Almost automorphic synchronization of quaternion-valued high-order Hopfield neural networks with time-varying and distributed delays

Almost automorphic synchronization of quaternion-valued high-order Hopfield neural networks with... Abstract In this paper, we consider the problem of the almost automorphic synchronization of quaternion-valued high-order Hopfield neural networks (QVHHNNs) with time-varying and distributed delays. Firstly, to avoid the non-commutativity of quaternion multiplication, we decompose QVHHNNs into an equivalent real-valued system. Secondly, we use the Banach fixed point theorem to obtain the existence of almost automorphic solutions of QVHHNNs. Thirdly, by designing a novel state-feedback controller and constructing suitable Lyapunov functions, we obtain that the drive-response structure of QVHHNNs with almost automorphic coefficients can realize the exponential synchronization. Our results are completely new. Finally, a numerical example is given to illustrate the feasibility of our results. 1. Introduction As is well known, synchronization is an extensive phenomenon in real systems, which means that two or more systems adjust each other to result in a common dynamical behavior. Experimental and theoretical studies have reported that synchronization phenomenon could be seen in various real systems. By synchronization, we can understand an unknown system from the well-known systems. So it has played a significant role in network control and system design as a powerful tool (Tang et al., 2015, 2016). The analysis of synchronization of neural networks has attracted a lot of scholars from various fields such as information science, secure communication and chemical reactions (Masoller & Zanette, 2001; Wu & Chen, 2008; Zhang & Zhao, 2012; Wong et al., 2013; Yu et al., 2013; Chai & Chen, 2014; Chen et al., 2014; Lü et al., 2014; Hong, 2014; Bao & Cao, 2015; Cai et al., 2015; Cai et al., 2015; Wu et al., 2015; Yang et al., 2017). Many good results have been obtained about the synchronization of neural networks including periodic (Chai & Chen, 2014; Hong, 2014; Cai et al., 2015; Cai et al., 2015; Wu et al., 2015), global (Wu & Chen, 2008; Chen et al., 2014), stochastic (Wong et al., 2013), combinatorial, cluster, adaptive (Yu et al., 2013), projective (Zhang & Zhao, 2012; Bao & Cao, 2015), lag (Lü et al., 2014; Yang et al., 2017) and anticipated synchronization (Masoller & Zanette, 2001). The well-known Hopfield neural network, as a form of recurrent artificial neural networks popularized by John Hopfield in 1982 (Hopfield, 1984), has been widely studied in recent years (Xiao, 2009; Zhang & Jin, 2000; Zhang et al., 2003), especially the high-order Hopfield neural networks (HHNNs), because of its faster convergence rate, stronger approximation property, greater capacity and higher fault tolerance. Many scholars have made extensive research on the existence, uniqueness and stability of equilibrium points, periodic solutions, almost periodic solutions and almost automorphic solutions of HHNNs, see Xiao (2009), Xiao & Meng (2009), Zhang & Gui (2009), Li & Yang (2014), Arbi et al. (2015), Li & Yang (2016), Li & Meng (2017), Zhao et al. (2018) and the references therein. And there are also a few results about the periodic synchronization of the HHNNs. However, there has been no paper published on the problem of the almost periodic synchronization of neural networks. Due to the finite switching speed of neurons and amplifiers, time delays inevitably exist in biological and artificial neural network models. And, as is known to all, time delays may change the dynamical behaviors of neural networks under consideration. Therefore, the dynamics of neural networks with various delays have been a long-term focus issue (Zhang & Jin, 2000; Zhang et al., 2003; Xiao, 2009; Zhang & Gui, 2009; Wu et al., 2012; Li & Yang, 2014; Arbi et al., 2015; Li & Yang, 2016; Arbi & Cao, 2017; Li & Meng, 2017; Arbi & Cao, 2018; Xiong et al., 2018; Zhao et al., 2018). However, in these recent publications, most results on the synchronization of delayed neural networks have been restricted to simple cases of discrete delays. Since the neural networks usually have a spatial nature due to the presence of an amount of paralleled pathway of a variety of axon sizes and lengths, it is desired to model them by introducing distributed delays (Zhang et al., 2003). Therefore, both discrete and distributed delays should be taken into account when modeling more realistic HHNNs. On the one hand, the concept of almost automorphy, which is much more general than the almost periodicity, was introduced in the literature by Bochner in 1955 (Bochner, 1964) in the context of differential geometry (Bochner, 1964) and plays a very important role in understanding the almost periodicity. Moreover, in reality, almost periodicity is universal than periodicity. So almost automorphic oscillation of neural networks has been studied by several authors, see Bochner (1964), Li & Yang (2014), Yang et al. (2017) and the references therein. On the other hand, the quaternion algebra was first proposed by Hamilton (Sudbery, 1979) in 1843. A quaternion consists of a real and three imaginary parts. The three imaginary units i, j and k obey the Hamilton’s multiplication rules:   \begin{equation*} ij=-ji=k,\quad jk=-kj=i,\quad ki=-ik=j,\quad i^2=j^2=k^2=-1. \end{equation*}The skew field of quaternions is denoted by $$\mathbb{Q}:=\{q=q^R+iq^I+jq^J+kq^K\}$$, where $$q^R, q^I, q^J, q^K$$ are real numbers. The quaternion-valued neural networks (QVNNs), which are a generalization of the real-valued and complex-valued neural networks (Wang et al., 2017; Li et al., 2018), can be extensively applied in the fields such as aerospace, satellite tracking, processing of polarized waves and image processing (Matsui et al., 2004; Yoshida et al., 2005; Luo et al., 2010). One of the benefits of using quaternion is the three-dimensional geometrical affine transformation that can be represented efficiently and compactly. So, the study of QVNNs has received much attention of many scholars and some good results have been obtained for the stability, dissipativity, periodicity and pseudo almost periodicity of QVNNs (Liu et al., 2016; Chen et al., 2017; Chen et al., 2017; Li & Meng, 2017; Liu et al., 2017; Liu et al., 2017; Valle & de Castro, 2017; Zhang et al., 2017; Li & Qin, 2018). For example, in Zhang et al. (2017), the global exponential stability for recurrent neural networks with asynchronous time delays is investigated in the quaternion field; in Liu et al. (2016), some sufficient conditions on the global $$\mu $$-stability of the QVNNs with unbounded time-varying delays were obtained; in Chen et al. (2017), the author dealt with the problem of robust stability for QVNNs with leakage delay, discrete delay and parameter uncertainties. However, as far as we know, there is no result about the almost automorphic synchronization of QVNNs. Motivated by the considerations mentioned above, in the present work, we are concerned with the following quaternion-valued high-order Hopfield neural network (QVHHNN) with time-varying and distributed delays:   \begin{eqnarray} X^{\prime}_{p}(t)&=&-c_{p}(t)X_{p}(t)+\sum_{q=1}^{n}a_{pq}(t)f_{q}(X_{q}(t-\tau_{pq}(t))) +\sum_{q=1}^{n}b_{pq}(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}(X_{q}(t-u))\,\mathrm{d}u\nonumber\\ &&+\sum_{q=1}^{n}\sum_{r=1}^{n}d_{pqr}(t)h_{q}(X_{q}(t-\sigma_{pqr}(t)))h_{r}(X_{r}(t-\xi_{pqr}(t))) +I_{p}(t),\,\,t\in \mathbb{R}, \end{eqnarray} (1.1)where p ∈{1, 2, …, n} := N, n corresponds to the number of units in the neural network; $$X_p(t)\in \mathbb{Q}$$ denotes the activation of the pth neuron at time t; $$c_{p}(t)>0$$ represents the rate with which the pth unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs at time t; $$a_{pq}(t), b_{pq}(t)\in \mathbb{Q}$$ represent the delayed strengths of connectivity between cell p and q at time t, respectively; $$d_{pqr}(t)\in \mathbb{Q}$$ denotes the second-order connection weights of the neural network; $$f_{q}, g_{q}, h_{q},h_{r}: \mathbb{Q}\rightarrow \mathbb{Q}$$ are the activation functions of signal transmission; $$I_{p}(t)\in \mathbb{Q}$$ is an external input on the pth unit at time t; $$\tau _{pq}(t)$$, $$\sigma _{pqr}(t)$$ and $$\xi _{pqr}(t)$$ are transmission delays at time t. The initial conditions associated with system (1.1) are of the form   \begin{equation*} X_p(s)=\Phi_p (s),\,\, s\in(-\infty, 0],\,\, p\in N, \end{equation*}where $$\Phi _p(s)=\varphi _p^R(s)+i\varphi _p^I(s)+j\varphi _p^J(s)+k\varphi _p^K(s)$$ is the quaternion-valued bounded continuous function defined on $$(-\infty , 0]$$. Our main purpose of this paper is to study the problem of the almost automorphic synchronization of (1.1). To the best of our knowledge, this is the first paper to study the almost automorphic synchronization of neural networks. Our result of this paper is completely new and our method can be used to study the problem of the periodic, almost periodic and almost automorphic synchronization for other types of neural networks. Besides, QVHHNN (1.1) contains real-valued and complex-valued HHNNs as its special cases. Remark 1.1 In system (1.1), if $$X_{q}=x_{q}^{R}+x_{q}^{I}i\in \mathbb{C}$$, where $$x_{q}^{R},x_{q}^{I} \in \mathbb{R}$$ and the activation functions $$f_{q},g_{q}: \mathbb{C}\rightarrow \mathbb{C}$$ are complex variable functions, that is, $$ f_{q}(X_{q})=f_{q}^{R}(x_{q}^{R},x_{q}^{I})+if_{q}^{I}(x_{q}^{R},x_{q}^{I}), g_{q}(X_{q})=g_{q}^{R}(x_{q}^{R},x_{q}^{I}) +ig_{q}^{I}(x_{q}^{R},x_{q}^{I}), h_{q}(X_{q})=h_{q}^{R}(x_{q}^{R},x_{q}^{I}) +ih_{q}^{I}(x_{q}^{R},x_{q}^{I}), $$ where $$f^{\nu }_{q}, g^{\nu }_{q}:\mathbb{R}^{2}\rightarrow \mathbb{R}$$, q ∈ N, $$\nu =R,I$$ and all the quaternion-valued coefficients of (1.1) are complex-valued coefficients, then system (1.1) degenerates to a complex-valued system; if all of the activation functions and coefficients of (1.1) are real variable functions, then system (1.1) degenerates to a real-valued system. This paper is organized as follows. In Section 2, some preliminaries and notations are given. In Section 3, sufficient conditions for the existence of almost automorphic solutions of system (1.1) are obtained. In Section 4, the exponential synchronization was investigated. In Section 5, the effectiveness and feasibility of the developed methods in this paper are shown by a numerical example. In Section 5, we draw a brief conclusion. 2. Preliminaries We will first introduce some definitions and recall some basic lemmas that will be used in this paper. Analogously with the definition of almost automorphic functions in the study by Diagana (2013), we gave the following. Definition 2.1 A function $$f\in C(\mathbb{R}, \mathbb{Q}^n)$$ is said to be almost automorphic in Bochner’s sense if for every sequence of real numbers $$\{s_n\}_{n\in \mathbb{N}}$$, there exists a subsequence $$\{\tau _n\}_{n\in \mathbb{N}}$$ such that   \begin{equation*} g(t)=\lim\limits_{n\rightarrow\infty}f(t+\tau_n) \end{equation*}is well defined for each $$t\in \mathbb{R}$$ and   \begin{equation*} \lim\limits_{n\rightarrow\infty}g(t-\tau_n)=f(t) \end{equation*}for each $$t\in \mathbb{R}$$. Remark 2.1 Obviously, function $$f=f^R+if^I+jf^J+kf^K\in C(\mathbb{R}, \mathbb{Q}^n)$$ is almost automorphic if and only if $$f^\nu \in C(\mathbb{R}, \mathbb{R}^n)$$ is almost automorphic, where $$\nu \in \Lambda :=\{R, I,J, K\}$$. Denote all the almost automorphic functions by $$AA(\mathbb{R}, \mathbb{Q}^n)$$, which is a Banach space when it is endowed with the supremum norm. Lemma 2.1 (N’Guérékata, 2001; Diagana, 2013) Let X, Y be Banach spaces, and if $$f, f_1, f_2 \in AA(\mathbb{R}, X)$$, then (i) $$f_1+f_2\in AA(\mathbb{R}, X)$$; (ii) $$\alpha f\in AA(\mathbb{R}, X)$$ for any constant $$\alpha \in \mathbb{R}$$; (iii) if $$\varphi :X\rightarrow Y$$ is a continuous function, then the composite function $$f\circ \varphi : \mathbb{R}\rightarrow Y$$ is almost automorphic. Definition 2.2 (Li & Yang, 2014) Let A(t) be an n × n matrix function on $$\mathbb{R}$$. Then the linear system   \begin{equation} x^{\prime}(t)=A(t)x(t),\,\, t\in\mathbb{R} \end{equation} (2.1)is said to admit an exponential dichotomy on $$\mathbb{R}$$ if there exist positive constants $$k_i, \alpha _i, i=1, 2$$, projection P and the fundamental solution matrix X(t) of (2.1), satisfying   \begin{align*} \big|X(t)PX^{-1}(s)\big| &\leq k_1e^{-\alpha_1(t-s)}, \quad s, t \in\mathbb{R},\,\, t \geq s, \\ \big|X(t)(I-P)X^{-1}(s)\big|&\leq k_2e^{-\alpha_2(s-t)},\quad s, t \in\mathbb{R},\,\, t \leq s, \end{align*}where |⋅| is a matrix norm on $$\mathbb{R}$$. Consider the following almost automorphic system   \begin{equation} x^{\prime}(t)=A(t)x(t)+f(t),\,\, t \in \mathbb{R}, \end{equation} (2.2)where A(t) is an almost automorphic matrix function and f(t) is an almost automorphic vector function. Lemma 2.2 (Li & Yang, 2014) Suppose that $$A(t)\in AA(\mathbb{R}, \mathbb{R}^{n\times n})$$, such that $$\{A^{-1}(t)\}$$ is bounded. Moreover, suppose that $$f\in AA(\mathbb{R}, \mathbb{R}^n)$$ and (2.1) admits an exponential dichotomy, then (2.2) has a solution $$x(t)\in AA(\mathbb{R}, \mathbb{R}^n)$$ that can be expressed as   $$ x(t)=\int_{-\infty}^{t}X(t)PX^{-1}(s)\,f(s)\,\mathrm{d}s -\int_t^{+\infty}X(t)(I-P)X^{-1}(s)\,f(s)\,\mathrm{d}s, $$where X(t) is the fundamental solution matrix of (2.1), I denotes the n × n-identity matrix. Lemma 2.3 (Li & Yang, 2014) Let $$c_{p}(t)>0$$ be an almost automorphic function on $$\mathbb{R}$$ and   $$ \min_{1\leq p \leq n}\{\inf_{t\in \mathbb{R}}c_p(t)\}>0,$$then the linear system   \begin{equation*} x^{\prime}(t)=\textrm{diag}(-c_{1}(t),-c_{2}(t),\dots,-c_{n}(t))x(t) \end{equation*}admits an exponential dichotomy on $$\mathbb{R}$$. Let $$X=x^R+ix^I+jx^J+kx^K\in \mathbb{Q},$$ where $$x^\nu \in \mathbb{R},\nu \in \Lambda $$ and assume that $$f_q:\mathbb{Q}\rightarrow \mathbb{Q}$$ can be expressed as   \begin{align*} f_q(x)&=f^R(x^R, x^I, x^J, x^K)+if^I(x^R, x^I, x^J, x^K)+jf^J(x^R, x^I, x^J, x^K)\\ &\quad+kf^K(x^R, x^I, x^J, x^K), \end{align*}where $$f_q^\nu : \mathbb{R}^4\rightarrow \mathbb{R}$$, $$\nu \in \Lambda $$, q ∈ N. To avoid the non-commutativity of quaternion multiplication, according to Hamilton rules, we decompose system (1.1) into an equivalent real-valued system:   \begin{eqnarray*} \left(x_{p}^R\right)^{\prime}(t)&=&-c_p(t) x_p^R(t)+\sum_{q=1}^{n}a_{pq}^R(t)f_q^R[t, x] -\sum_{q=1}^{n}a_{pq}^I(t)f_q^I[t, x] -\sum_{q=1}^{n}a_{pq}^J(t)f_q^J[t, x]\nonumber\\ &&-\sum_{q=1}^{n}a_{pq}^K(t)f_q^K[t, x] +\sum_{q=1}^{n}b_{pq}^R(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}^R[t, u, x]\,\mathrm{d}u -\sum_{q=1}^{n}b_{pq}^I(t)\nonumber\\ &&\times\int_{0}^{+\infty}K_{pq}(u) g_{q}^I[t, u, x]\,\mathrm{d}u-\sum_{q=1}^{n}b_{pq}^J(t) \int_{0}^{+\infty}K_{pq}(u)g_{q}^J[t, u, x]\,\mathrm{d}u\nonumber\\ &&-\sum_{q=1}^{n}b_{pq}^K(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}^K[t, u, x]\,\mathrm{d}u +\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}\bigg(d_{pqr}^R(t)h_q^R[t, x]\tilde{h}_r^R[t, x]\nonumber\\ &&-d_{pqr}^I(t)h_q^I[t, x]\tilde{h}_r^R[t, x]-d_{pqr}^J(t)h_q^J[t, x]\tilde{h}_r^R[t, x] -d_{pqr}^K(t)h_q^K[t, x]\tilde{h}_r^R[t, x]\nonumber\\ &&-d_{pqr}^I(t)h_q^R[t, x]\tilde{h}_r^I[t, x]-d_{pqr}^R(t)h_q^I[t, x]\tilde{h}_r^I[t, x] +d_{pqr}^K(t)h_q^J[t, x]\tilde{h}_r^I[t, x]\nonumber\\ &&-d_{pqr}^J(t)h_q^K[t, x]\tilde{h}_r^I[t, x]-d_{pqr}^J(t)h_q^R[t, x]\tilde{h}_r^J[t, x] -d_{pqr}^K(t)h_q^I[t, x]\tilde{h}_r^J[t, x]\nonumber\\ &&-d_{pqr}^R(t)h_q^J[t, x]h_q^J[t, x]+d_{pqr}^I(t)h_q^K[t, x]\tilde{h}_r^J[t, x]-d_{pqr}^K(t)h_q^R[t, x]\tilde{h}_r^K[t, x]\nonumber\\ &&+d_{pqr}^J(t)h_q^I[t, x]\tilde{h}_r^K[t, x]-d_{pqr}^I(t)h_q^J[t, x]\tilde{h}_r^K[t, x] -d_{pqr}^R(t)h_q^K[t, x]\tilde{h}_r^K[t, x]\bigg)+I_p^R(t)\nonumber\\ &=:& -c_p(t) x_p^R(t)+\Pi_p^R(t,x)+I_p^R(t), \,\, p\in N, \end{eqnarray*}  \begin{eqnarray*} \left(x_{p}^I\right)^{\prime}(t)&=&-c_p(t) x_p^I(t)+\sum_{q=1}^{n}a_{pq}^I(t)f_q^R[t, x] +\sum_{q=1}^{n}a_{pq}^R(t)f_q^I[t, x]-\sum_{q=1}^{n}a_{pq}^K(t)f_q^J[t, x]\nonumber\\[-7pt] &&+\sum_{q=1}^{n}a_{pq}^J(t)f_q^K[t, u, x] +\sum_{q=1}^{n}b_{pq}^I(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}^R[t, u, x]\,\mathrm{d}u +\sum_{q=1}^{n}b_{pq}^R(t)\nonumber\\[-5pt] &&\times\int_{0}^{+\infty}K_{pq}(u) g_{q}^I[t, u, x]\,\mathrm{d}u-\sum_{q=1}^{n}b_{pq}^K(t) \int_{0}^{+\infty}K_{pq}(u)g_{q}^J[t, u, x]\,\mathrm{d}u\nonumber\\[-5pt] &&+\sum_{q=1}^{n}b_{pq}^J(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}^K[t, u, x]\,\mathrm{d}u +\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}\bigg(d_{pqr}^I(t) h_q^R[t, x]\tilde{h}_r^R[t, x]\nonumber\\[-5pt] &&+d_{pqr}^R(t)h_q^I[t, x] \tilde{h}_r^R[t, x]-d_{pqr}^K(t)h_q^J[t, x]\tilde{h}_r^R[t,x] +d_{pqr}^J(t)h_q^K[t, x]\tilde{h}_r^R[t, x]\nonumber\\ &&+d_{pqr}^R(t)h_q^R[t, x]\tilde{h}_r^I[t, x]-d_{pqr}^I(t)h_q^I[t, x]\tilde{h}_r^I[t,x]-d_{pqr}^J(t)h_q^J[t, x]\tilde{h}_r^I[t, x]\nonumber\\ &&-d_{pqr}^K(t)h_q^K[t, x]\tilde{h}_r^I[t, x]-d_{pqr}^K(t)h_q^R[t, x]\tilde{h}_r^J[t, x] +d_{pqr}^J(t)h_q^I[t, x]\tilde{h}_r^J[t, x]\nonumber\\ &&-d_{pqr}^I(t)h_q^J[t, x] h_q^J[t, x]-d_{pqr}^R(t)h_q^K[t, x]\tilde{h}_r^J[t, x] +d_{pqr}^J(t)h_q^R[t, x]\tilde{h}_r^K[t, x]\nonumber\\[-5pt] &&+d_{pqr}^K(t)h_q^I[t, x]\tilde{h}_r^K[t, x] +d_{pqr}^R(t)h_q^J[t, x]\tilde{h}_r^K[t, x] -d_{pqr}^I(t)h_q^K[t, x]\tilde{h}_r^K[t, x]\bigg)+I_p^I(t)\nonumber\\[-5pt] &=:& -c_p(t) x_p^I(t)+\Pi_p^I(t,x)+I_p^I(t), \,\, p\in N, \end{eqnarray*}  \begin{eqnarray*} \left(x_{p}^J\right)^{\prime}(t)&=&-c_p(t) x_p^J(t)+\sum_{q=1}^{n}a_{pq}^J(t)f_q^R[t, x] +\sum_{q=1}^{n}a_{pq}^K(t)f_q^I[t, x]+\sum_{q=1}^{n}a_{pq}^R(t)f_q^J[t, x]\nonumber\\[-5pt] &&-\sum_{q=1}^{n}a_{pq}^I(t)f_q^K[t, x] +\sum_{q=1}^{n}b_{pq}^J(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}^R[t, u, x]\,\mathrm{d}u +\sum_{q=1}^{n}b_{pq}^K(t)\nonumber\\[-5pt] &&\times\int_{0}^{+\infty}K_{pq}(u) g_{q}^I[t, u, x]\,\mathrm{d}u+\sum_{q=1}^{n}b_{pq}^R(t) \int_{0}^{+\infty}K_{pq}(u)g_{q}^J[t, u, x]\,\mathrm{d}u\nonumber\\ &&-\sum_{q=1}^{n}b_{pq}^I(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}^K[t, ux]\,\mathrm{d}u+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n} \bigg(d_{pqr}^J(t) h_q^R[t, x]\tilde{h}_r^R[t, x]\nonumber\\ &&+d_{pqr}^K(t)h_q^I[t, x]\tilde{h}_r^R[t, x]+d_{pqr}^R(t)h_q^J[t, x]\tilde{h}_r^R[t, x] -d_{pqr}^I(t)h_q^K[t, x]\tilde{h}_r^R[t, x]\nonumber\\[3pt] &&-d_{pqr}^K(t)h_q^R[t, x]\tilde{h}_r^I[t, x]-d_{pqr}^J(t)h_q^I[t, x]\tilde{h}_r^I[t, x]+d_{pqr}^I(t)h_q^J[t, x]\tilde{h}_r^I[t, x]\nonumber\\ &&+d_{pqr}^R(t)h_q^K[t, x]\tilde{h}_r^I[t, x]+d_{pqr}^R(t)h_q^R[t, x]\tilde{h}_r^J[t, x] -d_{pqr}^I(t)h_q^I[t, x]\tilde{h}_r^J[t, x]\nonumber\\ &&-d_{pqr}^J(t)h_q^J[t, x]h_q^J[t, x]-d_{pqr}^K(t)h_q^K[t, x]\tilde{h}_r^J[t, x] -d_{pqr}^I(t)h_q^R[t, x]\tilde{h}_r^K[t, x]\nonumber\\ &&-d_{pqr}^R(t)h_q^I[t, x]\tilde{h}_r^K[t, x]+d_{pqr}^K(t)h_q^J[t, x]\tilde{h}_r^K[t, x] -d_{pqr}^J(t)h_q^K[t, x]\tilde{h}_r^K[t, x]\bigg)+I_p^J(t)\nonumber\\ &=:& -c_p(t) x_p^J(t)+\Pi_p^J(t,x)+I_p^J(t), \,\, p\in N, \end{eqnarray*}  \begin{eqnarray*} \left(x_{p}^K\right)^{\prime}(t)&=&-c_p(t) x_p^K(t)+\sum_{q=1}^{n}a_{pq}^K(t)f_q^R[t, x] -\sum_{q=1}^{n}a_{pq}^J(t)f_q^I[t, x] +\sum_{q=1}^{n}a_{pq}^I(t)f_q^J[t, x]\nonumber\\ &&+\sum_{q=1}^{n}a_{pq}^R(t)f_q^K[t, x] +\sum_{q=1}^{n}b_{pq}^K(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}^R[t, u, x]\,\mathrm{d}u -\sum_{q=1}^{n}b_{pq}^J(t)\nonumber\\ &&\times \int_{0}^{+\infty}K_{pq}(u)g_{q}^I[t, u, x]\,\mathrm{d}u+\sum_{q=1}^{n}b_{pq}^I(t) \int_{0}^{+\infty}K_{pq}(u)g_{q}^J[t, u, x]\,\mathrm{d}u\nonumber\\[-5pt] &&+\sum_{q=1}^{n}b_{pq}^R(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}^K[t, u, x]\,\mathrm{d}u +\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}\bigg(d_{pqr}^K(t) h_q^R[t, x]\tilde{h}_r^R[t, x]\nonumber\\[-5pt] &&-d_{pqr}^I(t)h_q^I[t, x] \tilde{h}_r^R[t, x]+d_{pqr}^I(t)h_q^J[t, x]\tilde{h}_r^R[t, x] +d_{pqr}^R(t)h_q^K[t, x]\tilde{h}_r^R[t, x]\nonumber\\ &&-d_{pqr}^J(t)h_q^R[t, x]\tilde{h}_r^I[t, x] -d_{pqr}^K(t)h_q^I[t, x]\tilde{h}_r^I[t, x]-d_{pqr}^R(t)h_q^J[t, x]\tilde{h}_r^I[t, x]\nonumber\\ &&-d_{pqr}^I(t)h_q^K[t, x]\tilde{h}_r^I[t, x] +d_{pqr}^I(t)h_q^R[t, x]\tilde{h}_r^J[t, x] +d_{pqr}^R(t)h_q^I[t, x]\tilde{h}_r^J[t, x]\nonumber\\ &&-d_{pqr}^K(t)h_q^J[t, x]h_q^J[t, x]+d_{pqr}^J(t)h_q^K[t, x]\tilde{h}_r^J[t, x] -d_{pqr}^R(t)h_q^R[t, x]\tilde{h}_r^K[t, x]\nonumber\\ &&-d_{pqr}^I(t)h_q^I[t, x]\tilde{h}_r^K[t, x]-d_{pqr}^J(t)h_q^J[t, x]\tilde{h}_r^K[t, x] -d_{pqr}^J(t)h_q^K[t, x]\tilde{h}_r^K[t, x]\bigg)+I_p^K(t)\nonumber\\ &=:& -c_p(t) x_p^K(t)+\Pi_p^K(t,x)+I_p^K(t), \,\, p\in N, \end{eqnarray*} where   \begin{align*} f_q^l[t, x]&\triangleq f_q^l(x_q^R(t-\tau_{pq}(t)), x_q^I(t-\tau_{pq}(t)), x_q^J(t-\tau_{pq}(t)), x_q^K(t-\tau_{pq}(t))),\\ g_q^l[t, u, x]&\triangleq g_q^l(x_q^R(t-u), x_q^I(t-u), x_q^J(t-u), x_q^K(t-u)), \\ h_q^l[t, x]&\triangleq h_q^l(x_q^R(t-\sigma_{pqr}(t)), x_q^I(t-\sigma_{pqr}(t)), x_q^J(t-\sigma_{pqr}(t)), x_q^K(t-\sigma_{pqr}(t))), \\ \tilde{h}_r^l[t, x]&\triangleq \tilde{h}_r^l(x_r^R(t-\xi_{pqr}(t)), x_r^I(t-\xi_{pqr}(t)), x_r^J(t-\xi_{pqr}(t)), x_r^K(t-\xi_{pqr}(t))). \end{align*} Hence, system (1.1) is transformed into the following equivalent real-valued system:   \begin{equation} \left(x_{p}^\nu\right)^{\prime}(t)= -c_p(t) x_p^\nu(t)+\Pi_p^\nu(t,x)+I_p^\nu(t), \,\, p\in N, \nu\in \Lambda. \end{equation} (2.3) Remark 2.2 If $$x=(x_1^R, x_2^R, \ldots , x_n^R, x_1^I, x_2^I, \ldots , x_n^I, x_1^J, x_2^J, \ldots , x_n^J, x_1^K, x_2^K, \ldots , x_n^K)$$ is a solution to system (2.3), then $$X(t)=(X_{1}(t),X_{2}(t),\ldots , X_{n}(t))^{T}$$ must be a solution of (1.1), where $$X_{l}(t)=x^{R}_{l}(t)+ix^{I}_{l}(t)+jx^{J}_{l}(t)+kx^{K}_{l}(t),l\in N$$. Thus, the problem of finding a solution for (1.1) is reduced to finding one for system (2.3). For considering the stability of solutions of (1.1), we just need to consider the stability of solutions of system (2.3). For convenience, in the following, we introduce the following notation:   $$ w^-=\inf\limits_{t\in\mathbb{R}}\big|w(t)\big|,\,\,w^+=\sup\limits_{t\in\mathbb{R}}\big|w(t)\big|, $$where $$w: \mathbb{R}\rightarrow \mathbb{R}$$ is a bounded function, and for p, q, r ∈ N, we denote   $$ \sigma^+=\max\limits_{p, q, r\in N}\big\{\sigma^+_{pqr}\big\},\,\, \xi^+=\max\limits_{ p, q, r\in N}\big\{\xi^+_{pqr}\big\},\,\,c^-=\max\limits_{p\in N}\big\{c_p^-\big\}, \,\,\tau^{+}=\max\limits_{p, q \in N}\big\{\tau^+_{pq}\big\},$$  $$ \alpha_{pq}=\inf\limits_{t\in \mathbb{R}}\{1-\tau_{pq}^{\prime}(t)\},\,\, \beta_{pqr}=\inf\limits_{t\in \mathbb{R}}\{1-\sigma_{pqr}^{\prime}(t)\},\,\, \gamma_{pqr}=\inf\limits_{t\in \mathbb{R}}\{1-\xi_{pqr}^{\prime}(t)\}.\quad\quad\;\,$$ Throughout the paper, we assume that the following conditions hold: $$(A_1)$$ Function $$c_{p}\in AA(\mathbb{R},\mathbb{R}^{+})$$ with $$\min _{1\leq p \leq n}\{\inf _{t\in \mathbb{R}}c_p(t)\}>0$$, $$a_{pq}, b_{pq}, d_{pqr}, I_p\in AA(\mathbb{R}, \mathbb{Q})$$ and $$\tau _{pq}, \sigma _{pqr}, \xi _{pqr}\in C^1(\mathbb{R}, \mathbb{R}^+)\cap AA(\mathbb{R},\mathbb{R})$$ with $$\inf \limits _{t\in R}\{1-\tau _{pq}^{\prime}(t)\}>0$$, $$\inf \limits _{t\in R}\{1-\sigma _{pqr}^{\prime}(t)\}>0$$, $$\inf \limits _{t\in R}\{1-\xi _{pqr}^{\prime}(t)\}>0$$, where p, q, r ∈ N. $$(A_2)$$ Functions $$f_q, g_q, h_q, h_r\in C(\mathbb{R}, \mathbb{Q})$$ and for any $$u^\nu , v^\nu \in \mathbb{R}$$, there exist positive constants $$\lambda _q^{\nu }$$, $$\delta _q^{\nu }$$, $$\rho _q^{\nu }$$, $$N_q$$ such that $$|h_q^\nu (u^R, u^I, u^J, u^K)|\leq N_q$$,   \begin{eqnarray*} \big|\,f_q^\nu(u^R, u^I, u^J, u^K)-f_q^\nu(v^R, v^I, v^J, v^K)\big| &\leq&\lambda_q^{R}|u^R-v^R|+\lambda_q^{I}|u^I-v^I|+ \lambda_q^{J}|u^J-v^J|\\ &&+\lambda_q^{K}|u^K-v^K|,\\ \big|g_q^\nu(u^R, u^I, u^J, u^K)-g_q^\nu(v^R, v^I, v^J, v^K)\big| &\leq& \delta_q^{R}|u^R-v^R|+\delta_q^{I}|u^I-v^I|+\delta_q^{J}|u^J-v^J|\\ &&+\delta_q^{K}|u^K-v^K|,\\ \big|h_q^\nu(u^R, u^I, u^J, u^K)-h_q^\nu(v^R, v^I, v^J, v^K)\big| &\leq& \rho_q^{R}|u^R-v^R|+\rho_q^{I}|u^I-v^I|+\rho_q^{J}|u^J-v^J|\\ &&+\rho_q^{K}|u^K-v^K| \end{eqnarray*}and $$f_q^\nu (0, 0, 0, 0)=g_q^\nu (0, 0, 0, 0)=h_q^\nu (0, 0, 0, 0)=0$$, where $$\nu \in \Lambda $$, q ∈ N. $$(A_{3})$$ For p, q ∈ N, the delay kernels $$K_{pq}: [0, \infty )\rightarrow \mathbb{R}$$ are continuous and $$|K_{pq}(t)|e^{\iota t}$$ are integrable on $$[0, \infty )$$ for a certain positive constant $$\iota $$. $$(A_{4})$$ There exists a positive constant $$\kappa $$ such that   \begin{equation*} \max\limits_{p\in E}\bigg\{\max\limits_{\nu\in\Lambda}\bigg\{\frac{\Theta_p \kappa +I_{p}^{\nu^{+}}}{c_p^-} \bigg\}\bigg\}\leq \kappa,\quad \max\limits_{p\in E}\bigg\{\frac{\Xi_p}{c_p^-}\bigg\}:=\mu<1, \end{equation*}where   \begin{align*} \Theta_p & =A_{p}^{\ast}+B_{p}^{\ast}+D_{p}^{\ast},\quad p\in N,\\ \Xi_p & =A_{p}^{\ast}+B_{p}^{\ast}+D_{p}^{\ast}+\hat{D}_{p}^{\ast},\quad p\in N,\\ A_{p}^{\ast}&=\sum_{q=1}^{n}4\lambda_q\Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big),\quad p\in N,\\ B_{p}^{\ast}&=\sum_{q=1}^{n}4\delta_q\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big)\int_{0}^{+\infty}\big|K_{pq}(u)\big|\,\mathrm{d}u ,\quad p\in N,\\ D_{p}^{\ast}&=\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}16N_q\rho_r \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big),\quad p\in N,\\ \hat{D}_{p}^{\ast}&=\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}16N_{r}\rho_{q} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big),\quad p\in N,\\ \lambda_q&=\max\left\{\lambda^R_q,\lambda^I_q, \lambda^J_q, \lambda^K_q\right\}, \delta_q=\max\left\{\delta^R_q, \delta^I_q, \delta^J_q, \delta^K_q\right\}, \rho_q=\max\left\{\rho^R_q, \rho^I_q, \rho^J_q, \rho^K_q\right\}. \end{align*} 3. The existence of almost automorphic solutions In this section, we shall state and prove the sufficient conditions for the existence of almost automorphic solutions of system (1.1). In view of Remark 2.7, we only need to prove that (2.3) has an almost automorphic solution. To this end, set $$\mathbb{Y}\!=\!\big \{\varphi \!=\!(\varphi _1^R, \varphi _2^R,\! \ldots , \varphi _n^R, \varphi _1^I, \varphi _2^I,\! \ldots , \varphi _n^I, \varphi _1^J, \varphi _2^J,\! \ldots , \varphi _n^J, \varphi _1^K, \varphi _2^K, \ldots , \varphi _n^K)^T \!\!\in AA(\mathbb{R}, \mathbb{R}^{4n}) \big \}$$ with the norm $$\|\varphi \|=\max _{p\in N}\{\sup _{t\in \mathbb{R}}|\varphi _p^\nu (t)|, \nu \in \Lambda \}$$, then $$\mathbb{Y}$$ is a Banach space. Theorem 3.1 Assume that $$(A_1)$$–$$(A_4)$$ hold, then system (2.3) has a unique almost automorphic solution in $$\mathbb{Y}_0=\{\varphi \in \mathbb{Y}: ||\varphi ||\leq \kappa \}$$. Proof. For any given $$\varphi \in \mathbb{Y}$$, consider the following system:   \begin{equation} \left(x_{p}^\nu\right)^{\prime}(t)=-c_p(t) x_p^\nu(t)+\Pi^\nu_p(t, \varphi(t))+I_p^\nu(t),\,\, p\in N,\nu\in \Lambda. \end{equation} (3.1)According to Lemma 2.3, the linear system   \begin{equation*} \left(x_p^\nu\right)^{\prime}(t)=-c_p(t)x_p^\nu(t),\,\, p\in N, \nu\in \Lambda \end{equation*}admits an exponential dichotomy. Then by Lemma 2.2, we obtain that system (3.1) has a unique almost automorphic solution   $$ x^\varphi=((x^\varphi)_1^R, \ldots, (x^\varphi)_n^R, (x^\varphi)_1^I, \ldots, (x^\varphi)_n^I, (x^\varphi)_1^J, \ldots, (x^\varphi)_n^J, (x^\varphi)_1^K, \ldots, (x^\varphi)_n^K)^T,$$where   \begin{equation*} (x^{\varphi})_p^l(t)=\int_{-\infty}^{t}e^{-\int_s^{t}c_p(u)\,\mathrm{d}u}\left[\Pi_p^l(s, \varphi )+I_p^l(t)\right]\,\mathrm{d}s,\quad p\in N, l\in \Lambda. \end{equation*} Define an operator $$\Phi :\mathbb{Y}\rightarrow \mathbb{Y}$$ by setting   \begin{equation*} \varphi\rightarrow x^\varphi,\quad \varphi\in \mathbb{Y}. \end{equation*} At first, we will prove that for any $$ \varphi \in \mathbb{Y}_0$$, $$\Phi \varphi \in \mathbb{Y}_0$$. For p ∈ N, we have   \begin{eqnarray*} \big|\Pi_p^R(t, \varphi(t))\big| &\leq& \sum_{q=1}^{n}\Big[\big|a_{pq}^{R}(t)\big|\big|f_{q}^R[t, \varphi]\big| +\big|a_{pq}^{I}(t)\big|\big|f_{q}^I[t, \varphi]\big|+\big|a_{pq}^{J}(t)\big|\big|f_{q}^J[t, \varphi]\big|\\[-9pt] &&+\big|a_{pq}^{K}(t)\big|\big|f_{q}^K[t, \varphi]\big|\Big] +\sum_{q=1}^{n}\Big[\big|b_{pq}^{R}(t)\big|\int_{0}^{+\infty}\big|K_{pq}(u)\big|\big|g_{q}^R[t, u, \varphi]\big|\,\mathrm{d}u\\[-5pt] &&+\big|b_{pq}^{I}(t)\big|\int_{0}^{+\infty}\big|K_{pq}(u)\big|\big|g_{q}^I[t, u, \varphi]\big|\,\mathrm{d}u+\big|b_{pq}^{J}(t)\big|\int_{0}^{+\infty}\big|K_{pq}(u)\big|\big|g_{q}^J[t, u, \varphi]\big|\,\mathrm{d}u\\[-5pt] &&+\big|b_{pq}^{K}(t)\big|\int_{0}^{+\infty}\big|K_{pq}(u)\big|\big|g_{q}^K[t, u, \varphi]\big|\,\mathrm{d}u\Big] +\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}N_q\bigg(\big|d_{pqr}^R(t)\big|\big|\tilde{h}_r^R[t, \varphi]\big|\\[-5pt] &&+\big|d_{pqr}^I(t)\big|\big|\tilde{h}_r^R[t, \varphi]\big| +\big|d_{pqr}^J(t)\big|\big|\tilde{h}_r^R[t, \varphi]\big|+\big|d_{pqr}^K(t)\big|\big|\tilde{h}_r^R[t, \varphi]\big| +\big|d_{pqr}^I(t)\big|\big|\tilde{h}_r^I[t, \varphi]\big|\\ &&+\big|d_{pqr}^R(t)\big|\big|\tilde{h}_r^I[t, \varphi]\big| +\big|d_{pqr}^K(t)\big|\big|\tilde{h}_r^I[t, \varphi]\big| +\big|d_{pqr}^J(t)\big|\big|\tilde{h}_r^I[t, \varphi]\big| +\big|d_{pqr}^J(t)\big|\big|\tilde{h}_r^J[t, \varphi]\big|\\ &&+\big|d_{pqr}^K(t)\big|\big|\tilde{h}_r^J[t, \varphi]\big| +\big|d_{pqr}^R(t)\big|\big|\tilde{h}_r^J[t, \varphi]\big| +\big|d_{pqr}^I(t)\big|\big|\tilde{h}_r^J[t, \varphi]\big| +\big|d_{pqr}^K(t)\big|\big|\tilde{h}_r^K[t, \varphi]\big|\\ &&+\big|d_{pqr}^J(t)\big|\tilde{h}_r^K[t, \varphi]\big| +\big|d_{pqr}^I(t)\big|\big|\tilde{h}_r^K[t, \varphi]\big| +\big|d_{pqr}^R(t)\big|\big|\tilde{h}_r^K[t, \varphi]\big|\bigg)\\ &\leq&\sum_{q=1}^{n}\Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+ a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big) \Big[\lambda_q^R\big|\varphi_q^R(t-\tau_{pq}(t))\big|+ \lambda_q^I\big|\varphi_q^I(t-\tau_{pq}(t))\big|\\ &&+\lambda_q^J\big|\varphi_q^J(t-\tau_{pq}(t))\big| +\lambda_q^K\big|\varphi_q^K(t-\tau_{pq}(t))\big|\Big] +\sum_{q=1}^{n}\int_{0}^{+\infty}\big|K_{pq}(u)\big|\\ &&\times\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big) \Big[\delta_q^R\big|\varphi_q^R(t-u)\big| +\delta_q^I\big|\varphi_q^I(t-u)\big|\\ &&+\delta_q^J\big|\varphi_q^J(t-u)\big| +\delta_q^K\big|\varphi_q^K(t-u)\big|\Big]\,\mathrm{d}u +\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}4N_q\Big(d_{pqr}^{R^{+}} +d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ &&\times\Big[\rho_r^R\big|\varphi_r^R(t-\xi_{pqr}(t))\big| +\rho_r^I\big|\varphi_r^I(t-\xi_{pqr}(t))\big|+\rho_r^J\big|\varphi_r^J(t-\xi_{pqr}(t))\big|\\ &&+\rho_r^K\big|\varphi_r^K(t-\xi_{pqr}(t))\big|\Big]. \end{eqnarray*}Hence,   \begin{eqnarray} \big|(\Phi\varphi)^R_p(t)\big|&=&\bigg|\int_{-\infty}^te^{-\int_s^{t}c_p(u)\,\mathrm{d}u}\Big(\Pi_p^R(s, \varphi)+I_p^R(s)\Big)\bigg|\,\mathrm{d}s\nonumber\\ &\leq&\int_{-\infty}^te^{-\int_s^{t}c_p(u)\,\mathrm{d}u}\bigg\{\bigg(\sum_{q=1}^{n} 4\lambda_q\Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\nonumber\\ &&+\sum_{q=1}^{n}4\delta_q\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big) \int_{0}^{+\infty}\big|K_{pq}(u)\big|\,\mathrm{d}u\nonumber\\ &&+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}16N_q\rho_r \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\bigg)\kappa +\big|I_p^R(s)\big|\bigg\}\mathrm{d}s\nonumber\\ &\leq& \frac{1}{c_p^{-}}\bigg(A_{p}^{\ast}\kappa+B_{p}^{\ast}\kappa +D_{p}^{\ast}\kappa+I_p^{R^{+}}\bigg)\nonumber\\ &=&\frac{\Theta_p \kappa+I_p^{R^{+}}}{c_p^{-}} \leq \kappa,\quad p\in E. \end{eqnarray} (3.2)Repeating the same calculation, from system (3.1), we obtain   \begin{equation} \big|(\Phi\varphi)^\nu_p(t)\big| \leq\frac{\Theta_p \kappa+I_{p}^{\nu^{+}}}{c_p^{-}}\leq \kappa,\quad p\in N, \,\,\nu=I,J,K. \end{equation} (3.3)Together with the above inequalities (3.2) and (3.3) and Assumption $$(A_4)$$, we obtain that $$\Phi $$ is a self mapping in $$\mathbb{Y}_0$$. Next, we will show that $$\Phi $$ is a contraction mapping in $$\mathbb{Y}_0$$. For any $$\varphi , \psi \in \mathbb{Y}_0$$ and p ∈ N, we have   \begin{align*} \big|\Pi^R_p(t, \varphi)-\Pi_p^R(t, \psi)\big| \leq&\sum_{q=1}^{n}\Big(\big|a_{pq}^{R}(t)\big|\big|f_{q}^R[t, \varphi] -f_q^R[t, \psi]\big|+\big|a_{pq}^{I}(t)\big|\big|f_{q}^I[t, \varphi]-f_q^I[t, \psi]\big|\qquad\qquad\\[-9pt] &+\big|a_{pq}^{J}(t)\big|\big|f_{q}^J[t, \varphi]-f_q^J[t, \psi]\big|+\big|a_{pq}^{K}(t)\big|\big|f_{q}^K[t, \varphi]-f_q^K[t, \psi]\big|\Big)\\[-9pt] &+\sum_{q=1}^{n}\bigg(\big|b_{pq}^{R}(t)\big|\int_0^{+\infty}\big|K_{pq}(u)\big| \big|g_{q}^R[t, u, \varphi]-g_q^R[t, u, \psi]\big|\,\mathrm{d}u\\[-9pt] &+\big|b_{pq}^{I}(t)\big|\int_0^{+\infty}\big|K_{pq}(u)\big| \big|g_{q}^I[t, u, \varphi]-g_q^I[t, u, \psi]\big|\,\mathrm{d}u\\[-2pt] &+\big|b_{pq}^{J}(t)\big|\int_0^{+\infty}\big|K_{pq}(u)\big|\big|g_{q}^J[t, u, \varphi] -g_q^J[t, u, \psi]\big|\,\mathrm{d}u\\[-2pt] &+\big|b_{pq}^{K}(t)\big|\int_0^{+\infty}\big|K_{pq}(u)\big|\big|g_{q}^K[t, u, \varphi] -g_q^K[t, u, \psi]\big|\,\mathrm{d}u\bigg) \end{align*}  \begin{eqnarray*} &&\qquad\qquad\qquad\qquad\qquad+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}\bigg[N_q\Big(\big|d_{pqr}^R(t)\big| \big|\tilde{h}_r^R[t, \varphi]-\tilde{h}_r^R[t, \psi]\big|\! +\!\big|d_{pqr}^I(t)\big|\big|\tilde{h}_r^R[t, \varphi]-\tilde{h}_r^R[t, \psi]\big|\\[-5pt] &&\qquad\qquad\qquad\qquad\qquad+\big|d_{pqr}^J(t)\big|\big|\tilde{h}_r^R[t, \varphi]-\tilde{h}_r^R[t, \psi]\big| +\big|d_{pqr}^K(t)\big|\big|\tilde{h}_r^R[t, \varphi]-\tilde{h}_r^R[t, \psi]\big| \end{eqnarray*}  \begin{eqnarray*} &&+\big|d_{pqr}^I(t)\big|\big|\tilde{h}_r^I[t, \varphi]-\tilde{h}_r^I[t, \psi]\big| +\big|d_{pqr}^R(t)\big| \big|\tilde{h}_r^I[t, \varphi]-\tilde{h}_r^I[t, \psi]\big|\\[2pt] &&+\big|d_{pqr}^K(t)\big|\big|\tilde{h}_r^I[t, \varphi]-\tilde{h}_r^I[t, \psi]\big|+\big|d_{pqr}^J(t)\big|\big|\tilde{h}_r^I[t, \varphi]-\tilde{h}_r^I[t, \psi]\big|\\ &&+\big|d_{pqr}^J(t)\big| \big|\tilde{h}_r^J[t, \varphi]-\tilde{h}_r^J[t, \psi]\big|+\big|d_{pqr}^K(t)\big|\big|\tilde{h}_r^J[t, \varphi]-\tilde{h}_r^J[t, \psi]\big|\\ &&+\big|d_{pqr}^R(t)\big|\big|\tilde{h}_r^J[t, \varphi]-\tilde{h}_r^J[t, \psi]\big| +\big|d_{pqr}^I(t)\big|\big|\tilde{h}_r^J[t, \varphi]-\tilde{h}_r^J[t, \psi]\big|\\ &&+\big|d_{pqr}^K(t)\big|\big|\tilde{h}_r^K[t, \varphi]-\tilde{h}_r^K[t, \psi]\big| +\big|d_{pqr}^J(t)\big|\big|\tilde{h}_r^K[t, \varphi]-\tilde{h}_r^K[t, \psi]\big|\\ &&+\big|d_{pqr}^I(t)\big|\big|\tilde{h}_r^K[t, \varphi]-\tilde{h}_r^K[t, \psi]\big| +\big|d_{pqr}^R(t)\big|\big|\tilde{h}_r^K[t, \varphi]-\tilde{h}_r^K[t, \psi]\big|\Big)\\[-2pt] &&+N_r\Big(\big|d_{pqr}^R(t)\big| \big|h_q^R[t, \varphi]-h_q^R[t, \psi]\big| +\big|d_{pqr}^I(t)\big|\big|h_q^R[t, \varphi]-h_q^R[t, \psi]\big|\\[-5pt] &&+\big|d_{pqr}^J(t)\big|\big|h_q^R[t, \varphi]-h_q^R[t, \psi]\big| +\big|d_{pqr}^K(t)\big|\big|h_q^R[t, \varphi]-h_q^R[t, \psi]\big|\\ &&+\big|d_{pqr}^I(t)\big|\big|h_q^I[t, \varphi]-h_q^I[t, \psi]\big| +\big|d_{pqr}^R(t)\big| \big|h_q^I[t, \varphi]-h_q^I[t, \psi]\big|\\ &&+\big|d_{pqr}^K(t)\big|\big|h_q^I[t, \varphi]-h_q^I[t, \psi]\big|+\big|d_{pqr}^J(t)\big|\big|h_q^I[t, \varphi]-h_q^I[t, \psi]\big|\\ &&+\big|d_{pqr}^J(t)\big| \big|h_q^J[t, \varphi]-h_q^J[t, \psi]\big|+\big|d_{pqr}^K(t)\big|\big|h_q^J[t, \varphi]-h_q^J[t, \psi]\big|\\ &&+\big|d_{pqr}^R(t)\big|\big|h_q^J[t, \varphi]-h_q^J[t, \psi]\big| +\big|d_{pqr}^I(t)\big|\big|h_q^J[t, \varphi]-h_q^J[t, \psi]\big|\\ &&+\big|d_{pqr}^K(t)\big|\big|h_q^K[t, \varphi]-h_q^K[t, \psi]\big| +\big|d_{pqr}^J(t)\big|\big|h_q^K[t, \varphi]-h_q^K[t, \psi]\big|\\[-2pt] &&+\big|d_{pqr}^I(t)\big|\big|h_q^K[t, \varphi]-h_q^K[t, \psi]\big| +\big|d_{pqr}^R(t)\big|\big|h_q^K[t, \varphi]-h_q^K[t, \psi]\big|\Big)\bigg]\\ &\leq&\sum_{q=1}^{n}\lambda_q\Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}} +a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big) \Big[\big|\varphi_q^R(t-\tau_{pq}(t))-\psi_q^R(t-\tau_{pq}(t))\big|\\[-4pt] &&\big|\varphi_q^I(t-\tau_{pq}(t))-\psi_q^I(t-\tau_{pq}(t))\big| +\big|\varphi_q^J(t-\tau_{pq}(t))-\psi_q^J(t-\tau_{pq}(t))\big|\\[-5pt] &&+\big|\varphi_q^K(t-\tau_{pq}(t))-\psi_q^K(t-\tau_{pq}(t))\big|\Big] +\sum_{q=1}^{n}\delta_q\int_{0}^{+\infty}\big|K_{pq}(u)\big|\\[-9pt] &&\times\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}} +b_{pq}^{K^{+}}\Big)\Big[\big|\varphi_q^R(t-u)-\psi_q^R(t-u)\big| +\big|\varphi_q^I(t-u)\\[-5pt] &&-\psi_q^I(t-u)\big|+\big|\varphi_q^J(t-u)-\psi_q^J(t-u)\big| +\big|\varphi_q^K(t-u)-\psi_q^K(t-u)\big|\Big]\,\mathrm{d}u\\[-5pt] &&+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}\bigg\{4N_q\rho_r \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \Big[\big|\varphi_r^R(t-\xi_{pqr}(t))-\psi_r^R(t-\xi_{pqr}(t))\big|\\[-5pt] &&+\big|\varphi_r^I(t-\xi_{pqr}(t))-\psi_r^I(t-\xi_{pqr}(t))\big| +\big|\varphi_r^J(t-\xi_{pqr}(t))-\psi_r^J(t-\xi_{pqr}(t))\big|\\[-3pt] &&+\big|\varphi_r^K(t-\xi_{pqr}(t))-\psi_r^K(t-\xi_{pqr}(t))\big|\Big] +4N_r\rho_q\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\[-5pt] &&\times\Big[|\varphi_q^R(t-\sigma_{pqr}(t))-\psi_q^R(t-\sigma_{pqr}(t))\big| +\big|\varphi_q^I(t-\sigma_{pqr}(t))-\psi_q^I(t-\sigma_{pqr}(t))\big|\\[-5pt] &&+\big|\varphi_q^J(t-\sigma_{pqr}(t))-\psi_q^J(t-\sigma_{pqr}(t))\big| +\big|\varphi_q^K(t-\sigma_{pqr}(t))-\psi_q^K(t-\sigma_{pqr}(t))\big|\Big]\bigg\}. \end{eqnarray*} From the above inequality, we obtain   \begin{eqnarray*} \big|(\Phi\varphi)_{p}^R(t)-(\Phi\psi)_{p}^R(t)\big| &\leq&\int_{-\infty}^{t}e^{-\int_s^{t}c_p(u)\,\mathrm{d}u} \bigg[\sum_{q=1}^{n}4\lambda_q\Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\\ &&+\sum_{q=1}^{n}4\delta_q\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big) \int_{0}^{+\infty}\big|K_{pq}(u)\big|\,\mathrm{d}u\\ &&+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}16\big(N_q\rho_r+N_r\rho_q\big) \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\bigg]\,\mathrm{d}s\|\varphi-\psi\|\\ &\leq&\frac{1}{c_p^-}\bigg(A_p^{\ast}+B_p^{\ast} +D_{p}^{\ast}+\hat{D}_{p}^{\ast}\bigg)\|\varphi-\psi\|\\ &=&\frac{\Xi_p}{c_p^-}\|\varphi-\psi\|,\quad p\in N. \end{eqnarray*} By the same way, we have   \begin{equation*} \big|(\Phi\varphi)_{p}^\nu(t)-(\Phi\psi)_{p}^\nu(t)\big|\leq\frac{\Xi_p}{c_p^-}\|\varphi-\psi\|,\quad p\in N,\,\, \nu=I, J, K. \end{equation*}Therefore, we get   \begin{equation*} \|\Phi\varphi-\Phi\psi\|\leq \mu\|\varphi-\psi\|. \end{equation*}It follows from Assumption $$(A_4)$$ that $$\Phi $$ is a contraction mapping. Based on the Banach fixed theorem, we obtain that $$\Phi $$ has a fixed point in $$\mathbb{Y}_0$$, which means that system (2.3) has a unique almost automorphic solution in $$\mathbb{Y}_0$$. This completes the proof. 4. Almost automorphic synchronization In this section, by designing a novel state-feedback controller, utilizing some analytic techniques and constructing a suitable Lyapunov function, we will investigate the exponential synchronization problem of QVHHNNs with time-varying and distributed delays and almost automorphic coefficients. For this purpose, we consider the system (1.1) as drive system, and a response system is designed as   \begin{eqnarray} Y^{\prime}_{p}(t)&=&-c_{p}(t)Y_{p}(t)+\sum_{q=1}^{n}a_{pq}(t)f_{q}(Y_{q}(t-\tau_{pq}(t))) +\sum_{q=1}^{n}b_{pq}(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}(Y_{q}(t-u))\,\mathrm{d}u\nonumber\\ &&+\sum_{q=1}^{n}\sum_{r=1}^{m}d_{pqr}(t)h_{q}(Y_{q}(t-\sigma_{pqr}(t))) h_{r}(Y_{r}(t-\sigma_{pqr}(t)))+I_{p}(t)+U_p(t), \end{eqnarray} (4.1)where p ∈ N, $$Y_p(t)=y_p^R(t)+iy_p^I(t)+jy_p^J(t)+ky_p^K(t)$$ denotes the state of the response system, $$\theta _p(t)$$ is a state-feedback controller, the rest notation is the same as those in system (1.1) and the initial condition is as follows:   \begin{equation*} Y_p(s)=\Psi_p(s),\,\, s\in(-\infty, 0],\,\, p\in N, \end{equation*}where $$\Psi _p(s)=\psi _p^R(s)+i\psi _p^I(s)+j\psi _p^J(s)+k\psi _p^K(s)$$ is the quaternion-valued bounded continuous function defined on $$(-\infty , 0]$$. Put $$z_p(t)=y_p(t)-x_p(t)$$, subtracting (1.1) from (4.1) yields the following error system:   \begin{eqnarray} Z^{\prime}_{p}(t)&=&-c_{p}(t)Z_{p}(t)+\sum_{q=1}^{n}a_{pq}(t)F_{q}(Z_{q}(t-\tau_{pq}(t))) +\sum_{q=1}^{n}b_{pq}(t)\int_{0}^{+\infty}K_{pq}(u)G_{q}(Z_{q}(t-u))\,\mathrm{d}u\nonumber\\ &&+\sum_{q=1}^{n}\sum_{r=1}^{m}d_{pqr}(t)H_{q}(Z_{q}(t-\sigma_{pqr}(t))) H_{r}(Z_{r}(t-\xi_{pqr}(t)))+U_{p}(t), \end{eqnarray} (4.2)where $$F_{q}(Z_{q}(t-\delta _{pq}(t)))=f_{q}(y_{q}(t-\delta _{pq}(t)))-f_{q}(x_{q}(t-\delta _{pq}(t)))$$, $$G_{q}(Z_{q}(t-u)=g_{q}(y_{q}(t-u)-g_{q}(x_{q}(t-u))$$, $$H_{q}(Z_{q}(t-\sigma _{pqr}(t)))H_{r}(Z_{r}(t-\xi _{pqr}(t)))=h_{q}(y_{q}(t-\sigma _{pqr}(t))) h_{r}(y_{r}(t-\xi _{pqr}(t)))-h_{q}(x_{q}(t-\sigma _{pqr}(t)))h_{r}(x_{r}(t-\xi _{pqr}(t)))$$, p ∈ N. In order to realize the almost automorphic synchronization of the drive-response system, we choose the following state-feedback controller   \begin{equation} U_{p}(t)=-\theta_{p}(t)Z_p(t)+\sum_{q=1}^{n}e_{pq}(t)l_q(Z_q(t-\vartheta_{pq}(t))),\quad p\in N, \end{equation} (4.3)where $$\theta _p,\vartheta _{pq}:\mathbb{R}\rightarrow \mathbb{R}^+, e_{pq}: \mathbb{R}\rightarrow \mathbb{Q}, l_q:\mathbb{Q}\rightarrow \mathbb{Q}$$, p, q ∈ N. Similarly, according to Hamilton rules, we can decompose system (4.2) into an equivalent real-valued system:   \begin{eqnarray*} \left(z_{p}^{R}\right)^{\prime}(t)&=&-c_{p}(t)z_{p}^{R}(t)+\sum_{q=1}^{n}\Big(a_{pq}^{R}(t)\big(f_{q}^{R}[t,y]-f_{q}^{R}[t,x]\big) -a_{pq}^{I}(t)\big(f_{q}^{I}[t,y]-f_{q}^{I}[t,x]\big)\\ &&-a_{pq}^{J}(t)\big(f_{q}^{J}[t,y]-f_{q}^{J}[t,x]\big) -a_{pq}^{K}(t)\big(f_{q}^{K}[t,y]-f_{q}^{K}[t,x]\big)\Big)\\ &&+\sum_{q=1}^{n}\bigg(b_{pq}^R(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^R[t,u,y]-g_{q}^R[t,u,x]\big)\,\mathrm{d}u\\ &&-b_{pq}^I(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^I[t,u,y]-g_{q}^I[t,u,x]\big)\,\mathrm{d}u -b_{pq}^J(t)\int_{0}^{+\infty}K_{pq}(u)\\ &&\times\big(g_{q}^J[t,u,y] -g_{q}^J[t,u,x]\big)\,\mathrm{d}u-b_{pq}^K(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^K[t, u, y]-g_{q}^K[t, u, x]\big)\,\mathrm{d}u\bigg) \end{eqnarray*}  \begin{eqnarray*} &&+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}\bigg(d_{pqr}^R(t) \big(h_q^R[t, y]\tilde{h}_r^R[t, y]-h_q^R[t, x]\tilde{h}_r^R[t, x]\big)-d_{pqr}^I(t)\big(h_q^I[t, y]\tilde{h}_r^R[t, y]\\ &&-h_q^I[t, x]\tilde{h}_r^R[t, x]\big) -d_{pqr}^J(t)\big(h_q^J[t, y]\tilde{h}_r^R[t, y]-h_q^J[t, x]\tilde{h}_r^R[t, x]\big)\\ &&-d_{pqr}^K(t)\big(h_q^K[t, y]\tilde{h}_r^R[t, y]-h_q^K[t, x]\tilde{h}_r^R[t, x]\big) -d_{pqr}^I(t)\big(h_q^R[t, y]\tilde{h}_r^I[t, y]\\ &&-h_q^R[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^R(t)\big(h_q^I[t, y]\tilde{h}_r^I[t, y]-h_q^I[t, x]\tilde{h}_r^I[t, x]\big)\\ &&+d_{pqr}^K(t)\big(h_q^J[t, y]\tilde{h}_r^I[t, y]-h_q^J[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^J(t)\big(h_q^K[t, y]\tilde{h}_r^I[t, y]\\ &&-h_q^K[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^J(t)\big(h_q^R[t, y]\tilde{h}_r^J[t, y]-h_q^R[t, x]\tilde{h}_r^J[t, x]\big)\\ &&-d_{pqr}^K(t)\big(h_q^I[t, y]\tilde{h}_r^J[t, y]-h_q^I[t, x]\tilde{h}_r^J[t, x]\big) -d_{pqr}^R(t)\big(h_q^J[t, y]\tilde{h}_r^J[t, y]\\ &&-h_q^J[t, x]\tilde{h}_r^J[t, x]\big) +d_{pqr}^I(t)\big(h_q^K[t, y]\tilde{h}_r^J[t, y]-h_q^K[t,x]\tilde{h}_r^J[t,x]\big)\\ &&-d_{pqr}^K(t)\big(h_q^R[t, y]\tilde{h}_r^K[t, y]-h_q^R[t, x]\tilde{h}_r^K[t, x]\big) +d_{pqr}^J(t)\big(h_q^I[t, y]\tilde{h}_r^K[t, y]\\ &&-h_q^I[t, x]\tilde{h}_r^K[t, x]\big) -d_{pqr}^I(t)\big(h_q^J[t, y]\tilde{h}_r^K[t, y]-h_q^J[t, x]\tilde{h}_r^K[t, x]\big)\\ &&-d_{pqr}^R(t)\big(h_q^K[t, y]\tilde{h}_r^K[t, y]-h_q^K[t, x]\tilde{h}_r^K[t, x]\big)\bigg)-\theta_p(t)z^R_p(t)\\ &&+\sum_{q=1}^{n}\Big(e_{pq}^{R}(t)l_{q}^{R}[t,z] -e_{pq}^{I}(t)l_{q}^{I}[t,z] -e_{pq}^{J}(t)l_{q}^{J}[t,z]-e_{pq}^{K}(t)l_{q}^{K}[t,z]\Big),\,\, p\in N, \end{eqnarray*}  \begin{align*} \left(z_{p}^{I}\right)^{\prime}(t)=&-c_{p}(t)z_{p}^{I}(t)+\sum_{q=1}^{n} \Big(a_{pq}^{R}(t)\big(f_{q}^{I}[t,y]-f_{q}^{I}[t,x]\big) +a_{pq}^{I}(t)\big(f_{q}^{R}[t,y]-f_{q}^{R}[t,x]\big)&\\[4pt] &+a_{pq}^{J}(t)\big(f_{q}^{K}[t,y]-f_{q}^{K}[t,x]\big) -a_{pq}^{K}(t)\big(f_{q}^{J}[t,y]-f_{q}^{J}[t,x]\big)\Big)\\[4pt] &+\sum_{q=1}^{n}\bigg(b_{pq}^R(t) \int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^I[t,u,y]-g_{q}^I[t,u,x]\big)\,\mathrm{d}u\\[3pt] &+b_{pq}^I(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^R[t,u,y]-g_{q}^R[t,u,x]\big)\,\mathrm{d}u +b_{pq}^J(t)\int_{0}^{+\infty}K_{pq}(u)\\[3pt] &\times\big(g_{q}^K[t,u,y] -g_{q}^K[t,u,x]\big)\,\mathrm{d}u-b_{pq}^K(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^J[t, u, y]-g_{q}^J[t, u, x]\big)\,\mathrm{d}u\bigg) \\[3pt] &+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}\bigg(d_{pqr}^I(t) \big(h_q^R[t, y]\tilde{h}_r^R[t, y]-h_q^R[t, x]\tilde{h}_r^R[t, x]\big) +d_{pqr}^R(t)\big(h_q^I[t, y] \tilde{h}_r^R[t, y]\\[2pt] &-h_q^I[t, x] \tilde{h}_r^R[t,x]\big) -d_{pqr}^K(t)\big(h_q^J[t, y]\tilde{h}_r^R[t,y]-h_q^J[t, x]\tilde{h}_r^R[t,x]\big)\end{align*}  \begin{eqnarray*} &&+d_{pqr}^J(t)\big(h_q^K[t, y]\tilde{h}_r^R[t, y]-h_q^K[t, x]\tilde{h}_r^R[t, x]\big) +d_{pqr}^R(t)\big(h_q^R[t, y]\tilde{h}_r^I[t, y]\\ &&-h_q^R[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^I(t)\big(h_q^I[t, y]\tilde{h}_r^I[t,y]-h_q^I[t, x]\tilde{h}_r^I[t,x]\big)\\ &&-d_{pqr}^J(t)\big(h_q^J[t, y]\tilde{h}_r^I[t, y]-h_q^J[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^K(t)\big(h_q^K[t, y]\tilde{h}_r^I[t, y]\\ &&-h_q^K[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^K(t)\big(h_q^R[t, y]\tilde{h}_r^J[t, y]-h_q^R[t, x]\tilde{h}_r^J[t, x]\big)\\ &&+d_{pqr}^J(t)\big(h_q^I[t, y]\tilde{h}_r^J[t, y]-h_q^I[t, x]\tilde{h}_r^J[t, x]\big) -d_{pqr}^I(t)\big(h_q^J[t, y]\tilde{h}_r^J[t, y]\\ &&-h_q^J[t, x]\tilde{h}_r^J[t, x]\big) -d_{pqr}^R(t)\big(h_q^K[t, y]\tilde{h}_r^J[t, y]-h_q^K[t, x]\tilde{h}_r^J[t, x]\big)\\ &&+d_{pqr}^J(t)\big(h_q^R[t, y]\tilde{h}_r^K[t, y] -h_q^R[t, x]\tilde{h}_r^K[t, x]\big) +d_{pqr}^K(t)\big(h_q^I[t, y]\tilde{h}_r^K[t, y]\\ &&-h_q^I[t, x]\tilde{h}_r^K[t, x]\big) +d_{pqr}^R(t)\big(h_q^J[t, y]\tilde{h}_r^K[t, y]-h_q^J[t, x]\tilde{h}_r^K[t, x]\big)\\ &&-d_{pqr}^I(t)\big(h_q^K[t, y]\tilde{h}_r^K[t, y]-h_q^K[t, x]\tilde{h}_r^K[t, x]\big)\bigg)-\theta_{p}(t)z_{p}^{I}(t)\\[-9pt] &&+\sum_{q=1}^{n}\Big(e_{pq}^{R}(t)l_{q}^{I}[t,z] +e_{pq}^{I}(t)l_{q}^{R}[t,z] +e_{pq}^{J}(t)l_{q}^{K}[t,z]-e_{pq}^{K}(t)l_{q}^{J}[t,z]\Big),\,\, p\in N, \end{eqnarray*}  \begin{align*} \left(z_{p}^{J}\right)^{\prime}(t)=&-c_{p}(t)z_{p}^{J}(t)+\sum_{q=1}^{n} \Big(a_{pq}^{R}(t)\big(f_{q}^{J}[t,y]-f_{q}^{J}[t,x]\big) +a_{pq}^{J}(t)\big(f_{q}^{R}[t,y]-f_{q}^{R}[t,x]\big)\\[4pt] &-a_{pq}^{I}(t)\big(f_{q}^{K}[t,y]-f_{q}^{K}[t,x]\big) +a_{pq}^{K}(t)\big(f_{q}^{I}[t,y]-f_{q}^{I}[t,x]\big)\Big)\\[4pt] &+\sum_{q=1}^{n}\bigg(b_{pq}^R(t) \int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^J[t,u,y]-g_{q}^J[t,u,x]\big)\,\mathrm{d}u\\[4pt] &+\,b_{pq}^J(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^R[t,u,y]-g_{q}^R[t,u,x]\big)\,\mathrm{d}u -b_{pq}^I(t)\int_{0}^{+\infty}K_{pq}(u)\\[4pt] &\times\big(g_{q}^K[t,u,y]-g_{q}^K[t,u,x]\big)\,\mathrm{d}u +b_{pq}^K(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^I[t, u, y]-g_{q}^I[t, u, x]\big)\,\mathrm{d}u\bigg)\\[4pt] &+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n} \bigg(d_{pqr}^J(t)\big(h_q^R[t, y]\tilde{h}_r^R[t,y]-h_q^R[t, x]\tilde{h}_r^R[t, x]\big) +d_{pqr}^K(t)\big(h_q^I[t, y]\tilde{h}_r^R[t, y]\\[6pt] &-h_q^I[t, x]\tilde{h}_r^R[t, x]\big) +d_{pqr}^R(t)\big(h_q^J[t, y]\tilde{h}_r^R[t, y]-h_q^J[t, x]\tilde{h}_r^R[t, x]\big)\\[6pt] &-d_{pqr}^I(t)\big(h_q^K[t, y]\tilde{h}_r^R[t, y]-h_q^K[t, x]\tilde{h}_r^R[t, x]\big) -d_{pqr}^K(t)\big(h_q^R[t, y]\tilde{h}_r^I[t, y]\\[6pt] &-h_q^R[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^J(t)\big(h_q^I[t, y]\tilde{h}_r^I[t, y]-h_q^I[t, x]\tilde{h}_r^I[t, x]\big) \end{align*}  \begin{eqnarray*} &&+d_{pqr}^I(t)\big(h_q^J[t, y]\tilde{h}_r^I[t, y]-h_q^J[t, x]\tilde{h}_r^I[t, x]\big) +d_{pqr}^R(t)\big(h_q^K[t, y]\tilde{h}_r^I[t, y]\\ &&-h_q^K[t, x]\tilde{h}_r^I[t, x]\big) +d_{pqr}^R(t)\big(h_q^R[t, y]\tilde{h}_r^J[t, y]-h_q^R[t, x]\tilde{h}_r^J[t, x]\big)\\ &&-d_{pqr}^I(t)\big(h_q^I[t, y]\tilde{h}_r^J[t, y]-h_q^I[t, x]\tilde{h}_r^J[t, x]\big) -d_{pqr}^J(t)\big(h_q^J[t, y]\tilde{h}_r^J[t, y]\\ &&-h_q^J[t, x]\tilde{h}_r^J[t, x]\big) -d_{pqr}^K(t)\big(h_q^K[t, y]\tilde{h}_r^J[t, y]-h_q^K[t, x]\tilde{h}_r^J[t, x]\big)\\ &&-d_{pqr}^I(t)\big(h_q^R[t, y]\tilde{h}_r^K[t, y]-h_q^R[t, x]\tilde{h}_r^K[t, x]\big) -d_{pqr}^R(t)\big(h_q^I[t,y]\tilde{h}_r^K[t,y]\\ &&-h_q^I[t, x]\tilde{h}_r^K[t, x]\big) +d_{pqr}^K(t)\big(h_q^J[t, y]\tilde{h}_r^K[t, y]-h_q^J[t, x]\tilde{h}_r^K[t, x]\big)\\ &&-d_{pqr}^J(t)\big(h_q^K[t, y]\tilde{h}_r^K[t, y]-h_q^K[t, x]\tilde{h}_r^K[t, x]\big)\bigg) -\theta_{p}(t)z_{p}^{J}(t)\\ &&+\sum_{q=1}^{n}\Big(e_{pq}^{R}(t)l_{q}^{J}[t,z] +e_{pq}^{J}(t)l_{q}^{R}[t,z] -e_{pq}^{I}(t)l_{q}^{K}[t,z]+e_{pq}^{K}(t)l_{q}^{I}[t,z]\Big),\,\, p\in N, \end{eqnarray*}  \begin{align*} \left(z_{p}^{K}\right)^{\prime}(t)=&-c_{p}(t)z_{p}^{K}(t)+\sum_{q=1}^{n} \Big(a_{pq}^{R}(t)\big(f_{q}^{K}[t,y]-f_{q}^{K}[t,x]\big) +a_{pq}^{K}(t)\big(f_{q}^{R}[t,y]-f_{q}^{R}[t,x]\big)&\\ &+a_{pq}^{I}(t)\big(f_{q}^{J}[t,y]-f_{q}^{J}[t,x]\big) -a_{pq}^{J}(t)\big(f_{q}^{I}[t,y]-f_{q}^{I}[t,x]\big)\Big)\\ &+\sum_{q=1}^{n}\bigg(b_{pq}^R(t) \int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^K[t,u,y]-g_{q}^K[t,u,x]\big)\,\mathrm{d}u\\ &+b_{pq}^K(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^R[t,u,y]-g_{q}^R[t,u,x]\big)\,\mathrm{d}u +b_{pq}^I(t)\int_{0}^{+\infty}K_{pq}(u)\\ &\times\big(g_{q}^J[t,u,y]-g_{q}^J[t,u,x]\big)\,\mathrm{d}u -b_{pq}^J(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^I[t, u, y]-g_{q}^I[t, u, x]\big)\,\mathrm{d}u\bigg) \end{align*}  \begin{eqnarray*} &&+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n} \bigg(d_{pqr}^K(t)\big(h_q^R[t, y]\tilde{h}_r^R[t, y]-h_q^R[t, x]\tilde{h}_r^R[t, x]\big) -d_{pqr}^I(t)\big(h_q^I[t, y] \tilde{h}_r^R[t, y]\\ &&-h_q^I[t, x] \tilde{h}_r^R[t, x]\big) +d_{pqr}^I(t)\big(h_q^J[t, y]\tilde{h}_r^R[t, y]-h_q^J[t, x]\tilde{h}_r^R[t, x]\big)\\ &&+d_{pqr}^R(t)\big(h_q^K[t, y]\tilde{h}_r^R[t, y]-h_q^K[t, x]\tilde{h}_r^R[t, x]\big) -d_{pqr}^J(t)\big(h_q^R[t, y]\tilde{h}_r^I[t, y]\\ &&-h_q^R[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^K(t)\big(h_q^I[t, y]\tilde{h}_r^I[t, y]-h_q^I[t, x]\tilde{h}_r^I[t, x]\big)\\ &&-d_{pqr}^R(t)\big(h_q^J[t, y]\tilde{h}_r^I[t, y]-h_q^J[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^I(t)\big(h_q^K[t, y]\tilde{h}_r^I[t, y]\\ &&-h_q^K[t, x]\tilde{h}_r^I[t, x]\big) +d_{pqr}^I(t)\big(h_q^R[t, y]\tilde{h}_r^J[t, y]-h_q^R[t, x]\tilde{h}_r^J[t, x]\big)\\ &&+d_{pqr}^R(t)\big(h_q^I[t, y]\tilde{h}_r^J[t, y]-h_q^I[t, x]\tilde{h}_r^J[t, x]\big) -d_{pqr}^K(t)\big(h_q^J[t, y]\tilde{h}_r^J[t, y] \end{eqnarray*}  \begin{eqnarray*} &&-h_q^J[t, x]\tilde{h}_r^J[t, x]\big) +d_{pqr}^J(t)\big(h_q^K[t, y]\tilde{h}_r^J[t, y]-h_q^K[t, x]\tilde{h}_r^J[t, x]\big)\\ &&-d_{pqr}^R(t)\big(h_q^R[t, y]\tilde{h}_r^K[t, y]-h_q^R[t, x]\tilde{h}_r^K[t, x]\big) -d_{pqr}^I(t)\big(h_q^I[t, y]\tilde{h}_r^K[t, y]\\ &&-h_q^I[t, x]\tilde{h}_r^K[t, x]\big) -d_{pqr}^J(t)\big(h_q^J[t, y]\tilde{h}_r^K[t, y]-h_q^J[t, x]\tilde{h}_r^K[t, x]\big)\\ &&-d_{pqr}^J(t)\big(h_q^K[t, y]\tilde{h}_r^K[t, y]-h_q^K[t, x]\tilde{h}_r^K[t, x]\big)\bigg) -\theta_{p}(t)z_{p}^{K}(t)\\ &&+\sum_{q=1}^{n}\Big(e_{pq}^{R}(t)l_{q}^{K}[t,z] +e_{pq}^{K}(t)l_{q}^{R}[t,z] +e_{pq}^{I}(t)l_{q}^{J}[t,z]-e_{pq}^{J}(t)l_{q}^{I}[t,z]\Big),\,\, p\in N, \end{eqnarray*} where $$l_q^\nu [t, z]\triangleq l_q^\nu \big (z_q^R(t-\vartheta _{pq}(t)), z_q^I(t-\vartheta _{pq}(t)), z_q^J(t-\vartheta _{pq}(t)), z_q^K(t-\vartheta _{pq}(t))\big ), \nu \in \Lambda $$. Definition 4.1 The response system (4.1) and the drive system (1.1) can be globally, exponentially synchronized if there exist positive constants M and $$\omega $$ such that   \begin{equation*} \|y(t)-x(t)\|_0\leq M\|\psi-\varphi\|e^{-\omega t}, \end{equation*}where $$x=(x_1^R,\, x_2^R, \ldots , x_n^R\,, x_1^I\,, x_2^I, \ldots , x_n^I,\, x_1^J,\, x_2^J, \ldots , x_n^J,\, x_1^K,\, x_2^K, \ldots , x_n^K)$$ and $$y=(y_1^R, y_2^R, \ldots , y_n^R, y_1^I, y_2^I,\\ \ldots , y_n^I, y_1^J, y_2^J, \ldots , y_n^J, y_1^K, y_2^K, \ldots , y_n^K)$$ are solutions of the equivalent real-valued systems of (1.1) and (4.1) with initial values $$\varphi $$ and $$\psi $$, respectively, $$\|y(t)-x(t)\|_0=\max \limits _{p\in N,\nu \in \Lambda }\{|y_p^\nu (t)-x_p^\nu (t)|\},\|\psi -\varphi \|= \max \limits _{p\in N,\nu \in \Lambda }\Big \{\sup \limits _{t\in \mathbb{R}}|\psi _p^\nu (t)-\varphi _p^\nu (t)|\Big \}$$. Theorem 4.1 Let $$(A_1)$$–$$(A_4)$$ hold. Suppose further that: $$(A_2)$$ Function $$\theta _p\in AA(\mathbb{R}, \mathbb{R}^+), e_{pq}\in AA(\mathbb{R}, \mathbb{Q}), \vartheta _{pq}\in AA(\mathbb{R}, \mathbb{R}^+)\cap C^1(\mathbb{R},\mathbb{R})$$ with $$\inf \limits _{t\in R}\{1-\vartheta _{pq}^{\prime}(t)\}:=\varsigma _{pq}>0, p,q\in N$$. $$(A_2)$$ Function $$l_q \in C(\mathbb{Q}, \mathbb{Q})$$ and for any $$u^\nu , v^\nu \in \mathbb{R}$$, there exist positive constants $$\eta _{q}^{\nu }$$ such that   \begin{equation*} \left|l_q^\nu(u^R, u^I, u^J, u^K)-l_q^\nu(v^R, v^I, v^J, v^K)\right| \leq\eta_q^{R}|u^R-v^R|+\eta_q^{I}|u^I-v^I|+ \eta_q^{J}|u^J-v^J|+\eta_q^{K}|u^K-v^K| \end{equation*}and $$l_q^\nu (0, 0, 0, 0)=0$$, where $$\nu \in \Lambda $$, q ∈ N. $$(A_5)$$ For p ∈ N, there exists a positive constant $$\omega $$ such that   \begin{eqnarray*} &&\sum\limits_{p=1}^n\bigg\{\big(\omega-c_p^{-}-\theta_{p}^{-}\big) +B^{\ast}_p+\sum\limits_{q=1}^n\frac{4\lambda_qe^{\omega\tau^+}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\nonumber\\&& +\sum\limits_{q=1}^n\sum\limits_{r=1}^n\bigg( \frac{16N_r \rho_qe^{\omega \sigma^+}}{\beta_{pqr}}+\frac{16N_q \rho_r e^{\omega \xi^+}}{\gamma_{pqr}}\bigg)\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \nonumber\\&& \sum\limits_{q=1}^n\frac{4\eta_{q}e^{\omega\vartheta^+}}{\delta_{pq}}\Big(e_{pq}^{R^{+}} +e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{k^{+}}\Big)\bigg\} <0, \end{eqnarray*}where $$\eta _q=\max \big \{\eta ^R_q, \eta ^I_q, \eta ^J_q, \eta ^K_q\big \}$$.Then the drive system (1.1) and response system (4.1) are globally, exponentially synchronized. Proof. In view of the error system (4.2), for any t > 0, $$\nu \in \Lambda $$, we have   \begin{eqnarray*} D^{+}\big|z_p^\nu(t)\big| &\leq&-\big(c_p^- +\theta_{p}^{-}\big)\big|z_p^\nu(t)\big| +\sum\limits_{q=1}^n\lambda_{q}\Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{k^{+}}\Big) \Big[\big|z_q^R(t-\tau_{pq}(t))\big|\\ &&+\big|z_q^I(t-\tau_{pq}(t))\big| +\big|z_q^J(t-\tau_{pq}(t))\big|+\big|z_q^K(t-\tau_{pq}(t))\big|\Big]\\ &&+\sum\limits_{q=1}^n\int_0^\infty\big| K_{pq}(u)\big| \delta_{q}\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big) \Big[\big|z_q^R(t-u)\big|+\big|z_q^I(t-u)\big|\\ &&+\big|z_q^J(t-u)\big| +\big|z_q^K(t-u)\big|\Big]\,\mathrm{d}u +\sum\limits_{q=1}^n\sum\limits_{r=1}^n4N_q \rho_r\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ &&\times\Big[\big|z_r^R(t-\xi_{pqr}(t))\big|+\big|z_r^I(t-\xi_{pqr}(t))\big| +\big|z_r^J(t-\xi_{pqr}(t))\big|\\ &&+\big|z_r^K(t-\xi_{pqr}(t))\big|\Big] +\sum\limits_{q=1}^n\sum\limits_{r=1}^n4N_r\rho_q\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}} +d_{pqr}^{K^{+}}\Big)\\ &&\times\Big[\big|z_q^R(t-\sigma_{pqr}(t))\big| +\big|z_q^I(t-\sigma_{pqr}(t))\big| +\big|z_q^J(t-\sigma_{pqr}(t))\big|\\ &&+\big|z_q^K(t-\sigma_{pqr}(t))\big|\Big] +\sum\limits_{q=1}^n\eta_{q}\Big(e_{pq}^{R^{+}}+e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{K^{+}}\Big) \Big[\big|z_q^R(t-\vartheta_{pq}(t))\big|\\ &&+\big|z_q^I(t-\vartheta_{pq}(t))\big| +\big|z_q^J(t-\vartheta_{pq}(t))\big|+\big|z_q^K(t-\vartheta_{pq}(t))\big|\Big]. \end{eqnarray*} We consider the Lyapunov function as follows:   \begin{equation*} V(t)=V^R(t)+V^I(t)+V^J(t)+V^K(t), \end{equation*}where $$V^\nu (t)=\sum \nolimits _{p=1}^{n} ( |z_{p}^{\nu }(t) |e^{\omega t}+\Gamma _{p} )$$, $$\nu \in \Lambda $$ and   \begin{eqnarray*} \Gamma_{p}&=&\sum\limits_{q=1}^n\frac{\lambda_qe^{\omega\tau^{+}}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\int_{t-\tau_{pq}(t)}^{t} \Big[\big|z_q^R(s)\big|+\big|z_q^I(s)\big| +\big|z_q^J(s)\big|\\ &&+\big|z_q^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s+\sum\limits_{q=1}^n\delta_q \Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big)\int_0^{\infty}\big|K_{pq}(u)\big|e^{\omega u}\int_{t-u}^{t}\Big[\big|z_q^R(s)\big| \end{eqnarray*}  \begin{eqnarray*} &&+\big|z_q^I(s)\big|+\big|z_q^J(s)\big|+\big|z_q^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s\,\mathrm{d}u +\sum\limits_{q=1}^n\sum\limits_{r=1}^n \frac{4N_r \rho_qe^{\omega \sigma^+}}{\beta_{pqr}}\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}\\ &&+d_{pqr}^{K^{+}}\Big) \int_{t-\sigma_{pqr}(t)}^t \Big[\big|z_q^R(s)\big|+\big|z_q^I(s)\big|+\big|z_q^J(s)\big| +\big|z_q^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s\\ &&+\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{4N_q \rho_re^{\omega \xi^+}}{\gamma_{pqr}} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \int_{t-\xi_{pqr}(t)}^t\Big[\big|z_r^R(s)\big|+\big|z_r^I(s)\big|\\ &&+\big|z_r^J(s)\big|+\big|z_r^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s +\sum\limits_{q=1}^n\frac{\eta_qe^{\omega\vartheta^{+}}}{\delta_{pq}} \Big(e_{pq}^{R^{+}}+e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{K^{+}}\Big) \int_{t-\vartheta_{pq}(t)}^{t} \Big[\big|z_q^R(s)\big|\\ &&+\big|z_q^I(s)\big| +\big|z_q^J(s)\big|+\big|z_q^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s,\quad p\in N. \end{eqnarray*} Computing the derivative of V (t) along the solutions of the error system (4.2), we can get   \begin{eqnarray} D^{+}V^R(t) &=&\sum\limits_{p=1}^n\bigg\{\omega e^{\omega t}\big|z_p^R(t)\big|+e^{\omega t}D^+\big|z_p^R(t)\big| +\sum\limits_{q=1}^n\frac{\lambda_qe^{\omega\tau^+}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\nonumber\\ &&\times\Big[\big|z_q^R(t)\big|+\big|z_q^I(t)\big|+\big|z_q^J(t)\big|+\big|z_q^K(t)\big|\Big]e^{\omega t}-\sum\limits_{q=1}^n\frac{\lambda_qe^{\omega\tau^+}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\nonumber\\ &&\times\Big[\big|z_q^R(t-\tau_{pq}(t))\big| +\big|z_q^I(t-\tau_{pq}(t))\big|+\big|z_q^J(t-\tau_{pq}(t))\big| +\big|z_q^K(t-\tau_{pq}(t))\big|\Big]\nonumber\\ &&\times\big(1-\tau_{pq}^{\prime}(t)\big)e^{\omega (t-\tau_{pq}(t))} +\sum\limits_{q=1}^n\delta_q\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big) \int_0^\infty \big|K_{pq}(u)\big|e^{\omega u}\nonumber\\ &&\times\Big[\big|z_q^R(t)\big| +\big|z_q^I(t)\big|+\big|z_q^J(t)\big|+\big|z_q^K(t)\big|\Big]e^{\omega t}\,\mathrm{d}u -\sum\limits_{q=1}^n\delta_q\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big) \nonumber\\ &&\times\int_0^\infty \big|K_{pq}(u)\big|e^{\omega u} \Big[\big|z_q^R(t-u)\big|+\big|z_q^I(t-u)\big|+\big|z_q^J(t-u)\big|\nonumber\\ &&+\big|z_q^K(t-u)\big|\Big]e^{\omega (t-u)}\,\mathrm{d}u +\sum\limits_{q=1}^n\sum\limits_{r=1}^n \frac{4N_r \rho_q e^{\omega \sigma^+}}{\beta_{pqr}}\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\nonumber\\ &&\Big[\big|z_q^R(t)\big|+\big|z_q^I(t)\big|+\big|z_q^J(t)\big| +\big|z_q^K(t)\big|\Big]e^{\omega t}-\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{4N_r \rho_qe^{\omega \sigma^+}}{\beta_{pqr}} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}\nonumber\\ &&+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \Big[\big|z_q^R(t-\sigma_{pqr}(t))\big| +\big|z_q^I(t-\sigma_{pqr}(t))\big|+\big|z_q^J(t-\sigma_{pqr}(t))\big|\nonumber\\ &&+\big|z_q^K(t-\sigma_{pqr}(t))\big|\Big]\big(1-\sigma_{pqr}^{\prime}(t)\big)e^{\omega (t-\sigma_{pqr}(t))} +\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{4N_q \rho_r e^{\omega \xi^+}}{\gamma_{pqr}}\nonumber\end{eqnarray}  \begin{eqnarray*} &&\times\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \Big[\big|z_r^R(t)\big|+\big|z_r^I(t)\big|+\big|z_r^J(t)\big| +\big|z_r^K(t)\big|\Big]e^{\omega t}\nonumber\\[2pt] &&-\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{4N_q \rho_r e^{\omega \xi^+}}{\gamma_{pqr}} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \Big[\big|z_r^R(t-\xi_{pqr}(t))\big|\nonumber\\[2pt] &&+\big|z_r^I(t-\xi_{pqr}(t))\big|+\big|z_r^J(t-\xi_{pqr}(t))\big| +\big|z_r^K(t-\xi_{pqr}(t))\big|\Big]\nonumber\\[2pt] &&\times\big(1-\xi_{pqr}^{\prime}(t)\big)e^{\omega (t-\xi_{pqr}(t))} +\sum_{q=1}^{n}\frac{\eta_{q}e^{\omega\vartheta^+}}{\delta_{pq}} \Big(e_{pq}^{R^{+}}+e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{K^{+}}\Big)\nonumber\\[2pt] &&\times\Big[\big|z_{q}^{R}(t)\big|+\big|z_{q}^{I}(t)\big|+\big|z_{q}^{J}(t)\big| +\big|z_{q}^{K}(t)\big|\Big]e^{\omega t}-\sum_{q=1}^{n}\frac{\eta_{q}e^{\omega\vartheta^+}}{\delta_{pq}} \Big(e_{pq}^{R^{+}}+e_{pq}^{I^{+}}\nonumber\\[2pt] &&+e_{pq}^{J^{+}}+e_{pq}^{K^{+}}\Big) \Big[\big|z_{q}^{R}(t-\vartheta_{pq}(t))\big| +\big|z_{q}^{I}(t-\vartheta_{pq}(t))\big| +\big|z_{q}^{J}(t-\vartheta_{pq}(t))\big|\nonumber\\ &&+\big|z_{q}^{K}(t-\vartheta_{pq}(t))\big|\Big] e^{\omega(t-\vartheta_{pq}(t))}(1-\vartheta_{pq}^{\prime}(t))\Big]\bigg\}\nonumber\\[2pt] &\leq&\sum\limits_{p=1}^n\bigg\{\big(\omega-c_p^{-}-\theta_p^{-}\big)e^{\omega t}\big|z_p^R(t)\big| +e^{\omega t}\bigg(\sum\limits_{q=1}^n\frac{\lambda_qe^{\omega\tau^+}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\nonumber\\[2pt] &&\times\Big[\big|z_p^R(t)\big| +\big|z_p^I(t)\big|+\big|z_p^J(t)\big|+\big|z_p^K(t)\big|\Big] +\sum\limits_{q=1}^n\delta_q\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big)\nonumber\\ &&\times\int_0^\infty \big|K_{pq}(u)\big|e^{u}\Big[\big|z_q^R(t)\big| +\big|z_q^I(t)\big|+\big|z_q^J(t)\big| +\big|z_q^K(t)\big|\Big]\,\mathrm{d}u\nonumber\\[2pt] &&+\sum\limits_{q=1}^n\sum\limits_{r=1}^n \frac{4N_r \rho_qe^{\omega \sigma^+}}{\beta_{pqr}}\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \Big[\big|z_q^R(t)\big|+\big|z_q^I(t)\big|\nonumber\\[2pt] &&+\big|z_q^J(t)\big|+\big|z_q^K(t)\big|\Big] +\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{4N_q \rho_r e^{\omega \xi^+}}{\gamma_{pqr}} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\nonumber\\[2pt] &&\times\Big[\big|z_r^R(t)\big|+\big|z_r^I(t)\big|+\big|z_r^J(t)\big| +\big|z_r^K(t)\big|\Big]+ \sum\limits_{q=1}^n\frac{\eta_{q}e^{\omega\vartheta^+}}{\delta_{pq}}\Big(e_{pq}^{R^{+}} +e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{k^{+}}\Big)\nonumber\\[2pt] &&\times\Big[\big|z_q^R(t)\big| +\big|z_q^I(t)\big| +\big|z_q^J(t)\big|+\big|z_q^K(t)\big|\Big]\bigg)\bigg\}\nonumber\\[2pt] &\leq&e^{\omega t}\|z(t)\|_{0}\sum\limits_{p=1}^n\bigg\{\big(\omega-c_p^{-}-\theta_{p}^{-}\big) +B^{\ast}_p+\sum\limits_{q=1}^n\frac{4\lambda_qe^{\omega\tau^+}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\nonumber \end{eqnarray*}  \begin{eqnarray*} && +\sum\limits_{q=1}^n\sum\limits_{r=1}^n\bigg( \frac{16N_r \rho_qe^{\omega \sigma^+}}{\beta_{pqr}}+\frac{16N_q \rho_r e^{\omega \xi^+}}{\gamma_{pqr}}\bigg)\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \nonumber\\ && \sum\limits_{q=1}^n\frac{4\eta_{q}e^{\omega\vartheta^+}}{\delta_{pq}}\Big(e_{pq}^{R^{+}} +e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{k^{+}}\Big)\bigg\}. \end{eqnarray*} Repeating the similar calculation, for $$\nu =I,J,K$$, we can obtain   \begin{eqnarray} D^{+}V^\nu(t) &\leq&e^{\omega t}\|z(t)\|_{0}\sum\limits_{p=1}^n\bigg\{\big(\omega-c_p^{-}-\theta_{p}^{-}\big) +B^{\ast}_p+\sum\limits_{q=1}^n\frac{4\lambda_qe^{\omega\tau^+}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\nonumber\\ && +\sum\limits_{q=1}^n\sum\limits_{r=1}^n\bigg( \frac{16N_r \rho_qe^{\omega \sigma^+}}{\beta_{pqr}}+\frac{16N_q \rho_r e^{\omega \xi^+}}{\gamma_{pqr}}\bigg)\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \nonumber\\ && \sum\limits_{q=1}^n\frac{4\eta_{q}e^{\omega\vartheta^+}}{\delta_{pq}}\Big(e_{pq}^{R^{+}} +e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{k^{+}}\Big)\bigg\}. \end{eqnarray} (4.5)From $$(A_{5})$$, (4.4) and (4.5) it follows that   \begin{equation*} V^{\prime}(t)\leq0, \end{equation*}which implies that V (t) ≤ V (0) for all t ≥ 0. On the other hand, we have   \begin{eqnarray*} V^{R}(0)&\leq&\sum_{p=1}^{n}\bigg\{\big|z_{p}^{R}(0)\big| +\sum\limits_{q=1}^n\frac{\lambda_qe^{\omega\tau^{+}}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\int_{-\tau_{pq}(0)}^{0} \Big[\big|z_q^R(s)\big|\\ &&+\big|z_q^I(s)\big|+\big|z_q^J(s)\big| +\big|z_q^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s +\sum\limits_{q=1}^n\delta_q\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big)\\ &&\times\int_0^{\infty}\big|K_{pq}(u)\big|e^{\omega u}\int_{-u}^{0}\Big[\big|z_q^R(s)\big| +\big|z_q^I(s)\big|+\big|z_q^J(s)\big|+\big|z_q^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s\,\mathrm{d}u\\ &&+\sum\limits_{q=1}^n\sum\limits_{r=1}^n \frac{4N_r \rho_qe^{\omega \sigma^+}}{\beta_{pqr}}\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \int_{-\sigma_{pqr}(0)}^0 \Big[\big|z_q^R(s)\big|+\big|z_q^I(s)\big|\\ &&+\big|z_q^J(s)\big| +\big|z_q^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s+\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{4N_q \rho_re^{\omega \xi^+}}{\gamma_{pqr}} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ &&\times\int_{-\xi_{pqr}(0)}^0\Big[\big|z_r^R(s)\big|+\big|z_r^I(s)\big|+\big|z_r^J(s)\big| +\big|z_r^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s +\sum\limits_{q=1}^n\frac{\eta_qe^{\omega\vartheta^{+}}}{\delta_{pq}} \Big(e_{pq}^{R^{+}}\\ &&+e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{K^{+}}\Big) \int_{-\vartheta_{pq}(0)}^{0}\Big[\big|z_q^R(s)\big|+\big|z_q^I(s)\big|+\big|z_q^J(s)\big| +\big|z_q^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s\bigg\} \end{eqnarray*}  \begin{eqnarray*} &\leq&\sum_{p=1}^{n}\bigg\{1 +\sum\limits_{q=1}^n\frac{4\lambda_q(e^{\omega\tau^{+}}-1)}{\omega\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big) +\frac{1}{\omega}B^{\ast}_p\\ &&+\sum\limits_{q=1}^n\sum\limits_{r=1}^n \frac{16N_r \rho_q(e^{\omega \sigma^+}-1)}{\omega\beta_{pqr}}\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ && +\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{16N_q (\rho_re^{\omega \xi^+}-1)}{\omega\gamma_{pqr}} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ && +\sum\limits_{q=1}^n\frac{4\eta_q(e^{\omega\vartheta^{+}}-1)}{\omega\delta_{pq}} \Big(e_{pq}^{R^{+}}+e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{K^{+}}\Big)\bigg\} \|\psi-\varphi\|. \end{eqnarray*} Similarly, for $$\nu =I,J,K$$, we can get   \begin{eqnarray*} V^{\nu}(0)&\leq&\sum_{p=1}^{n}\bigg\{1 +\sum\limits_{q=1}^n\frac{4\lambda_q(e^{\omega\tau^{+}}-1)}{\omega\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big) +\frac{1}{\omega}B^{\ast}_p\\ &&+\sum\limits_{q=1}^n\sum\limits_{r=1}^n \frac{16N_r \rho_q(e^{\omega \sigma^+}-1)}{\omega\beta_{pqr}}\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ && +\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{16N_q (\rho_re^{\omega \xi^+}-1)}{\omega\gamma_{pqr}} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ && +\sum\limits_{q=1}^n\frac{4\eta_q(e^{\omega\vartheta^{+}}-1)}{\omega\delta_{pq}} \Big(e_{pq}^{R^{+}}+e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{K^{+}}\Big)\bigg\}\|\psi-\varphi\|. \end{eqnarray*} It is obvious that $$\|y(t)-x(t)\|_{0}e^{\omega t}=\|z(t)\|_{0}e^{\omega t}\leq V(t), t\geq 0$$, hence, we have   \begin{equation*} \|y(t)-x(t)\|_{0}\leq V(t)e^{-\omega t}\leq V(0)e^{-\omega t}\leq M\|\psi-\varphi\|e^{-\omega t},\quad t\geq 0, \end{equation*} where   \begin{eqnarray*} M&=&\sum_{p=1}^{n}\bigg\{1 +\sum\limits_{q=1}^n\frac{4\lambda_q(e^{\omega\tau^{+}}-1)}{\omega\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big) +\frac{1}{\omega}B^{\ast}_p\\ &&+\sum\limits_{q=1}^n\sum\limits_{r=1}^n \frac{16N_r \rho_q(e^{\omega \sigma^+}-1)}{\omega\beta_{pqr}}\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ && +\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{16N_q (\rho_re^{\omega \xi^+}-1)}{\omega\gamma_{pqr}} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ && +\sum\limits_{q=1}^n\frac{4\eta_q(e^{\omega\vartheta^{+}}-1)}{\omega\delta_{pq}} \Big(e_{pq}^{R^{+}}+e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{K^{+}}\Big)\bigg\}\gt 0. \end{eqnarray*} Therefore, the drive system (1.1) and the response system (4.1) are almost automorphic global exponentially synchronization. The proof is completed. Remark 4.4 From the proofs of Theorems 3.1 and 4.1, one can easily see that if all the coefficients of (1.1) are periodic and almost periodic, respectively, then, similar to the proofs of Theorems 3.1 and 4.1 and under the same corresponding conditions, one can show that the similar results of Theorems 3.1 and 4.1 are still valid for both cases of the periodicity and almost periodicity. 5. A numerical example In this section, an example was given to show the effectiveness of the result obtained in this paper. Example 5.1 Consider the following QVHHNN as the drive system   \begin{eqnarray} x^{\prime}_{p}(t)&=&-c_{p}(t)x_{p}(t)+\sum_{q=1}^{2}a_{pq}(t)f_{q}(x_{q}(t-\tau_{pq}(t))) +\sum_{q=1}^{2}b_{pq}(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}(x_{q}(t-u))\,\mathrm{d}u\nonumber\\ &&+\sum_{q=1}^{2}\sum_{r=1}^{2}d_{pqr}(t)h_{q}(x_{q}(t-\sigma_{pqr}(t)))h_{r}(x_{r}(t-\xi_{pqr}(t))) +I_{p}(t),\,\,t\in \mathbb{R} \end{eqnarray} (5.1)and the corresponding response system is given by   \begin{eqnarray} y^{\prime}_{p}(t)&=&-c_{p}(t)y_{p}(t)+\sum_{q=1}^{2}a_{pq}(t)f_{q}(y_{q}(t-\tau_{pq}(t))) +\sum_{q=1}^{2}b_{pq}(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}(y_{q}(t-u))\,\mathrm{d}u\nonumber\\ &&+\sum_{q=1}^{2}\sum_{r=1}^{2}d_{pqr}(t)h_{q}(y_{q}(t-\sigma_{pqr}(t))) h_{r}(y_{r}(t-\sigma_{pqr}(t)))+I_{p}(t)+U_p(t), \end{eqnarray} (5.2)where $$K_{pq}(u)=(\cos u)e^{-u},p,q=1,2$$ and the coefficients are as follows:   $$ f_{q}(x_{q})=\frac{1}{16}\left|x_{q}^{R}\right|+i\frac{1}{12}\sin\left(x_{q}^{I}+x_{q}^{J}\right) +j\frac{1}{20}\sin^{2}x_{q}^{J}+k\frac{1}{16}\tanh x_{q}^{K},$$  $$ g_{q}(x_{q})=\frac{1}{18}\sin\left(x_{q}^{R}+x_{q}^{J}\right)+i\frac{1}{14}\left|x_{q}^{I}\right|+j\frac{1}{12}\sin x_{q}^{J}+k\frac{1}{15}\sin^{2}\left(x_{q}^{J}+x_{q}^{K}\right),$$  $$ h_{q}(x_{q})=\frac{1}{20}\sin x_{q}^{R}+i\frac{1}{24}\left|x_{q}^{R}+x_{q}^{I}\right| +j\frac{1}{18}\sin x_{q}^{J}+k\frac{1}{20}\sin x_{q}^{K},$$  $$ h_{r}(x_{r})=\frac{1}{18}\left|x_{r}^{R}\right|+i\frac{1}{20}\sin\left(x_{r}^{I}+x_{r}^{J}\right) +j\frac{1}{18}\left|x_{r}^{J}\right|+k\frac{1}{24}\tanh x_{r}^{K},$$  $$ l_{q}(z_{q})=\frac{1}{20}\left|z_{q}^{R}+z_{q}^{I}\right|+i\frac{1}{16}\sin^{2}z_{q}^{I} +j\frac{1}{18}\left|z_{q}^{J}\right|+k\frac{1}{14}\sin \left(z_{q}^{I}+z_{q}^{K}\right),$$  $$ c_{1}(t)=3+|\sin(\sqrt{2}t)|,\quad c_{2}(t)=5+|\cos (2t)|,\quad \theta_{1}(t)=2-|\cos t|,\quad \theta_{2}(t)=1-|\sin t|, $$  $$ a_{11}(t)=a_{12}(t)=0.1\sin \sqrt{2}t+i0.2\cos 2t+j0.1\sin t+k0.15\cos \sqrt{3}t,$$  $$ a_{21}(t)=a_{22}(t)=0.2\cos t+i0.1\sin \sqrt{2}t+j0.25\sin 2t+k0.3\cos t, $$  $$ b_{11}(t)=b_{12}(t)=0.3\cos 2t+i0.25\sin t+j0.15\cos t+k0.3\sin \sqrt{2}t,$$  $$ b_{21}(t)=b_{22}(t)=0.2\sin t+i0.1\sin\sqrt{2} t+j0.2\sin t+k0.4\cos \sqrt{3}t,$$  $$ e_{11}(t)=e_{12}(t)=0.15\cos t+i0.3\sin 3t+j0.1\cos\sqrt{2} t+k0.25\sin2t,$$  $$ e_{21}(t)=e_{22}(t)=0.3\sin 2t+i0.35\cos \sqrt{3}t+j0.25\cos \sqrt{2}t+k0.3\sin2 t, $$  $$ I_{1}(t)=I_{2}(t)=3\sin \sqrt{3}t+i2\cos \sqrt{2}t+j2.5\sin 3t+k3.5\sin 2t,$$  $$ \tau_{pq}(t)=\frac{1}{4}\sin^{2}t,\quad \sigma_{pq}(t)=\frac{1}{5}\sin^{2}t,\quad \xi_{pq}(t)=\frac{1}{5}\cos^{2}t,\quad\vartheta_{pq}(t)=\frac{1}{3}\cos^{2}t,$$  $$ d_{pqr}(t)=0.05\sin \sqrt{2}t+ i0.05\cos \sqrt{3}t +j 0.05\cos t +k0.05\sin \sqrt{3}t,\quad p,q,r=1,2.$$By a simple computing, we have   \begin{equation*}\,c_{1}^{-}=3,\quad c_{2}^{-}=5,\quad \theta_{1}^{-}=2,\quad \theta_{2}^{-}=1,\quad \lambda_q=\frac{1}{12},\quad \delta_q=\frac{1}{12}, \quad \rho_q=\frac{1}{18},\quad\end{equation*}   \begin{equation*}\qquad\rho_r=\frac{1}{18},\quad N_q=\frac{1}{18},\quad N_r=\frac{1}{18},\eta_q=\frac{1}{14},\quad d_{pqr}^{\nu}=0.05,\quad p,q,r=1,2,\,\,\nu\in\Lambda,\end{equation*}   \begin{equation*}a_{11}^{R^{+}}=a_{12}^{R^{+}}=0.1,\quad a_{11}^{I^{+}}=a_{12}^{I^{+}}=0.2,\quad a_{11}^{J^{+}}= a_{12}^{J^{+}}=0.1,\quad a_{11}^{K^{+}}=a_{12}^{K^{+}}=0.15,\;\end{equation*}   \begin{equation*}a_{21}^{R^{+}}=a_{22}^{R^{+}}=0.2,\quad a_{21}^{I^{+}}=a_{22}^{I^{+}}=0.1,\quad a_{21}^{J^{+}}=a_{22}^{J^{+}}=0.25,\quad a_{21}^{K^{+}}=a_{22}^{K^{+}}=0.3,\;\end{equation*}   \begin{equation*}\;b_{11}^{R^{+}}=a_{12}^{R^{+}}=0.3,\quad b_{11}^{I^{+}}=b_{12}^{I^{+}}=0.25,\quad b_{11}^{J^{+}}= b_{12}^{J^{+}}=0.15,\quad b_{11}^{K^{+}}=b_{12}^{K^{+}}=0.3,\end{equation*}   \begin{equation*}\;\,b_{21}^{R^{+}}=b_{22}^{R^{+}}=0.2,\quad b_{21}^{I^{+}}=b_{22}^{I^{+}}=0.1,\quad b_{21}^{J^{+}}=b_{22}^{J^{+}}=0.2,\quad b_{21}^{K^{+}}=b_{22}^{K^{+}}=0.4,\,\quad\end{equation*}   \begin{equation*}\,e_{11}^{R^{+}}=e_{12}^{R^{+}}=0.15,\quad e_{11}^{I^{+}}=e_{12}^{I^{+}}=0.3,\quad e_{11}^{J^{+}}= e_{12}^{J^{+}}=0.1,\quad e_{11}^{K^{+}}=e_{12}^{K^{+}}=0.25,\; \end{equation*}   \begin{equation*}\;e_{21}^{R^{+}}=e_{22}^{R^{+}}=0.3,\quad e_{21}^{I^{+}}=e_{22}^{I^{+}}=0.35,\quad e_{21}^{J^{+}}=e_{22}^{J^{+}}=0.25,\quad e_{21}^{K^{+}}=e_{22}^{K^{+}}=0.3,\,\;\end{equation*}   \begin{equation*}\;\,I_{1}^{R^{+}}=I_{2}^{R^{+}}=3,\quad I_{1}^{I^{+}}=I_{2}^{I^{+}}=2,\quad I_{1}^{J^{+}}=I_{2}^{J^{+}}=2.5,\quad I_{1}^{K^{+}}=I_{2}^{K^{+}}=3.5,\;\,\qquad\quad\end{equation*}   \begin{equation*}\,\;\;\;\;\,\tau^{+}=\frac{1}{4},\quad\sigma^{+}=\frac{1}{5},\quad\xi^{+}=\frac{1}{5},\quad\vartheta^{+}=\frac{1}{3},\quad \int_0^{+\infty}|K_{pq}(u)|\,\mathrm{d}u\leq 1,\quad p,q=1,2,\end{equation*}   \begin{equation*}\Theta_1\approx1.0729,\quad \Theta_2\approx1.2062,\quad \Xi_1\approx1.1124,\quad \Xi_2\approx1.2457.\end{equation*}Take $$\kappa =2$$, then   \begin{align*} &\max\bigg\{\frac{\Theta_1 \kappa +I_{1}^{R^{+}}}{c_1^-}\,,\, \frac{\Theta_1 \kappa +I_{1}^{I^{+}}}{c_1^-}\,,\, \frac{\Theta_1 \kappa+I_{1}^{J^{+}}}{c_1^-}\,,\, \frac{\Theta_1 \kappa+I_{1}^{K^{+}}}{c_1^-},\\ &\qquad\qquad\qquad\qquad\frac{\Theta_2 \kappa +I_{2}^{R^{+}}}{c_2^-}\,,\, \frac{\Theta_2 \kappa +I_{2}^{I^{+}}}{c_2^-}\,,\, \frac{\Theta_2 \kappa+I_{2}^{J^{+}}}{c_2^-}\,,\, \frac{\Theta_2 \kappa+I_{2}^{K^{+}}}{c_2^-}\bigg\}\approx1.8819 < \kappa=2,& \end{align*}and   \begin{equation*} \max\bigg\{\frac{\Xi_1}{c_1^-}\,,\,\frac{\Xi_2}{c_2^-}\bigg\}\approx0.3708=\mu<1. \end{equation*}Moreover, take $$\omega =1$$, $$\alpha _{pq}=\frac{3}{4}$$, $$\beta _{pqr}=\gamma _{pqr}=\frac{4}{5}$$, $$\delta _{pq}=\frac{2}{3}$$, we have   \begin{align*} &\sum\limits_{p=1}^2\bigg\{\big(\omega-c_p^{-}-\theta_{p}^{-}\big) +B^{\ast}_p+\sum\limits_{q=1}^2\frac{4\lambda_qe^{\omega\tau^+}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\nonumber\\ &\qquad +\sum\limits_{q=1}^2\sum\limits_{r=1}^2\bigg( \frac{16N_r \rho_qe^{\omega \sigma^+}}{\beta_{pqr}}+\frac{16N_q \rho_r e^{\omega \xi^+}}{\gamma_{pqr}}\bigg)\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \nonumber\\ &\qquad \sum\limits_{q=1}^2\frac{4\eta_{q}e^{\omega\vartheta^+}}{\delta_{pq}}\Big(e_{pq}^{R^{+}} +e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{k^{+}}\Big)\bigg\} \approx -3.9059<0. \end{align*}We can verify that all the assumptions of Theorem 4.1 are satisfied. So by Theorem 4.1, systems (5.1) and (5.2) are global exponentially synchronization about the almost automorphic solution (see Figs 1–5). Fig. 1. View largeDownload slide Curves of $$z_{1}^{\nu }$$$$(\nu \in \Lambda )$$ in three-dimensional space for synchronization case. Fig. 1. View largeDownload slide Curves of $$z_{1}^{\nu }$$$$(\nu \in \Lambda )$$ in three-dimensional space for synchronization case. Fig. 2. View largeDownload slide Curves of $$z_{2}^{\nu }$$$$(\nu \in \Lambda )$$ in three-dimensional space for synchronization case. Fig. 2. View largeDownload slide Curves of $$z_{2}^{\nu }$$$$(\nu \in \Lambda )$$ in three-dimensional space for synchronization case. Fig. 3. View largeDownload slide The states of four parts of $$x_{1}$$ and $$x_{2}$$. Fig. 3. View largeDownload slide The states of four parts of $$x_{1}$$ and $$x_{2}$$. Fig. 4. View largeDownload slide The states of four parts of $$y_{1}$$ and $$y_{2}$$. Fig. 4. View largeDownload slide The states of four parts of $$y_{1}$$ and $$y_{2}$$. Fig. 5. View largeDownload slide Synchronization. Fig. 5. View largeDownload slide Synchronization. 6. Conclusion In this paper, we consider the problem of the almost automorphic synchronization of QVHHNNs with time-varying and distributed delays. By applying the Banach fixed point theorem, constructing a suitable Lyapunov function and designing a state-feedback controller, we obtain that the drive-response structure of QVHHNNs with almost automorphic coefficients can realize the exponential synchronization. To the best of our knowledge, this is the first paper to study the automorphic synchronization of neural networks. Our result of this paper is completely new and our method can be used to study the problem of the periodic, almost periodic and automorphic synchronization for other types of neural networks. Besides, the study of QVHHNNs can unify the study of real-valued and complex-valued neural networks. 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Almost automorphic synchronization of quaternion-valued high-order Hopfield neural networks with time-varying and distributed delays

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Abstract

Abstract In this paper, we consider the problem of the almost automorphic synchronization of quaternion-valued high-order Hopfield neural networks (QVHHNNs) with time-varying and distributed delays. Firstly, to avoid the non-commutativity of quaternion multiplication, we decompose QVHHNNs into an equivalent real-valued system. Secondly, we use the Banach fixed point theorem to obtain the existence of almost automorphic solutions of QVHHNNs. Thirdly, by designing a novel state-feedback controller and constructing suitable Lyapunov functions, we obtain that the drive-response structure of QVHHNNs with almost automorphic coefficients can realize the exponential synchronization. Our results are completely new. Finally, a numerical example is given to illustrate the feasibility of our results. 1. Introduction As is well known, synchronization is an extensive phenomenon in real systems, which means that two or more systems adjust each other to result in a common dynamical behavior. Experimental and theoretical studies have reported that synchronization phenomenon could be seen in various real systems. By synchronization, we can understand an unknown system from the well-known systems. So it has played a significant role in network control and system design as a powerful tool (Tang et al., 2015, 2016). The analysis of synchronization of neural networks has attracted a lot of scholars from various fields such as information science, secure communication and chemical reactions (Masoller & Zanette, 2001; Wu & Chen, 2008; Zhang & Zhao, 2012; Wong et al., 2013; Yu et al., 2013; Chai & Chen, 2014; Chen et al., 2014; Lü et al., 2014; Hong, 2014; Bao & Cao, 2015; Cai et al., 2015; Cai et al., 2015; Wu et al., 2015; Yang et al., 2017). Many good results have been obtained about the synchronization of neural networks including periodic (Chai & Chen, 2014; Hong, 2014; Cai et al., 2015; Cai et al., 2015; Wu et al., 2015), global (Wu & Chen, 2008; Chen et al., 2014), stochastic (Wong et al., 2013), combinatorial, cluster, adaptive (Yu et al., 2013), projective (Zhang & Zhao, 2012; Bao & Cao, 2015), lag (Lü et al., 2014; Yang et al., 2017) and anticipated synchronization (Masoller & Zanette, 2001). The well-known Hopfield neural network, as a form of recurrent artificial neural networks popularized by John Hopfield in 1982 (Hopfield, 1984), has been widely studied in recent years (Xiao, 2009; Zhang & Jin, 2000; Zhang et al., 2003), especially the high-order Hopfield neural networks (HHNNs), because of its faster convergence rate, stronger approximation property, greater capacity and higher fault tolerance. Many scholars have made extensive research on the existence, uniqueness and stability of equilibrium points, periodic solutions, almost periodic solutions and almost automorphic solutions of HHNNs, see Xiao (2009), Xiao & Meng (2009), Zhang & Gui (2009), Li & Yang (2014), Arbi et al. (2015), Li & Yang (2016), Li & Meng (2017), Zhao et al. (2018) and the references therein. And there are also a few results about the periodic synchronization of the HHNNs. However, there has been no paper published on the problem of the almost periodic synchronization of neural networks. Due to the finite switching speed of neurons and amplifiers, time delays inevitably exist in biological and artificial neural network models. And, as is known to all, time delays may change the dynamical behaviors of neural networks under consideration. Therefore, the dynamics of neural networks with various delays have been a long-term focus issue (Zhang & Jin, 2000; Zhang et al., 2003; Xiao, 2009; Zhang & Gui, 2009; Wu et al., 2012; Li & Yang, 2014; Arbi et al., 2015; Li & Yang, 2016; Arbi & Cao, 2017; Li & Meng, 2017; Arbi & Cao, 2018; Xiong et al., 2018; Zhao et al., 2018). However, in these recent publications, most results on the synchronization of delayed neural networks have been restricted to simple cases of discrete delays. Since the neural networks usually have a spatial nature due to the presence of an amount of paralleled pathway of a variety of axon sizes and lengths, it is desired to model them by introducing distributed delays (Zhang et al., 2003). Therefore, both discrete and distributed delays should be taken into account when modeling more realistic HHNNs. On the one hand, the concept of almost automorphy, which is much more general than the almost periodicity, was introduced in the literature by Bochner in 1955 (Bochner, 1964) in the context of differential geometry (Bochner, 1964) and plays a very important role in understanding the almost periodicity. Moreover, in reality, almost periodicity is universal than periodicity. So almost automorphic oscillation of neural networks has been studied by several authors, see Bochner (1964), Li & Yang (2014), Yang et al. (2017) and the references therein. On the other hand, the quaternion algebra was first proposed by Hamilton (Sudbery, 1979) in 1843. A quaternion consists of a real and three imaginary parts. The three imaginary units i, j and k obey the Hamilton’s multiplication rules:   \begin{equation*} ij=-ji=k,\quad jk=-kj=i,\quad ki=-ik=j,\quad i^2=j^2=k^2=-1. \end{equation*}The skew field of quaternions is denoted by $$\mathbb{Q}:=\{q=q^R+iq^I+jq^J+kq^K\}$$, where $$q^R, q^I, q^J, q^K$$ are real numbers. The quaternion-valued neural networks (QVNNs), which are a generalization of the real-valued and complex-valued neural networks (Wang et al., 2017; Li et al., 2018), can be extensively applied in the fields such as aerospace, satellite tracking, processing of polarized waves and image processing (Matsui et al., 2004; Yoshida et al., 2005; Luo et al., 2010). One of the benefits of using quaternion is the three-dimensional geometrical affine transformation that can be represented efficiently and compactly. So, the study of QVNNs has received much attention of many scholars and some good results have been obtained for the stability, dissipativity, periodicity and pseudo almost periodicity of QVNNs (Liu et al., 2016; Chen et al., 2017; Chen et al., 2017; Li & Meng, 2017; Liu et al., 2017; Liu et al., 2017; Valle & de Castro, 2017; Zhang et al., 2017; Li & Qin, 2018). For example, in Zhang et al. (2017), the global exponential stability for recurrent neural networks with asynchronous time delays is investigated in the quaternion field; in Liu et al. (2016), some sufficient conditions on the global $$\mu $$-stability of the QVNNs with unbounded time-varying delays were obtained; in Chen et al. (2017), the author dealt with the problem of robust stability for QVNNs with leakage delay, discrete delay and parameter uncertainties. However, as far as we know, there is no result about the almost automorphic synchronization of QVNNs. Motivated by the considerations mentioned above, in the present work, we are concerned with the following quaternion-valued high-order Hopfield neural network (QVHHNN) with time-varying and distributed delays:   \begin{eqnarray} X^{\prime}_{p}(t)&=&-c_{p}(t)X_{p}(t)+\sum_{q=1}^{n}a_{pq}(t)f_{q}(X_{q}(t-\tau_{pq}(t))) +\sum_{q=1}^{n}b_{pq}(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}(X_{q}(t-u))\,\mathrm{d}u\nonumber\\ &&+\sum_{q=1}^{n}\sum_{r=1}^{n}d_{pqr}(t)h_{q}(X_{q}(t-\sigma_{pqr}(t)))h_{r}(X_{r}(t-\xi_{pqr}(t))) +I_{p}(t),\,\,t\in \mathbb{R}, \end{eqnarray} (1.1)where p ∈{1, 2, …, n} := N, n corresponds to the number of units in the neural network; $$X_p(t)\in \mathbb{Q}$$ denotes the activation of the pth neuron at time t; $$c_{p}(t)>0$$ represents the rate with which the pth unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs at time t; $$a_{pq}(t), b_{pq}(t)\in \mathbb{Q}$$ represent the delayed strengths of connectivity between cell p and q at time t, respectively; $$d_{pqr}(t)\in \mathbb{Q}$$ denotes the second-order connection weights of the neural network; $$f_{q}, g_{q}, h_{q},h_{r}: \mathbb{Q}\rightarrow \mathbb{Q}$$ are the activation functions of signal transmission; $$I_{p}(t)\in \mathbb{Q}$$ is an external input on the pth unit at time t; $$\tau _{pq}(t)$$, $$\sigma _{pqr}(t)$$ and $$\xi _{pqr}(t)$$ are transmission delays at time t. The initial conditions associated with system (1.1) are of the form   \begin{equation*} X_p(s)=\Phi_p (s),\,\, s\in(-\infty, 0],\,\, p\in N, \end{equation*}where $$\Phi _p(s)=\varphi _p^R(s)+i\varphi _p^I(s)+j\varphi _p^J(s)+k\varphi _p^K(s)$$ is the quaternion-valued bounded continuous function defined on $$(-\infty , 0]$$. Our main purpose of this paper is to study the problem of the almost automorphic synchronization of (1.1). To the best of our knowledge, this is the first paper to study the almost automorphic synchronization of neural networks. Our result of this paper is completely new and our method can be used to study the problem of the periodic, almost periodic and almost automorphic synchronization for other types of neural networks. Besides, QVHHNN (1.1) contains real-valued and complex-valued HHNNs as its special cases. Remark 1.1 In system (1.1), if $$X_{q}=x_{q}^{R}+x_{q}^{I}i\in \mathbb{C}$$, where $$x_{q}^{R},x_{q}^{I} \in \mathbb{R}$$ and the activation functions $$f_{q},g_{q}: \mathbb{C}\rightarrow \mathbb{C}$$ are complex variable functions, that is, $$ f_{q}(X_{q})=f_{q}^{R}(x_{q}^{R},x_{q}^{I})+if_{q}^{I}(x_{q}^{R},x_{q}^{I}), g_{q}(X_{q})=g_{q}^{R}(x_{q}^{R},x_{q}^{I}) +ig_{q}^{I}(x_{q}^{R},x_{q}^{I}), h_{q}(X_{q})=h_{q}^{R}(x_{q}^{R},x_{q}^{I}) +ih_{q}^{I}(x_{q}^{R},x_{q}^{I}), $$ where $$f^{\nu }_{q}, g^{\nu }_{q}:\mathbb{R}^{2}\rightarrow \mathbb{R}$$, q ∈ N, $$\nu =R,I$$ and all the quaternion-valued coefficients of (1.1) are complex-valued coefficients, then system (1.1) degenerates to a complex-valued system; if all of the activation functions and coefficients of (1.1) are real variable functions, then system (1.1) degenerates to a real-valued system. This paper is organized as follows. In Section 2, some preliminaries and notations are given. In Section 3, sufficient conditions for the existence of almost automorphic solutions of system (1.1) are obtained. In Section 4, the exponential synchronization was investigated. In Section 5, the effectiveness and feasibility of the developed methods in this paper are shown by a numerical example. In Section 5, we draw a brief conclusion. 2. Preliminaries We will first introduce some definitions and recall some basic lemmas that will be used in this paper. Analogously with the definition of almost automorphic functions in the study by Diagana (2013), we gave the following. Definition 2.1 A function $$f\in C(\mathbb{R}, \mathbb{Q}^n)$$ is said to be almost automorphic in Bochner’s sense if for every sequence of real numbers $$\{s_n\}_{n\in \mathbb{N}}$$, there exists a subsequence $$\{\tau _n\}_{n\in \mathbb{N}}$$ such that   \begin{equation*} g(t)=\lim\limits_{n\rightarrow\infty}f(t+\tau_n) \end{equation*}is well defined for each $$t\in \mathbb{R}$$ and   \begin{equation*} \lim\limits_{n\rightarrow\infty}g(t-\tau_n)=f(t) \end{equation*}for each $$t\in \mathbb{R}$$. Remark 2.1 Obviously, function $$f=f^R+if^I+jf^J+kf^K\in C(\mathbb{R}, \mathbb{Q}^n)$$ is almost automorphic if and only if $$f^\nu \in C(\mathbb{R}, \mathbb{R}^n)$$ is almost automorphic, where $$\nu \in \Lambda :=\{R, I,J, K\}$$. Denote all the almost automorphic functions by $$AA(\mathbb{R}, \mathbb{Q}^n)$$, which is a Banach space when it is endowed with the supremum norm. Lemma 2.1 (N’Guérékata, 2001; Diagana, 2013) Let X, Y be Banach spaces, and if $$f, f_1, f_2 \in AA(\mathbb{R}, X)$$, then (i) $$f_1+f_2\in AA(\mathbb{R}, X)$$; (ii) $$\alpha f\in AA(\mathbb{R}, X)$$ for any constant $$\alpha \in \mathbb{R}$$; (iii) if $$\varphi :X\rightarrow Y$$ is a continuous function, then the composite function $$f\circ \varphi : \mathbb{R}\rightarrow Y$$ is almost automorphic. Definition 2.2 (Li & Yang, 2014) Let A(t) be an n × n matrix function on $$\mathbb{R}$$. Then the linear system   \begin{equation} x^{\prime}(t)=A(t)x(t),\,\, t\in\mathbb{R} \end{equation} (2.1)is said to admit an exponential dichotomy on $$\mathbb{R}$$ if there exist positive constants $$k_i, \alpha _i, i=1, 2$$, projection P and the fundamental solution matrix X(t) of (2.1), satisfying   \begin{align*} \big|X(t)PX^{-1}(s)\big| &\leq k_1e^{-\alpha_1(t-s)}, \quad s, t \in\mathbb{R},\,\, t \geq s, \\ \big|X(t)(I-P)X^{-1}(s)\big|&\leq k_2e^{-\alpha_2(s-t)},\quad s, t \in\mathbb{R},\,\, t \leq s, \end{align*}where |⋅| is a matrix norm on $$\mathbb{R}$$. Consider the following almost automorphic system   \begin{equation} x^{\prime}(t)=A(t)x(t)+f(t),\,\, t \in \mathbb{R}, \end{equation} (2.2)where A(t) is an almost automorphic matrix function and f(t) is an almost automorphic vector function. Lemma 2.2 (Li & Yang, 2014) Suppose that $$A(t)\in AA(\mathbb{R}, \mathbb{R}^{n\times n})$$, such that $$\{A^{-1}(t)\}$$ is bounded. Moreover, suppose that $$f\in AA(\mathbb{R}, \mathbb{R}^n)$$ and (2.1) admits an exponential dichotomy, then (2.2) has a solution $$x(t)\in AA(\mathbb{R}, \mathbb{R}^n)$$ that can be expressed as   $$ x(t)=\int_{-\infty}^{t}X(t)PX^{-1}(s)\,f(s)\,\mathrm{d}s -\int_t^{+\infty}X(t)(I-P)X^{-1}(s)\,f(s)\,\mathrm{d}s, $$where X(t) is the fundamental solution matrix of (2.1), I denotes the n × n-identity matrix. Lemma 2.3 (Li & Yang, 2014) Let $$c_{p}(t)>0$$ be an almost automorphic function on $$\mathbb{R}$$ and   $$ \min_{1\leq p \leq n}\{\inf_{t\in \mathbb{R}}c_p(t)\}>0,$$then the linear system   \begin{equation*} x^{\prime}(t)=\textrm{diag}(-c_{1}(t),-c_{2}(t),\dots,-c_{n}(t))x(t) \end{equation*}admits an exponential dichotomy on $$\mathbb{R}$$. Let $$X=x^R+ix^I+jx^J+kx^K\in \mathbb{Q},$$ where $$x^\nu \in \mathbb{R},\nu \in \Lambda $$ and assume that $$f_q:\mathbb{Q}\rightarrow \mathbb{Q}$$ can be expressed as   \begin{align*} f_q(x)&=f^R(x^R, x^I, x^J, x^K)+if^I(x^R, x^I, x^J, x^K)+jf^J(x^R, x^I, x^J, x^K)\\ &\quad+kf^K(x^R, x^I, x^J, x^K), \end{align*}where $$f_q^\nu : \mathbb{R}^4\rightarrow \mathbb{R}$$, $$\nu \in \Lambda $$, q ∈ N. To avoid the non-commutativity of quaternion multiplication, according to Hamilton rules, we decompose system (1.1) into an equivalent real-valued system:   \begin{eqnarray*} \left(x_{p}^R\right)^{\prime}(t)&=&-c_p(t) x_p^R(t)+\sum_{q=1}^{n}a_{pq}^R(t)f_q^R[t, x] -\sum_{q=1}^{n}a_{pq}^I(t)f_q^I[t, x] -\sum_{q=1}^{n}a_{pq}^J(t)f_q^J[t, x]\nonumber\\ &&-\sum_{q=1}^{n}a_{pq}^K(t)f_q^K[t, x] +\sum_{q=1}^{n}b_{pq}^R(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}^R[t, u, x]\,\mathrm{d}u -\sum_{q=1}^{n}b_{pq}^I(t)\nonumber\\ &&\times\int_{0}^{+\infty}K_{pq}(u) g_{q}^I[t, u, x]\,\mathrm{d}u-\sum_{q=1}^{n}b_{pq}^J(t) \int_{0}^{+\infty}K_{pq}(u)g_{q}^J[t, u, x]\,\mathrm{d}u\nonumber\\ &&-\sum_{q=1}^{n}b_{pq}^K(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}^K[t, u, x]\,\mathrm{d}u +\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}\bigg(d_{pqr}^R(t)h_q^R[t, x]\tilde{h}_r^R[t, x]\nonumber\\ &&-d_{pqr}^I(t)h_q^I[t, x]\tilde{h}_r^R[t, x]-d_{pqr}^J(t)h_q^J[t, x]\tilde{h}_r^R[t, x] -d_{pqr}^K(t)h_q^K[t, x]\tilde{h}_r^R[t, x]\nonumber\\ &&-d_{pqr}^I(t)h_q^R[t, x]\tilde{h}_r^I[t, x]-d_{pqr}^R(t)h_q^I[t, x]\tilde{h}_r^I[t, x] +d_{pqr}^K(t)h_q^J[t, x]\tilde{h}_r^I[t, x]\nonumber\\ &&-d_{pqr}^J(t)h_q^K[t, x]\tilde{h}_r^I[t, x]-d_{pqr}^J(t)h_q^R[t, x]\tilde{h}_r^J[t, x] -d_{pqr}^K(t)h_q^I[t, x]\tilde{h}_r^J[t, x]\nonumber\\ &&-d_{pqr}^R(t)h_q^J[t, x]h_q^J[t, x]+d_{pqr}^I(t)h_q^K[t, x]\tilde{h}_r^J[t, x]-d_{pqr}^K(t)h_q^R[t, x]\tilde{h}_r^K[t, x]\nonumber\\ &&+d_{pqr}^J(t)h_q^I[t, x]\tilde{h}_r^K[t, x]-d_{pqr}^I(t)h_q^J[t, x]\tilde{h}_r^K[t, x] -d_{pqr}^R(t)h_q^K[t, x]\tilde{h}_r^K[t, x]\bigg)+I_p^R(t)\nonumber\\ &=:& -c_p(t) x_p^R(t)+\Pi_p^R(t,x)+I_p^R(t), \,\, p\in N, \end{eqnarray*}  \begin{eqnarray*} \left(x_{p}^I\right)^{\prime}(t)&=&-c_p(t) x_p^I(t)+\sum_{q=1}^{n}a_{pq}^I(t)f_q^R[t, x] +\sum_{q=1}^{n}a_{pq}^R(t)f_q^I[t, x]-\sum_{q=1}^{n}a_{pq}^K(t)f_q^J[t, x]\nonumber\\[-7pt] &&+\sum_{q=1}^{n}a_{pq}^J(t)f_q^K[t, u, x] +\sum_{q=1}^{n}b_{pq}^I(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}^R[t, u, x]\,\mathrm{d}u +\sum_{q=1}^{n}b_{pq}^R(t)\nonumber\\[-5pt] &&\times\int_{0}^{+\infty}K_{pq}(u) g_{q}^I[t, u, x]\,\mathrm{d}u-\sum_{q=1}^{n}b_{pq}^K(t) \int_{0}^{+\infty}K_{pq}(u)g_{q}^J[t, u, x]\,\mathrm{d}u\nonumber\\[-5pt] &&+\sum_{q=1}^{n}b_{pq}^J(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}^K[t, u, x]\,\mathrm{d}u +\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}\bigg(d_{pqr}^I(t) h_q^R[t, x]\tilde{h}_r^R[t, x]\nonumber\\[-5pt] &&+d_{pqr}^R(t)h_q^I[t, x] \tilde{h}_r^R[t, x]-d_{pqr}^K(t)h_q^J[t, x]\tilde{h}_r^R[t,x] +d_{pqr}^J(t)h_q^K[t, x]\tilde{h}_r^R[t, x]\nonumber\\ &&+d_{pqr}^R(t)h_q^R[t, x]\tilde{h}_r^I[t, x]-d_{pqr}^I(t)h_q^I[t, x]\tilde{h}_r^I[t,x]-d_{pqr}^J(t)h_q^J[t, x]\tilde{h}_r^I[t, x]\nonumber\\ &&-d_{pqr}^K(t)h_q^K[t, x]\tilde{h}_r^I[t, x]-d_{pqr}^K(t)h_q^R[t, x]\tilde{h}_r^J[t, x] +d_{pqr}^J(t)h_q^I[t, x]\tilde{h}_r^J[t, x]\nonumber\\ &&-d_{pqr}^I(t)h_q^J[t, x] h_q^J[t, x]-d_{pqr}^R(t)h_q^K[t, x]\tilde{h}_r^J[t, x] +d_{pqr}^J(t)h_q^R[t, x]\tilde{h}_r^K[t, x]\nonumber\\[-5pt] &&+d_{pqr}^K(t)h_q^I[t, x]\tilde{h}_r^K[t, x] +d_{pqr}^R(t)h_q^J[t, x]\tilde{h}_r^K[t, x] -d_{pqr}^I(t)h_q^K[t, x]\tilde{h}_r^K[t, x]\bigg)+I_p^I(t)\nonumber\\[-5pt] &=:& -c_p(t) x_p^I(t)+\Pi_p^I(t,x)+I_p^I(t), \,\, p\in N, \end{eqnarray*}  \begin{eqnarray*} \left(x_{p}^J\right)^{\prime}(t)&=&-c_p(t) x_p^J(t)+\sum_{q=1}^{n}a_{pq}^J(t)f_q^R[t, x] +\sum_{q=1}^{n}a_{pq}^K(t)f_q^I[t, x]+\sum_{q=1}^{n}a_{pq}^R(t)f_q^J[t, x]\nonumber\\[-5pt] &&-\sum_{q=1}^{n}a_{pq}^I(t)f_q^K[t, x] +\sum_{q=1}^{n}b_{pq}^J(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}^R[t, u, x]\,\mathrm{d}u +\sum_{q=1}^{n}b_{pq}^K(t)\nonumber\\[-5pt] &&\times\int_{0}^{+\infty}K_{pq}(u) g_{q}^I[t, u, x]\,\mathrm{d}u+\sum_{q=1}^{n}b_{pq}^R(t) \int_{0}^{+\infty}K_{pq}(u)g_{q}^J[t, u, x]\,\mathrm{d}u\nonumber\\ &&-\sum_{q=1}^{n}b_{pq}^I(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}^K[t, ux]\,\mathrm{d}u+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n} \bigg(d_{pqr}^J(t) h_q^R[t, x]\tilde{h}_r^R[t, x]\nonumber\\ &&+d_{pqr}^K(t)h_q^I[t, x]\tilde{h}_r^R[t, x]+d_{pqr}^R(t)h_q^J[t, x]\tilde{h}_r^R[t, x] -d_{pqr}^I(t)h_q^K[t, x]\tilde{h}_r^R[t, x]\nonumber\\[3pt] &&-d_{pqr}^K(t)h_q^R[t, x]\tilde{h}_r^I[t, x]-d_{pqr}^J(t)h_q^I[t, x]\tilde{h}_r^I[t, x]+d_{pqr}^I(t)h_q^J[t, x]\tilde{h}_r^I[t, x]\nonumber\\ &&+d_{pqr}^R(t)h_q^K[t, x]\tilde{h}_r^I[t, x]+d_{pqr}^R(t)h_q^R[t, x]\tilde{h}_r^J[t, x] -d_{pqr}^I(t)h_q^I[t, x]\tilde{h}_r^J[t, x]\nonumber\\ &&-d_{pqr}^J(t)h_q^J[t, x]h_q^J[t, x]-d_{pqr}^K(t)h_q^K[t, x]\tilde{h}_r^J[t, x] -d_{pqr}^I(t)h_q^R[t, x]\tilde{h}_r^K[t, x]\nonumber\\ &&-d_{pqr}^R(t)h_q^I[t, x]\tilde{h}_r^K[t, x]+d_{pqr}^K(t)h_q^J[t, x]\tilde{h}_r^K[t, x] -d_{pqr}^J(t)h_q^K[t, x]\tilde{h}_r^K[t, x]\bigg)+I_p^J(t)\nonumber\\ &=:& -c_p(t) x_p^J(t)+\Pi_p^J(t,x)+I_p^J(t), \,\, p\in N, \end{eqnarray*}  \begin{eqnarray*} \left(x_{p}^K\right)^{\prime}(t)&=&-c_p(t) x_p^K(t)+\sum_{q=1}^{n}a_{pq}^K(t)f_q^R[t, x] -\sum_{q=1}^{n}a_{pq}^J(t)f_q^I[t, x] +\sum_{q=1}^{n}a_{pq}^I(t)f_q^J[t, x]\nonumber\\ &&+\sum_{q=1}^{n}a_{pq}^R(t)f_q^K[t, x] +\sum_{q=1}^{n}b_{pq}^K(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}^R[t, u, x]\,\mathrm{d}u -\sum_{q=1}^{n}b_{pq}^J(t)\nonumber\\ &&\times \int_{0}^{+\infty}K_{pq}(u)g_{q}^I[t, u, x]\,\mathrm{d}u+\sum_{q=1}^{n}b_{pq}^I(t) \int_{0}^{+\infty}K_{pq}(u)g_{q}^J[t, u, x]\,\mathrm{d}u\nonumber\\[-5pt] &&+\sum_{q=1}^{n}b_{pq}^R(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}^K[t, u, x]\,\mathrm{d}u +\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}\bigg(d_{pqr}^K(t) h_q^R[t, x]\tilde{h}_r^R[t, x]\nonumber\\[-5pt] &&-d_{pqr}^I(t)h_q^I[t, x] \tilde{h}_r^R[t, x]+d_{pqr}^I(t)h_q^J[t, x]\tilde{h}_r^R[t, x] +d_{pqr}^R(t)h_q^K[t, x]\tilde{h}_r^R[t, x]\nonumber\\ &&-d_{pqr}^J(t)h_q^R[t, x]\tilde{h}_r^I[t, x] -d_{pqr}^K(t)h_q^I[t, x]\tilde{h}_r^I[t, x]-d_{pqr}^R(t)h_q^J[t, x]\tilde{h}_r^I[t, x]\nonumber\\ &&-d_{pqr}^I(t)h_q^K[t, x]\tilde{h}_r^I[t, x] +d_{pqr}^I(t)h_q^R[t, x]\tilde{h}_r^J[t, x] +d_{pqr}^R(t)h_q^I[t, x]\tilde{h}_r^J[t, x]\nonumber\\ &&-d_{pqr}^K(t)h_q^J[t, x]h_q^J[t, x]+d_{pqr}^J(t)h_q^K[t, x]\tilde{h}_r^J[t, x] -d_{pqr}^R(t)h_q^R[t, x]\tilde{h}_r^K[t, x]\nonumber\\ &&-d_{pqr}^I(t)h_q^I[t, x]\tilde{h}_r^K[t, x]-d_{pqr}^J(t)h_q^J[t, x]\tilde{h}_r^K[t, x] -d_{pqr}^J(t)h_q^K[t, x]\tilde{h}_r^K[t, x]\bigg)+I_p^K(t)\nonumber\\ &=:& -c_p(t) x_p^K(t)+\Pi_p^K(t,x)+I_p^K(t), \,\, p\in N, \end{eqnarray*} where   \begin{align*} f_q^l[t, x]&\triangleq f_q^l(x_q^R(t-\tau_{pq}(t)), x_q^I(t-\tau_{pq}(t)), x_q^J(t-\tau_{pq}(t)), x_q^K(t-\tau_{pq}(t))),\\ g_q^l[t, u, x]&\triangleq g_q^l(x_q^R(t-u), x_q^I(t-u), x_q^J(t-u), x_q^K(t-u)), \\ h_q^l[t, x]&\triangleq h_q^l(x_q^R(t-\sigma_{pqr}(t)), x_q^I(t-\sigma_{pqr}(t)), x_q^J(t-\sigma_{pqr}(t)), x_q^K(t-\sigma_{pqr}(t))), \\ \tilde{h}_r^l[t, x]&\triangleq \tilde{h}_r^l(x_r^R(t-\xi_{pqr}(t)), x_r^I(t-\xi_{pqr}(t)), x_r^J(t-\xi_{pqr}(t)), x_r^K(t-\xi_{pqr}(t))). \end{align*} Hence, system (1.1) is transformed into the following equivalent real-valued system:   \begin{equation} \left(x_{p}^\nu\right)^{\prime}(t)= -c_p(t) x_p^\nu(t)+\Pi_p^\nu(t,x)+I_p^\nu(t), \,\, p\in N, \nu\in \Lambda. \end{equation} (2.3) Remark 2.2 If $$x=(x_1^R, x_2^R, \ldots , x_n^R, x_1^I, x_2^I, \ldots , x_n^I, x_1^J, x_2^J, \ldots , x_n^J, x_1^K, x_2^K, \ldots , x_n^K)$$ is a solution to system (2.3), then $$X(t)=(X_{1}(t),X_{2}(t),\ldots , X_{n}(t))^{T}$$ must be a solution of (1.1), where $$X_{l}(t)=x^{R}_{l}(t)+ix^{I}_{l}(t)+jx^{J}_{l}(t)+kx^{K}_{l}(t),l\in N$$. Thus, the problem of finding a solution for (1.1) is reduced to finding one for system (2.3). For considering the stability of solutions of (1.1), we just need to consider the stability of solutions of system (2.3). For convenience, in the following, we introduce the following notation:   $$ w^-=\inf\limits_{t\in\mathbb{R}}\big|w(t)\big|,\,\,w^+=\sup\limits_{t\in\mathbb{R}}\big|w(t)\big|, $$where $$w: \mathbb{R}\rightarrow \mathbb{R}$$ is a bounded function, and for p, q, r ∈ N, we denote   $$ \sigma^+=\max\limits_{p, q, r\in N}\big\{\sigma^+_{pqr}\big\},\,\, \xi^+=\max\limits_{ p, q, r\in N}\big\{\xi^+_{pqr}\big\},\,\,c^-=\max\limits_{p\in N}\big\{c_p^-\big\}, \,\,\tau^{+}=\max\limits_{p, q \in N}\big\{\tau^+_{pq}\big\},$$  $$ \alpha_{pq}=\inf\limits_{t\in \mathbb{R}}\{1-\tau_{pq}^{\prime}(t)\},\,\, \beta_{pqr}=\inf\limits_{t\in \mathbb{R}}\{1-\sigma_{pqr}^{\prime}(t)\},\,\, \gamma_{pqr}=\inf\limits_{t\in \mathbb{R}}\{1-\xi_{pqr}^{\prime}(t)\}.\quad\quad\;\,$$ Throughout the paper, we assume that the following conditions hold: $$(A_1)$$ Function $$c_{p}\in AA(\mathbb{R},\mathbb{R}^{+})$$ with $$\min _{1\leq p \leq n}\{\inf _{t\in \mathbb{R}}c_p(t)\}>0$$, $$a_{pq}, b_{pq}, d_{pqr}, I_p\in AA(\mathbb{R}, \mathbb{Q})$$ and $$\tau _{pq}, \sigma _{pqr}, \xi _{pqr}\in C^1(\mathbb{R}, \mathbb{R}^+)\cap AA(\mathbb{R},\mathbb{R})$$ with $$\inf \limits _{t\in R}\{1-\tau _{pq}^{\prime}(t)\}>0$$, $$\inf \limits _{t\in R}\{1-\sigma _{pqr}^{\prime}(t)\}>0$$, $$\inf \limits _{t\in R}\{1-\xi _{pqr}^{\prime}(t)\}>0$$, where p, q, r ∈ N. $$(A_2)$$ Functions $$f_q, g_q, h_q, h_r\in C(\mathbb{R}, \mathbb{Q})$$ and for any $$u^\nu , v^\nu \in \mathbb{R}$$, there exist positive constants $$\lambda _q^{\nu }$$, $$\delta _q^{\nu }$$, $$\rho _q^{\nu }$$, $$N_q$$ such that $$|h_q^\nu (u^R, u^I, u^J, u^K)|\leq N_q$$,   \begin{eqnarray*} \big|\,f_q^\nu(u^R, u^I, u^J, u^K)-f_q^\nu(v^R, v^I, v^J, v^K)\big| &\leq&\lambda_q^{R}|u^R-v^R|+\lambda_q^{I}|u^I-v^I|+ \lambda_q^{J}|u^J-v^J|\\ &&+\lambda_q^{K}|u^K-v^K|,\\ \big|g_q^\nu(u^R, u^I, u^J, u^K)-g_q^\nu(v^R, v^I, v^J, v^K)\big| &\leq& \delta_q^{R}|u^R-v^R|+\delta_q^{I}|u^I-v^I|+\delta_q^{J}|u^J-v^J|\\ &&+\delta_q^{K}|u^K-v^K|,\\ \big|h_q^\nu(u^R, u^I, u^J, u^K)-h_q^\nu(v^R, v^I, v^J, v^K)\big| &\leq& \rho_q^{R}|u^R-v^R|+\rho_q^{I}|u^I-v^I|+\rho_q^{J}|u^J-v^J|\\ &&+\rho_q^{K}|u^K-v^K| \end{eqnarray*}and $$f_q^\nu (0, 0, 0, 0)=g_q^\nu (0, 0, 0, 0)=h_q^\nu (0, 0, 0, 0)=0$$, where $$\nu \in \Lambda $$, q ∈ N. $$(A_{3})$$ For p, q ∈ N, the delay kernels $$K_{pq}: [0, \infty )\rightarrow \mathbb{R}$$ are continuous and $$|K_{pq}(t)|e^{\iota t}$$ are integrable on $$[0, \infty )$$ for a certain positive constant $$\iota $$. $$(A_{4})$$ There exists a positive constant $$\kappa $$ such that   \begin{equation*} \max\limits_{p\in E}\bigg\{\max\limits_{\nu\in\Lambda}\bigg\{\frac{\Theta_p \kappa +I_{p}^{\nu^{+}}}{c_p^-} \bigg\}\bigg\}\leq \kappa,\quad \max\limits_{p\in E}\bigg\{\frac{\Xi_p}{c_p^-}\bigg\}:=\mu<1, \end{equation*}where   \begin{align*} \Theta_p & =A_{p}^{\ast}+B_{p}^{\ast}+D_{p}^{\ast},\quad p\in N,\\ \Xi_p & =A_{p}^{\ast}+B_{p}^{\ast}+D_{p}^{\ast}+\hat{D}_{p}^{\ast},\quad p\in N,\\ A_{p}^{\ast}&=\sum_{q=1}^{n}4\lambda_q\Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big),\quad p\in N,\\ B_{p}^{\ast}&=\sum_{q=1}^{n}4\delta_q\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big)\int_{0}^{+\infty}\big|K_{pq}(u)\big|\,\mathrm{d}u ,\quad p\in N,\\ D_{p}^{\ast}&=\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}16N_q\rho_r \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big),\quad p\in N,\\ \hat{D}_{p}^{\ast}&=\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}16N_{r}\rho_{q} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big),\quad p\in N,\\ \lambda_q&=\max\left\{\lambda^R_q,\lambda^I_q, \lambda^J_q, \lambda^K_q\right\}, \delta_q=\max\left\{\delta^R_q, \delta^I_q, \delta^J_q, \delta^K_q\right\}, \rho_q=\max\left\{\rho^R_q, \rho^I_q, \rho^J_q, \rho^K_q\right\}. \end{align*} 3. The existence of almost automorphic solutions In this section, we shall state and prove the sufficient conditions for the existence of almost automorphic solutions of system (1.1). In view of Remark 2.7, we only need to prove that (2.3) has an almost automorphic solution. To this end, set $$\mathbb{Y}\!=\!\big \{\varphi \!=\!(\varphi _1^R, \varphi _2^R,\! \ldots , \varphi _n^R, \varphi _1^I, \varphi _2^I,\! \ldots , \varphi _n^I, \varphi _1^J, \varphi _2^J,\! \ldots , \varphi _n^J, \varphi _1^K, \varphi _2^K, \ldots , \varphi _n^K)^T \!\!\in AA(\mathbb{R}, \mathbb{R}^{4n}) \big \}$$ with the norm $$\|\varphi \|=\max _{p\in N}\{\sup _{t\in \mathbb{R}}|\varphi _p^\nu (t)|, \nu \in \Lambda \}$$, then $$\mathbb{Y}$$ is a Banach space. Theorem 3.1 Assume that $$(A_1)$$–$$(A_4)$$ hold, then system (2.3) has a unique almost automorphic solution in $$\mathbb{Y}_0=\{\varphi \in \mathbb{Y}: ||\varphi ||\leq \kappa \}$$. Proof. For any given $$\varphi \in \mathbb{Y}$$, consider the following system:   \begin{equation} \left(x_{p}^\nu\right)^{\prime}(t)=-c_p(t) x_p^\nu(t)+\Pi^\nu_p(t, \varphi(t))+I_p^\nu(t),\,\, p\in N,\nu\in \Lambda. \end{equation} (3.1)According to Lemma 2.3, the linear system   \begin{equation*} \left(x_p^\nu\right)^{\prime}(t)=-c_p(t)x_p^\nu(t),\,\, p\in N, \nu\in \Lambda \end{equation*}admits an exponential dichotomy. Then by Lemma 2.2, we obtain that system (3.1) has a unique almost automorphic solution   $$ x^\varphi=((x^\varphi)_1^R, \ldots, (x^\varphi)_n^R, (x^\varphi)_1^I, \ldots, (x^\varphi)_n^I, (x^\varphi)_1^J, \ldots, (x^\varphi)_n^J, (x^\varphi)_1^K, \ldots, (x^\varphi)_n^K)^T,$$where   \begin{equation*} (x^{\varphi})_p^l(t)=\int_{-\infty}^{t}e^{-\int_s^{t}c_p(u)\,\mathrm{d}u}\left[\Pi_p^l(s, \varphi )+I_p^l(t)\right]\,\mathrm{d}s,\quad p\in N, l\in \Lambda. \end{equation*} Define an operator $$\Phi :\mathbb{Y}\rightarrow \mathbb{Y}$$ by setting   \begin{equation*} \varphi\rightarrow x^\varphi,\quad \varphi\in \mathbb{Y}. \end{equation*} At first, we will prove that for any $$ \varphi \in \mathbb{Y}_0$$, $$\Phi \varphi \in \mathbb{Y}_0$$. For p ∈ N, we have   \begin{eqnarray*} \big|\Pi_p^R(t, \varphi(t))\big| &\leq& \sum_{q=1}^{n}\Big[\big|a_{pq}^{R}(t)\big|\big|f_{q}^R[t, \varphi]\big| +\big|a_{pq}^{I}(t)\big|\big|f_{q}^I[t, \varphi]\big|+\big|a_{pq}^{J}(t)\big|\big|f_{q}^J[t, \varphi]\big|\\[-9pt] &&+\big|a_{pq}^{K}(t)\big|\big|f_{q}^K[t, \varphi]\big|\Big] +\sum_{q=1}^{n}\Big[\big|b_{pq}^{R}(t)\big|\int_{0}^{+\infty}\big|K_{pq}(u)\big|\big|g_{q}^R[t, u, \varphi]\big|\,\mathrm{d}u\\[-5pt] &&+\big|b_{pq}^{I}(t)\big|\int_{0}^{+\infty}\big|K_{pq}(u)\big|\big|g_{q}^I[t, u, \varphi]\big|\,\mathrm{d}u+\big|b_{pq}^{J}(t)\big|\int_{0}^{+\infty}\big|K_{pq}(u)\big|\big|g_{q}^J[t, u, \varphi]\big|\,\mathrm{d}u\\[-5pt] &&+\big|b_{pq}^{K}(t)\big|\int_{0}^{+\infty}\big|K_{pq}(u)\big|\big|g_{q}^K[t, u, \varphi]\big|\,\mathrm{d}u\Big] +\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}N_q\bigg(\big|d_{pqr}^R(t)\big|\big|\tilde{h}_r^R[t, \varphi]\big|\\[-5pt] &&+\big|d_{pqr}^I(t)\big|\big|\tilde{h}_r^R[t, \varphi]\big| +\big|d_{pqr}^J(t)\big|\big|\tilde{h}_r^R[t, \varphi]\big|+\big|d_{pqr}^K(t)\big|\big|\tilde{h}_r^R[t, \varphi]\big| +\big|d_{pqr}^I(t)\big|\big|\tilde{h}_r^I[t, \varphi]\big|\\ &&+\big|d_{pqr}^R(t)\big|\big|\tilde{h}_r^I[t, \varphi]\big| +\big|d_{pqr}^K(t)\big|\big|\tilde{h}_r^I[t, \varphi]\big| +\big|d_{pqr}^J(t)\big|\big|\tilde{h}_r^I[t, \varphi]\big| +\big|d_{pqr}^J(t)\big|\big|\tilde{h}_r^J[t, \varphi]\big|\\ &&+\big|d_{pqr}^K(t)\big|\big|\tilde{h}_r^J[t, \varphi]\big| +\big|d_{pqr}^R(t)\big|\big|\tilde{h}_r^J[t, \varphi]\big| +\big|d_{pqr}^I(t)\big|\big|\tilde{h}_r^J[t, \varphi]\big| +\big|d_{pqr}^K(t)\big|\big|\tilde{h}_r^K[t, \varphi]\big|\\ &&+\big|d_{pqr}^J(t)\big|\tilde{h}_r^K[t, \varphi]\big| +\big|d_{pqr}^I(t)\big|\big|\tilde{h}_r^K[t, \varphi]\big| +\big|d_{pqr}^R(t)\big|\big|\tilde{h}_r^K[t, \varphi]\big|\bigg)\\ &\leq&\sum_{q=1}^{n}\Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+ a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big) \Big[\lambda_q^R\big|\varphi_q^R(t-\tau_{pq}(t))\big|+ \lambda_q^I\big|\varphi_q^I(t-\tau_{pq}(t))\big|\\ &&+\lambda_q^J\big|\varphi_q^J(t-\tau_{pq}(t))\big| +\lambda_q^K\big|\varphi_q^K(t-\tau_{pq}(t))\big|\Big] +\sum_{q=1}^{n}\int_{0}^{+\infty}\big|K_{pq}(u)\big|\\ &&\times\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big) \Big[\delta_q^R\big|\varphi_q^R(t-u)\big| +\delta_q^I\big|\varphi_q^I(t-u)\big|\\ &&+\delta_q^J\big|\varphi_q^J(t-u)\big| +\delta_q^K\big|\varphi_q^K(t-u)\big|\Big]\,\mathrm{d}u +\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}4N_q\Big(d_{pqr}^{R^{+}} +d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ &&\times\Big[\rho_r^R\big|\varphi_r^R(t-\xi_{pqr}(t))\big| +\rho_r^I\big|\varphi_r^I(t-\xi_{pqr}(t))\big|+\rho_r^J\big|\varphi_r^J(t-\xi_{pqr}(t))\big|\\ &&+\rho_r^K\big|\varphi_r^K(t-\xi_{pqr}(t))\big|\Big]. \end{eqnarray*}Hence,   \begin{eqnarray} \big|(\Phi\varphi)^R_p(t)\big|&=&\bigg|\int_{-\infty}^te^{-\int_s^{t}c_p(u)\,\mathrm{d}u}\Big(\Pi_p^R(s, \varphi)+I_p^R(s)\Big)\bigg|\,\mathrm{d}s\nonumber\\ &\leq&\int_{-\infty}^te^{-\int_s^{t}c_p(u)\,\mathrm{d}u}\bigg\{\bigg(\sum_{q=1}^{n} 4\lambda_q\Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\nonumber\\ &&+\sum_{q=1}^{n}4\delta_q\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big) \int_{0}^{+\infty}\big|K_{pq}(u)\big|\,\mathrm{d}u\nonumber\\ &&+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}16N_q\rho_r \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\bigg)\kappa +\big|I_p^R(s)\big|\bigg\}\mathrm{d}s\nonumber\\ &\leq& \frac{1}{c_p^{-}}\bigg(A_{p}^{\ast}\kappa+B_{p}^{\ast}\kappa +D_{p}^{\ast}\kappa+I_p^{R^{+}}\bigg)\nonumber\\ &=&\frac{\Theta_p \kappa+I_p^{R^{+}}}{c_p^{-}} \leq \kappa,\quad p\in E. \end{eqnarray} (3.2)Repeating the same calculation, from system (3.1), we obtain   \begin{equation} \big|(\Phi\varphi)^\nu_p(t)\big| \leq\frac{\Theta_p \kappa+I_{p}^{\nu^{+}}}{c_p^{-}}\leq \kappa,\quad p\in N, \,\,\nu=I,J,K. \end{equation} (3.3)Together with the above inequalities (3.2) and (3.3) and Assumption $$(A_4)$$, we obtain that $$\Phi $$ is a self mapping in $$\mathbb{Y}_0$$. Next, we will show that $$\Phi $$ is a contraction mapping in $$\mathbb{Y}_0$$. For any $$\varphi , \psi \in \mathbb{Y}_0$$ and p ∈ N, we have   \begin{align*} \big|\Pi^R_p(t, \varphi)-\Pi_p^R(t, \psi)\big| \leq&\sum_{q=1}^{n}\Big(\big|a_{pq}^{R}(t)\big|\big|f_{q}^R[t, \varphi] -f_q^R[t, \psi]\big|+\big|a_{pq}^{I}(t)\big|\big|f_{q}^I[t, \varphi]-f_q^I[t, \psi]\big|\qquad\qquad\\[-9pt] &+\big|a_{pq}^{J}(t)\big|\big|f_{q}^J[t, \varphi]-f_q^J[t, \psi]\big|+\big|a_{pq}^{K}(t)\big|\big|f_{q}^K[t, \varphi]-f_q^K[t, \psi]\big|\Big)\\[-9pt] &+\sum_{q=1}^{n}\bigg(\big|b_{pq}^{R}(t)\big|\int_0^{+\infty}\big|K_{pq}(u)\big| \big|g_{q}^R[t, u, \varphi]-g_q^R[t, u, \psi]\big|\,\mathrm{d}u\\[-9pt] &+\big|b_{pq}^{I}(t)\big|\int_0^{+\infty}\big|K_{pq}(u)\big| \big|g_{q}^I[t, u, \varphi]-g_q^I[t, u, \psi]\big|\,\mathrm{d}u\\[-2pt] &+\big|b_{pq}^{J}(t)\big|\int_0^{+\infty}\big|K_{pq}(u)\big|\big|g_{q}^J[t, u, \varphi] -g_q^J[t, u, \psi]\big|\,\mathrm{d}u\\[-2pt] &+\big|b_{pq}^{K}(t)\big|\int_0^{+\infty}\big|K_{pq}(u)\big|\big|g_{q}^K[t, u, \varphi] -g_q^K[t, u, \psi]\big|\,\mathrm{d}u\bigg) \end{align*}  \begin{eqnarray*} &&\qquad\qquad\qquad\qquad\qquad+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}\bigg[N_q\Big(\big|d_{pqr}^R(t)\big| \big|\tilde{h}_r^R[t, \varphi]-\tilde{h}_r^R[t, \psi]\big|\! +\!\big|d_{pqr}^I(t)\big|\big|\tilde{h}_r^R[t, \varphi]-\tilde{h}_r^R[t, \psi]\big|\\[-5pt] &&\qquad\qquad\qquad\qquad\qquad+\big|d_{pqr}^J(t)\big|\big|\tilde{h}_r^R[t, \varphi]-\tilde{h}_r^R[t, \psi]\big| +\big|d_{pqr}^K(t)\big|\big|\tilde{h}_r^R[t, \varphi]-\tilde{h}_r^R[t, \psi]\big| \end{eqnarray*}  \begin{eqnarray*} &&+\big|d_{pqr}^I(t)\big|\big|\tilde{h}_r^I[t, \varphi]-\tilde{h}_r^I[t, \psi]\big| +\big|d_{pqr}^R(t)\big| \big|\tilde{h}_r^I[t, \varphi]-\tilde{h}_r^I[t, \psi]\big|\\[2pt] &&+\big|d_{pqr}^K(t)\big|\big|\tilde{h}_r^I[t, \varphi]-\tilde{h}_r^I[t, \psi]\big|+\big|d_{pqr}^J(t)\big|\big|\tilde{h}_r^I[t, \varphi]-\tilde{h}_r^I[t, \psi]\big|\\ &&+\big|d_{pqr}^J(t)\big| \big|\tilde{h}_r^J[t, \varphi]-\tilde{h}_r^J[t, \psi]\big|+\big|d_{pqr}^K(t)\big|\big|\tilde{h}_r^J[t, \varphi]-\tilde{h}_r^J[t, \psi]\big|\\ &&+\big|d_{pqr}^R(t)\big|\big|\tilde{h}_r^J[t, \varphi]-\tilde{h}_r^J[t, \psi]\big| +\big|d_{pqr}^I(t)\big|\big|\tilde{h}_r^J[t, \varphi]-\tilde{h}_r^J[t, \psi]\big|\\ &&+\big|d_{pqr}^K(t)\big|\big|\tilde{h}_r^K[t, \varphi]-\tilde{h}_r^K[t, \psi]\big| +\big|d_{pqr}^J(t)\big|\big|\tilde{h}_r^K[t, \varphi]-\tilde{h}_r^K[t, \psi]\big|\\ &&+\big|d_{pqr}^I(t)\big|\big|\tilde{h}_r^K[t, \varphi]-\tilde{h}_r^K[t, \psi]\big| +\big|d_{pqr}^R(t)\big|\big|\tilde{h}_r^K[t, \varphi]-\tilde{h}_r^K[t, \psi]\big|\Big)\\[-2pt] &&+N_r\Big(\big|d_{pqr}^R(t)\big| \big|h_q^R[t, \varphi]-h_q^R[t, \psi]\big| +\big|d_{pqr}^I(t)\big|\big|h_q^R[t, \varphi]-h_q^R[t, \psi]\big|\\[-5pt] &&+\big|d_{pqr}^J(t)\big|\big|h_q^R[t, \varphi]-h_q^R[t, \psi]\big| +\big|d_{pqr}^K(t)\big|\big|h_q^R[t, \varphi]-h_q^R[t, \psi]\big|\\ &&+\big|d_{pqr}^I(t)\big|\big|h_q^I[t, \varphi]-h_q^I[t, \psi]\big| +\big|d_{pqr}^R(t)\big| \big|h_q^I[t, \varphi]-h_q^I[t, \psi]\big|\\ &&+\big|d_{pqr}^K(t)\big|\big|h_q^I[t, \varphi]-h_q^I[t, \psi]\big|+\big|d_{pqr}^J(t)\big|\big|h_q^I[t, \varphi]-h_q^I[t, \psi]\big|\\ &&+\big|d_{pqr}^J(t)\big| \big|h_q^J[t, \varphi]-h_q^J[t, \psi]\big|+\big|d_{pqr}^K(t)\big|\big|h_q^J[t, \varphi]-h_q^J[t, \psi]\big|\\ &&+\big|d_{pqr}^R(t)\big|\big|h_q^J[t, \varphi]-h_q^J[t, \psi]\big| +\big|d_{pqr}^I(t)\big|\big|h_q^J[t, \varphi]-h_q^J[t, \psi]\big|\\ &&+\big|d_{pqr}^K(t)\big|\big|h_q^K[t, \varphi]-h_q^K[t, \psi]\big| +\big|d_{pqr}^J(t)\big|\big|h_q^K[t, \varphi]-h_q^K[t, \psi]\big|\\[-2pt] &&+\big|d_{pqr}^I(t)\big|\big|h_q^K[t, \varphi]-h_q^K[t, \psi]\big| +\big|d_{pqr}^R(t)\big|\big|h_q^K[t, \varphi]-h_q^K[t, \psi]\big|\Big)\bigg]\\ &\leq&\sum_{q=1}^{n}\lambda_q\Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}} +a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big) \Big[\big|\varphi_q^R(t-\tau_{pq}(t))-\psi_q^R(t-\tau_{pq}(t))\big|\\[-4pt] &&\big|\varphi_q^I(t-\tau_{pq}(t))-\psi_q^I(t-\tau_{pq}(t))\big| +\big|\varphi_q^J(t-\tau_{pq}(t))-\psi_q^J(t-\tau_{pq}(t))\big|\\[-5pt] &&+\big|\varphi_q^K(t-\tau_{pq}(t))-\psi_q^K(t-\tau_{pq}(t))\big|\Big] +\sum_{q=1}^{n}\delta_q\int_{0}^{+\infty}\big|K_{pq}(u)\big|\\[-9pt] &&\times\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}} +b_{pq}^{K^{+}}\Big)\Big[\big|\varphi_q^R(t-u)-\psi_q^R(t-u)\big| +\big|\varphi_q^I(t-u)\\[-5pt] &&-\psi_q^I(t-u)\big|+\big|\varphi_q^J(t-u)-\psi_q^J(t-u)\big| +\big|\varphi_q^K(t-u)-\psi_q^K(t-u)\big|\Big]\,\mathrm{d}u\\[-5pt] &&+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}\bigg\{4N_q\rho_r \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \Big[\big|\varphi_r^R(t-\xi_{pqr}(t))-\psi_r^R(t-\xi_{pqr}(t))\big|\\[-5pt] &&+\big|\varphi_r^I(t-\xi_{pqr}(t))-\psi_r^I(t-\xi_{pqr}(t))\big| +\big|\varphi_r^J(t-\xi_{pqr}(t))-\psi_r^J(t-\xi_{pqr}(t))\big|\\[-3pt] &&+\big|\varphi_r^K(t-\xi_{pqr}(t))-\psi_r^K(t-\xi_{pqr}(t))\big|\Big] +4N_r\rho_q\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\[-5pt] &&\times\Big[|\varphi_q^R(t-\sigma_{pqr}(t))-\psi_q^R(t-\sigma_{pqr}(t))\big| +\big|\varphi_q^I(t-\sigma_{pqr}(t))-\psi_q^I(t-\sigma_{pqr}(t))\big|\\[-5pt] &&+\big|\varphi_q^J(t-\sigma_{pqr}(t))-\psi_q^J(t-\sigma_{pqr}(t))\big| +\big|\varphi_q^K(t-\sigma_{pqr}(t))-\psi_q^K(t-\sigma_{pqr}(t))\big|\Big]\bigg\}. \end{eqnarray*} From the above inequality, we obtain   \begin{eqnarray*} \big|(\Phi\varphi)_{p}^R(t)-(\Phi\psi)_{p}^R(t)\big| &\leq&\int_{-\infty}^{t}e^{-\int_s^{t}c_p(u)\,\mathrm{d}u} \bigg[\sum_{q=1}^{n}4\lambda_q\Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\\ &&+\sum_{q=1}^{n}4\delta_q\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big) \int_{0}^{+\infty}\big|K_{pq}(u)\big|\,\mathrm{d}u\\ &&+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}16\big(N_q\rho_r+N_r\rho_q\big) \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\bigg]\,\mathrm{d}s\|\varphi-\psi\|\\ &\leq&\frac{1}{c_p^-}\bigg(A_p^{\ast}+B_p^{\ast} +D_{p}^{\ast}+\hat{D}_{p}^{\ast}\bigg)\|\varphi-\psi\|\\ &=&\frac{\Xi_p}{c_p^-}\|\varphi-\psi\|,\quad p\in N. \end{eqnarray*} By the same way, we have   \begin{equation*} \big|(\Phi\varphi)_{p}^\nu(t)-(\Phi\psi)_{p}^\nu(t)\big|\leq\frac{\Xi_p}{c_p^-}\|\varphi-\psi\|,\quad p\in N,\,\, \nu=I, J, K. \end{equation*}Therefore, we get   \begin{equation*} \|\Phi\varphi-\Phi\psi\|\leq \mu\|\varphi-\psi\|. \end{equation*}It follows from Assumption $$(A_4)$$ that $$\Phi $$ is a contraction mapping. Based on the Banach fixed theorem, we obtain that $$\Phi $$ has a fixed point in $$\mathbb{Y}_0$$, which means that system (2.3) has a unique almost automorphic solution in $$\mathbb{Y}_0$$. This completes the proof. 4. Almost automorphic synchronization In this section, by designing a novel state-feedback controller, utilizing some analytic techniques and constructing a suitable Lyapunov function, we will investigate the exponential synchronization problem of QVHHNNs with time-varying and distributed delays and almost automorphic coefficients. For this purpose, we consider the system (1.1) as drive system, and a response system is designed as   \begin{eqnarray} Y^{\prime}_{p}(t)&=&-c_{p}(t)Y_{p}(t)+\sum_{q=1}^{n}a_{pq}(t)f_{q}(Y_{q}(t-\tau_{pq}(t))) +\sum_{q=1}^{n}b_{pq}(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}(Y_{q}(t-u))\,\mathrm{d}u\nonumber\\ &&+\sum_{q=1}^{n}\sum_{r=1}^{m}d_{pqr}(t)h_{q}(Y_{q}(t-\sigma_{pqr}(t))) h_{r}(Y_{r}(t-\sigma_{pqr}(t)))+I_{p}(t)+U_p(t), \end{eqnarray} (4.1)where p ∈ N, $$Y_p(t)=y_p^R(t)+iy_p^I(t)+jy_p^J(t)+ky_p^K(t)$$ denotes the state of the response system, $$\theta _p(t)$$ is a state-feedback controller, the rest notation is the same as those in system (1.1) and the initial condition is as follows:   \begin{equation*} Y_p(s)=\Psi_p(s),\,\, s\in(-\infty, 0],\,\, p\in N, \end{equation*}where $$\Psi _p(s)=\psi _p^R(s)+i\psi _p^I(s)+j\psi _p^J(s)+k\psi _p^K(s)$$ is the quaternion-valued bounded continuous function defined on $$(-\infty , 0]$$. Put $$z_p(t)=y_p(t)-x_p(t)$$, subtracting (1.1) from (4.1) yields the following error system:   \begin{eqnarray} Z^{\prime}_{p}(t)&=&-c_{p}(t)Z_{p}(t)+\sum_{q=1}^{n}a_{pq}(t)F_{q}(Z_{q}(t-\tau_{pq}(t))) +\sum_{q=1}^{n}b_{pq}(t)\int_{0}^{+\infty}K_{pq}(u)G_{q}(Z_{q}(t-u))\,\mathrm{d}u\nonumber\\ &&+\sum_{q=1}^{n}\sum_{r=1}^{m}d_{pqr}(t)H_{q}(Z_{q}(t-\sigma_{pqr}(t))) H_{r}(Z_{r}(t-\xi_{pqr}(t)))+U_{p}(t), \end{eqnarray} (4.2)where $$F_{q}(Z_{q}(t-\delta _{pq}(t)))=f_{q}(y_{q}(t-\delta _{pq}(t)))-f_{q}(x_{q}(t-\delta _{pq}(t)))$$, $$G_{q}(Z_{q}(t-u)=g_{q}(y_{q}(t-u)-g_{q}(x_{q}(t-u))$$, $$H_{q}(Z_{q}(t-\sigma _{pqr}(t)))H_{r}(Z_{r}(t-\xi _{pqr}(t)))=h_{q}(y_{q}(t-\sigma _{pqr}(t))) h_{r}(y_{r}(t-\xi _{pqr}(t)))-h_{q}(x_{q}(t-\sigma _{pqr}(t)))h_{r}(x_{r}(t-\xi _{pqr}(t)))$$, p ∈ N. In order to realize the almost automorphic synchronization of the drive-response system, we choose the following state-feedback controller   \begin{equation} U_{p}(t)=-\theta_{p}(t)Z_p(t)+\sum_{q=1}^{n}e_{pq}(t)l_q(Z_q(t-\vartheta_{pq}(t))),\quad p\in N, \end{equation} (4.3)where $$\theta _p,\vartheta _{pq}:\mathbb{R}\rightarrow \mathbb{R}^+, e_{pq}: \mathbb{R}\rightarrow \mathbb{Q}, l_q:\mathbb{Q}\rightarrow \mathbb{Q}$$, p, q ∈ N. Similarly, according to Hamilton rules, we can decompose system (4.2) into an equivalent real-valued system:   \begin{eqnarray*} \left(z_{p}^{R}\right)^{\prime}(t)&=&-c_{p}(t)z_{p}^{R}(t)+\sum_{q=1}^{n}\Big(a_{pq}^{R}(t)\big(f_{q}^{R}[t,y]-f_{q}^{R}[t,x]\big) -a_{pq}^{I}(t)\big(f_{q}^{I}[t,y]-f_{q}^{I}[t,x]\big)\\ &&-a_{pq}^{J}(t)\big(f_{q}^{J}[t,y]-f_{q}^{J}[t,x]\big) -a_{pq}^{K}(t)\big(f_{q}^{K}[t,y]-f_{q}^{K}[t,x]\big)\Big)\\ &&+\sum_{q=1}^{n}\bigg(b_{pq}^R(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^R[t,u,y]-g_{q}^R[t,u,x]\big)\,\mathrm{d}u\\ &&-b_{pq}^I(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^I[t,u,y]-g_{q}^I[t,u,x]\big)\,\mathrm{d}u -b_{pq}^J(t)\int_{0}^{+\infty}K_{pq}(u)\\ &&\times\big(g_{q}^J[t,u,y] -g_{q}^J[t,u,x]\big)\,\mathrm{d}u-b_{pq}^K(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^K[t, u, y]-g_{q}^K[t, u, x]\big)\,\mathrm{d}u\bigg) \end{eqnarray*}  \begin{eqnarray*} &&+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}\bigg(d_{pqr}^R(t) \big(h_q^R[t, y]\tilde{h}_r^R[t, y]-h_q^R[t, x]\tilde{h}_r^R[t, x]\big)-d_{pqr}^I(t)\big(h_q^I[t, y]\tilde{h}_r^R[t, y]\\ &&-h_q^I[t, x]\tilde{h}_r^R[t, x]\big) -d_{pqr}^J(t)\big(h_q^J[t, y]\tilde{h}_r^R[t, y]-h_q^J[t, x]\tilde{h}_r^R[t, x]\big)\\ &&-d_{pqr}^K(t)\big(h_q^K[t, y]\tilde{h}_r^R[t, y]-h_q^K[t, x]\tilde{h}_r^R[t, x]\big) -d_{pqr}^I(t)\big(h_q^R[t, y]\tilde{h}_r^I[t, y]\\ &&-h_q^R[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^R(t)\big(h_q^I[t, y]\tilde{h}_r^I[t, y]-h_q^I[t, x]\tilde{h}_r^I[t, x]\big)\\ &&+d_{pqr}^K(t)\big(h_q^J[t, y]\tilde{h}_r^I[t, y]-h_q^J[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^J(t)\big(h_q^K[t, y]\tilde{h}_r^I[t, y]\\ &&-h_q^K[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^J(t)\big(h_q^R[t, y]\tilde{h}_r^J[t, y]-h_q^R[t, x]\tilde{h}_r^J[t, x]\big)\\ &&-d_{pqr}^K(t)\big(h_q^I[t, y]\tilde{h}_r^J[t, y]-h_q^I[t, x]\tilde{h}_r^J[t, x]\big) -d_{pqr}^R(t)\big(h_q^J[t, y]\tilde{h}_r^J[t, y]\\ &&-h_q^J[t, x]\tilde{h}_r^J[t, x]\big) +d_{pqr}^I(t)\big(h_q^K[t, y]\tilde{h}_r^J[t, y]-h_q^K[t,x]\tilde{h}_r^J[t,x]\big)\\ &&-d_{pqr}^K(t)\big(h_q^R[t, y]\tilde{h}_r^K[t, y]-h_q^R[t, x]\tilde{h}_r^K[t, x]\big) +d_{pqr}^J(t)\big(h_q^I[t, y]\tilde{h}_r^K[t, y]\\ &&-h_q^I[t, x]\tilde{h}_r^K[t, x]\big) -d_{pqr}^I(t)\big(h_q^J[t, y]\tilde{h}_r^K[t, y]-h_q^J[t, x]\tilde{h}_r^K[t, x]\big)\\ &&-d_{pqr}^R(t)\big(h_q^K[t, y]\tilde{h}_r^K[t, y]-h_q^K[t, x]\tilde{h}_r^K[t, x]\big)\bigg)-\theta_p(t)z^R_p(t)\\ &&+\sum_{q=1}^{n}\Big(e_{pq}^{R}(t)l_{q}^{R}[t,z] -e_{pq}^{I}(t)l_{q}^{I}[t,z] -e_{pq}^{J}(t)l_{q}^{J}[t,z]-e_{pq}^{K}(t)l_{q}^{K}[t,z]\Big),\,\, p\in N, \end{eqnarray*}  \begin{align*} \left(z_{p}^{I}\right)^{\prime}(t)=&-c_{p}(t)z_{p}^{I}(t)+\sum_{q=1}^{n} \Big(a_{pq}^{R}(t)\big(f_{q}^{I}[t,y]-f_{q}^{I}[t,x]\big) +a_{pq}^{I}(t)\big(f_{q}^{R}[t,y]-f_{q}^{R}[t,x]\big)&\\[4pt] &+a_{pq}^{J}(t)\big(f_{q}^{K}[t,y]-f_{q}^{K}[t,x]\big) -a_{pq}^{K}(t)\big(f_{q}^{J}[t,y]-f_{q}^{J}[t,x]\big)\Big)\\[4pt] &+\sum_{q=1}^{n}\bigg(b_{pq}^R(t) \int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^I[t,u,y]-g_{q}^I[t,u,x]\big)\,\mathrm{d}u\\[3pt] &+b_{pq}^I(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^R[t,u,y]-g_{q}^R[t,u,x]\big)\,\mathrm{d}u +b_{pq}^J(t)\int_{0}^{+\infty}K_{pq}(u)\\[3pt] &\times\big(g_{q}^K[t,u,y] -g_{q}^K[t,u,x]\big)\,\mathrm{d}u-b_{pq}^K(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^J[t, u, y]-g_{q}^J[t, u, x]\big)\,\mathrm{d}u\bigg) \\[3pt] &+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n}\bigg(d_{pqr}^I(t) \big(h_q^R[t, y]\tilde{h}_r^R[t, y]-h_q^R[t, x]\tilde{h}_r^R[t, x]\big) +d_{pqr}^R(t)\big(h_q^I[t, y] \tilde{h}_r^R[t, y]\\[2pt] &-h_q^I[t, x] \tilde{h}_r^R[t,x]\big) -d_{pqr}^K(t)\big(h_q^J[t, y]\tilde{h}_r^R[t,y]-h_q^J[t, x]\tilde{h}_r^R[t,x]\big)\end{align*}  \begin{eqnarray*} &&+d_{pqr}^J(t)\big(h_q^K[t, y]\tilde{h}_r^R[t, y]-h_q^K[t, x]\tilde{h}_r^R[t, x]\big) +d_{pqr}^R(t)\big(h_q^R[t, y]\tilde{h}_r^I[t, y]\\ &&-h_q^R[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^I(t)\big(h_q^I[t, y]\tilde{h}_r^I[t,y]-h_q^I[t, x]\tilde{h}_r^I[t,x]\big)\\ &&-d_{pqr}^J(t)\big(h_q^J[t, y]\tilde{h}_r^I[t, y]-h_q^J[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^K(t)\big(h_q^K[t, y]\tilde{h}_r^I[t, y]\\ &&-h_q^K[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^K(t)\big(h_q^R[t, y]\tilde{h}_r^J[t, y]-h_q^R[t, x]\tilde{h}_r^J[t, x]\big)\\ &&+d_{pqr}^J(t)\big(h_q^I[t, y]\tilde{h}_r^J[t, y]-h_q^I[t, x]\tilde{h}_r^J[t, x]\big) -d_{pqr}^I(t)\big(h_q^J[t, y]\tilde{h}_r^J[t, y]\\ &&-h_q^J[t, x]\tilde{h}_r^J[t, x]\big) -d_{pqr}^R(t)\big(h_q^K[t, y]\tilde{h}_r^J[t, y]-h_q^K[t, x]\tilde{h}_r^J[t, x]\big)\\ &&+d_{pqr}^J(t)\big(h_q^R[t, y]\tilde{h}_r^K[t, y] -h_q^R[t, x]\tilde{h}_r^K[t, x]\big) +d_{pqr}^K(t)\big(h_q^I[t, y]\tilde{h}_r^K[t, y]\\ &&-h_q^I[t, x]\tilde{h}_r^K[t, x]\big) +d_{pqr}^R(t)\big(h_q^J[t, y]\tilde{h}_r^K[t, y]-h_q^J[t, x]\tilde{h}_r^K[t, x]\big)\\ &&-d_{pqr}^I(t)\big(h_q^K[t, y]\tilde{h}_r^K[t, y]-h_q^K[t, x]\tilde{h}_r^K[t, x]\big)\bigg)-\theta_{p}(t)z_{p}^{I}(t)\\[-9pt] &&+\sum_{q=1}^{n}\Big(e_{pq}^{R}(t)l_{q}^{I}[t,z] +e_{pq}^{I}(t)l_{q}^{R}[t,z] +e_{pq}^{J}(t)l_{q}^{K}[t,z]-e_{pq}^{K}(t)l_{q}^{J}[t,z]\Big),\,\, p\in N, \end{eqnarray*}  \begin{align*} \left(z_{p}^{J}\right)^{\prime}(t)=&-c_{p}(t)z_{p}^{J}(t)+\sum_{q=1}^{n} \Big(a_{pq}^{R}(t)\big(f_{q}^{J}[t,y]-f_{q}^{J}[t,x]\big) +a_{pq}^{J}(t)\big(f_{q}^{R}[t,y]-f_{q}^{R}[t,x]\big)\\[4pt] &-a_{pq}^{I}(t)\big(f_{q}^{K}[t,y]-f_{q}^{K}[t,x]\big) +a_{pq}^{K}(t)\big(f_{q}^{I}[t,y]-f_{q}^{I}[t,x]\big)\Big)\\[4pt] &+\sum_{q=1}^{n}\bigg(b_{pq}^R(t) \int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^J[t,u,y]-g_{q}^J[t,u,x]\big)\,\mathrm{d}u\\[4pt] &+\,b_{pq}^J(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^R[t,u,y]-g_{q}^R[t,u,x]\big)\,\mathrm{d}u -b_{pq}^I(t)\int_{0}^{+\infty}K_{pq}(u)\\[4pt] &\times\big(g_{q}^K[t,u,y]-g_{q}^K[t,u,x]\big)\,\mathrm{d}u +b_{pq}^K(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^I[t, u, y]-g_{q}^I[t, u, x]\big)\,\mathrm{d}u\bigg)\\[4pt] &+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n} \bigg(d_{pqr}^J(t)\big(h_q^R[t, y]\tilde{h}_r^R[t,y]-h_q^R[t, x]\tilde{h}_r^R[t, x]\big) +d_{pqr}^K(t)\big(h_q^I[t, y]\tilde{h}_r^R[t, y]\\[6pt] &-h_q^I[t, x]\tilde{h}_r^R[t, x]\big) +d_{pqr}^R(t)\big(h_q^J[t, y]\tilde{h}_r^R[t, y]-h_q^J[t, x]\tilde{h}_r^R[t, x]\big)\\[6pt] &-d_{pqr}^I(t)\big(h_q^K[t, y]\tilde{h}_r^R[t, y]-h_q^K[t, x]\tilde{h}_r^R[t, x]\big) -d_{pqr}^K(t)\big(h_q^R[t, y]\tilde{h}_r^I[t, y]\\[6pt] &-h_q^R[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^J(t)\big(h_q^I[t, y]\tilde{h}_r^I[t, y]-h_q^I[t, x]\tilde{h}_r^I[t, x]\big) \end{align*}  \begin{eqnarray*} &&+d_{pqr}^I(t)\big(h_q^J[t, y]\tilde{h}_r^I[t, y]-h_q^J[t, x]\tilde{h}_r^I[t, x]\big) +d_{pqr}^R(t)\big(h_q^K[t, y]\tilde{h}_r^I[t, y]\\ &&-h_q^K[t, x]\tilde{h}_r^I[t, x]\big) +d_{pqr}^R(t)\big(h_q^R[t, y]\tilde{h}_r^J[t, y]-h_q^R[t, x]\tilde{h}_r^J[t, x]\big)\\ &&-d_{pqr}^I(t)\big(h_q^I[t, y]\tilde{h}_r^J[t, y]-h_q^I[t, x]\tilde{h}_r^J[t, x]\big) -d_{pqr}^J(t)\big(h_q^J[t, y]\tilde{h}_r^J[t, y]\\ &&-h_q^J[t, x]\tilde{h}_r^J[t, x]\big) -d_{pqr}^K(t)\big(h_q^K[t, y]\tilde{h}_r^J[t, y]-h_q^K[t, x]\tilde{h}_r^J[t, x]\big)\\ &&-d_{pqr}^I(t)\big(h_q^R[t, y]\tilde{h}_r^K[t, y]-h_q^R[t, x]\tilde{h}_r^K[t, x]\big) -d_{pqr}^R(t)\big(h_q^I[t,y]\tilde{h}_r^K[t,y]\\ &&-h_q^I[t, x]\tilde{h}_r^K[t, x]\big) +d_{pqr}^K(t)\big(h_q^J[t, y]\tilde{h}_r^K[t, y]-h_q^J[t, x]\tilde{h}_r^K[t, x]\big)\\ &&-d_{pqr}^J(t)\big(h_q^K[t, y]\tilde{h}_r^K[t, y]-h_q^K[t, x]\tilde{h}_r^K[t, x]\big)\bigg) -\theta_{p}(t)z_{p}^{J}(t)\\ &&+\sum_{q=1}^{n}\Big(e_{pq}^{R}(t)l_{q}^{J}[t,z] +e_{pq}^{J}(t)l_{q}^{R}[t,z] -e_{pq}^{I}(t)l_{q}^{K}[t,z]+e_{pq}^{K}(t)l_{q}^{I}[t,z]\Big),\,\, p\in N, \end{eqnarray*}  \begin{align*} \left(z_{p}^{K}\right)^{\prime}(t)=&-c_{p}(t)z_{p}^{K}(t)+\sum_{q=1}^{n} \Big(a_{pq}^{R}(t)\big(f_{q}^{K}[t,y]-f_{q}^{K}[t,x]\big) +a_{pq}^{K}(t)\big(f_{q}^{R}[t,y]-f_{q}^{R}[t,x]\big)&\\ &+a_{pq}^{I}(t)\big(f_{q}^{J}[t,y]-f_{q}^{J}[t,x]\big) -a_{pq}^{J}(t)\big(f_{q}^{I}[t,y]-f_{q}^{I}[t,x]\big)\Big)\\ &+\sum_{q=1}^{n}\bigg(b_{pq}^R(t) \int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^K[t,u,y]-g_{q}^K[t,u,x]\big)\,\mathrm{d}u\\ &+b_{pq}^K(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^R[t,u,y]-g_{q}^R[t,u,x]\big)\,\mathrm{d}u +b_{pq}^I(t)\int_{0}^{+\infty}K_{pq}(u)\\ &\times\big(g_{q}^J[t,u,y]-g_{q}^J[t,u,x]\big)\,\mathrm{d}u -b_{pq}^J(t)\int_{0}^{+\infty}K_{pq}(u)\big(g_{q}^I[t, u, y]-g_{q}^I[t, u, x]\big)\,\mathrm{d}u\bigg) \end{align*}  \begin{eqnarray*} &&+\sum\limits_{q=1}^{n}\sum\limits_{r=1}^{n} \bigg(d_{pqr}^K(t)\big(h_q^R[t, y]\tilde{h}_r^R[t, y]-h_q^R[t, x]\tilde{h}_r^R[t, x]\big) -d_{pqr}^I(t)\big(h_q^I[t, y] \tilde{h}_r^R[t, y]\\ &&-h_q^I[t, x] \tilde{h}_r^R[t, x]\big) +d_{pqr}^I(t)\big(h_q^J[t, y]\tilde{h}_r^R[t, y]-h_q^J[t, x]\tilde{h}_r^R[t, x]\big)\\ &&+d_{pqr}^R(t)\big(h_q^K[t, y]\tilde{h}_r^R[t, y]-h_q^K[t, x]\tilde{h}_r^R[t, x]\big) -d_{pqr}^J(t)\big(h_q^R[t, y]\tilde{h}_r^I[t, y]\\ &&-h_q^R[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^K(t)\big(h_q^I[t, y]\tilde{h}_r^I[t, y]-h_q^I[t, x]\tilde{h}_r^I[t, x]\big)\\ &&-d_{pqr}^R(t)\big(h_q^J[t, y]\tilde{h}_r^I[t, y]-h_q^J[t, x]\tilde{h}_r^I[t, x]\big) -d_{pqr}^I(t)\big(h_q^K[t, y]\tilde{h}_r^I[t, y]\\ &&-h_q^K[t, x]\tilde{h}_r^I[t, x]\big) +d_{pqr}^I(t)\big(h_q^R[t, y]\tilde{h}_r^J[t, y]-h_q^R[t, x]\tilde{h}_r^J[t, x]\big)\\ &&+d_{pqr}^R(t)\big(h_q^I[t, y]\tilde{h}_r^J[t, y]-h_q^I[t, x]\tilde{h}_r^J[t, x]\big) -d_{pqr}^K(t)\big(h_q^J[t, y]\tilde{h}_r^J[t, y] \end{eqnarray*}  \begin{eqnarray*} &&-h_q^J[t, x]\tilde{h}_r^J[t, x]\big) +d_{pqr}^J(t)\big(h_q^K[t, y]\tilde{h}_r^J[t, y]-h_q^K[t, x]\tilde{h}_r^J[t, x]\big)\\ &&-d_{pqr}^R(t)\big(h_q^R[t, y]\tilde{h}_r^K[t, y]-h_q^R[t, x]\tilde{h}_r^K[t, x]\big) -d_{pqr}^I(t)\big(h_q^I[t, y]\tilde{h}_r^K[t, y]\\ &&-h_q^I[t, x]\tilde{h}_r^K[t, x]\big) -d_{pqr}^J(t)\big(h_q^J[t, y]\tilde{h}_r^K[t, y]-h_q^J[t, x]\tilde{h}_r^K[t, x]\big)\\ &&-d_{pqr}^J(t)\big(h_q^K[t, y]\tilde{h}_r^K[t, y]-h_q^K[t, x]\tilde{h}_r^K[t, x]\big)\bigg) -\theta_{p}(t)z_{p}^{K}(t)\\ &&+\sum_{q=1}^{n}\Big(e_{pq}^{R}(t)l_{q}^{K}[t,z] +e_{pq}^{K}(t)l_{q}^{R}[t,z] +e_{pq}^{I}(t)l_{q}^{J}[t,z]-e_{pq}^{J}(t)l_{q}^{I}[t,z]\Big),\,\, p\in N, \end{eqnarray*} where $$l_q^\nu [t, z]\triangleq l_q^\nu \big (z_q^R(t-\vartheta _{pq}(t)), z_q^I(t-\vartheta _{pq}(t)), z_q^J(t-\vartheta _{pq}(t)), z_q^K(t-\vartheta _{pq}(t))\big ), \nu \in \Lambda $$. Definition 4.1 The response system (4.1) and the drive system (1.1) can be globally, exponentially synchronized if there exist positive constants M and $$\omega $$ such that   \begin{equation*} \|y(t)-x(t)\|_0\leq M\|\psi-\varphi\|e^{-\omega t}, \end{equation*}where $$x=(x_1^R,\, x_2^R, \ldots , x_n^R\,, x_1^I\,, x_2^I, \ldots , x_n^I,\, x_1^J,\, x_2^J, \ldots , x_n^J,\, x_1^K,\, x_2^K, \ldots , x_n^K)$$ and $$y=(y_1^R, y_2^R, \ldots , y_n^R, y_1^I, y_2^I,\\ \ldots , y_n^I, y_1^J, y_2^J, \ldots , y_n^J, y_1^K, y_2^K, \ldots , y_n^K)$$ are solutions of the equivalent real-valued systems of (1.1) and (4.1) with initial values $$\varphi $$ and $$\psi $$, respectively, $$\|y(t)-x(t)\|_0=\max \limits _{p\in N,\nu \in \Lambda }\{|y_p^\nu (t)-x_p^\nu (t)|\},\|\psi -\varphi \|= \max \limits _{p\in N,\nu \in \Lambda }\Big \{\sup \limits _{t\in \mathbb{R}}|\psi _p^\nu (t)-\varphi _p^\nu (t)|\Big \}$$. Theorem 4.1 Let $$(A_1)$$–$$(A_4)$$ hold. Suppose further that: $$(A_2)$$ Function $$\theta _p\in AA(\mathbb{R}, \mathbb{R}^+), e_{pq}\in AA(\mathbb{R}, \mathbb{Q}), \vartheta _{pq}\in AA(\mathbb{R}, \mathbb{R}^+)\cap C^1(\mathbb{R},\mathbb{R})$$ with $$\inf \limits _{t\in R}\{1-\vartheta _{pq}^{\prime}(t)\}:=\varsigma _{pq}>0, p,q\in N$$. $$(A_2)$$ Function $$l_q \in C(\mathbb{Q}, \mathbb{Q})$$ and for any $$u^\nu , v^\nu \in \mathbb{R}$$, there exist positive constants $$\eta _{q}^{\nu }$$ such that   \begin{equation*} \left|l_q^\nu(u^R, u^I, u^J, u^K)-l_q^\nu(v^R, v^I, v^J, v^K)\right| \leq\eta_q^{R}|u^R-v^R|+\eta_q^{I}|u^I-v^I|+ \eta_q^{J}|u^J-v^J|+\eta_q^{K}|u^K-v^K| \end{equation*}and $$l_q^\nu (0, 0, 0, 0)=0$$, where $$\nu \in \Lambda $$, q ∈ N. $$(A_5)$$ For p ∈ N, there exists a positive constant $$\omega $$ such that   \begin{eqnarray*} &&\sum\limits_{p=1}^n\bigg\{\big(\omega-c_p^{-}-\theta_{p}^{-}\big) +B^{\ast}_p+\sum\limits_{q=1}^n\frac{4\lambda_qe^{\omega\tau^+}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\nonumber\\&& +\sum\limits_{q=1}^n\sum\limits_{r=1}^n\bigg( \frac{16N_r \rho_qe^{\omega \sigma^+}}{\beta_{pqr}}+\frac{16N_q \rho_r e^{\omega \xi^+}}{\gamma_{pqr}}\bigg)\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \nonumber\\&& \sum\limits_{q=1}^n\frac{4\eta_{q}e^{\omega\vartheta^+}}{\delta_{pq}}\Big(e_{pq}^{R^{+}} +e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{k^{+}}\Big)\bigg\} <0, \end{eqnarray*}where $$\eta _q=\max \big \{\eta ^R_q, \eta ^I_q, \eta ^J_q, \eta ^K_q\big \}$$.Then the drive system (1.1) and response system (4.1) are globally, exponentially synchronized. Proof. In view of the error system (4.2), for any t > 0, $$\nu \in \Lambda $$, we have   \begin{eqnarray*} D^{+}\big|z_p^\nu(t)\big| &\leq&-\big(c_p^- +\theta_{p}^{-}\big)\big|z_p^\nu(t)\big| +\sum\limits_{q=1}^n\lambda_{q}\Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{k^{+}}\Big) \Big[\big|z_q^R(t-\tau_{pq}(t))\big|\\ &&+\big|z_q^I(t-\tau_{pq}(t))\big| +\big|z_q^J(t-\tau_{pq}(t))\big|+\big|z_q^K(t-\tau_{pq}(t))\big|\Big]\\ &&+\sum\limits_{q=1}^n\int_0^\infty\big| K_{pq}(u)\big| \delta_{q}\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big) \Big[\big|z_q^R(t-u)\big|+\big|z_q^I(t-u)\big|\\ &&+\big|z_q^J(t-u)\big| +\big|z_q^K(t-u)\big|\Big]\,\mathrm{d}u +\sum\limits_{q=1}^n\sum\limits_{r=1}^n4N_q \rho_r\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ &&\times\Big[\big|z_r^R(t-\xi_{pqr}(t))\big|+\big|z_r^I(t-\xi_{pqr}(t))\big| +\big|z_r^J(t-\xi_{pqr}(t))\big|\\ &&+\big|z_r^K(t-\xi_{pqr}(t))\big|\Big] +\sum\limits_{q=1}^n\sum\limits_{r=1}^n4N_r\rho_q\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}} +d_{pqr}^{K^{+}}\Big)\\ &&\times\Big[\big|z_q^R(t-\sigma_{pqr}(t))\big| +\big|z_q^I(t-\sigma_{pqr}(t))\big| +\big|z_q^J(t-\sigma_{pqr}(t))\big|\\ &&+\big|z_q^K(t-\sigma_{pqr}(t))\big|\Big] +\sum\limits_{q=1}^n\eta_{q}\Big(e_{pq}^{R^{+}}+e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{K^{+}}\Big) \Big[\big|z_q^R(t-\vartheta_{pq}(t))\big|\\ &&+\big|z_q^I(t-\vartheta_{pq}(t))\big| +\big|z_q^J(t-\vartheta_{pq}(t))\big|+\big|z_q^K(t-\vartheta_{pq}(t))\big|\Big]. \end{eqnarray*} We consider the Lyapunov function as follows:   \begin{equation*} V(t)=V^R(t)+V^I(t)+V^J(t)+V^K(t), \end{equation*}where $$V^\nu (t)=\sum \nolimits _{p=1}^{n} ( |z_{p}^{\nu }(t) |e^{\omega t}+\Gamma _{p} )$$, $$\nu \in \Lambda $$ and   \begin{eqnarray*} \Gamma_{p}&=&\sum\limits_{q=1}^n\frac{\lambda_qe^{\omega\tau^{+}}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\int_{t-\tau_{pq}(t)}^{t} \Big[\big|z_q^R(s)\big|+\big|z_q^I(s)\big| +\big|z_q^J(s)\big|\\ &&+\big|z_q^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s+\sum\limits_{q=1}^n\delta_q \Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big)\int_0^{\infty}\big|K_{pq}(u)\big|e^{\omega u}\int_{t-u}^{t}\Big[\big|z_q^R(s)\big| \end{eqnarray*}  \begin{eqnarray*} &&+\big|z_q^I(s)\big|+\big|z_q^J(s)\big|+\big|z_q^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s\,\mathrm{d}u +\sum\limits_{q=1}^n\sum\limits_{r=1}^n \frac{4N_r \rho_qe^{\omega \sigma^+}}{\beta_{pqr}}\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}\\ &&+d_{pqr}^{K^{+}}\Big) \int_{t-\sigma_{pqr}(t)}^t \Big[\big|z_q^R(s)\big|+\big|z_q^I(s)\big|+\big|z_q^J(s)\big| +\big|z_q^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s\\ &&+\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{4N_q \rho_re^{\omega \xi^+}}{\gamma_{pqr}} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \int_{t-\xi_{pqr}(t)}^t\Big[\big|z_r^R(s)\big|+\big|z_r^I(s)\big|\\ &&+\big|z_r^J(s)\big|+\big|z_r^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s +\sum\limits_{q=1}^n\frac{\eta_qe^{\omega\vartheta^{+}}}{\delta_{pq}} \Big(e_{pq}^{R^{+}}+e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{K^{+}}\Big) \int_{t-\vartheta_{pq}(t)}^{t} \Big[\big|z_q^R(s)\big|\\ &&+\big|z_q^I(s)\big| +\big|z_q^J(s)\big|+\big|z_q^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s,\quad p\in N. \end{eqnarray*} Computing the derivative of V (t) along the solutions of the error system (4.2), we can get   \begin{eqnarray} D^{+}V^R(t) &=&\sum\limits_{p=1}^n\bigg\{\omega e^{\omega t}\big|z_p^R(t)\big|+e^{\omega t}D^+\big|z_p^R(t)\big| +\sum\limits_{q=1}^n\frac{\lambda_qe^{\omega\tau^+}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\nonumber\\ &&\times\Big[\big|z_q^R(t)\big|+\big|z_q^I(t)\big|+\big|z_q^J(t)\big|+\big|z_q^K(t)\big|\Big]e^{\omega t}-\sum\limits_{q=1}^n\frac{\lambda_qe^{\omega\tau^+}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\nonumber\\ &&\times\Big[\big|z_q^R(t-\tau_{pq}(t))\big| +\big|z_q^I(t-\tau_{pq}(t))\big|+\big|z_q^J(t-\tau_{pq}(t))\big| +\big|z_q^K(t-\tau_{pq}(t))\big|\Big]\nonumber\\ &&\times\big(1-\tau_{pq}^{\prime}(t)\big)e^{\omega (t-\tau_{pq}(t))} +\sum\limits_{q=1}^n\delta_q\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big) \int_0^\infty \big|K_{pq}(u)\big|e^{\omega u}\nonumber\\ &&\times\Big[\big|z_q^R(t)\big| +\big|z_q^I(t)\big|+\big|z_q^J(t)\big|+\big|z_q^K(t)\big|\Big]e^{\omega t}\,\mathrm{d}u -\sum\limits_{q=1}^n\delta_q\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big) \nonumber\\ &&\times\int_0^\infty \big|K_{pq}(u)\big|e^{\omega u} \Big[\big|z_q^R(t-u)\big|+\big|z_q^I(t-u)\big|+\big|z_q^J(t-u)\big|\nonumber\\ &&+\big|z_q^K(t-u)\big|\Big]e^{\omega (t-u)}\,\mathrm{d}u +\sum\limits_{q=1}^n\sum\limits_{r=1}^n \frac{4N_r \rho_q e^{\omega \sigma^+}}{\beta_{pqr}}\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\nonumber\\ &&\Big[\big|z_q^R(t)\big|+\big|z_q^I(t)\big|+\big|z_q^J(t)\big| +\big|z_q^K(t)\big|\Big]e^{\omega t}-\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{4N_r \rho_qe^{\omega \sigma^+}}{\beta_{pqr}} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}\nonumber\\ &&+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \Big[\big|z_q^R(t-\sigma_{pqr}(t))\big| +\big|z_q^I(t-\sigma_{pqr}(t))\big|+\big|z_q^J(t-\sigma_{pqr}(t))\big|\nonumber\\ &&+\big|z_q^K(t-\sigma_{pqr}(t))\big|\Big]\big(1-\sigma_{pqr}^{\prime}(t)\big)e^{\omega (t-\sigma_{pqr}(t))} +\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{4N_q \rho_r e^{\omega \xi^+}}{\gamma_{pqr}}\nonumber\end{eqnarray}  \begin{eqnarray*} &&\times\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \Big[\big|z_r^R(t)\big|+\big|z_r^I(t)\big|+\big|z_r^J(t)\big| +\big|z_r^K(t)\big|\Big]e^{\omega t}\nonumber\\[2pt] &&-\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{4N_q \rho_r e^{\omega \xi^+}}{\gamma_{pqr}} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \Big[\big|z_r^R(t-\xi_{pqr}(t))\big|\nonumber\\[2pt] &&+\big|z_r^I(t-\xi_{pqr}(t))\big|+\big|z_r^J(t-\xi_{pqr}(t))\big| +\big|z_r^K(t-\xi_{pqr}(t))\big|\Big]\nonumber\\[2pt] &&\times\big(1-\xi_{pqr}^{\prime}(t)\big)e^{\omega (t-\xi_{pqr}(t))} +\sum_{q=1}^{n}\frac{\eta_{q}e^{\omega\vartheta^+}}{\delta_{pq}} \Big(e_{pq}^{R^{+}}+e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{K^{+}}\Big)\nonumber\\[2pt] &&\times\Big[\big|z_{q}^{R}(t)\big|+\big|z_{q}^{I}(t)\big|+\big|z_{q}^{J}(t)\big| +\big|z_{q}^{K}(t)\big|\Big]e^{\omega t}-\sum_{q=1}^{n}\frac{\eta_{q}e^{\omega\vartheta^+}}{\delta_{pq}} \Big(e_{pq}^{R^{+}}+e_{pq}^{I^{+}}\nonumber\\[2pt] &&+e_{pq}^{J^{+}}+e_{pq}^{K^{+}}\Big) \Big[\big|z_{q}^{R}(t-\vartheta_{pq}(t))\big| +\big|z_{q}^{I}(t-\vartheta_{pq}(t))\big| +\big|z_{q}^{J}(t-\vartheta_{pq}(t))\big|\nonumber\\ &&+\big|z_{q}^{K}(t-\vartheta_{pq}(t))\big|\Big] e^{\omega(t-\vartheta_{pq}(t))}(1-\vartheta_{pq}^{\prime}(t))\Big]\bigg\}\nonumber\\[2pt] &\leq&\sum\limits_{p=1}^n\bigg\{\big(\omega-c_p^{-}-\theta_p^{-}\big)e^{\omega t}\big|z_p^R(t)\big| +e^{\omega t}\bigg(\sum\limits_{q=1}^n\frac{\lambda_qe^{\omega\tau^+}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\nonumber\\[2pt] &&\times\Big[\big|z_p^R(t)\big| +\big|z_p^I(t)\big|+\big|z_p^J(t)\big|+\big|z_p^K(t)\big|\Big] +\sum\limits_{q=1}^n\delta_q\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big)\nonumber\\ &&\times\int_0^\infty \big|K_{pq}(u)\big|e^{u}\Big[\big|z_q^R(t)\big| +\big|z_q^I(t)\big|+\big|z_q^J(t)\big| +\big|z_q^K(t)\big|\Big]\,\mathrm{d}u\nonumber\\[2pt] &&+\sum\limits_{q=1}^n\sum\limits_{r=1}^n \frac{4N_r \rho_qe^{\omega \sigma^+}}{\beta_{pqr}}\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \Big[\big|z_q^R(t)\big|+\big|z_q^I(t)\big|\nonumber\\[2pt] &&+\big|z_q^J(t)\big|+\big|z_q^K(t)\big|\Big] +\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{4N_q \rho_r e^{\omega \xi^+}}{\gamma_{pqr}} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\nonumber\\[2pt] &&\times\Big[\big|z_r^R(t)\big|+\big|z_r^I(t)\big|+\big|z_r^J(t)\big| +\big|z_r^K(t)\big|\Big]+ \sum\limits_{q=1}^n\frac{\eta_{q}e^{\omega\vartheta^+}}{\delta_{pq}}\Big(e_{pq}^{R^{+}} +e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{k^{+}}\Big)\nonumber\\[2pt] &&\times\Big[\big|z_q^R(t)\big| +\big|z_q^I(t)\big| +\big|z_q^J(t)\big|+\big|z_q^K(t)\big|\Big]\bigg)\bigg\}\nonumber\\[2pt] &\leq&e^{\omega t}\|z(t)\|_{0}\sum\limits_{p=1}^n\bigg\{\big(\omega-c_p^{-}-\theta_{p}^{-}\big) +B^{\ast}_p+\sum\limits_{q=1}^n\frac{4\lambda_qe^{\omega\tau^+}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\nonumber \end{eqnarray*}  \begin{eqnarray*} && +\sum\limits_{q=1}^n\sum\limits_{r=1}^n\bigg( \frac{16N_r \rho_qe^{\omega \sigma^+}}{\beta_{pqr}}+\frac{16N_q \rho_r e^{\omega \xi^+}}{\gamma_{pqr}}\bigg)\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \nonumber\\ && \sum\limits_{q=1}^n\frac{4\eta_{q}e^{\omega\vartheta^+}}{\delta_{pq}}\Big(e_{pq}^{R^{+}} +e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{k^{+}}\Big)\bigg\}. \end{eqnarray*} Repeating the similar calculation, for $$\nu =I,J,K$$, we can obtain   \begin{eqnarray} D^{+}V^\nu(t) &\leq&e^{\omega t}\|z(t)\|_{0}\sum\limits_{p=1}^n\bigg\{\big(\omega-c_p^{-}-\theta_{p}^{-}\big) +B^{\ast}_p+\sum\limits_{q=1}^n\frac{4\lambda_qe^{\omega\tau^+}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\nonumber\\ && +\sum\limits_{q=1}^n\sum\limits_{r=1}^n\bigg( \frac{16N_r \rho_qe^{\omega \sigma^+}}{\beta_{pqr}}+\frac{16N_q \rho_r e^{\omega \xi^+}}{\gamma_{pqr}}\bigg)\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \nonumber\\ && \sum\limits_{q=1}^n\frac{4\eta_{q}e^{\omega\vartheta^+}}{\delta_{pq}}\Big(e_{pq}^{R^{+}} +e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{k^{+}}\Big)\bigg\}. \end{eqnarray} (4.5)From $$(A_{5})$$, (4.4) and (4.5) it follows that   \begin{equation*} V^{\prime}(t)\leq0, \end{equation*}which implies that V (t) ≤ V (0) for all t ≥ 0. On the other hand, we have   \begin{eqnarray*} V^{R}(0)&\leq&\sum_{p=1}^{n}\bigg\{\big|z_{p}^{R}(0)\big| +\sum\limits_{q=1}^n\frac{\lambda_qe^{\omega\tau^{+}}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\int_{-\tau_{pq}(0)}^{0} \Big[\big|z_q^R(s)\big|\\ &&+\big|z_q^I(s)\big|+\big|z_q^J(s)\big| +\big|z_q^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s +\sum\limits_{q=1}^n\delta_q\Big(b_{pq}^{R^{+}}+b_{pq}^{I^{+}}+b_{pq}^{J^{+}}+b_{pq}^{K^{+}}\Big)\\ &&\times\int_0^{\infty}\big|K_{pq}(u)\big|e^{\omega u}\int_{-u}^{0}\Big[\big|z_q^R(s)\big| +\big|z_q^I(s)\big|+\big|z_q^J(s)\big|+\big|z_q^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s\,\mathrm{d}u\\ &&+\sum\limits_{q=1}^n\sum\limits_{r=1}^n \frac{4N_r \rho_qe^{\omega \sigma^+}}{\beta_{pqr}}\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \int_{-\sigma_{pqr}(0)}^0 \Big[\big|z_q^R(s)\big|+\big|z_q^I(s)\big|\\ &&+\big|z_q^J(s)\big| +\big|z_q^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s+\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{4N_q \rho_re^{\omega \xi^+}}{\gamma_{pqr}} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ &&\times\int_{-\xi_{pqr}(0)}^0\Big[\big|z_r^R(s)\big|+\big|z_r^I(s)\big|+\big|z_r^J(s)\big| +\big|z_r^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s +\sum\limits_{q=1}^n\frac{\eta_qe^{\omega\vartheta^{+}}}{\delta_{pq}} \Big(e_{pq}^{R^{+}}\\ &&+e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{K^{+}}\Big) \int_{-\vartheta_{pq}(0)}^{0}\Big[\big|z_q^R(s)\big|+\big|z_q^I(s)\big|+\big|z_q^J(s)\big| +\big|z_q^K(s)\big|\Big]e^{\omega s}\,\mathrm{d}s\bigg\} \end{eqnarray*}  \begin{eqnarray*} &\leq&\sum_{p=1}^{n}\bigg\{1 +\sum\limits_{q=1}^n\frac{4\lambda_q(e^{\omega\tau^{+}}-1)}{\omega\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big) +\frac{1}{\omega}B^{\ast}_p\\ &&+\sum\limits_{q=1}^n\sum\limits_{r=1}^n \frac{16N_r \rho_q(e^{\omega \sigma^+}-1)}{\omega\beta_{pqr}}\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ && +\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{16N_q (\rho_re^{\omega \xi^+}-1)}{\omega\gamma_{pqr}} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ && +\sum\limits_{q=1}^n\frac{4\eta_q(e^{\omega\vartheta^{+}}-1)}{\omega\delta_{pq}} \Big(e_{pq}^{R^{+}}+e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{K^{+}}\Big)\bigg\} \|\psi-\varphi\|. \end{eqnarray*} Similarly, for $$\nu =I,J,K$$, we can get   \begin{eqnarray*} V^{\nu}(0)&\leq&\sum_{p=1}^{n}\bigg\{1 +\sum\limits_{q=1}^n\frac{4\lambda_q(e^{\omega\tau^{+}}-1)}{\omega\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big) +\frac{1}{\omega}B^{\ast}_p\\ &&+\sum\limits_{q=1}^n\sum\limits_{r=1}^n \frac{16N_r \rho_q(e^{\omega \sigma^+}-1)}{\omega\beta_{pqr}}\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ && +\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{16N_q (\rho_re^{\omega \xi^+}-1)}{\omega\gamma_{pqr}} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ && +\sum\limits_{q=1}^n\frac{4\eta_q(e^{\omega\vartheta^{+}}-1)}{\omega\delta_{pq}} \Big(e_{pq}^{R^{+}}+e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{K^{+}}\Big)\bigg\}\|\psi-\varphi\|. \end{eqnarray*} It is obvious that $$\|y(t)-x(t)\|_{0}e^{\omega t}=\|z(t)\|_{0}e^{\omega t}\leq V(t), t\geq 0$$, hence, we have   \begin{equation*} \|y(t)-x(t)\|_{0}\leq V(t)e^{-\omega t}\leq V(0)e^{-\omega t}\leq M\|\psi-\varphi\|e^{-\omega t},\quad t\geq 0, \end{equation*} where   \begin{eqnarray*} M&=&\sum_{p=1}^{n}\bigg\{1 +\sum\limits_{q=1}^n\frac{4\lambda_q(e^{\omega\tau^{+}}-1)}{\omega\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big) +\frac{1}{\omega}B^{\ast}_p\\ &&+\sum\limits_{q=1}^n\sum\limits_{r=1}^n \frac{16N_r \rho_q(e^{\omega \sigma^+}-1)}{\omega\beta_{pqr}}\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ && +\sum\limits_{q=1}^n\sum\limits_{r=1}^n\frac{16N_q (\rho_re^{\omega \xi^+}-1)}{\omega\gamma_{pqr}} \Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big)\\ && +\sum\limits_{q=1}^n\frac{4\eta_q(e^{\omega\vartheta^{+}}-1)}{\omega\delta_{pq}} \Big(e_{pq}^{R^{+}}+e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{K^{+}}\Big)\bigg\}\gt 0. \end{eqnarray*} Therefore, the drive system (1.1) and the response system (4.1) are almost automorphic global exponentially synchronization. The proof is completed. Remark 4.4 From the proofs of Theorems 3.1 and 4.1, one can easily see that if all the coefficients of (1.1) are periodic and almost periodic, respectively, then, similar to the proofs of Theorems 3.1 and 4.1 and under the same corresponding conditions, one can show that the similar results of Theorems 3.1 and 4.1 are still valid for both cases of the periodicity and almost periodicity. 5. A numerical example In this section, an example was given to show the effectiveness of the result obtained in this paper. Example 5.1 Consider the following QVHHNN as the drive system   \begin{eqnarray} x^{\prime}_{p}(t)&=&-c_{p}(t)x_{p}(t)+\sum_{q=1}^{2}a_{pq}(t)f_{q}(x_{q}(t-\tau_{pq}(t))) +\sum_{q=1}^{2}b_{pq}(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}(x_{q}(t-u))\,\mathrm{d}u\nonumber\\ &&+\sum_{q=1}^{2}\sum_{r=1}^{2}d_{pqr}(t)h_{q}(x_{q}(t-\sigma_{pqr}(t)))h_{r}(x_{r}(t-\xi_{pqr}(t))) +I_{p}(t),\,\,t\in \mathbb{R} \end{eqnarray} (5.1)and the corresponding response system is given by   \begin{eqnarray} y^{\prime}_{p}(t)&=&-c_{p}(t)y_{p}(t)+\sum_{q=1}^{2}a_{pq}(t)f_{q}(y_{q}(t-\tau_{pq}(t))) +\sum_{q=1}^{2}b_{pq}(t)\int_{0}^{+\infty}K_{pq}(u)g_{q}(y_{q}(t-u))\,\mathrm{d}u\nonumber\\ &&+\sum_{q=1}^{2}\sum_{r=1}^{2}d_{pqr}(t)h_{q}(y_{q}(t-\sigma_{pqr}(t))) h_{r}(y_{r}(t-\sigma_{pqr}(t)))+I_{p}(t)+U_p(t), \end{eqnarray} (5.2)where $$K_{pq}(u)=(\cos u)e^{-u},p,q=1,2$$ and the coefficients are as follows:   $$ f_{q}(x_{q})=\frac{1}{16}\left|x_{q}^{R}\right|+i\frac{1}{12}\sin\left(x_{q}^{I}+x_{q}^{J}\right) +j\frac{1}{20}\sin^{2}x_{q}^{J}+k\frac{1}{16}\tanh x_{q}^{K},$$  $$ g_{q}(x_{q})=\frac{1}{18}\sin\left(x_{q}^{R}+x_{q}^{J}\right)+i\frac{1}{14}\left|x_{q}^{I}\right|+j\frac{1}{12}\sin x_{q}^{J}+k\frac{1}{15}\sin^{2}\left(x_{q}^{J}+x_{q}^{K}\right),$$  $$ h_{q}(x_{q})=\frac{1}{20}\sin x_{q}^{R}+i\frac{1}{24}\left|x_{q}^{R}+x_{q}^{I}\right| +j\frac{1}{18}\sin x_{q}^{J}+k\frac{1}{20}\sin x_{q}^{K},$$  $$ h_{r}(x_{r})=\frac{1}{18}\left|x_{r}^{R}\right|+i\frac{1}{20}\sin\left(x_{r}^{I}+x_{r}^{J}\right) +j\frac{1}{18}\left|x_{r}^{J}\right|+k\frac{1}{24}\tanh x_{r}^{K},$$  $$ l_{q}(z_{q})=\frac{1}{20}\left|z_{q}^{R}+z_{q}^{I}\right|+i\frac{1}{16}\sin^{2}z_{q}^{I} +j\frac{1}{18}\left|z_{q}^{J}\right|+k\frac{1}{14}\sin \left(z_{q}^{I}+z_{q}^{K}\right),$$  $$ c_{1}(t)=3+|\sin(\sqrt{2}t)|,\quad c_{2}(t)=5+|\cos (2t)|,\quad \theta_{1}(t)=2-|\cos t|,\quad \theta_{2}(t)=1-|\sin t|, $$  $$ a_{11}(t)=a_{12}(t)=0.1\sin \sqrt{2}t+i0.2\cos 2t+j0.1\sin t+k0.15\cos \sqrt{3}t,$$  $$ a_{21}(t)=a_{22}(t)=0.2\cos t+i0.1\sin \sqrt{2}t+j0.25\sin 2t+k0.3\cos t, $$  $$ b_{11}(t)=b_{12}(t)=0.3\cos 2t+i0.25\sin t+j0.15\cos t+k0.3\sin \sqrt{2}t,$$  $$ b_{21}(t)=b_{22}(t)=0.2\sin t+i0.1\sin\sqrt{2} t+j0.2\sin t+k0.4\cos \sqrt{3}t,$$  $$ e_{11}(t)=e_{12}(t)=0.15\cos t+i0.3\sin 3t+j0.1\cos\sqrt{2} t+k0.25\sin2t,$$  $$ e_{21}(t)=e_{22}(t)=0.3\sin 2t+i0.35\cos \sqrt{3}t+j0.25\cos \sqrt{2}t+k0.3\sin2 t, $$  $$ I_{1}(t)=I_{2}(t)=3\sin \sqrt{3}t+i2\cos \sqrt{2}t+j2.5\sin 3t+k3.5\sin 2t,$$  $$ \tau_{pq}(t)=\frac{1}{4}\sin^{2}t,\quad \sigma_{pq}(t)=\frac{1}{5}\sin^{2}t,\quad \xi_{pq}(t)=\frac{1}{5}\cos^{2}t,\quad\vartheta_{pq}(t)=\frac{1}{3}\cos^{2}t,$$  $$ d_{pqr}(t)=0.05\sin \sqrt{2}t+ i0.05\cos \sqrt{3}t +j 0.05\cos t +k0.05\sin \sqrt{3}t,\quad p,q,r=1,2.$$By a simple computing, we have   \begin{equation*}\,c_{1}^{-}=3,\quad c_{2}^{-}=5,\quad \theta_{1}^{-}=2,\quad \theta_{2}^{-}=1,\quad \lambda_q=\frac{1}{12},\quad \delta_q=\frac{1}{12}, \quad \rho_q=\frac{1}{18},\quad\end{equation*}   \begin{equation*}\qquad\rho_r=\frac{1}{18},\quad N_q=\frac{1}{18},\quad N_r=\frac{1}{18},\eta_q=\frac{1}{14},\quad d_{pqr}^{\nu}=0.05,\quad p,q,r=1,2,\,\,\nu\in\Lambda,\end{equation*}   \begin{equation*}a_{11}^{R^{+}}=a_{12}^{R^{+}}=0.1,\quad a_{11}^{I^{+}}=a_{12}^{I^{+}}=0.2,\quad a_{11}^{J^{+}}= a_{12}^{J^{+}}=0.1,\quad a_{11}^{K^{+}}=a_{12}^{K^{+}}=0.15,\;\end{equation*}   \begin{equation*}a_{21}^{R^{+}}=a_{22}^{R^{+}}=0.2,\quad a_{21}^{I^{+}}=a_{22}^{I^{+}}=0.1,\quad a_{21}^{J^{+}}=a_{22}^{J^{+}}=0.25,\quad a_{21}^{K^{+}}=a_{22}^{K^{+}}=0.3,\;\end{equation*}   \begin{equation*}\;b_{11}^{R^{+}}=a_{12}^{R^{+}}=0.3,\quad b_{11}^{I^{+}}=b_{12}^{I^{+}}=0.25,\quad b_{11}^{J^{+}}= b_{12}^{J^{+}}=0.15,\quad b_{11}^{K^{+}}=b_{12}^{K^{+}}=0.3,\end{equation*}   \begin{equation*}\;\,b_{21}^{R^{+}}=b_{22}^{R^{+}}=0.2,\quad b_{21}^{I^{+}}=b_{22}^{I^{+}}=0.1,\quad b_{21}^{J^{+}}=b_{22}^{J^{+}}=0.2,\quad b_{21}^{K^{+}}=b_{22}^{K^{+}}=0.4,\,\quad\end{equation*}   \begin{equation*}\,e_{11}^{R^{+}}=e_{12}^{R^{+}}=0.15,\quad e_{11}^{I^{+}}=e_{12}^{I^{+}}=0.3,\quad e_{11}^{J^{+}}= e_{12}^{J^{+}}=0.1,\quad e_{11}^{K^{+}}=e_{12}^{K^{+}}=0.25,\; \end{equation*}   \begin{equation*}\;e_{21}^{R^{+}}=e_{22}^{R^{+}}=0.3,\quad e_{21}^{I^{+}}=e_{22}^{I^{+}}=0.35,\quad e_{21}^{J^{+}}=e_{22}^{J^{+}}=0.25,\quad e_{21}^{K^{+}}=e_{22}^{K^{+}}=0.3,\,\;\end{equation*}   \begin{equation*}\;\,I_{1}^{R^{+}}=I_{2}^{R^{+}}=3,\quad I_{1}^{I^{+}}=I_{2}^{I^{+}}=2,\quad I_{1}^{J^{+}}=I_{2}^{J^{+}}=2.5,\quad I_{1}^{K^{+}}=I_{2}^{K^{+}}=3.5,\;\,\qquad\quad\end{equation*}   \begin{equation*}\,\;\;\;\;\,\tau^{+}=\frac{1}{4},\quad\sigma^{+}=\frac{1}{5},\quad\xi^{+}=\frac{1}{5},\quad\vartheta^{+}=\frac{1}{3},\quad \int_0^{+\infty}|K_{pq}(u)|\,\mathrm{d}u\leq 1,\quad p,q=1,2,\end{equation*}   \begin{equation*}\Theta_1\approx1.0729,\quad \Theta_2\approx1.2062,\quad \Xi_1\approx1.1124,\quad \Xi_2\approx1.2457.\end{equation*}Take $$\kappa =2$$, then   \begin{align*} &\max\bigg\{\frac{\Theta_1 \kappa +I_{1}^{R^{+}}}{c_1^-}\,,\, \frac{\Theta_1 \kappa +I_{1}^{I^{+}}}{c_1^-}\,,\, \frac{\Theta_1 \kappa+I_{1}^{J^{+}}}{c_1^-}\,,\, \frac{\Theta_1 \kappa+I_{1}^{K^{+}}}{c_1^-},\\ &\qquad\qquad\qquad\qquad\frac{\Theta_2 \kappa +I_{2}^{R^{+}}}{c_2^-}\,,\, \frac{\Theta_2 \kappa +I_{2}^{I^{+}}}{c_2^-}\,,\, \frac{\Theta_2 \kappa+I_{2}^{J^{+}}}{c_2^-}\,,\, \frac{\Theta_2 \kappa+I_{2}^{K^{+}}}{c_2^-}\bigg\}\approx1.8819 < \kappa=2,& \end{align*}and   \begin{equation*} \max\bigg\{\frac{\Xi_1}{c_1^-}\,,\,\frac{\Xi_2}{c_2^-}\bigg\}\approx0.3708=\mu<1. \end{equation*}Moreover, take $$\omega =1$$, $$\alpha _{pq}=\frac{3}{4}$$, $$\beta _{pqr}=\gamma _{pqr}=\frac{4}{5}$$, $$\delta _{pq}=\frac{2}{3}$$, we have   \begin{align*} &\sum\limits_{p=1}^2\bigg\{\big(\omega-c_p^{-}-\theta_{p}^{-}\big) +B^{\ast}_p+\sum\limits_{q=1}^2\frac{4\lambda_qe^{\omega\tau^+}}{\alpha_{pq}} \Big(a_{pq}^{R^{+}}+a_{pq}^{I^{+}}+a_{pq}^{J^{+}}+a_{pq}^{K^{+}}\Big)\nonumber\\ &\qquad +\sum\limits_{q=1}^2\sum\limits_{r=1}^2\bigg( \frac{16N_r \rho_qe^{\omega \sigma^+}}{\beta_{pqr}}+\frac{16N_q \rho_r e^{\omega \xi^+}}{\gamma_{pqr}}\bigg)\Big(d_{pqr}^{R^{+}}+d_{pqr}^{I^{+}}+d_{pqr}^{J^{+}}+d_{pqr}^{K^{+}}\Big) \nonumber\\ &\qquad \sum\limits_{q=1}^2\frac{4\eta_{q}e^{\omega\vartheta^+}}{\delta_{pq}}\Big(e_{pq}^{R^{+}} +e_{pq}^{I^{+}}+e_{pq}^{J^{+}}+e_{pq}^{k^{+}}\Big)\bigg\} \approx -3.9059<0. \end{align*}We can verify that all the assumptions of Theorem 4.1 are satisfied. So by Theorem 4.1, systems (5.1) and (5.2) are global exponentially synchronization about the almost automorphic solution (see Figs 1–5). Fig. 1. View largeDownload slide Curves of $$z_{1}^{\nu }$$$$(\nu \in \Lambda )$$ in three-dimensional space for synchronization case. Fig. 1. View largeDownload slide Curves of $$z_{1}^{\nu }$$$$(\nu \in \Lambda )$$ in three-dimensional space for synchronization case. Fig. 2. View largeDownload slide Curves of $$z_{2}^{\nu }$$$$(\nu \in \Lambda )$$ in three-dimensional space for synchronization case. Fig. 2. View largeDownload slide Curves of $$z_{2}^{\nu }$$$$(\nu \in \Lambda )$$ in three-dimensional space for synchronization case. Fig. 3. View largeDownload slide The states of four parts of $$x_{1}$$ and $$x_{2}$$. Fig. 3. View largeDownload slide The states of four parts of $$x_{1}$$ and $$x_{2}$$. Fig. 4. View largeDownload slide The states of four parts of $$y_{1}$$ and $$y_{2}$$. Fig. 4. View largeDownload slide The states of four parts of $$y_{1}$$ and $$y_{2}$$. Fig. 5. View largeDownload slide Synchronization. Fig. 5. View largeDownload slide Synchronization. 6. Conclusion In this paper, we consider the problem of the almost automorphic synchronization of QVHHNNs with time-varying and distributed delays. By applying the Banach fixed point theorem, constructing a suitable Lyapunov function and designing a state-feedback controller, we obtain that the drive-response structure of QVHHNNs with almost automorphic coefficients can realize the exponential synchronization. To the best of our knowledge, this is the first paper to study the automorphic synchronization of neural networks. Our result of this paper is completely new and our method can be used to study the problem of the periodic, almost periodic and automorphic synchronization for other types of neural networks. Besides, the study of QVHHNNs can unify the study of real-valued and complex-valued neural networks. 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IMA Journal of Mathematical Control and InformationOxford University Press

Published: May 3, 2018

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