Nonseparation Approach to General-Decay Synchronization of Quaternion-Valued Neural Networks with Mixed Time Delays

Abstract

In this paper, the general-decay synchronization issue of a class of quaternion-valued neural networks with mixed time delays is investigated. Firstly, unlike some previous works where the quaternion-valued model is separated into four real-valued networks or two complex-valued networks, we consider the mixed-delayed quaternion-valued neural network model as a whole and introduce a novel nonlinear feedback controller for the corresponding response system. Then, by introducing a suitable Lyapunov–Krasovskii functional and employing a novel inequality technique, some easily verifiable sufficient conditions are obtained to ensure the general-decay synchronization for the considered drive-response networks. Finally, the feasibility of the established theoretical results is verified by carrying out Matlab numerical simulations.

1. Introduction

Synchronization, as an important dynamic characteristic of neural networks (NNs), has been widely studied by scholars during the past three decades. Roughly speaking, synchronization is the realization of the same dynamic behavior between two or more dynamical systems through mutual coupling or external forces [1,2,3]. Also it has been found that the synchronization is a widespread phenomenon in nature and manmade systems, such as the synchronous glow of fireflies, the homophony of various instruments in an orchestra, and the relative stationarity of communication satellites and the Earth in space. Both from theoretical and application aspect, various types of synchronization have been introduced and extensively studied in recent decades. For example, finite-time synchronization [4], fixed-time synchronization [5,6], exponential synchronization [7], asymptotic synchronization [8], partial synchronization [9] and multiclustering [10].

It is inevitable to encounter time delay in the implementation of NNs, which will lead to some unexpected dynamic behavior. For example, in hardware implementation of NNs, time delay usually leads to bad characteristics such as oscillation, instability, divergence and chaos. In theoretically, time delay is usually divided into two categories: discrete time delay and distributed time delay. So the synchronization and stability of NNs with delays, especially those with both discrete and distributed time delays have been studied extensively [11,12,13].

In the study of synchronization, estimating the convergence rate of synchronization error approaching to zero solution is an interesting and meaningful task. However, in some cases, the convergence rate of synchronization is very difficult to evaluate or it can not be obtained by traditional means of stability concept. Therefore some scholars tried to solve this challenging issue by introducing new concept of stability, such as general decay stability, which provides a wider convergence rate compared to asymptotic stability and exponential stability. In [14], the authors studied the general decay synchronization of delayed NNs with mismatched coefficients by designing a hybrid switching controller which composed of linear and nonlinear terms. In [15], the authors considered the general decay synchronization issue of chaotic NNs with discontinuous activation functions. After that general decay stability and synchronization of various types of nonlinear systems have been studied, and some interesting results were established [16,17,18,19]. Ref. [16] investigated general decay synchronization of a class of chaotic NNs with general activation functions via improving the controllers of [14,15]. The authors in [17] investigated the general decay synchronization and $H_{\infty }$ synchronization issue of multi-weighted coupled NNs with reaction-diffusion terms. The general decay stability of inverse Euler-Maruyama method for nonlinear stochastic integral-differential equations was investigated in [18]. The authors in [19] studied the general decay stability analysis of impulsive NNs with mixed time delays.

It’s worth nothing that all of the above results are obtained for real-valued or complex-valued NNs. However, in practical applications, some data need to be processed are usually high-dimensional, and the correlation within the data may be lost when using real-valued or complex-valued NNs to process these multidimensional data. It is well known that quaternion-valued systems can effectively handle multi-dimensional data and can improve computational efficiency and accuracy. Therefore, its applications such as 3D rotation, pose estimation, signal processing and multimodal data fusion have received extensive attention from many researchers [20,21]. In recent years, scholars introduced quaternion-valued numbers into NNs to construct quaternion-valued neural netwoks (QVNNs). Due to the non-exchangeability of quaternion-valued numbers, the methods used to deal with the synchronization of real-valued or complex-valued NNs cannot be directly extended to synchronization study of QVNNs. Thus, most of the existing research results on the stability and synchronization of QVNNs are based on the quaternion-valued decomposition method, which equivalently decomposes the QVNN into two complex-valued NNs or four real-valued NNs [22,23,24,25]. For example, ref. [22] analyzed the stability of continuous time and discrete-time quaternion NNs with linear threshold neurons by decomposition method. By using the same method, in [23], the authors considered the global $\mu$-stability of QVNNs with non-differentiable time-varying delays. Again, in [24,25], the authors analyzed the global exponential stability and $\mu$-stability of quaternion-valued recursive NNs with time-varying delays respectively by using decomposition method. Because the decomposition method is complex and computationally heavy, it brings great inconvenience to theoretical analysis and practical application. Up to now, there are few results on the synchronization of QVNNs which not uses the decomposition approach. Therefore, it is an interesting and challenging task to study the general decay synchronization QVNNs by direct analysis method.

Inspired by the above analysis, this paper tries the cope the general decay synchronization issue of QVNNs with mixed delays. The main contribution of of this paper are as follows:

1. In this paper, the problem of general decay synchronization of QVNNs with mixed delays is discussed by using the concept of $\psi$ type function.

2. By designing a new nonlinear feedback controller, some sufficient conditions for general decay synchronization of QVNNs with mixed delays is obtained. The classical exponential synchronization, polynomial synchronization and logarithmic synchronization can be seen as the special forms of general decay synchronization.

3. In this paper, nonseperation method is used to analyse the synchronization performance of QVNNs, which is more natural and compact than the decomposition method.

The rest of the article is structured as follows. Section 2 provides the system model and some preliminary knowledge. In Section 3, the sufficient conditions for determining general decay synchronization of QVNNs are given. Section 4 carries on the numerical simulations using Matlab 2020A (Appendix A).

2. Preliminaries

In order to enhance the readability of this paper, some definitions and symbols are briefly described. A quaternion-valued number is a hypercomplex number with one real part and three imaginary parts denoted as $\mathrm{i},\mathrm{j},$ and k. A quaternion-valued z can be denoted as

$$z=z^{\left(r\right)}+z^{\left(i\right)}\mathrm{i}+z^{\left(j\right)}\mathrm{j}+z^{\left(k\right)}\mathrm{k},$$

where $z^{\left(r\right)}$, $z^{\left(i\right)}$, $z^{\left(j\right)}$, $z^{\left(k\right)}\in \mathbb{R}$, $\mathrm{i},\mathrm{j},\mathrm{k}$ satisfies the $Hamilton$ rule, i.e.,

$${\mathrm{i}}^2={\mathrm{j}}^2={\mathrm{k}}^2=\mathrm{ijk}=-1,\mathrm{ij}=-\mathrm{ji}=\mathrm{k},\mathrm{jk}=-\mathrm{kj}=\mathrm{i},\mathrm{ki}=-\mathrm{ik}=\mathrm{j}.$$

For a quaternion-valued number z, $\overline{z}$ denotes the conjugate of z, and it is defined as $\overline{z}=z^{\left(r\right)}-z^{\left(i\right)}\mathrm{i}-z^{\left(j\right)}\mathrm{j}-z^{\left(k\right)}\mathrm{k}$. The modulus of z is denoted as $\left|z\right|$

$$\left|z\right|=\sqrt{z\overline{z}}=\sqrt{{z^{\left(r\right)}}^2+{z^{\left(i\right)}}^2+{z^{\left(j\right)}}^2+{z^{\left(k\right)}}^2}.$$

For $z={(z_1,z_2,\cdots ,z_n)}^T\in {\mathbb{Q}}^n$, $\parallel z\parallel =({\sum }_{i=1}^n|z_i{{|}^2)}^{1/2}$ is the norm of z.

In this paper, the following QVNN model with time delays is considered:

$${\dot{z}}_i\left(t\right)=-c_i{z}_i\left(t\right)+\sum _{j=1}^n{a}_{ij}{f}_j\left(z_j\left(t\right)\right)+\sum _{j=1}^n{b}_{ij}{f}_j\left(z_j(t-{\tau }_{ij}\left(t\right))\right)+\sum _{j=1}^n{d}_{ij}{\int }_{t-{\sigma }_{ij}\left(t\right)}^t{f}_j\left(z_j\left(s\right)\right)ds+I_i,$$

(1)

where $i,j\in \mathcal{J}\triangleq \{1,2,\cdots ,n\}$, $n\ge 2$ represents the number of neurons in the NN; $z_i\left(t\right)\in \mathbb{Q}$ represents the state variables of the ith neurons at time t; $c_i>0$ is the passive decay rate of the ith neuron; $a_{ij}\in \mathbb{Q}$, $b_{ij}\in \mathbb{Q}$, and $d_{ij}\in \mathbb{Q}$ represent the connection strength, discrete delayed connection strength, and distributed delayed connection strength between neurons j and i, respectively; $f_j\left(z_j\left(t\right)\right)\in \mathbb{Q}$ is a neuron activation function; ${\tau }_{ij}\left(t\right)$ represents the discrete delay of the ith neuron along the axon of the j neuron and satisfies $0\le {\tau }_{ij}\left(t\right)\le {\tau }_{ij}$; ${\sigma }_{ij}\left(t\right)$ represents the distributed delay and satisfies $0\le {\sigma }_{ij}\left(t\right)\le {\sigma }_{ij}$; and $I_i\in \mathbb{Q}$ corresponds to the external input on neuron i.

The initial condition of the drive system (1) is

$$z_i\left(s\right)={\phi }_i\left(s\right),s\in [-\tau ,0],i\in \mathcal{J},$$

where $\tau ={max}_{1\le i,j\le n}\{{\tau }_{ij},{\sigma }_{ij}\}$, $\phi \left(s\right)={({\phi }_1\left(s\right),\dots ,{\phi }_n\left(s\right))}^T\in \mathcal{M}=C([-\tau ,0],{\mathbb{Q}}^n)$, $\mathcal{M}$ represents the set of all continuous functions from $[-\tau ,0]$ to ${\mathbb{Q}}^n$.

The response system of the drive system (1) is expressed as follows:

$${\dot{r}}_i\left(t\right)=-c_i{r}_i\left(t\right)+\sum _{j=1}^n{a}_{ij}{f}_j\left(r_j\left(t\right)\right)+\sum _{j=1}^n{b}_{ij}{f}_j\left(r_j(t-{\tau }_{ij}\left(t\right))\right)+\sum _{j=1}^n{d}_{ij}{\int }_{t-{\sigma }_{ij}\left(t\right)}^t{f}_j\left(r_j\left(s\right)\right)ds+I_i+u_i\left(t\right),$$

(2)

where $i\in \mathcal{J}$; $r_i\in \mathbb{Q}$ represents the state variable of the responding system; $u_i\left(t\right)$ is the controller that needs to be designed; and the other parameters are defined in the drive system (1).

The initial value of the drive system (2) is expressed as

$$r_i\left(s\right)={\varphi }_i\left(s\right),s\in [-\tau ,0],i\in \mathcal{J},$$

where $\varphi \left(s\right)={({\varphi }_1\left(s\right),\dots ,{\varphi }_n\left(s\right))}^T\in \mathcal{M}$.

Let ${\omega }_i\left(t\right)=r_i\left(t\right)-z_i\left(t\right)$ be the corresponding synchronization error. Then, we have the following error system

$${\dot{\omega }}_i\left(t\right)=-c_i{\omega }_i\left(t\right)+\sum _{j=1}^n{a}_{ij}{\tilde{f}}_j\left({\omega }_j\left(t\right)\right)+\sum _{j=1}^n{b}_{ij}{\tilde{f}}_j\left({\omega }_j(t-{\tau }_{ij}\left(t\right))\right)+\sum _{j=1}^n{d}_{ij}{\int }_{t-{\sigma }_{ij}\left(t\right)}^t{\tilde{f}}_j\left({\omega }_j\left(s\right)\right)ds+u_i\left(t\right),$$

(3)

where ${\tilde{f}}_j\left({\omega }_j\left(t\right)\right)=f_j({\omega }_j\left(t\right)+z_j\left(t\right))-f_j\left(z_j\left(t\right)\right)$ and ${\tilde{f}}_j\left({\omega }_j(t-{\tau }_{ij}\left(t\right))\right)=f_j({\omega }_j(t-{\tau }_{ij}\left(t\right))+z_j(t-{\tau }_{ij}\left(t\right)))-f_j\left(z_j(t-{\tau }_{ij}\left(t\right))\right)$.

We need the following assumptions in later study.

Assumption 1.

There exists a positive real number $L_j$ such that the activation function $f_j(·)$ satisfies the following inequality

$$\begin{array}{cc}\hfill |f_j\left(r_j\left(t\right)\right)-f_j\left(z_j\left(t\right)\right)|& \le L_j|r_j\left(t\right)-z_j\left(t\right)|.\hfill \end{array}$$

Assumption 2.

${\tau }_{ij}\left(t\right)$ and ${\sigma }_{ij}\left(t\right)$ are differentiable, and there are real numbers $0\le {\kappa }_{ij}<1$ and $0\le {\upsilon }_{ij}\le 0.5$ such that

$${\dot{\tau }}_{ij}\left(t\right)\le {\kappa }_{ij},{\dot{\sigma }}_{ij}\left(t\right)\le {\upsilon }_{ij}.$$

Definition 1

([16]). The function Ψ: ${\mathbb{R}}^{+}\to [1,+\infty )$ is called the Ψ-function if it satisfies the following four conditions:

(1) Ψ is non-decreasing and differentiable;

(2) $\mathrm{\Psi }\left(0\right)=1$ and $\mathrm{\Psi }(+\infty )=+\infty$;

(3) $\tilde{\mathrm{\Psi }}\left(t\right)=\dot{\mathrm{\Psi }}\left(t\right)/\mathrm{\Psi }\left(t\right)\ge 0$ and ${\mathrm{\Psi }}^{*}={sup}_{t\ge 0}\tilde{\mathrm{\Psi }}\left(t\right)<+\infty$;

(4) For any t, $s\ge 0$, $\mathrm{\Psi }(t+s)\le \mathrm{\Psi }\left(t\right)\mathrm{\Psi }\left(s\right)$.

Definition 2

([16]). If there is a constant $\epsilon >0$, such that

$$\underset{t\to +\infty }{lim\; sup}\frac{log\parallel \omega \left(t\right)\parallel }{log\mathrm{\Psi }\left(t\right)}\le -\epsilon ,$$

then the error system (3) can be general decay stable with the convergence rate of $\epsilon >0$.

Assumption 3

([16]). For the functions Ψ and $\tilde{\mathrm{\Psi }}$ given in Definition 1, there exists a positive function $\varrho \in C(\mathbb{R},{\mathbb{R}}^{+})$ and a real number $\theta >0$ such that

$$\tilde{\mathrm{\Psi }}\left(t\right)\le 1,\underset{t\in [0,+\infty )}{sup}{\int }_0^t{\mathrm{\Psi }}^{\theta }\left(s\right)\varrho \left(s\right)ds<+\infty ,\forall t\ge 0.$$

(4)

Lemma 1

([16]). Suppose that the Assumption 3 hold true, and real-valued function $\upsilon \left(t\right)\in {\mathbb{R}}^n$ satisfies $\dot{\upsilon }\left(t\right)=\mathcal{H}(t,{\upsilon }_t)$, where ${\upsilon }_t\left(s\right)=\upsilon (t+s)$ for $s\in [-\tau ,0]$, where $\mathcal{H}(t,{\upsilon }_t)$ is a locally bounded function. If there exist a differential functional $\mathcal{V}(·,{\upsilon }_t):{\mathbb{R}}^{+}\times {\mathbb{R}}^n\to {\mathbb{R}}^{+}$, and positive scalers ${\chi }_1$, ${\chi }_2$ such that for any $(t,{\upsilon }_t)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^n$

$$({\chi }_1{\parallel \upsilon \left(t\right)\parallel )}^2\le \mathcal{V}(t,{\upsilon }_t),$$

(5)

$$\frac{d\mathcal{V}(t,{\upsilon }_t)}{dt}\le -\theta \mathcal{V}(t,{\upsilon }_t)+{\chi }_2\varrho \left(t\right),$$

(6)

where $\theta >0$ and $\varrho \left(t\right)$ are introduced in above Assumption 3. Then $\upsilon \left(t\right)$ will be decay stable in the sense of Definition 2, and its convergence rate of approaching zero is $\theta /2$.

Lemma 2

([26]). For any $x,y\in \mathbb{Q}$, the following equation holds:

$$\begin{array}{cc}\hfill \overline{x+y}& =\overline{x}+\overline{y},\hfill \\ \hfill \overline{xy}& =\overline{y}\overline{x}.\hfill \end{array}$$

Lemma 3

([26]). For any $x,y\in \mathbb{Q}$, $\epsilon \in \mathbb{R}$ is the normal number, then $yx+\overline{x}\overline{y}\le \epsilon \overline{x}x+\frac{1}{\epsilon }y\overline{y}$.

3. Analysis Process

In this section, we will derive some sufficient criteria for general decay synchronization of (1) and (2) for driver-response systems. First the controller $u_i\left(t\right)$ is designed as follows:

$$u_i\left(t\right)=-\frac{{\eta }_i{\parallel \omega \left(t\right)\parallel }^2{\omega }_i\left(t\right)}{{\parallel \omega \left(t\right)\parallel }^2+\varrho \left(t\right)},$$

(7)

where ${\eta }_i$ is positive control gain and satisfies:

$$2c_i+{\eta }_i-\sum _{j=1}^n\left({\epsilon }_{ji}^1{L}_i^2+\frac{1}{{\epsilon }_{ij}^1}{a}_{ij}\overline{a_{ij}}+\frac{1}{{\epsilon }_{ij}^2}{b}_{ij}\overline{b_{ij}}+\frac{1}{{\epsilon }_{ij}^3}{d}_{ij}\overline{d_{ij}}+\frac{{\zeta }_{ji}}{1-{\kappa }_{ji}}+{\zeta }_{ji}{\tau }_{ji}+2{\pi }_{ji}{\sigma }_{ji}+\frac{1}{2}{\pi }_{ji}{\sigma }_{ji}^2\right)>0.$$

(8)

Theorem 1.

Suppose that the basic Assumptions 1–3 hold true. If the control gain ${\eta }_i$ satisfies inequality (8), then the systems (1) and (2) realize general decay synchronization via the nonlinear feedback controller (7).

Proof. 

Construct the following type Lyapunov-Krasovskii functional:

$$\begin{array}{cc}\hfill V\left(t\right)=& \sum _{i=1}^n\overline{{\omega }_i\left(t\right)}{\omega }_i\left(t\right)+\sum _{i=1}^n\sum _{j=1}^n\frac{{\zeta }_{ij}}{1-{\kappa }_{ij}}{\int }_{t-{\tau }_{ij}\left(t\right)}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^n{\zeta }_{ij}{\int }_{-{\tau }_{ij}}^0{\int }_{t+s}^t\overline{{\omega }_j\left(\varsigma \right)}{\omega }_j\left(\varsigma \right)d\varsigma ds+2\sum _{i=1}^n\sum _{j=1}^n{\pi }_{ij}{\int }_{-{\sigma }_{ij}\left(t\right)}^0{\int }_{t+s}^t\overline{{\omega }_j\left(\varsigma \right)}{\omega }_j\left(\varsigma \right)d\varsigma ds\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^n{\pi }_{ij}{\int }_{-{\sigma }_{ij}}^0{\int }_{ϵ}^0{\int }_{t+s}^t\overline{{\omega }_j\left(\varsigma \right)}{\omega }_j\left(\varsigma \right)d\varsigma dsdϵ,\hfill \end{array}$$

(9)

where ${\zeta }_{ij}$ and ${\pi }_{ij}$ are positive real numbers. Therefore, it is not difficult to find that there exists a positive scalar $\gamma >1$ such that

$$\begin{array}{cc}\hfill \sum _{i=1}^n\overline{{\omega }_i\left(t\right)}{\omega }_i\left(t\right)& \le V\left(t\right)\le \gamma \sum _{i=1}^n\overline{{\omega }_i\left(t\right)}{\omega }_i\left(t\right)+\frac{\gamma }{h}(\sum _{i=1}^n\sum _{j=1}^n{\zeta }_{ij}{\int }_{t-{\tau }_{ij}}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^n{\pi }_{ij}{\int }_{-{\sigma }_{ij}}^t{\int }_{t+s}^0\overline{{\omega }_j\left(\varsigma \right)}{\omega }_j\left(\varsigma \right)d\varsigma ds),\hfill \end{array}$$

(10)

where $h={min}_{i\in \mathcal{J}}\left\{h_i\right\}$ and

$$h_i\triangleq 2c_i+{\eta }_i-\sum _{j=1}^n\left({\epsilon }_{ji}^1{L}_i^2+\frac{1}{{\epsilon }_{ij}^1}{a}_{ij}\overline{a_{ij}}+\frac{1}{{\epsilon }_{ij}^2}{b}_{ij}\overline{b_{ij}}+\frac{1}{{\epsilon }_{ij}^3}{d}_{ij}\overline{d_{ij}}+\frac{{\zeta }_{ji}}{1-{\kappa }_{ji}}+{\zeta }_{ji}{\tau }_{ji}+2{\pi }_{ji}{\sigma }_{ji}+\frac{1}{2}{\pi }_{ji}{\sigma }_{ji}^2\right).$$

Calculate the time derivative of $V\left(t\right)$ yields

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)=& \sum _{i=1}^n\overline{{\dot{\omega }}_i\left(t\right)}{\omega }_i\left(t\right)+\sum _{i=1}^n\overline{{\omega }_i\left(t\right)}{\dot{\omega }}_i\left(t\right)\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^n\frac{{\zeta }_{ij}}{1-{\kappa }_{ij}}\left(\overline{{\omega }_j\left(t\right)}{\omega }_j\left(t\right)-(1-{\dot{\tau }}_{ij}\left(t\right))\overline{{\omega }_j(t-{\tau }_{ij}\left(t\right))}{\omega }_j(t-{\tau }_{ij}\left(t\right))\right)\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^n{\zeta }_{ij}\left({\tau }_{ij}\overline{{\omega }_j\left(t\right)}{\omega }_j\left(t\right)-{\int }_{t-{\tau }_{ij}}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds\right)\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^{n}2{\pi }_{ij}\left({\sigma }_{ij}\left(t\right)\overline{{\omega }_j\left(t\right)}{\omega }_j\left(t\right)-(1-{\dot{\sigma }}_{ij}\left(t\right)){\int }_{t-{\sigma }_{ij}\left(t\right)}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds\right)\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^n{\pi }_{ij}\left(\frac{1}{2}{\sigma }_{ij}^2\overline{{\omega }_j\left(t\right)}{\omega }_j\left(t\right)-{\int }_{-{\sigma }_{ij}}^0{\int }_{t+s}^t\overline{{\omega }_j\left(\varsigma \right)}{\omega }_j\left(\varsigma \right)d\varsigma ds\right)\hfill \\ \hfill =& \sum _{i=1}^n(-c_i\overline{{\omega }_i\left(t\right)}+\sum _{j=1}^n\overline{a_{ij}{\tilde{f}}_j\left({\omega }_j\left(t\right)\right)}+\sum _{j=1}^n\overline{b_{ij}{\tilde{f}}_j\left({\omega }_j(t-{\tau }_{ij}\left(t\right))\right)}\hfill \\ \hfill & +\sum _{j=1}^n\overline{d_{ij}{\int }_{t-{\sigma }_{ij}\left(t\right)}^t{\tilde{f}}_j\left({\omega }_j\left(s\right)\right)ds}+\overline{u_i\left(t\right)}){\omega }_i\left(t\right)\hfill \\ \hfill & +\sum _{i=1}^n\overline{{\omega }_i\left(t\right)}(-c_i{\omega }_i\left(t\right)+\sum _{j=1}^n{a}_{ij}{\tilde{f}}_j\left({\omega }_j\left(t\right)\right)+\sum _{j=1}^n{b}_{ij}{\tilde{f}}_j\left({\omega }_j(t-{\tau }_{ij}\left(t\right))\right)\hfill \\ \hfill & +\sum _{j=1}^n{d}_{ij}{\int }_{t-{\sigma }_{ij}\left(t\right)}^t{\tilde{f}}_j\left({\omega }_j\left(s\right)\right)ds+u_i\left(t\right))\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^n\frac{{\zeta }_{ij}}{1-{\kappa }_{ij}}\left(\overline{{\omega }_j\left(t\right)}{\omega }_j\left(t\right)-(1-{\dot{\tau }}_{ij}\left(t\right))\overline{{\omega }_j(t-{\tau }_{ij}\left(t\right))}{\omega }_j(t-{\tau }_{ij}\left(t\right))\right)\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^n{\zeta }_{ij}\left({\tau }_{ij}\overline{{\omega }_j\left(t\right)}{\omega }_j\left(t\right)-{\int }_{t-{\tau }_{ij}}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds\right)\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^{n}2{\pi }_{ij}\left({\sigma }_{ij}\left(t\right)\overline{{\omega }_j\left(t\right)}{\omega }_j\left(t\right)-(1-{\dot{\sigma }}_{ij}\left(t\right)){\int }_{t-{\sigma }_{ij}\left(t\right)}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds\right)\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^n{\pi }_{ij}\left(\frac{1}{2}{\sigma }_{ij}^2\overline{{\omega }_j\left(t\right)}{\omega }_j\left(t\right)-{\int }_{-{\sigma }_{ij}}^0{\int }_{t+s}^t\overline{{\omega }_j\left(\varsigma \right)}{\omega }_j\left(\varsigma \right)d\varsigma ds\right)\hfill \end{array}$$

According to the Assumption 1, it follows from Lemma 3 that

$$\begin{array}{cc}\hfill \overline{{\tilde{f}}_j\left({\omega }_j\left(t\right)\right)}\overline{a_{ij}}{\omega }_i\left(t\right)+\overline{{\omega }_i\left(t\right)}{a}_{ij}{\tilde{f}}_j\left({\omega }_j\left(t\right)\right)\le & {\epsilon }_{ij}^1\overline{{\tilde{f}}_j\left({\omega }_j\left(t\right)\right)}{\tilde{f}}_j\left({\omega }_j\left(t\right)\right)+\frac{1}{{\epsilon }_{ij}^1}\overline{{\omega }_i\left(t\right)}{a}_{ij}\overline{a_{ij}}{\omega }_i\left(t\right)\hfill \\ \hfill \le & {\epsilon }_{ij}^1{L}_j^2\overline{{\omega }_j\left(t\right)}{\omega }_j\left(t\right)+\frac{1}{{\epsilon }_{ij}^1}\overline{{\omega }_i\left(t\right)}{a}_{ij}\overline{a_{ij}}{\omega }_i\left(t\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \overline{{\tilde{f}}_j\left({\omega }_j(t-{\tau }_{ij}\left(t\right))\right)}& \overline{b_{ij}}{\omega }_i\left(t\right)+\overline{{\omega }_i\left(t\right)}{b}_{ij}{\tilde{f}}_j\left({\omega }_j(t-{\tau }_{ij}\left(t\right))\right)\hfill \\ \hfill \le & {\epsilon }_{ij}^2\overline{{\tilde{f}}_j\left({\omega }_j(t-{\tau }_{ij}\left(t\right))\right)}{\tilde{f}}_j\left({\omega }_j(t-{\tau }_{ij}\left(t\right))\right)+\frac{1}{{\epsilon }_{ij}^2}\overline{{\omega }_i\left(t\right)}{b}_{ij}\overline{b_{ij}}{\omega }_i\left(t\right)\hfill \\ \hfill \le & {\epsilon }_{ij}^2{L}_j^2\overline{{\omega }_j(t-{\tau }_{ij}\left(t\right))}{\omega }_j(t-{\tau }_{ij}\left(t\right))+\frac{1}{{\epsilon }_{ij}^2}\overline{{\omega }_i\left(t\right)}{b}_{ij}\overline{b_{ij}}{\omega }_i\left(t\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \overline{{\int }_{t-{\sigma }_{ij}\left(t\right)}^t{\tilde{f}}_j\left({\omega }_j\left(s\right)\right)ds}& \overline{d_{ij}}{\omega }_i\left(t\right)+\overline{{\omega }_i\left(t\right)}{d}_{ij}{\int }_{t-{\sigma }_{ij}\left(t\right)}^t{\tilde{f}}_j\left({\omega }_j\left(s\right)\right)ds\hfill \\ \hfill =& {\int }_{t-{\sigma }_{ij}\left(t\right)}^t\overline{{\tilde{f}}_j\left({\omega }_j\left(s\right)\right)d_{ij}}{\omega }_i\left(t\right)+\overline{{\omega }_i\left(t\right)}{d}_{ij}{\tilde{f}}_j\left({\omega }_j\left(s\right)\right)ds\hfill \\ \hfill \le & {\epsilon }_{ij}^3{\int }_{t-{\sigma }_{ij}\left(t\right)}^t\overline{{\tilde{f}}_j\left({\omega }_j\left(t\right)\right)}{\tilde{f}}_j\left({\omega }_j\left(t\right)\right)ds+\frac{{\sigma }_{ij}\left(t\right)}{{\epsilon }_{ij}^3}\overline{{\omega }_i\left(t\right)}{d}_{ij}\overline{d_{ij}}{\omega }_i\left(t\right)\hfill \\ \hfill \le & L_j^2{\epsilon }_{ij}^3{\int }_{t-{\sigma }_{ij}\left(t\right)}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds+\frac{{\sigma }_{ij}}{{\epsilon }_{ij}^3}\overline{{\omega }_i\left(t\right)}{d}_{ij}\overline{d_{ij}}{\omega }_i\left(t\right),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill -\sum _{i=1}^n\left(\overline{u_i\left(t\right)}{\omega }_i\left(t\right)+\overline{{\omega }_i\left(t\right)}{u}_i\left(t\right)\right)=-& \sum _{i=1}^n\frac{{\eta }_i{\parallel \omega \left(t\right)\parallel }^2}{{\parallel \omega \left(t\right)\parallel }^2+\varrho \left(t\right)}\left(\overline{{\omega }_i\left(t\right)}{\omega }_i\left(t\right)+\overline{{\omega }_i\left(t\right)}{\omega }_i\left(t\right)\right)\hfill \\ \hfill =-& \sum _{i=1}^n\frac{2{\eta }_i{\parallel \omega \left(t\right)\parallel }^2}{{\parallel \omega \left(t\right)\parallel }^2+\varrho \left(t\right)}{\omega }_i\left(t\right)\overline{{\omega }_i\left(t\right)}+2\sum _{i=1}^n{\eta }_i{\omega }_i\left(t\right)\overline{{\omega }_i\left(t\right)}\hfill \\ \hfill & -2\sum _{i=1}^n{\eta }_i{\omega }_i\left(t\right)\overline{{\omega }_i\left(t\right)}\hfill \\ \hfill =& \sum _{i=1}^n\frac{2{\eta }_i\varrho \left(t\right)}{{\parallel \omega \left(t\right)\parallel }^2+\varrho \left(t\right)}{\omega }_i\left(t\right)\overline{{\omega }_i\left(t\right)}-2\sum _{i=1}^n{\eta }_i{\omega }_i\left(t\right)\overline{{\omega }_i\left(t\right)}\hfill \\ \hfill \le & 2\underset{1\le i\le n}{max}\left\{{\eta }_i\right\}\frac{{\varrho \left(t\right)\parallel \omega \left(t\right)\parallel }^2}{{\parallel \omega \left(t\right)\parallel }^2+\varrho \left(t\right)}-2\sum _{i=1}^n{\eta }_i{\omega }_i\left(t\right)\overline{{\omega }_i\left(t\right)}\hfill \\ \hfill \le & 2\underset{1\le i\le n}{max}\left\{{\eta }_i\right\}\varrho \left(t\right)-2\sum _{i=1}^n{\eta }_i{\omega }_i\left(t\right)\overline{{\omega }_i\left(t\right)},\hfill \end{array}$$

(13)

where ${\epsilon }_{ij}^1,{\epsilon }_{ij}^2$ and ${\epsilon }_{ij}^3$ are the positive numbers.

Based on the inequality (11)–(12), the following relation can be obtained:

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)\le & -\sum _{i=1}^{n}2c_i\overline{{\omega }_i\left(t\right)}{\omega }_i\left(t\right)+\sum _{i=1}^n\sum _{j=1}^n{\epsilon }_{ij}^1{L}_j^2\overline{{\omega }_j\left(t\right)}{\omega }_j\left(t\right)+\sum _{i=1}^n\sum _{j=1}^n\frac{1}{{\epsilon }_{ij}^1}\overline{{\omega }_i\left(t\right)}{a}_{ij}\overline{a_{ij}}{\omega }_i\left(t\right)\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^n{\epsilon }_{ij}^2{L}_j^2\overline{{\omega }_j(t-{\tau }_{ij}\left(t\right))}{\omega }_j(t-{\tau }_{ij}\left(t\right))+\sum _{i=1}^n\sum _{j=1}^n\frac{1}{{\epsilon }_{ij}^2}\overline{{\omega }_i\left(t\right)}{b}_{ij}\overline{b_{ij}}{\omega }_i\left(t\right)\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^n{L}_j^2{\epsilon }_{ij}^3{\int }_{t-{\sigma }_{ij}\left(t\right)}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds+\sum _{i=1}^n\sum _{j=1}^n\frac{1}{{\epsilon }_{ij}^3}\overline{{\omega }_i\left(t\right)}{d}_{ij}\overline{d_{ij}}{\omega }_i\left(t\right)\hfill \\ \hfill & +2\underset{1\le i\le n}{max}\left\{{\eta }_i\right\}\varrho \left(t\right)-2\sum _{i=1}^n{\eta }_i{\omega }_i\left(t\right)\overline{{\omega }_i\left(t\right)}+\sum _{i=1}^n\sum _{j=1}^n\frac{{\zeta }_{ij}}{1-{\kappa }_{ij}}\overline{{\omega }_j\left(t\right)}{\omega }_j\left(t\right)\hfill \\ \hfill & -\sum _{i=1}^n\sum _{j=1}^n\frac{{\zeta }_{ij}}{1-{\kappa }_{ij}}(1-{\dot{\tau }}_{ij}\left(t\right))\overline{{\omega }_j(t-{\tau }_{ij}\left(t\right))}{\omega }_j(t-{\tau }_{ij}\left(t\right))+\sum _{i=1}^n\sum _{j=1}^n{\zeta }_{ij}{\tau }_{ij}\overline{{\omega }_j\left(t\right)}{\omega }_j\left(t\right)\hfill \\ \hfill & -\sum _{i=1}^n\sum _{j=1}^n{\zeta }_{ij}{\int }_{t-{\tau }_{ij}}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds+\sum _{i=1}^n\sum _{j=1}^{n}2{\pi }_{ij}{\sigma }_{ij}\overline{{\omega }_j\left(t\right)}{\omega }_j\left(t\right)\hfill \\ \hfill & -\sum _{i=1}^n\sum _{j=1}^{n}2{\pi }_{ij}(1-{\dot{\sigma }}_{ij}\left(t\right)){\int }_{t-{\sigma }_{ij}\left(t\right)}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds+\sum _{i=1}^n\sum _{j=1}^n{\pi }_{ij}\frac{1}{2}{\sigma }_{ij}^2\overline{{\omega }_j\left(t\right)}{\omega }_j\left(t\right)\hfill \\ \hfill & -\sum _{i=1}^n\sum _{j=1}^n{\pi }_{ij}{\int }_{-{\sigma }_{ij}}^0{\int }_{t+s}^t\overline{{\omega }_j\left(\varsigma \right)}{\omega }_j\left(\varsigma \right)d\varsigma ds.\hfill \end{array}$$

(14)

According to the assumption 2, the inequality (14) can be further written as:

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)\le & (-2c_i-{\eta }_i+\sum _{j=1}^n({\epsilon }_{ji}^1{L}_i^2+\frac{1}{{\epsilon }_{ij}^1}{a}_{ij}\overline{a_{ij}}+\frac{1}{{\epsilon }_{ij}^2}{b}_{ij}\overline{b_{ij}}+\frac{1}{{\epsilon }_{ij}^3}{d}_{ij}\overline{d_{ij}}+\frac{{\zeta }_{ji}}{1-{\kappa }_{ji}}\hfill \\ \hfill & +{\zeta }_{ji}{\tau }_{ji}+2{\pi }_{ji}{\sigma }_{ji}+\frac{1}{2}{\pi }_{ji}{\sigma }_{ji}^2\left)\right)\overline{{\omega }_i\left(t\right)}{\omega }_i\left(t\right)+2\underset{1\le i\le n}{max}\left\{{\eta }_i\right\}\varrho \left(t\right)\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^n{\epsilon }_{ij}^2{L}_j^2\overline{{\omega }_j(t-{\tau }_{ij}\left(t\right))}{\omega }_j(t-{\tau }_{ij}\left(t\right))+\sum _{i=1}^n\sum _{j=1}^n{\epsilon }_{ij}^3{L}_j^2{\int }_{t-{\sigma }_{ij}\left(t\right)}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds\hfill \\ \hfill & -\sum _{i=1}^n\sum _{j=1}^n{\zeta }_{ij}\overline{{\omega }_j(t-{\tau }_{ij}\left(t\right))}{\omega }_j(t-{\tau }_{ij}\left(t\right))-\sum _{i=1}^n\sum _{j=1}^n{\zeta }_{ij}{\int }_{t-{\tau }_{ij}}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds\hfill \\ \hfill & -\sum _{i=1}^n\sum _{j=1}^n{\pi }_{ij}{\int }_{t-{\sigma }_{ij}\left(t\right)}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds-\sum _{i=1}^n\sum _{j=1}^n{\pi }_{ij}{\int }_{-{\sigma }_{ij}}^0{\int }_{t+s}^t\overline{{\omega }_j\left(\varsigma \right)}{\omega }_j\left(\varsigma \right)d\varsigma ds\hfill \\ \hfill \le & -h_i\overline{{\omega }_i\left(t\right)}{\omega }_i\left(t\right)+2\underset{1\le i\le n}{max}\left\{{\eta }_i\right\}\varrho \left(t\right)-\sum _{i=1}^n\sum _{j=1}^n{\zeta }_{ij}{\int }_{t-{\tau }_{ij}}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds\hfill \\ \hfill & -\sum _{i=1}^n\sum _{j=1}^n{\pi }_{ij}{\int }_{-{\sigma }_{ij}}^0{\int }_{t+s}^t\overline{{\omega }_j\left(\varsigma \right)}{\omega }_j\left(\varsigma \right)d\varsigma ds\hfill \\ \hfill \le & -h_i\overline{{\omega }_i\left(t\right)}{\omega }_i\left(t\right)+\eta \varrho \left(t\right)-\sum _{i=1}^n\sum _{j=1}^n{\zeta }_{ij}{\int }_{t-{\tau }_{ij}}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds\hfill \\ \hfill & -\sum _{i=1}^n\sum _{j=1}^n{\pi }_{ij}{\int }_{-{\sigma }_{ij}}^0{\int }_{t+s}^t\overline{{\omega }_j\left(\varsigma \right)}{\omega }_j\left(\varsigma \right)d\varsigma ds,\hfill \end{array}$$

(15)

where $\eta =2{max}_{1\le i\le n}\left\{{\eta }_i\right\}>0$ and we take ${\zeta }_{ij}$ and ${\pi }_{ij}$ as ${\zeta }_{ij}={\epsilon }_{ij}^2{L}_j^2$ and ${\pi }_{ij}={\epsilon }_{ij}^3{L}_j^2$, respectively.

Taking a small enough $\alpha$ such that $\alpha \gamma <d$ and using the inequality (10) and (15) one get

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)+\alpha V\left(t\right)\le & -\sum _{i=1}^n{h}_i\overline{{\omega }_i\left(t\right)}{\omega }_i\left(t\right)+\eta \varrho \left(t\right)-\sum _{i=1}^n\sum _{j=1}^n{\zeta }_{ij}{\int }_{t-{\tau }_{ij}}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds\hfill \\ \hfill & -\sum _{i=1}^n\sum _{j=1}^n{\pi }_{ij}{\int }_{-{\sigma }_{ij}}^t{\int }_{t+s}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)dsd\varsigma \hfill \\ \hfill & +\alpha [\gamma \sum _{i=1}^n\overline{{\omega }_i\left(t\right)}{\omega }_i\left(t\right)+\frac{\gamma }{h}(\sum _{i=1}^n\sum _{j=1}^n{\zeta }_{ij}{\int }_{t-{\tau }_{ij}}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^n{\pi }_{ij}{\int }_{-{\sigma }_{ij}}^0{\int }_{t+s}^t\overline{{\omega }_j\left(\varsigma \right)}{\omega }_j\left(\varsigma \right)d\varsigma ds\left)\right]\hfill \\ \hfill \le & \left(\alpha \gamma -h\right)\sum _{i=1}^n\overline{{\omega }_i\left(t\right)}{\omega }_i\left(t\right)+\left(\frac{\alpha \gamma }{h}-1\right)(\sum _{i=1}^n\sum _{j=1}^n{\zeta }_{ij}{\int }_{t-{\tau }_{ij}}^t\overline{{\omega }_j\left(s\right)}{\omega }_j\left(s\right)ds\hfill \\ \hfill & +\sum _{i=1}^n\sum _{j=1}^n{\pi }_{ij}{\int }_{-{\sigma }_{ij}}^0{\int }_{t+s}^t\overline{{\omega }_j\left(\varsigma \right)}{\omega }_j\left(\varsigma \right)d\varsigma ds)+\eta \varrho \left(t\right)\hfill \\ \hfill \le & \eta \varrho \left(t\right).\hfill \end{array}$$

That is

$$\dot{V}\left(t\right)+\alpha V\left(t\right)\le \eta \varrho \left(t\right).$$

(16)

Therefore, from Lemma 1, the drive-response systems (1) and (2) achieve general decay synchronization under nonlinear feedback controller (7). The convergence rate of $\omega \left(t\right)$ to zero is $\alpha /2$. The certificate is completed.    □

Remark 1.

As we can see from the Lyapunov-Krasovskii functional (9) and proof of the theorem 1, the first term of (9) is used to achieve the asymptotic stability of the error system (3) while the second and third terms are designed to counteract the discrete time delay, and the fourth and fifth terms are designed to counteract the distributed time delay.

If system (1) has no distributed time delay, then system (1) degenerates into the following form:

$${\dot{z}}_i\left(t\right)=-c_i{z}_i\left(t\right)+\sum _{j=1}^n{a}_{ij}{f}_j\left(z_j\left(t\right)\right)+\sum _{j=1}^n{b}_{ij}{f}_j\left(z_j(t-{\tau }_{ij}\left(t\right))\right)+I_i.$$

(17)

The corresponding response system of drive system (17) is given as follows:

$${\dot{r}}_i\left(t\right)=-c_i{r}_i\left(t\right)+\sum _{j=1}^n{a}_{ij}{f}_j\left(r_j\left(t\right)\right)+\sum _{j=1}^n{b}_{ij}{f}_j\left(r_j(t-{\tau }_{ij}\left(t\right))\right)+I_i+u_i\left(t\right).$$

(18)

In this case, the following Corollary can be obtained from Theorem 1.

Corollary 1.

Under the Assumptions 1–3, if the control gain ${\eta }_i$ satisfies the following inequality,

$$2c_i+{\eta }_i-\sum _{j=1}^n\left({\epsilon }_{ji}^1{L}_i^2+\frac{1}{{\epsilon }_{ij}^1}{a}_{ij}\overline{a_{ij}}+\frac{1}{{\epsilon }_{ij}^2}{b}_{ij}\overline{b_{ij}}+\frac{{\zeta }_{ji}}{1-{\kappa }_{ji}}+{\zeta }_{ji}{\tau }_{ji}\right)>0.$$

(19)

Then the drive-response systems (17) and (18) realize general decay synchronization via nonlinear feedback controller (7).

Remark 2.

In works [22,23,24,25], the stability and synchronization of QVNNs are studied by decomposing the original model into two complex-valued NNs or four real-valued NNs. Although the decomposition method is feasible, it evidently increases the dimension of the system under consideration and brings extra the amount of computation. Inspired from the recent work [26,27,28], this paper directly discusses the general decay synchronization of QVNNs, which is more natural and compact than decompressing QVNNs into two complex-valued NNs or four real-valued NNs.

Remark 3.

This paper focuses on the general decay synchronization of QVNNs with time delay. In [22], the stability of continuous QVNNs and discrete QVNNs is considered. In [24,25], the global μ-stability of QVNNs with unbounded delays is discussed. In [29], the boundedness and periodicity of discrete QVNNs with time delays are studied. In [26], the dissipative degree of QVNN is studied. However, as far as we know, there are few results on general decay synchronization of QVNNs with discrete and distributed time delays. Therefore, the research results of this paper are innovative and contribute to the further research of general decay synchronization of NNs.

Remark 4.

From Definition 1 and 2, it is clear that exponential synchronization and polynomial synchronization can be seen as the special forms of general decay synchronization, and the corresponding synchronization can be obtained by choosing the proper $\mathrm{\Psi }\left(t\right)$ functions as $\mathrm{\Psi }\left(t\right)=r^{\beta t}$, $\mathrm{\Psi }\left(t\right)={(1+t)}^{\beta }$, thus the driving-response systems (1) and (2) realize exponential synchronization and polynomial synchronization respectively. From this aspect, the results of this paper are more general and have a better applicability.

4. Numerical Examples and Simulations

Example 1.

For $n=2$, consider the following quaternion-valued NN system with time-varying delays

$${\dot{z}}_i\left(t\right)=-c_i{z}_i\left(t\right)+\sum _{j=1}^2{a}_{ij}{f}_j\left(z_j\left(t\right)\right)+\sum _{j=1}^2{b}_{ij}{f}_j\left(z_j(t-{\tau }_{ij}\left(t\right))\right)+I_i,$$

(20)

where $f_1\left(u\right)=f_2\left(u\right)=tanh\left(u\right)$. The parameters of system (20) are taken as $c_1=1.2+1\mathrm{i}+1\mathrm{j}+1.2\mathrm{k}$, $c_2=1.1+1\mathrm{i}+1\mathrm{j}+1.2\mathrm{k}$, $a_{11}=-1.3+1.8\mathrm{i}+1.2\mathrm{j}+1.1\mathrm{k}$, $a_{12}=1+1.5\mathrm{i}+1.5\mathrm{j}-0.12\mathrm{k}$, $a_{21}=-2.1+0.1\mathrm{i}-0.9\mathrm{j}-4.8\mathrm{k}$, $a_{22}=-1.6+1.8\mathrm{i}-1.7\mathrm{j}+2.2\mathrm{k}$, $b_{11}=-0.6+1.4\mathrm{i}+0.8\mathrm{j}+1.5\mathrm{k}$, $b_{12}=1.3+1.1\mathrm{i}+1.9\mathrm{j}+2\mathrm{k}$, $b_{21}=-1.7+0.1\mathrm{i}+2.5\mathrm{j}+3\mathrm{k}$, $b_{22}=-3.2+1.4\mathrm{i}+2.1\mathrm{j}+2.5\mathrm{k}$, ${\tau }_{ij}\left(t\right)=e^t/(1+e^t)$, and $I_i=0$ for $i=1,2$.

The initial conditions of the primary system (20) are given as $z_1\left(s\right)=1.2-0.5\mathrm{i}+0.9\mathrm{j}-0.2\mathrm{k}$ and $z_2\left(s\right)=0.7+0.2\mathrm{i}-0.8\mathrm{j}+0.3\mathrm{k}$ to $s\in [-1,0]$. Its corresponding Matlab simulation is shown in Figure 1 and Figure 2. We can see that the drive system (20) has a chaotic attractor.

The corresponding response system can be expressed as

$${\dot{r}}_i\left(t\right)=-c_i{r}_i\left(t\right)+\sum _{j=1}^2{a}_{ij}{f}_j\left(r_j\left(t\right)\right)+\sum _{j=1}^2{b}_{ij}{f}_j\left(r_j(t-{\tau }_{ij}\left(t\right))\right)+I_i+u_i\left(t\right),$$

(21)

where $c_i,a_{ij},b_{ij}$ and $I_i$ are the same as in system (20). The nonlinear feedback controller $u_i\left(t\right)$ is designed as follows

$$u_i\left(t\right)=-\frac{{\eta }_i{\parallel \omega \left(t\right)\parallel }^2{\omega }_i\left(t\right)}{{\parallel \omega \left(t\right)\parallel }^2+\varrho \left(t\right)},i=1,2.$$

(22)

where ${\omega }_i\left(t\right)=r_i\left(t\right)-z_i\left(t\right)$. By simple computation, we can obtain that $L_j=1$, ${\tau }_{ij}=1$. Therefore, Assumptions 1 and 2 are satisfied. Let $\varrho \left(t\right)=e^{-8.8t}$, $\psi \left(t\right)=e^t$, and choosing ${\xi }_1=1$, ${\xi }_2=2$, ${\eta }_1=1$ and ${\eta }_2=2$. Then all the conditions of Theorem 1 are satisfied, and thus the general decay synchronization between system (20) and system (21) is guaranteed. The time evolution of the synchronization error of the drive-response systems (20) and (21) is provided in Figure 3. The initial conditions of system (21) are taken as $z_1\left(s\right)=(-1.2-0.5i+0.9j-0.2k)+0.2h$ and $z_2\left(s\right)=(0.7+0.2i-0.8j+0.3k)+0.2h$ for $h=\{-6,-4,-2,0,2,4\}$ and $s\in [-1,0]$. The synchronization curves between systems (20) and (21) are given in Figure 4.

5. Conclusions

This paper focuses on the general-decay synchronization of QVNNs with discrete and distributed time delays. The general-decay synchronization property of QVNNs is directly analyzed without decomposing the QVNN into two complex-valued or four real-valued neural networks. By introducing the appropriate Lyapunov functional and designing a new nonlinear feedback controller, some sufficient conditions for the system to achieve general decay synchronization are established. Also, the feasibility of the obtained results is verified using one numerical example.