (t))} \right) }\nonumber \\&+ {U_i}\left( t \right) \end{aligned}$$
(4)

The controller under random deception cyber attacks is

$$\begin{aligned} {U_i}\left( t \right) = {\varepsilon _i}\left( t \right) + {\alpha }\left( t \right) \left( {C_i\left( {{{\varepsilon }_i}\left( t \right) } \right) } \right) , \end{aligned}$$
(5)

where \(C_i\left( {{{\varepsilon }_i}\left( t \right) } \right) \) characterizes the signal of the cyber attacks, and

$$\begin{aligned} C_i\left( {{{\varepsilon }_i}\left( t \right) } \right) = {k_1}{\varepsilon _i}\left( t \right) + {k_2}h\left( {{k_3}{{\varepsilon }_i}\left( t \right) } \right) , \end{aligned}$$
(6)

where \(h\left( \cdot \right) \) is a norm bounded nonlinear function, and \({k_1} \ge 0,{k_2} \ge 0,{k_3} \ge 0\).

$$\begin{aligned} {\varepsilon _i}\left( t \right) = - {\gamma _i}\left( t \right) \left( {Q\left( {{\omega _i}\left( t \right) } \right) } \right) - {m_i}sign\left( {Q\left( {{\omega _i}\left( t \right) } \right) } \right) , \end{aligned}$$
(7)

where \({m_i} > 0\),

and

$$\begin{aligned} {\dot{\gamma }_i}(t) = \left( {1\mathrm{{ + }}\alpha {k_1}} \right) \left( {1 - \sigma } \right) {e^{{2\eta }t}}{\left( {{\omega _i}\left( t \right) } \right) ^T}{\omega _i}\left( t \right) , \end{aligned}$$
(8)

where \(\eta > 0\).

Theorem 1

There are \({L}> 0\), \(k_1> 0\), \(\eta > 0\), \(k_2> 0\), \({{\hat{\gamma }}_{\min }} > 0\), \(m_{min} >0\) and \(\xi >0\) satisfing

$$\begin{aligned} \left\{ \begin{array}{l} L + N(1-\alpha ) p\left| {{b_M}} \right| \left\| \Gamma \right\| \mathrm{{ + }}\frac{N}{2}\ \alpha q\left| {{d_M}} \right| \left\| \Gamma \right\| \\ \mathrm{{ + }}\eta \mathrm{{ + }}\frac{{\xi {e^{{2\eta }{\tau }}}}}{{2\left( {1 - {\tau _M}} \right) }} - {{\hat{\gamma } }_{\min }}\left( {1\mathrm{{ + }}\alpha {k_1}} \right) \left( {1 - \sigma } \right) \le 0\\ N \alpha q\left| {{d_M}} \right| \left\| \Gamma \right\| - \xi \le 0\\ {M_{\max }}\alpha {k_2} - {m_{\min }}\left( {1+\alpha {k_1}} \right) \le 0, \end{array} \right. \end{aligned}$$

where \({L} = \max \left\{ L_i\right\} \), \({\hat{\gamma } _{\min }}{= }\min \left\{ {\hat{\gamma } }_i \right\} \), \({M_{\max }} = \max \left\{ M_i\right\} \),

\({m_{\min }} =\min \left\{ m_i\right\} \), \(\left| {{b_M}} \right| = \max \{ \left| {{b_{ij}}} \right| \}\), \(0 < \tau (t)\le {\tau }\), \(0< \dot{\tau }(t) \le {\tau _M} < 1\), \(\left| {{d_M}} \right| = \max \{ \left| {{d_{ij}}} \right| \}\), then the error systems (4) is exponential stable, i.e. exponentially asymptotical interlayer synchronization between DANNs (1) and (3) is achieved by adaptive controller (5) under random cyber attacks.

Proof

Lyapunov–Krasovskii functional (LKF) is

$$\begin{aligned} V\left( t \right) =&\frac{1}{2}{e^{2\eta t}}\sum \limits _{i = 1}^N {{{\left( {{\omega _i}\left( t \right) } \right) }^T}{\omega _i}\left( t \right) } \nonumber \\&+ \frac{\xi }{{2\left( {1 - {\tau _M}} \right) }}\nonumber \\&\times \sum \limits _{i = 1}^N {\int _{t - \tau \left( t \right) }^t {{{\left( {{\omega _i}\left( s \right) } \right) }^T}{\omega _i}\left( s \right) {e^{{2\eta } \left( {s + \tau \left( t \right) } \right) }}ds } } \nonumber \\&+ \frac{(1 - \sigma )}{2}\sum \limits _{i = 1}^N {{{({\gamma _i}(t) - {{\hat{\gamma } }_i})}^2}}. \end{aligned}$$
(9)

After deriving the LKF, one gets

$$\begin{aligned}&E\left( {\dot{V}\left( t \right) } \right) \nonumber \\&\quad \le E\left\{ {e^{{2\eta }t}}\left[ \sum \limits _{i = 1}^N {{\left( {{\omega _i}\left( t \right) } \right) }^T}\right. \right. \nonumber \\&\quad \times \left. \left. \left( -Cy_i(t)+Cx_i(t)+f\left( {{y_i}\left( t \right) } \right) - f\left( {{x_i}\left( t \right) } \right) \right. \right. \right. \nonumber \\&\quad \left. \left. + (1-\alpha (t))p\sum \limits _{j = 1}^N {{b_{ij}}\Gamma \left( {{y_j}\left( t \right) - {x_j}\left( t \right) } \right) } \right. \right. \nonumber \\&\quad \left. \left. + \alpha (t) q\sum \limits _{j = 1}^N {{d_{ij}}\Gamma \left( {{y_j}\left( {t - \tau \left( t \right) } \right) - {x_j}\left( {t - \tau \left( t \right) } \right) } \right) }\right. \right. \nonumber \\&\quad \left. \left. +{U_i}(t))+\Xi _\omega \right] \right\} \end{aligned}$$
(10)

with

$$\begin{aligned} {\Xi _\omega } =&\sum \limits _{i = 1}^N {\eta {{({\omega _i}(t))}^T}{\omega _i}(t)} \\ {}&\quad + \frac{\xi }{{2\left( {1 - {\tau _M}} \right) }}\sum \limits _{i = 1}^N {{{\left( {{\omega _i}\left( t \right) } \right) }^T}{\omega _i}\left( t \right) {e^{{2\eta }{\tau }}}} \nonumber \\&- \frac{\xi }{2}\sum \limits _{i = 1}^N {{{\left( {{\omega _i}\left( {t - \tau \left( t \right) } \right) } \right) }^T}{\omega _i}\left( {t - \tau \left( t \right) } \right) }\nonumber \\&{{+}\sum \limits _{i = 1}^N {\left( {{\gamma _i}\left( t \right) {-} {{\hat{\gamma }}_i}} \right) \left( {1 {+} \alpha {k_1}} \right) (1 {-} \sigma ){{\left( {{\omega _i}\left( t \right) } \right) }^T}\!{\omega _i}\left( t \right) } }. \end{aligned}$$

Based on Assumption 1 and the controller (5), we obtain

$$\begin{aligned}&E\left( {\dot{V}\left( t \right) } \right) \nonumber \\&\quad \le E\left\{ {{e^{{2\eta }t}}\left[ {\sum \limits _{i = 1}^N {{{\left( {{\omega _i}\left( t \right) } \right) }^T}\left( {L_i{\omega _i}\left( t \right) } \right. } } \right. } \right. \nonumber \\&\quad + (1-\alpha (t)) p\sum \limits _{j = 1}^N {{b_{ij}}\Gamma {\omega _j}\left( t \right) } \nonumber \\&\quad + \alpha (t) q\sum \limits _{j = 1}^N {{d_{ij}}\Gamma {\omega _j}\left( {t - \tau \left( t \right) } \right) } \nonumber \\&\quad \left. { + {{\varepsilon }_i}\left( t \right) + {\alpha }(t)\left( {C\left( {{{\varepsilon }_i}\left( t \right) } \right) } \right) } \right) \nonumber \\&\quad +\Xi _\omega \Bigg ]\Bigg \}. \end{aligned}$$
(11)

Because of

$$\begin{aligned}&(1{-}\alpha ) p\sum \limits _{i {=} 1}^N {{{\left( {{y_i}\left( t \right) {-} {x_i}\left( t \right) } \right) }^T}} \sum \limits _{j = 1}^N {{b_{ij}}\Gamma \left( {{y_j}\left( t \right) { -} {x_j}\left( t \right) } \right) } \nonumber \\&\quad \le N(1-\alpha ) p\sum \limits _{i = 1}^N {\left| {{b_M}} \right| \left\| \Gamma \right\| } {\left( {{y_i}\left( t \right) - {x_i}\left( t \right) } \right) ^T}\left( {{y_i}\left( t \right) }\right. \nonumber \\ {}&\quad \left. { - {x_i}\left( t \right) } \right) , \end{aligned}$$
(12)

and

$$\begin{aligned}&\alpha q\sum \limits _{i = 1}^N {{{\left( {{y_i}\left( t \right) - {x_i}\left( t \right) } \right) }^T}} \sum \limits _{j = 1}^N {{d_{ij}}\Gamma \left( {{\omega _j}\left( {t - \tau \left( t \right) } \right) } \right) } \nonumber \\&{\le } \frac{N}{2}\alpha q\left| {{d_M}} \right| \left\| \Gamma \right\| \sum \limits _{i = 1}^N {{{\left( {{y_i}\left( t \right) {-} {x_i}\left( t \right) } \right) }^T}\left( {{y_i}\left( t \right) {-} {x_i}\left( t \right) } \right) } \nonumber \\&+\frac{N}{2} \alpha q\left| {{d_M}} \right| \left\| \Gamma \right\| \nonumber \\&\times \sum \limits _{i = 1}^N {{{\left( {{\omega _i}\left( {t - \tau \left( t \right) } \right) } \right) }^T}\left( {{\omega _i}\left( {t - \tau \left( t \right) } \right) } \right) }, \end{aligned}$$
(13)

After uniting (6)–(8), (11), one obtains

$$\begin{aligned}&E\left( {\dot{V}\left( t \right) } \right) \nonumber \\&\quad \le E\left\{ {{e^{{2\eta }t}}\left[ {\sum \limits _{i = 1}^N {{{\left( {{\omega _i}\left( t \right) } \right) }^T}\left( {L{\omega _i}\left( t \right) } \right. } } \right. } \right. \nonumber \\&\quad + (1-\alpha ) p\sum \limits _{j = 1}^N {{b_{ij}}\Gamma {\omega _j}\left( t \right) } \nonumber \\&\quad + \alpha q\sum \limits _{j = 1}^N {{d_{ij}}\Gamma {\omega _j}\left( {t - \tau \left( t \right) } \right) } \nonumber \\&\quad \mathrm{{ + }}{{\varepsilon }_i}\left( t \right) \nonumber \\&\quad \left. { + \alpha (t)\left( {{k_1}{{\varepsilon }_i}\left( t \right) + {k_2}h\left( {{k_3}{{\varepsilon }_i}\left( t \right) } \right) } \right) } \right) \nonumber \\&\quad + \Xi _\omega \Bigg ]\Bigg \}. \end{aligned}$$
(14)

After adjustment, one obtains

$$\begin{aligned}&E\left( {\dot{V}\left( t \right) } \right) \nonumber \\&\quad \le E\left\{ {{e^{{2\eta }t}}\left[ {\sum \limits _{i = 1}^N {{{\left( {{\omega _i}\left( t \right) } \right) }^T}\left( {L{\omega _i}\left( t \right) } \right. } } \right. } \right. \nonumber \\&\quad + (1-\alpha ) p\sum \limits _{j = 1}^N {{b_{ij}}\Gamma {\omega _j}\left( t \right) } \nonumber \\&\quad + \alpha q\sum \limits _{j = 1}^N {{d_{ij}}\Gamma {\omega _j}\left( {t - \tau \left( t \right) } \right) }+\left( {1+\alpha {k_1}} \right) {{\varepsilon }_i}\left( t \right) \nonumber \\&\quad \left. { + \alpha {k_2}h\left( {{k_3}{{\varepsilon }_i}\left( t \right) } \right) } \right) \nonumber \\&\quad +\Xi _\omega \Bigg ]\Bigg \}. \end{aligned}$$
(15)

After uniting (12), (13) and (15), one has

$$\begin{aligned}&E\left( {\dot{V}\left( t \right) } \right) \nonumber \\&\quad \le E\left\{ {{e^{{2\eta }t}}\left\{ {\sum \limits _{i = 1}^N {{{\left( {{\omega _i}\left( t \right) } \right) }^T}L{\omega _i}\left( t \right) } } \right. } \right. \nonumber \\&\quad + N(1-\alpha ) p\sum \limits _{i = 1}^N {\left| {{b_M}} \right| \left\| \Gamma \right\| } {\left( {{\omega _i}\left( t \right) } \right) ^T}{\omega _i}\left( t \right) \nonumber \\&\quad +\frac{N}{2}\alpha q\left| {{d_M}} \right| \left\| \Gamma \right\| \sum \limits _{i = 1}^N {{{\left( {{\omega _i}\left( t \right) } \right) }^T}{\omega _i}\left( t \right) } \nonumber \\&\quad +\frac{N}{2} \alpha q\left| {{d_M}} \right| \left\| \Gamma \right\| \sum \limits _{i = 1}^N {{{\left( {{\omega _i}\left( {t - \tau \left( t \right) } \right) } \right) }^T}{\omega _i}\left( {t - \tau \left( t \right) } \right) } \nonumber \\&\quad - \left( {1 + \alpha {k_1}} \right) \sum \limits _{i = 1}^N {{\gamma _i}\left( t \right) {{\left( {{\omega _i}\left( t \right) } \right) }^T}\left( {Q\left( {{\omega _i}\left( t \right) } \right) } \right) } \nonumber \\&\quad - \left( {1\mathrm{{ + }}\alpha {k_1}} \right) \sum \limits _{i = 1}^N {{m_i}{{\left( {{\omega _i}\left( t \right) } \right) }^T}sign\left( {Q\left( {{\omega _i}\left( t \right) } \right) } \right) } \nonumber \\&\quad + \alpha {k_2}\sum \limits _{i = 1}^N {{{\left( {{\omega _i}\left( t \right) } \right) }^T}\left( {h\left( {{k_3}{{\varepsilon }_i}\left( t \right) } \right) } \right) }+\Xi _\omega \Bigg ]\Bigg \}. \end{aligned}$$
(16)

According to [33] and \(\sigma \in \left( {0,1} \right) \), one gets

$$\begin{aligned}&- \left( {1\mathrm{{ + }}\alpha {k_1}} \right) \sum \limits _{i = 1}^N {{\gamma _i}\left( t \right) {{\left( {{\omega _i}\left( t \right) } \right) }^T}\left( {Q\left( {{\omega _i}\left( t \right) } \right) } \right) } \nonumber \\&\quad \le - \left( {1\mathrm{{ + }}\alpha {k_1}} \right) \left( {1 - \sigma } \right) \sum \limits _{i = 1}^N {{\gamma _i}\left( t \right) {{\left( {{\omega _i}\left( t \right) } \right) }^T}{\omega _i}\left( t \right) }, \end{aligned}$$
(17)

and

$$\begin{aligned}&- \left( {1\mathrm{{ + }}\alpha {k_1}} \right) \sum \limits _{i = 1}^N {{m_i}{{\left( {{\omega _i}\left( t \right) } \right) }^T}sign\left( {Q\left( {{\omega _i}\left( t \right) } \right) } \right) } \nonumber \\&\quad = - \left( {1\mathrm{{ + }}\alpha {k_1}} \right) \sum \limits _{i = 1}^N {{m_i}{{\left( {{\omega _i}\left( t \right) } \right) }^T}sign\left( {{\omega _i}\left( t \right) } \right) }. \end{aligned}$$
(18)

Uniting (16)–(18), we can get

$$\begin{aligned}&E\left( {\dot{V}\left( t \right) } \right) \nonumber \\&\quad \le E\left\{ {{e^{{2\eta }t}}\left\{ {\sum \limits _{i = 1}^N {{{\left( {{\omega _i}\left( t \right) } \right) }^T}L{\omega _i}\left( t \right) } } \right. } \right. \nonumber \\&\quad + N(1-\alpha ) p\sum \limits _{i = 1}^N {\left| {{b_M}} \right| \left\| \Gamma \right\| } {\left( {{\omega _i}\left( t \right) } \right) ^T}{\omega _i}\left( t \right) \nonumber \\&\quad +\frac{N}{2}\alpha q\left| {{d_M}} \right| \left\| \Gamma \right\| \sum \limits _{i = 1}^N {{{\left( {{\omega _i}\left( t \right) } \right) }^T}{\omega _i}\left( t \right) } \nonumber \\&\quad \mathrm{{ + }}\frac{N}{2} \alpha q\left| {{d_M}} \right| \left\| \Gamma \right\| \sum \limits _{i = 1}^N {{{\left( {{\omega _i}\left( {t - \tau \left( t \right) } \right) } \right) }^T}{\omega _i}\left( {t - \tau \left( t \right) } \right) } \nonumber \\&\quad - \left( {1\mathrm{{ + }}\alpha {k_1}} \right) \left( {1 - \sigma } \right) \sum \limits _{i = 1}^N {{\gamma _i}\left( t \right) {{\left( {{\omega _i}\left( t \right) } \right) }^T}{\omega _i}\left( t \right) } \nonumber \\&\quad - \left( {1\mathrm{{ + }}\alpha {k_1}} \right) \sum \limits _{i = 1}^N {{m_i}{{\left( {{\omega _i}\left( t \right) } \right) }^T}sign\left( {{\omega _i}\left( t \right) } \right) } \nonumber \\&\quad + \alpha {k_2}\sum \limits _{i = 1}^N {\left\| {{{\left( {{\omega _i}\left( t \right) } \right) }^T}} \right\| \left\| {h\left( {{k_3}{{\varepsilon }_i}\left( t \right) } \right) } \right\| } +\Xi _\omega \Bigg ]\Bigg \}. \end{aligned}$$
(19)

According to Assumption 1, because of \(\left\| {h\left( {{k_3}{{\tilde{u}}_i}\left( t \right) } \right) } \right\| \le {M_i}\), we can get

$$\begin{aligned}&E\left( {\dot{V}\left( t \right) } \right) \nonumber \\&\quad \le E\left\{ {{e^{{2\eta }t}}\left\{ {\sum \limits _{i = 1}^N {{{\left( {{\omega _i}\left( t \right) } \right) }^T}\left( L \right. } } \right. } \right. \nonumber \\&\quad + N(1-\alpha ) p\left| {{b_M}} \right| \left\| \Gamma \right\| \mathrm{{ + }}\frac{N}{2} \alpha q\left| {{d_M}} \right| \left\| \Gamma \right\| \nonumber \\&\quad \left. {\mathrm{{ + }}\eta \mathrm{{ + }}\frac{{\xi {e^{{2\eta }{\tau }}}}}{{2\left( {1 - {\tau _M}} \right) }} - {{\hat{\gamma }}_i}\left( {1\mathrm{{ + }}\alpha {k_1}} \right) \left( {1 - \sigma } \right) } \right) {\omega _i}\left( t \right) \nonumber \\&\quad +\left( {\frac{N}{2} \alpha q\left| {{d_M}} \right| \left\| \Gamma \right\| - \frac{\xi }{2}} \right) \nonumber \\&\quad \times \sum \limits _{i = 1}^N {{{\left( {{\omega _i}\left( {t - \tau \left( t \right) } \right) } \right) }^T}{\omega _i}\left( {t - \tau \left( t \right) } \right) } \nonumber \\&\quad \left. {\left. { + \sum \limits _{i = 1}^N {\left( {{M_i}\alpha {k_2} - {m_i}\left( {1\mathrm{{ + }}\alpha {k_1}} \right) } \right) {{\left\| {{{\left( {{\omega _i}\left( t \right) } \right) }^T}} \right\| }_1}} } \right\} } \right\} . \end{aligned}$$
(20)

If

$$\begin{aligned} \left\{ \begin{array}{l} L + N(1-\alpha ) p\left| {{b_M}} \right| \left\| \Gamma \right\| \mathrm{{ + }}\frac{N}{2}\alpha q\left| {{d_M}} \right| \left\| \Gamma \right\| \\ \mathrm{{ + }}\eta \mathrm{{ + }}\frac{{\xi {e^{{2\eta }{\tau }}}}}{{2\left( {1 - {\tau _M}} \right) }} - {{\hat{\gamma }}_{\min }}\left( {1\mathrm{{ + }}\alpha {k_1}} \right) \left( {1 - \sigma } \right) \le 0\\ N \alpha q\left| {{d_M}} \right| \left\| \Gamma \right\| - \xi \le 0\\ {M_{\max }}\alpha {k_2} - {m_{\min }}\left( {1\mathrm{{ + }}\alpha {k_1}} \right) \le 0, \end{array} \right. , \end{aligned}$$

then \(\dot{V}\left( t \right) \le 0\), namely, \(V\left( t \right) \le V\left( 0 \right) ,\forall t \ge 0.\)

Because of \(\frac{1}{2}{e^{2\eta t}}{{{\left( {{\omega _i}\left( t \right) } \right) }^T}{\omega _i}\left( t \right) } \le V\left( t \right) \le V\left( 0 \right) \), we have \(\left\| {{\omega _i}\left( t \right) } \right\| \le \textrm{K}{e^{ - \eta t}}\), where \(\textrm{K} = \sqrt{2V\left( 0 \right) } > 0\), so that \(\mathop {\lim }\limits _{t \rightarrow \infty } \left\| {{\omega _i}\left( t \right) } \right\| = 0\). That is to say, under stochastic cyber attacks, systems (4) is exponential stable, i.e. exponentially asymptotical interlayer synchronization for networks (1) and (3) with random occuring delays is achieved. \(\square \)

3.2 Exponential interlayer synchronization

Error systems for networks (1) and (2) are as follows:

$$\begin{aligned}&{{\dot{\omega }}_i}\left( t \right) \nonumber \\&\quad =- C{\omega _i}(t) + Af({\omega _i}(t))\nonumber \\&\quad + \alpha \left( t \right) p\sum \limits _{j = 1}^N {{b_{ij}}\Gamma \left( {{\omega _j}\left( t \right) } \right) } \nonumber \\&\quad + \left( {1 - \alpha \left( t \right) } \right) q\sum \limits _{j = 1}^N {{d_{ij}}\Gamma \left( {{\omega _j}\left( {t - \tau \left( t \right) } \right) } \right) }\nonumber \\&\quad + {u_i}\left( t \right) . \end{aligned}$$
(21)

The adaptive quantitative controller \({u_i}\left( t \right) \) as

$$\begin{aligned} {u_i}\left( t \right) = - {\gamma _i}\left( t \right) \left( {Q\left( {{\omega _i}\left( t \right) } \right) } \right) \end{aligned}$$
(22)

with \({\gamma _i}\left( t \right) > 0\).

Design adaptive law is

$$\begin{aligned} {\dot{\gamma }_i}\left( t \right)= & {} {e^{{2\eta }t}}{\left( {{\omega _i}\left( t \right) } \right) ^T}{\omega _i}\left( t \right) , \end{aligned}$$
(23)

where \(\eta > 0\).

Theorem 2

There exist L, \(\eta > 0\), \({{\hat{\gamma }}_{\min }} > 0\) and \(\xi >0\) satisfy

$$\begin{aligned} \left\{ \begin{array}{l} L+N(1 - \alpha ) p\left| {{b_M}} \right| \left\| \Gamma \right\| \mathrm{{ + }}\frac{N}{2}\alpha q\left| {{d_M}} \right| \left\| \Gamma \right\| \\ \mathrm{{ + }}\eta \mathrm{{ + }}\frac{{\xi {e^{{2\eta }{\tau }}}}}{{2\left( {1 - {\tau _M}} \right) }} - \left( {1 - \sigma } \right) {{\hat{\gamma }}_{\min }} \le 0\\ N\alpha q\left| {{d_M}} \right| \left\| \Gamma \right\| - \xi \le 0 \end{array} \right. , \end{aligned}$$

then error systems are exponential stable, i.e. exponentially asymptotical interlayer synchronization under DANNs (1) and (2) is achieved by quantized adaptive controller (22).

Proof

The proof process is similar to Theorem 1 and is ignored here. \(\square \)

Remark 1

The DANNs in this paper can be directed or undirected, that is, the derived theorem strategies are for both exponentially interlayer synchronization of undirected DANNs and directed DANNs.

4 Secure communication

4.1 Secure synchronous

Simulations are given to illustrate the effective of the theoretical framework. Each system is 3-D cellular artificial neural networks in duplex networks [39,40,41,42].

$$\begin{aligned} \dot{\omega }(t) = - C\omega (t) + Af(\omega (t)) \end{aligned}$$
(24)

with \(C = {I_3}\). \(A = \left( {\begin{array}{*{20}{c}} {1.25}&{}\quad { - 3.2}&{}\quad { - 3.2}\\ { - 3.2}&{}\quad {1.1}&{}\quad { - 4.4}\\ { - 3.2}&{}\quad {4.4}&{}\quad 1 \end{array}} \right) \), \(f(\omega (t)) = \left( {\left| {\omega (t) + 1} \right| - \left| {\omega (t) - 1} \right| } \right) /2\).

Fig. 2
figure 2

Error \({\omega _{i}}(t)\)

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Driving DANNs with 4 3-D cellular neural networks can be

$$\begin{aligned} {\dot{x}_i}\left( t \right) =&-Cx_i(t)+Af( {{x_i}(t)}) \nonumber \\ {}&\quad + (1-\alpha (t))p\sum \limits _{j = 1}^4 {{b_{ij}}\Gamma {x_j}\left( t \right) }\nonumber \\&+ \alpha (t) q\sum \limits _{j = 1}^4 {{d_{ij}}\Gamma {x_j}\left( {t - \tau ( t)} \right) },\nonumber \\&i = 1,2,3,4, \end{aligned}$$
(25)
Fig. 3
figure 3

Adaptive quantitative controller \({U_{i}}(t)\)

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and response DANNs can be

$$\begin{aligned} {{\dot{y}}_i}\left( t \right) =&-Cy_i(t) + Af\left( {{y_i}\left( t \right) } \right) \nonumber \\ {}&\quad + (1-\alpha \left( t \right) )p\sum \limits _{j = 1}^4 {{b_{ij}}\Gamma {y_j}\left( t \right) }\nonumber \\&+ \alpha (t) q\sum \limits _{j = 1}^4 {{d_{ij}}\Gamma {y_j}\left( {t - \tau \left( t \right) } \right) } \nonumber \\&+ {{U}_i}\left( t \right) , \end{aligned}$$
(26)
Fig. 4
figure 4

Adaptive updating laws \({{\gamma _i}}(t)\)

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Fig. 5
figure 5

Signal \(C_{i}\) of random attacks

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where \(\alpha \mathrm{{ = }}0.5\), \(p=0.1\), \(q=0.2\), \(\tau \left( t \right) = \frac{{{e^t}}}{{2 + 2{e^t}}}\), \(\Gamma \mathrm{{ = }}{I_3}\), \({\delta _i} = 60\), \(\eta = 0.9\), \({m_i} = 0.005\), \({k_1} = 0.2\), \({k_2} = 0.2\), \({k_3} = 1\). Quantized density is \(\sigma = 0.5\). \(B{ = }\left[ {\begin{array}{*{20}{c}} { - 1}&{}\quad 1&{}\quad 0&{}\quad 0\\ 0&{}\quad { - 1}&{}\quad 1&{}\quad 0\\ 0&{}\quad 0&{}\quad { - 1}&{}\quad 1\\ 1&{}\quad 0&{}\quad 0&{}\quad { - 1} \end{array}} \right] \), \(\mathrm{{D = }}\left[ {\begin{array}{*{20}{c}} { - 3}&{}\quad 1&{}\quad 1&{}\quad 1\\ 0&{}\quad { - 2}&{}\quad 1&{}\quad 1\\ 1&{}\quad 0&{}\quad { - 1}&{}\quad 0\\ 0&{}\quad 1&{}\quad 0&{}\quad { - 1} \end{array}} \right] \).

Errors \({\omega _{ij}}(t)\) between DANNs (25) and (26) under random cyber attacks can be shown in Fig. 2a–c. From Fig. 2a–c, errors for synchronous tend to zero. That means exponentially asymptotical interlayer synchronization under DANNs (25) and (26) is achieved which can support Theorem 1, and the established theoretical framework for synchronous is effective. In Fig. 3, the trajectory of the adaptive law tends towards some stable values. The feedback coefficient of the controller has been adapted.

Fig. 6
figure 6

Signal \(\alpha (t) \) of random attacks

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\({\gamma _i}(t)\) in (7) can be seen in Fig. 4. Signal \(C_{i}\) of random attacks can be shown in Fig. 5. Comparing Figs. 3 and 5, it is shown that signal \(C_{i}\) of random attacks is similar to the control signal \(U_i(t)\), thus achieving the goal of hiding and conducting deception attacks. The signal \(\alpha (t)\) of random attacks randomly switches between 0 and 1 in Fig. 6.

4.2 Signal secure communication

The framework of signal secure communication based on secure synchronous for DANNs under cyber attacks can be shown in Fig. 7.

Fig. 7
figure 7

Secure communication framework

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Fig. 8
figure 8

Overlap between \(\vartheta _i(t)\) and \(M_i(t)\)

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The transmitted plaintext signals are \({{P}_i}(t)\). Steps are as follows:

Step 1: After mixing the random signal \({\mathcal {R}_i}(t)\) with the plaintext signals \({{P}_i}(t)\), it can be obtained that

$$\begin{aligned} {M_i}(t) = \left\{ {\begin{array}{*{20}{c}} {{\mathcal {R}_i}(t),t \in [0,0.05)}\\ {{{P}_i}(t - 0.05),t \in [0.05, + \infty )} \end{array}} \right. \end{aligned}$$
$$\begin{aligned} {P_i} = \left\{ {\begin{array}{*{20}{c}} { - 2\sin (2t) + 2\cos (0.2t)}\\ { - 2\sin (1.2t) + 2\cos (0.1t)}\\ { - 2\sin (2t) + 2\cos (0.1t)} \end{array}} \right. . \end{aligned}$$

Step 2: Generate encrypted signals \(\vartheta _i(t) = M_i(t) + {y_i}(t)\).

Step 3: The sender transmits the signals encrypted ciphertext signals \(\vartheta _i(t)\) from the second layer to the receiver in the first layer through a communication channel.

Step 4: When the secure exponential interlayer synchronous of DANNs under cyber attacks is achieved, the receiver can decrypt the ciphertext signals \(\vartheta _i(t)\) and obtain the correct plaintext signals \(Z_i(t)=P_i(t)=M_i(t)\).

In Fig. 8, the trajectories of \(\vartheta _i(t)\) and \(M_i(t)\) are overlap, which means correct plaintext signals are obtained, and satisfy \(Z_i(t)=P_i(t)=M_i(t)=\vartheta _i(t)\).

Remark 2

Compared with many existing reports, especially literatures [14, 15] and [21], this paper focuses on secure exponentially stable based synchronization of DANNs under cyber attacks; an adaptive control strategy is designed to resist cyber attacks in DANNs; the conditions of interlayer exponential synchronous are given in DANNs; the design method of the intelligent adaptive quantization controller are obtained in DANNs; a secure communication framework based on the synchronous policy is derived. However, in references [14, 15], cyber attacks, DANNs and secure communication frameworks based on the synchronous policy were not been considered. Compared with literature [21], the authors explored fractional order cyber physical systems under random cyber attacks in [21]. However, in this paper, we study coupled neural networks with random occuring time delays under random cyber deception attacks. In [21], researchers adopted feedback control strategy, however, to overcome random uncertainties, an adaptive quantization control strategy is designed in this paper.

Remark 3

In the designed security communication strategy, control signals for exponentially stable based synchronous and ciphertext signals are transmitted within the channel. If the control signals are attacked by random cyber attacks to a certain extent, cyber attacks result in severe distortion of control signals. It will lead to the inability to achieve exponential synchronization, and the ciphertext cannot be decrypted into plaintext. As a result, security communication cannot be achieved.

5 Conclusion

In this paper, a theoretical framework of secure exponential synchronous for DANNs and secure communication framework were derived under stochastic cyber attacks. The cyber attacks were considered in transmission of the intelligent adaptive controller signals. The criteria for the secure synchronous were got based on the Lyapunov’s second method. The design method for structure and parameters of the intelligent controller which can resist random cyber attacks was obtained via Lyapunov stable theory. Finally, a secure communication framework based on the synchronous policy was derived. Through a simulation example, secure synchronous and secure communication were achieved, which verified the effectiveness of the proposed theoretical framework and the application framework of secure communication.