Global attractive set for quaternion-valued neural networks with neutral items

Abstract

This paper investigated the global attractive set for quaternion-valued neural networks (QVNNs) with leakage delay, time-varying delay, and neutral items. Based on various basic conditions of activation function, the global attractive set and global exponential attractive set of QVNNs are given combined with novel analytical techniques and Lyapunov theory. The QVNNs are studied by a direct method, without any decomposition. The time delay can be non-differential, which makes the results more pragmatic. Restrictions on the activation function of the neutral item are relaxed. The neutral activation function can be bounded or unbounded, which makes the results more practical. Two simulation examples are given to verify the validity of the theory results.

Introduction

With the development of artificial intelligence science, scholars concentrate on the study of artificial neural network,which has been widely used in control, medical diagnosis, and other fields^1–5. In the existing literatures, neural networks are usually based on real-valued and complex-valued. However, they are limited in processing high-dimensional data. Quaternion was proposed by Hamilton in 1843, quaternion carry more information, containing real and imaginary parts. Nevertheless, its multiplication is not commutative. It is more difficult to study than real and complex numbers.

The literature⁶ shows that quaternion-valued neural networks (QVNNs) are more productive in solving multi-dimensional issues, and quaternion can better describe the spatial rotation state^7,8. QVNNs are widely used in optimization, image processing, and life sciences^9–12. It is important to study the characteristics and applications of QVNNs.

Decomposition methods are usually adopted to study QVNNs^13,14. However, the computational complexity of the decomposition method is doubled or even quadrupled. To overcome this shortcoming, scholars established a more direct approach, without any decomposition¹⁵.

The time delay in the practical systems usually leads to instability, oscillation, or chaos. Considering that appropriate time delay can enhance stability, many scholars have concentrated on the dynamic behavior of time delay systems^16–18. In the process of network transmission, signals between two network nodes may pass through multiple network segments with different transmission conditions, which can cause multiple time delays^19,20. It is no longer appropriate to describe the induced effect with a single time delay component. Scholars focus on multiple time delays in neural networks^21,22. Zhang et al.²³discusses the QVNNs estimation problem with both distribution and leakage delay. However, these results are always based on real or complex numbers. According to available literature, there are few relevant research results in quaternions.

The performance of the neurons is connected with the past state, and also the derivative of the past state. A delay in the derivative is called a neutral delay²⁴, the corresponding neural network is called a neutral-type neural network. In the literature, the function of the neutral term is usually set to be bounded^25–28. But in real systems, the activation function is usually nonlinear or unbounded. It is a challenging problem to study such activation functions.

Exploring the dissipative and global attractive set of QVNN, which incorporates leakage delay, time-varying delay, and neutral delay, presents an unparalleled research frontier. This paper employs novel analytical techniques and Lyapunov theory to delve into this intricate issue, with the following main contributions:

• I.The QVNN offers a broader range of applications and innovative techniques compared to real-valued and complex-valued neural network. The approach to quaternions is holistic, eschewing any decomposition. Several novel Lyapunov-Krasovskii function are devised to ascertain the global attractive set and global exponential attractive set of QVNN, providing a comprehensive analysis.
• II.By incorporating the leakage delay term, the results are not only theoretically but also highly relevant and practical in real-world applications.
• III.This paper broadens the perspective on the activation function types of neutral items. By dispensing with traditional assumptions, the neutral activation function can to be either bounded or unbounded, thereby enhancing the applicability and practical value of the findings.

The subsequent sections of this paper are structured as follows: “Preliminaries” section presents preliminaries and model descriptions, “Main results” section delves into the detailed discussion and proofs of the global exponential dissipation and global dissipation of QVNN(1), “Discussion of results” section explores the global dissipation of QVNN(1) under varying activation function conditions, “Illustrative examples” section verifies the results through numerical simulations, and “Conclusions” section concludes the paper.

Preliminaries

Notations $\mathbb{R}$ stands for the set of real numbers, ${\mathbb{R}}^{+}$ stands for the set of positive real numbers, $\mathbb{Q}$ represents the set of quaternions, ${\mathbb{Q}}^n$ denotes the n-dimensional quaternion space. Quaternion u can be denoted as

$$\begin{array}{c}\hfill u=u^{(r)}+u^{(i)}i+u^{(j)}j+u^{(k)}k,u^{(r)},u^{(i)},u^{(j)},u^{(k)}\in \mathbb{R},\end{array}$$

i, j, k are imaginary parts, and satisfy the Hamilton rule

$$\begin{array}{c}\hfill i^2=j^2=k^2=ijk=-1,-kj=jk=i,-ik=ki=j,-ji=ij=k.\end{array}$$

The conjugate of quaternion u is denoted as $\overline{u}=u^{(r)}-u^{(i)}i-u^{(j)}j-u^{(k)}k$. The modulus of $u\in \mathbb{Q}$ is specified as

$$\begin{array}{c}\hfill |u|=\sqrt{u\overline{u}}=\sqrt{{(u^{(r)})}^2+{(u^{(i)})}^2+{(u^{(j)})}^2+{(u^{(k)})}^2}.\end{array}$$

The norm of vector $\xi ={({\xi }_1,{\xi }_2,\cdots ,{\xi }_n)}^T\in {\mathbb{Q}}^n$ is defined as

$$\begin{array}{c}\hfill \Vert \xi \Vert \triangleq (\sum _{c=1}^n{|{\xi }_c|}^2{)}^{\frac{1}{2}}.\end{array}$$

$C([t_0-\mu ,t_0];{\mathbb{Q}}^n)$ represents a class of continuous mapping set from $[t_0-\mu ,t_0]$ to ${\mathbb{Q}}^n$, where $\mu$ in ${\mathbb{R}}^{+}$. For $\psi \in C([t_0-\mu ,t_0];{\mathbb{Q}}^n)$,

$$\begin{array}{c}\hfill \Vert \psi \Vert \triangleq \underset{t_0-\mu \le s\le t_0}{sup}|\psi (s)|.\end{array}$$

This paper investigates the QVNN model with neutral item²⁹ and leakage delays¹⁷,

$$\begin{array}{cc}\hfill \frac{\text{d}{\xi }_c(t)}{\text{d}t}=& -p_c{\xi }_c(t-\iota )+\sum _{d=1}^n(a_{\mathit{cd}}{f}_d({\xi }_d(t))+b_{\mathit{cd}}{f}_d({\xi }_d(t-{\mu }_1(t)-{\mu }_2(t)))+m_{\mathit{cd}}{g}_d(\dot{{\xi }_d}(t-\omega (t))))+I_c(t),\hfill \end{array}$$ 1

${\xi }_c(t),c=1,2,\dots ,n$ is the state of the $c-$th neuron. $p_c$ is the self-feedback coefficient, $p_c>0$. $a_{\mathit{cd}}\in \mathbb{Q}$, $b_{\mathit{cd}}\in \mathbb{Q}$ and $m_{\mathit{cd}}\in \mathbb{Q}$ are link weights. $f_d(·)$ and $g_d(·)$ are activation functions. $\iota$ is a positive real constant, and presents the leakage time delay. ${\mu }_1(t)$ and ${\mu }_2(t)$ are additive time-varying delay, $\omega (t)$ is the neutral delay. They satisfy ${\mu }_1(t)\le {\mu }_1,{\mu }_2(t)\le {\mu }_2$, $0\le \omega (t)\le \omega$. For convenience, the variables that satisfy the conditions are represented as ${\mu }_1(t)+{\mu }_2(t)=\mu (t)$, ${\mu }_1+{\mu }_2=\mu$, $\dot{\mu }(t)\le {\theta }_1<1$, $\dot{\omega }(t)\le {\theta }_2<1$, and $\iota ,\mu ,\omega ,{\theta }_1,{\theta }_2$ are real constants. $I_c(t)$ is the external input.

Definition 1

³⁰ If there exists $\mathrm{\Xi }\subset {\mathbb{Q}}^n$, for $\forall$ $\epsilon >0$ and $\psi (s)\in {\mathbb{Q}}^n\backslash \mathrm{\Xi }$, $s\in \left[t_0,-,\mu ,,,t_0\right],\exists$ $T>0$, s.t.

$$\begin{array}{c}\hfill \xi \left(t,,,t_0,,,\psi ,(s)\right)\subseteq {\mathrm{\Xi }}_{\epsilon },t\ge T+t_0,\end{array}$$

then QVNN(1) is a globally dissipative system(GDS), and $\mathrm{\Xi }$ is the globally attractive set(GAS) of (1).

Definition 2

³⁰ If there exists $\mathrm{\Xi }\subset {\mathbb{Q}}^n$, for $\forall$ $\epsilon >0$ and $\psi (s)\in$ ${\mathbb{Q}}^n\backslash \mathrm{\Xi },s\in \left[t_0,-,\mu ,,,t_0\right]$, $\exists$ $\alpha >0,M(\Vert \psi \Vert )>0$, s.t.

$$\begin{array}{c}\hfill \underset{\tilde{\xi }\in {\mathrm{\Xi }}_{\epsilon }}{inf}\left\{∥\xi ,\left(t,,,t_0,,,\psi ,(s)\right),-,\tilde{\xi }∥\right\}\le M(\Vert \psi \Vert )exp\{-\alpha t\},\end{array}$$

then the system (1) is a globally exponentially dissipative system (GEDS), and $\mathrm{\Xi }$ is the globally exponentially attractive set(GEAS) of (1).

Lemma 1

³¹ For $\forall$ $u,v\in \mathbb{Q}$, the following conclusions can be drawn:

$$\begin{array}{c}\hfill \text{(I)}\overline{u+v}=\overline{u}+\overline{v};\text{(II)}\overline{\mathit{uv}}=\overline{v}\overline{u}.\end{array}$$

Remark 1

For general case, $u,v\in \mathbb{Q},uv\ne vu$. That is, quaternion multiplication generally does not satisfy the commutativity. The method that decomposes quaternion into real and complex numbers is more complicated, which increases the complexity of calculation, this paper tries to use some new methods for analysis.

Lemma 2

³⁰ If $u,v\in \mathbb{Q}$, then $\exists$ $\epsilon \in {\mathbb{R}}^{+}$, s.t.

$$\begin{array}{c}\hfill vu+\overline{u}\overline{v}\le \frac{1}{\epsilon }v\overline{v}+\epsilon \overline{u}u.\end{array}$$

Main results

This article considers the following two types of activation functions:

• I.
  $$\begin{array}{cc}& \mathcal{B}\triangleq \left\{f(·)||,f_d,\left({\xi }_d\right),\mid ,\le ,k_d,,,k_d,\in ,{\mathbb{R}}^{+}\right\},f(·),g(·)\in \mathcal{B}\text{represents}\text{that}\left|f_d,({\xi }_d)\right|\le k_d,\left|g_d,({\xi }_d)\right|\le {\tilde{k}}_d,{\tilde{k}}_d\in {\mathbb{R}}^{+},\hfill \\ \hfill & d=1,2,\dots ,n;\hfill \end{array}$$
• II.
  $$\begin{array}{cc}& \mathcal{L}\triangleq \left\{f(·)||,f_d,({\xi }_d),-,f_d,(\widetilde{{\xi }_d}),\left|\le ,l_d\right|,{\xi }_d,-,\widetilde{{\xi }_d},\mid ,,,l_d,,\in ,,{\mathbb{R}}^{+}\right\},f(·),g(·)\in \mathcal{L}\text{represents}\text{that}|f_d\left({\xi }_d\right)-f_d(\widetilde{{\xi }_d})|\le l_d|{\xi }_d-\widetilde{{\xi }_d}|,\hfill \\ \hfill & |g_d\left({\xi }_d\right)-g_d(\widetilde{{\xi }_d})|\le {\tilde{l}}_d|{\xi }_d-\widetilde{{\xi }_d}|,{\tilde{l}}_d\in {\mathbb{R}}^{+},d=1,2,\dots ,n.\hfill \end{array}$$

Theorem 1

For $f(·)\in \mathcal{L},g(·)\in \mathcal{B}$, there exists positive real constant ${\lambda }_1,{\lambda }_2,{\lambda }_3$, ${\lambda }_4,{\lambda }_5,\alpha ,{\alpha }_c$, such that the QVNN(1) is a GEDS. Let

$$\begin{array}{c}\hfill \eta \triangleq \frac{p_c}{{\lambda }_1}-\frac{n}{{\lambda }_2}-\frac{n}{{\lambda }_3}-\frac{n}{{\lambda }_4}-\frac{1}{{\lambda }_5}-\alpha -{\alpha }_c-\sum _{d=1}^n{\lambda }_2{\overline{a}}_{\mathit{cd}}{a}_{\mathit{cd}}{l}_d^2,\end{array}$$

then the set

$$\begin{array}{c}\hfill {\mathrm{\Xi }}_1=\left\{\xi ,\in ,{\mathbb{Q}}^n,\mid ,{\overline{\xi }}_c,(t),{\xi }_c,(t),\le ,\frac{{\sum }_{d=1}^n{\lambda }_4{\overline{m}}_{\mathit{cd}}{m}_{\mathit{cd}}{k}_d^2+{\lambda }_5{\overline{I}}_c(t)I_c(t)}{\eta },,,c,,=,,1,,,2,,,,\dots ,,,,n,.\right\}\end{array}$$

is a GEAS of QVNN(1).

Proof

Construct the function

$$\begin{array}{c}\hfill V(t)=e^{\alpha t}\sum _{c=1}^n{\overline{\xi }}_c(t){\xi }_c(t)+\sum _{c=1}^n{\int }_{t-\mu (t)}^t{\alpha }_c{e}^{\alpha s}{\overline{\xi }}_c(s){\xi }_c(s)\text{d}s\text{,}\text{then}\end{array}$$

$$\begin{array}{cc}\hfill {\left(\frac{\text{d}V(t)}{\text{d}t}\right|}_{(1)}& =e^{\alpha t}\sum _{c=1}^n(\alpha {\overline{\xi }}_c(t){\xi }_c(t)+{\dot{\overline{\xi }}}_c(t){\xi }_c(t)+{\overline{\xi }}_c(t){\dot{\xi }}_c(t)+{\alpha }_c{\overline{\xi }}_c(t){\xi }_c(t)-{\alpha }_c{e}^{-\alpha \mu (t)}{\overline{\xi }}_c(t-\mu (t)){\xi }_c(t-\mu (t))(1-\dot{\mu }(t)))\hfill \\ \hfill & =e^{\alpha t}\sum _{c=1}^n(\alpha {\overline{\xi }}_c(t){\xi }_c(t)-(p_c{\overline{\xi }}_c(t-\iota )-\sum _{d=1}^n({\overline{f}}_d({\xi }_d(t)){\overline{a}}_{\mathit{cd}}-{\overline{f}}_d({\xi }_d(t-\mu (t))){\overline{b}}_{\mathit{cd}}-{\overline{g}}_d(\dot{{\xi }_d}(t-\omega (t))){\overline{m}}_{\mathit{cd}})-{\overline{I}}_c(t)){\xi }_c(t)\hfill \\ \hfill & +{\overline{\xi }}_c(t)(-p_c{\xi }_c(t-\iota )+\sum _{d=1}^n(a_{\mathit{cd}}{f}_d({\xi }_d(t))+b_{\mathit{cd}}{f}_d({\xi }_d(t-\mu (t)))+m_{\mathit{cd}}{g}_d(\dot{{\xi }_d}(t-\omega (t))))+I_c(t))\hfill \\ \hfill & +\sum _{c=1}^n{\alpha }_c{e}^{\alpha t}({\overline{\xi }}_c(t){\xi }_c(t)-e^{-\alpha \mu (t)}{\overline{\xi }}_c(t-\mu (t)){\xi }_c(t-\mu (t))(1-\dot{\mu }(t)))\hfill \\ \hfill & =e^{\alpha t}\sum _{c=1}^n(\alpha {\overline{\xi }}_c(t){\xi }_c(t)-p_c({\overline{\xi }}_c(t-\iota ){\xi }_c(t)+{\overline{\xi }}_c(t){\xi }_c(t-\iota ))+\sum _{d=1}^n({\overline{f}}_d({\xi }_d(t)){\overline{a}}_{\mathit{cd}}{\xi }_c(t)+{\overline{\xi }}_c(t)a_{\mathit{cd}}{f}_d({\xi }_d(t))\hfill \\ \hfill & +{\overline{f}}_d({\xi }_d(t-\mu (t))){\overline{b}}_{\mathit{cd}}{\xi }_c(t)+{\overline{\xi }}_c(t)b_{\mathit{cd}}{f}_d({\xi }_d(t-\mu (t)))+{\overline{g}}_d(\dot{{\xi }_d}(t-\omega (t))){\overline{m}}_{\mathit{cd}}{\xi }_c(t)+{\overline{\xi }}_c(t)m_{\mathit{cd}}{g}_d(\dot{{\xi }_d}(t-\omega (t))))\hfill \\ \hfill & +{\overline{I}}_c(t){\xi }_c(t)+{\overline{\xi }}_c(t)I_c(t))+\sum _{c=1}^n{\alpha }_c{e}^{\alpha t}({\overline{\xi }}_c(t){\xi }_c(t)-e^{-\alpha \mu (t)}{\overline{\xi }}_c(t-\mu (t)){\xi }_c(t-\mu (t))(1-\dot{\mu }(t)))\hfill \\ \hfill & \le e^{\alpha t}\sum _{c=1}^n(\alpha {\overline{\xi }}_c(t){\xi }_c(t)-p_c({\lambda }_1{\overline{\xi }}_c(t-\iota ){\xi }_c(t-\iota )+\frac{1}{{\lambda }_1}{\overline{\xi }}_c(t){\xi }_c(t))+\sum _{d=1}^n({\lambda }_2{\overline{f}}_d({\xi }_d(t)){\overline{a}}_{\mathit{cd}}{a}_{\mathit{cd}}{f}_d({\xi }_d(t))\hfill \\ \hfill & +\frac{1}{{\lambda }_2}{\overline{\xi }}_c(t){\xi }_c(t)+{\lambda }_3{\overline{f}}_d({\xi }_d(t-\mu (t))){\overline{b}}_{\mathit{cd}}{b}_{\mathit{cd}}{f}_d({\xi }_d(t-\mu (t)))+\frac{1}{{\lambda }_3}{\overline{\xi }}_c(t){\xi }_c(t)\hfill \\ \hfill & +{\lambda }_4{\overline{g}}_d(\dot{{\xi }_d}(t-\omega (t))){\overline{m}}_{\mathit{cd}}{m}_{\mathit{cd}}{g}_d(\dot{{\xi }_d}(t-\omega (t)))+\frac{1}{{\lambda }_4}{\overline{\xi }}_c(t){\xi }_c(t))+{\lambda }_5{\overline{I}}_c(t)I_c(t)+\frac{1}{{\lambda }_5}{\overline{\xi }}_c(t){\xi }_c(t))\hfill \\ \hfill & +\sum _{c=1}^n{\alpha }_c{e}^{\alpha t}({\overline{\xi }}_c(t){\xi }_c(t)-e^{-\alpha \mu (t)}{\overline{\xi }}_c(t-\mu (t)){\xi }_c(t-\mu (t))(1-\dot{\mu }(t)))\hfill \\ \hfill & =e^{\alpha t}\sum _{c=1}^n((-\frac{p_c}{{\lambda }_1}+\frac{n}{{\lambda }_2}+\frac{n}{{\lambda }_3}+\frac{n}{{\lambda }_4}+\frac{1}{{\lambda }_5}+\alpha +{\alpha }_c){\overline{\xi }}_c(t){\xi }_c(t)-p_c{\lambda }_1{\overline{\xi }}_c(t-\iota ){\xi }_c(t-\iota )\hfill \\ \hfill & +\sum _{d=1}^n({\lambda }_2{\overline{a}}_{\mathit{cd}}{a}_{\mathit{cd}}{\overline{f}}_d({\xi }_d(t))f_d({\xi }_d(t))+{\lambda }_3{\overline{b}}_{\mathit{cd}}{b}_{\mathit{cd}}{\overline{f}}_d({\xi }_d(t-\mu (t)))f_d({\xi }_d(t-\mu (t)))\hfill \\ \hfill & +{\lambda }_4{\overline{m}}_{\mathit{cd}}{m}_{\mathit{cd}}{\overline{g}}_d(\dot{{\xi }_d}(t-\omega (t)))g_d(\dot{{\xi }_d}(t-\omega (t))))+{\lambda }_5{\overline{I}}_c(t)I_c(t))\hfill \\ \hfill & -\sum _{c=1}^n{\alpha }_c{e}^{\alpha (t-\mu (t))}{\overline{\xi }}_c(t-\mu (t)){\xi }_c(t-\mu (t))(1-\dot{\mu }(t))\hfill \\ \hfill & \le e^{\alpha t}\sum _{c=1}^n((-\frac{p_c}{{\lambda }_1}+\frac{n}{{\lambda }_2}+\frac{n}{{\lambda }_3}+\frac{n}{{\lambda }_4}+\frac{1}{{\lambda }_5}+\alpha +{\alpha }_c){\overline{\xi }}_c(t){\xi }_c(t)+\sum _{d=1}^n({\lambda }_2{\overline{a}}_{\mathit{cd}}{a}_{\mathit{cd}}{l}_d^2{\overline{\xi }}_d(t){\xi }_d(t)\hfill \\ \hfill & +{\lambda }_3{\overline{b}}_{\mathit{cd}}{b}_{\mathit{cd}}{l}_d^2{\overline{\xi }}_d(t-\mu (t)){\xi }_d(t-\mu (t))+{\lambda }_4{\overline{m}}_{\mathit{cd}}{m}_{\mathit{cd}}{k}_d^2)+{\lambda }_5{\overline{I}}_c(t)I_c(t))\hfill \\ \hfill & -\sum _{c=1}^n{\alpha }_c{e}^{\alpha (t-\mu (t))}{\overline{\xi }}_c(t-\mu (t)){\xi }_c(t-\mu (t))(1-{\theta }_1).\hfill \end{array}$$

Let ${\sum }_{d=1}^n{\lambda }_3{\overline{b}}_{\mathit{cd}}{b}_{\mathit{cd}}{l}_d^2={\alpha }_c{e}^{-\alpha \mu (t)}(1-{\theta }_1)$, then

$$\begin{array}{c}\hfill {\alpha }_c=\frac{{\sum }_{d=1}^n{\lambda }_3{\overline{b}}_{\mathit{cd}}{b}_{\mathit{cd}}{l}_d^2}{e^{-\alpha \mu (t)}(1-{\theta }_1)},\end{array}$$

and

$$\begin{array}{cc}\hfill {\left(\frac{\text{d}V(t)}{\text{d}t}\right|}_{(1)}\le & e^{\alpha t}\sum _{c=1}^n((-\frac{p_c}{{\lambda }_1}+\frac{n}{{\lambda }_2}+\frac{n}{{\lambda }_3}+\frac{n}{{\lambda }_4}+\frac{1}{{\lambda }_5}+\alpha +{\alpha }_c\hfill \\ \hfill & +\sum _{d=1}^n{\lambda }_2{\overline{a}}_{\mathit{cd}}{a}_{\mathit{cd}}{l}_d^2){\overline{\xi }}_c(t){\xi }_c(t)+\sum _{d=1}^n{\lambda }_4{\overline{m}}_{\mathit{cd}}{m}_{\mathit{cd}}{k}_d^2+{\lambda }_5{\overline{I}}_c(t)I_c(t)).\hfill \end{array}$$

For $\xi (t)\notin {\mathrm{\Xi }}_1$, the following conclusion can be drawn

$$\begin{array}{c}\hfill {\left(\frac{\text{d}V(t)}{\text{d}t}\right|}_{(1)}\le 0,\end{array}$$

which implies that, for $\forall$ $t\ge t_0$, $\psi (s)\in {\mathrm{\Xi }}_1$, then $\xi (t)\subseteq {\mathrm{\Xi }}_1$. If $\psi (s)\notin {\mathrm{\Xi }}_1$, there exist $T>0$, for $\forall$ $t\ge T+t_0$, $\xi (t)\subseteq {\mathrm{\Xi }}_1$, that is, the QVNN(1) is GDS, and ${\mathrm{\Xi }}_1$ is a GAS.

Due to

$$\begin{array}{c}\hfill {\left(\frac{\text{d}V(t)}{\text{d}t}\right|}_{(1)}\le 0,\end{array}$$

one can obtain

$$\begin{array}{c}\hfill \sum _{c=1}^n{e}^{\alpha t}{\overline{\xi }}_c(t){\xi }_c(t)\le V(t)\le V\left(t_0\right)\text{,}\end{array}$$

implies that

$$\begin{array}{c}\hfill \sum _{c=1}^n{\overline{\xi }}_c(t){\xi }_c(t)\le M{e}^{-\alpha t},\end{array}$$

where $M=e^{\alpha t_0}{sup}_{t_0-\mu \le s\le t_0}V(s)$. One derives

$$\begin{array}{c}\hfill \underset{\tilde{\xi }\in {\mathrm{\Xi }}_1}{inf}\{\Vert \xi (t)-\tilde{\xi }\Vert \}\le M{e}^{-\alpha t}.\end{array}$$

By Definition 2, the QVNN(1) is a GEDS, and ${\mathrm{\Xi }}_1$ is a GEAS of the QVNN(1). $\square$

Theorem 2

If $f(·)$, $g(·)\in \mathcal{L}$, there are positive constants ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5$ and ${\alpha }_c,{\beta }_c,{\gamma }_c$, such that the QVNN(1) is a GDS. Let

$$\begin{array}{cc}\hfill {\eta }_1\triangleq & \frac{p_c}{{\lambda }_1}-\frac{n}{{\lambda }_2}-\frac{n}{{\lambda }_3}-\frac{n}{{\lambda }_4}-\frac{1}{{\lambda }_5}-\sum _{d=1}^n{\lambda }_2{\overline{a}}_{\mathit{cd}}{a}_{\mathit{cd}}{l}_d^2-{\beta }_c,\text{then}\hfill \end{array}$$

$$\begin{array}{c}\hfill {\mathrm{\Xi }}_2=\left\{\xi ,\in ,{\mathbb{Q}}^n,\mid ,{\overline{\xi }}_c,(t),{\xi }_c,(t),\le ,\frac{{\lambda }_5{\overline{I}}_c{I}_c}{{\eta }_1},,,c,,=,,1,,,2,,,,\dots ,,,,n,.\right\}\end{array}$$

is a GAS of the QVNN(1).

Proof

Considering the following function with positive constants ${\alpha }_c,{\beta }_c,{\gamma }_c$,

$$\begin{array}{cc}\hfill V(t)=& \sum _{c=1}^n({\overline{\xi }}_c(t){\xi }_c(t)+{\int }_0^{t-\iota }{\alpha }_c{\overline{f}}_c\left({\xi }_c,(s)\right)f_c\left({\xi }_c,(s)\right)\text{d}s+{\int }_{t-\mu (t)}^t{\beta }_c{\overline{\xi }}_c(s){\xi }_c(s)\text{d}s+{\int }_{t-\omega (t)}^0{\gamma }_c{\dot{\overline{\xi }}}_c(s){\dot{\xi }}_c(s)\text{d}s).\hfill \end{array}$$

By Lemma 2, one can obtain

$$\begin{array}{cc}\hfill {\left(\frac{\text{d}V(t)}{\text{d}t}\right|}_{(1)}& =\sum _{c=1}^n({\dot{\overline{\xi }}}_c(t){\xi }_c(t)+{\overline{\xi }}_c(t){\dot{\xi }}_c(t)+{\alpha }_c{\overline{f}}_c\left({\xi }_c,(t-\iota )\right)f_c\left({\xi }_c,(t-\iota )\right)+{\beta }_c{\overline{\xi }}_c(t){\xi }_c(t)\hfill \\ \hfill & -{\beta }_c{\overline{\xi }}_c(t-\mu (t)){\xi }_c(t-\mu (t))(1-\dot{\mu }(t))-{\gamma }_c{\dot{\overline{\xi }}}_c(t-\omega (t)){\dot{\xi }}_c(t-\omega (t))(1-\dot{\omega }(t)))\hfill \\ \hfill & =su{m}_{c=1}^n(-p_c{\overline{\xi }}_c(t-\iota )+\sum _{d=1}^n({\overline{f}}_d({\xi }_d(t)){\overline{a}}_{\mathit{cd}}+{\overline{f}}_d({\xi }_d(t-\mu (t))){\overline{b}}_{\mathit{cd}}+{\overline{g}}_d(\dot{{\xi }_d}(t-\omega (t))){\overline{m}}_{\mathit{cd}})\hfill \\ \hfill & +{\overline{I}}_c(t)){\xi }_c(t)+{\overline{\xi }}_c(t)(-p_c{\xi }_c(t-\iota )+\sum _{d=1}^n(a_{\mathit{cd}}{f}_d({\xi }_d(t))+b_{\mathit{cd}}{f}_d({\xi }_d(t-\mu (t)))\hfill \\ \hfill & +m_{\mathit{cd}}{g}_d(\dot{{\xi }_d}(t-\omega (t))))+I_c(t))+{\alpha }_c{\overline{f}}_c\left({\xi }_c,(t-\iota )\right)f_c\left({\xi }_c,(t-\iota )\right)+{\beta }_c{\overline{\xi }}_c(t){\xi }_c(t)\hfill \\ \hfill & -{\beta }_c{\overline{\xi }}_c(t-\mu (t)){\xi }_c(t-\mu (t))(1-\dot{\mu }(t))-{\gamma }_c{\dot{\overline{\xi }}}_c(t-\omega (t)){\dot{\xi }}_c(t-\omega (t))(1-\dot{\omega }(t)))\hfill \\ \hfill & =\sum _{c=1}^n(-p_c({\overline{\xi }}_c(t-\iota ){\xi }_c(t)+{\overline{\xi }}_c(t){\xi }_c(t-\iota ))+\sum _{d=1}^n({\overline{f}}_d({\xi }_d(t)){\overline{a}}_{\mathit{cd}}{\xi }_c(t)+{\overline{\xi }}_c(t)a_{\mathit{cd}}{f}_d({\xi }_d(t))\hfill \\ \hfill & +{\overline{f}}_d({\xi }_d(t-\mu (t))){\overline{b}}_{\mathit{cd}}{\xi }_c(t)+{\overline{\xi }}_c(t)b_{\mathit{cd}}{f}_d({\xi }_d(t-\mu (t)))+{\overline{g}}_d(\dot{{\xi }_d}(t-\omega (t))){\overline{m}}_{\mathit{cd}}{\xi }_c(t)\hfill \\ \hfill & +{\overline{\xi }}_c(t)m_{\mathit{cd}}{g}_d(\dot{{\xi }_d}(t-\omega (t))))+{\overline{I}}_c(t){\xi }_c(t)+{\overline{\xi }}_c(t)I_c(t)+{\alpha }_c{\overline{f}}_c({\xi }_c(t-\iota ))f_c\left({\xi }_c,(t-\iota )\right)\hfill \\ \hfill & +{\beta }_c{\overline{\xi }}_c(t){\xi }_c(t)-{\beta }_c{\overline{\xi }}_c(t-\mu (t)){\xi }_c(t-\mu (t))(1-\dot{\mu }(t))-{\gamma }_c{\dot{\overline{\xi }}}_c(t-\omega (t)){\dot{\xi }}_c(t-\omega (t))(1-\dot{\omega }(t)))\hfill \\ \hfill & \le \sum _{c=1}^n(-p_c({\lambda }_1{\overline{\xi }}_c(t-\iota ){\xi }_c(t-\iota )+\frac{1}{{\lambda }_1}{\overline{\xi }}_c(t){\xi }_c(t))+\sum _{d=1}^n({\lambda }_2{\overline{f}}_d({\xi }_d(t)){\overline{a}}_{\mathit{cd}}{a}_{\mathit{cd}}{f}_d({\xi }_d(t))\hfill \\ \hfill & +\frac{1}{{\lambda }_2}{\overline{\xi }}_c(t){\xi }_c(t)+{\lambda }_3{\overline{f}}_d({\xi }_d(t-\mu (t))){\overline{b}}_{\mathit{cd}}{b}_{\mathit{cd}}{f}_d({\xi }_d(t-\mu (t)))+\frac{1}{{\lambda }_3}{\overline{\xi }}_c(t){\xi }_c(t)\hfill \\ \hfill & +{\lambda }_4{\overline{g}}_d(\dot{{\xi }_d}(t-\omega (t))){\overline{m}}_{\mathit{cd}}{m}_{\mathit{cd}}{g}_d(\dot{{\xi }_d}(t-\omega (t)))+\frac{1}{{\lambda }_4}{\overline{\xi }}_c(t){\xi }_c(t))\hfill \\ \hfill & +{\lambda }_5{\overline{I}}_c(t)I_c(t)+\frac{1}{{\lambda }_5}{\overline{\xi }}_c(t){\xi }_c(t)+{\beta }_c{\overline{\xi }}_c(t){\xi }_c(t)+{\alpha }_c{\overline{f}}_c({\xi }_c(t-\iota ))f_c\left({\xi }_c,(t-\iota )\right)\hfill \\ \hfill & -{\beta }_c{\overline{\xi }}_c(t-\mu (t)){\xi }_c(t-\mu (t))(1-\dot{\mu }(t))-{\gamma }_c{\dot{\overline{\xi }}}_c(t-\omega (t)){\dot{\xi }}_c(t-\omega (t))(1-\dot{\omega }(t)))\hfill \\ \hfill & =\sum _{c=1}^n((\frac{-p_c}{{\lambda }_1}+\frac{n}{{\lambda }_2}+\frac{n}{{\lambda }_3}+\frac{n}{{\lambda }_4}+\frac{1}{{\lambda }_5}+{\beta }_c){\overline{\xi }}_c(t){\xi }_c(t)-p_c{\lambda }_1{\overline{\xi }}_c(t-\iota ){\xi }_c(t-\iota )\hfill \\ \hfill & +\sum _{d=1}^n({\lambda }_2{\overline{f}}_d({\xi }_d(t)){\overline{a}}_{\mathit{cd}}{a}_{\mathit{cd}}{f}_d({\xi }_d(t))+{\lambda }_3{\overline{f}}_d({\xi }_d(t-\mu (t))){\overline{b}}_{\mathit{cd}}{b}_{\mathit{cd}}{f}_d({\xi }_d(t-\mu (t)))\hfill \\ \hfill & +{\lambda }_4{\overline{g}}_d(\dot{{\xi }_d}(t-\omega (t))){\overline{m}}_{\mathit{cd}}{m}_{\mathit{cd}}{g}_d(\dot{{\xi }_d}(t-\omega (t))))+{\lambda }_5{\overline{I}}_c(t)I_c(t)+{\alpha }_c{\overline{f}}_c({\xi }_c(t-\iota ))f_c\left({\xi }_c,(t-\iota )\right)\hfill \\ \hfill & -{\beta }_c{\overline{\xi }}_c(t-\mu (t)){\xi }_c(t-\mu (t))(1-\dot{\mu }(t))-{\gamma }_c{\dot{\overline{\xi }}}_c(t-\omega (t)){\dot{\xi }}_c(t-\omega (t))(1-\dot{\omega }(t)))\hfill \\ \hfill & \le \sum _{c=1}^n((\frac{-p_c}{{\lambda }_1}+\frac{n}{{\lambda }_2}+\frac{n}{{\lambda }_3}+\frac{n}{{\lambda }_4}+\frac{1}{{\lambda }_5}+{\beta }_c){\overline{\xi }}_c(t){\xi }_c(t)-p_c{\lambda }_1{\overline{\xi }}_c(t-\iota ){\xi }_c(t-\iota )\hfill \\ \hfill & +\sum _{d=1}^n({\lambda }_2{\overline{a}}_{\mathit{cd}}{a}_{\mathit{cd}}{l}_d^2{\overline{\xi }}_d(t){\xi }_d(t)+{\lambda }_3{\overline{b}}_{\mathit{cd}}{b}_{\mathit{cd}}{l}_d^2{\overline{\xi }}_d(t-\mu (t)){\xi }_d(t-\mu (t))\hfill \\ \hfill & +{\lambda }_4{\overline{m}}_{\mathit{cd}}{m}_{\mathit{cd}}{k}_d^2{\dot{\overline{\xi }}}_d(t-\omega (t))\dot{{\xi }_d}(t-\omega (t)))+{\lambda }_5{\overline{I}}_c(t)I_c(t)+{\alpha }_c{\overline{f}}_c({\xi }_c(t-\iota ))f_c\left({\xi }_c,(t-\iota )\right)\hfill \\ \hfill & -{\beta }_c{\overline{\xi }}_c(t-\mu (t)){\xi }_c(t-\mu (t))(1-\dot{\mu }(t))-{\gamma }_c{\dot{\overline{\xi }}}_c(t-\omega (t)){\dot{\xi }}_c(t-\omega (t))(1-\dot{\omega }(t)))\hfill \\ \hfill & \le \sum _{c=1}^n((\frac{-p_c}{{\lambda }_1}+\frac{n}{{\lambda }_2}+\frac{n}{{\lambda }_3}+\frac{n}{{\lambda }_4}+\frac{1}{{\lambda }_5}+\sum _{d=1}^n{\lambda }_2{\overline{a}}_{\mathit{cd}}{a}_{\mathit{cd}}{l}_d^2+{\beta }_c){\overline{\xi }}_c(t){\xi }_c(t)\hfill \\ \hfill & +{\lambda }_5{\overline{I}}_c(t)I_c(t)-p_c{\lambda }_1{\overline{\xi }}_c(t-\iota ){\xi }_c(t-\iota )+{\alpha }_c{l}_c^2{\overline{\xi }}_c(t-\iota ){\xi }_c(t-\iota )\hfill \\ \hfill & +\sum _{d=1}^n({\lambda }_3{\overline{b}}_{\mathit{cd}}{b}_{\mathit{cd}}{l}_d^2{\mathrm{\Delta }}_d(t-\mu (t))+{\lambda }_4{\overline{m}}_{\mathit{cd}}{m}_{\mathit{cd}}{k}_d^2{\dot{\overline{\xi }}}_d(t-\omega (t))\dot{{\xi }_d}(t-\omega (t)))\hfill \\ \hfill & -{\beta }_c{\overline{\xi }}_c(t-\mu (t)){\xi }_c(t-\mu (t))(1-{\theta }_1)-{\gamma }_c{\dot{\overline{\xi }}}_c(t-\omega (t)){\dot{\xi }}_c(t-\omega (t))(1-{\theta }_2)).\hfill \end{array}$$

Let $p_c{\lambda }_1={\alpha }_c{l}_c^2,{\sum }_{d=1}^n{\lambda }_3{\overline{b}}_{\mathit{cd}}{b}_{\mathit{cd}}{l}_d^2={\beta }_c(1-{\theta }_1),{\sum }_{d=1}^n{\lambda }_4{\overline{m}}_{\mathit{cd}}{m}_{\mathit{cd}}{k}_d^2={\gamma }_c(1-{\theta }_2)$,then

$$\begin{array}{c}\hfill {\alpha }_c=\frac{p_c{\lambda }_1}{l_c^2},{\beta }_c=\frac{{\sum }_{d=1}^n{\lambda }_3{\overline{b}}_{\mathit{cd}}{b}_{\mathit{cd}}{l}_d^2}{(1-{\theta }_1)},{\gamma }_c=\frac{{\sum }_{d=1}^n{\lambda }_4{\overline{m}}_{\mathit{cd}}{m}_{\mathit{cd}}{k}_d^2}{(1-{\theta }_2)}.\end{array}$$

When $\xi (t)\in {\mathbb{Q}}^n\backslash {\mathrm{\Xi }}_2$, the following inequality holds:

$$\begin{array}{cc}\hfill {\left(\frac{\text{d}V(t)}{\text{d}t}\right|}_{(1)}\le & \sum _{c=1}^n(\frac{-p_c}{{\lambda }_1}+\frac{n}{{\lambda }_2}+\frac{n}{{\lambda }_3}+\frac{n}{{\lambda }_4}+\frac{1}{{\lambda }_5}+\sum _{d=1}^n{\lambda }_2{\overline{a}}_{\mathit{cd}}{a}_{\mathit{cd}}{l}_d^2+{\beta }_c){\overline{\xi }}_c(t){\xi }_c(t)+{\lambda }_5{\overline{I}}_c(t)I_c(t))\le 0.\hfill \end{array}$$ 2

Accordingly, for $\forall$ $t\ge t_0$, if $\psi (s)\in {\mathrm{\Xi }}_2$, then $\xi (t)\subseteq {\mathrm{\Xi }}_2$. If $\psi (s)\notin {\mathrm{\Xi }}_2$, $\exists$ $T>0$, for $\forall$ $t\ge T+t_0,\xi (t)\subseteq {\mathrm{\Xi }}_2$, then ${\mathrm{\Xi }}_2$ is a GAS of (1), the QVNN(1) is a GDS. $\square$

Discussion of results

In the previous section, we discussed and proved in detail the global exponential dissipation and global dissipation of QVNN(1) when the activation function satisfies different conditions. In fact, QVNN(1) can be similarly verified to be globally dissipative when the conditions of the activation function change to other conditions.

Corollary 1

If $f(·)\in \mathcal{L}$, $g(·)\in \mathcal{B}$, and $\exists$ ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5,{\alpha }_c,{\beta }_c$, let

$$\begin{array}{c}\hfill {\eta }_2\triangleq \frac{p_c}{{\lambda }_1}-\frac{n}{{\lambda }_2}-\frac{n}{{\lambda }_3}-\frac{n}{{\lambda }_4}-\frac{1}{{\lambda }_5}-\sum _{d=1}^n{\lambda }_2{\overline{a}}_{\mathit{cd}}{a}_{\mathit{cd}}{l}_d^2-{\beta }_c,\end{array}$$

then the set

$$\begin{array}{c}\hfill {\mathrm{\Xi }}_3=\left\{\xi ,\in ,{\mathbb{Q}}^n,\mid ,{\overline{\xi }}_c,(t),{\xi }_c,(t),\le ,\frac{{\sum }_{d=1}^n{\lambda }_4{\overline{m}}_{\mathit{cd}}{m}_{\mathit{cd}}{k}_c^2+{\lambda }_5{\overline{I}}_c{I}_c}{{\eta }_2}\right\}\end{array}$$

is a GAS of QVNN(1), and QVNN(1) is a GDS.

Proof

Construct the function

$$\begin{array}{c}\hfill V(t)=\sum _{c=1}^n({\overline{\xi }}_c(t){\xi }_c(t)+{\int }_0^{t-\iota }{\alpha }_c{\overline{f}}_c\left({\xi }_c,(s)\right)f_c\left({\xi }_c,(s)\right)\text{d}s+{\int }_{t-\mu (t)}^t{\beta }_c{\overline{\xi }}_c(s){\xi }_c(s)\text{d}s),\end{array}$$

following Theorem 2, the conclusion can be drawn. $\square$

Corollary 2

If $f(·)\in \mathcal{B}$, $g(·)\in \mathcal{B}$, and there exist positive constants ${\lambda }_1,{\lambda }_2,{\lambda }_3$, ${\lambda }_4,{\lambda }_5$, let

$$\begin{array}{cc}& \zeta \triangleq \sum _{d=1}^n({\lambda }_2{\overline{a}}_{\mathit{cd}}{a}_{\mathit{cd}}{l}_d^2+{\lambda }_3{\overline{b}}_{\mathit{cd}}{b}_{\mathit{cd}}{l}_d^2+{\lambda }_4{\overline{m}}_{\mathit{cd}}{m}_{\mathit{cd}}{k}_d^2)+{\lambda }_5{\overline{I}}_c{I}_c,\hfill \\ \hfill & {\eta }_3\triangleq \frac{p_c}{{\lambda }_1}-\frac{n}{{\lambda }_2}-\frac{n}{{\lambda }_3}-\frac{n}{{\lambda }_4}-\frac{1}{{\lambda }_5},\hfill \end{array}$$

then the set

$$\begin{array}{c}\hfill {\mathrm{\Xi }}_4=\left\{\xi ,\in ,{\mathbb{Q}}^n,\mid ,{\overline{\xi }}_c,(t),{\xi }_c,(t),\le ,\frac{\zeta }{{\eta }_3}\right\}\end{array}$$

is a GAS of QVNNs(1), and QVNN(1) is a GDS.

Proof

Construct the function

$$\begin{array}{c}\hfill V(t)=\sum _{c=1}^n{\overline{\xi }}_c(t){\xi }_c(t),\end{array}$$

following Theorem 2, the conclusion can be drawn. $\square$

Illustrative examples

Two simulation examples are given to test the validity of the theorems.

Example 1

Constructing the neural networks model:

$$\begin{array}{cc}\hfill \frac{\text{d}{\xi }_c(t)}{\text{d}t}=& -p_c{\xi }_c(t-\iota )+\sum _{d=1}^2(a_{\mathit{cd}}{f}_d({\xi }_d(t))+b_{\mathit{cd}}{f}_d({\xi }_d(t-\mu (t)))+m_{\mathit{cd}}{g}_d(\dot{{\xi }_d}(t-\omega (t))))+I_c(t),c=1,2,\hfill \end{array}$$ 3

$$\begin{array}{cc}& \text{where}{p}_1=9,p_2=9,a_{11}=0.3+0.3i-0.1j+1.05k,a_{12}=0.1-0.3i-1.05j-1.01k\text{,}{a}_{21}\hfill \\ \hfill & =0.2+0.2i+0.1j+1.01k,a_{22}\hfill \\ \hfill & =0.4+0.2i+1.03j+1.05k,b_{11}=0.5+1.02i+0.8j+1.01k,b_{12}=0.8+1.03i-0.6j\hfill \\ \hfill & +1.01k,b_{21}=0.3-1.01i-0.4j-0.1k,b_{22}\hfill \\ \hfill & =1.2+1.3i+0.7j+1.02k,m_{11}=0.6+1.05i+1.02j\hfill \\ \hfill & +1.15k,m_{12}=1.02+1.05i-1.04j+1.06k,m_{21}=1.02+1.05i\hfill \\ \hfill & +1.04j+1.06k,m_{22}=0.1+1.2i+1.01j-1.03k,\mu (t)=0.45{sin}^{2}t+0.55,\omega (t)=0.15{cos}^{2}t\hfill \\ \hfill & +0.85,I_1(t)=(sin2t;cos2t;cos3t;\hfill \\ \hfill & {sin3t)}^T,I_2(t)={(sin2t;cos2t;cos2t;sin2t)}^T,f_d({\xi }_d)=0.1(|{\xi }^{(r)}|+|{\xi }^{(i)}|i+|{\xi }^{(j)}|j+|{\xi }^{(k)}|k),g_d\left({\xi }_d\right)\hfill \\ \hfill & =\frac{1}{1+e^{-{\xi }_d^{(r)}}}+\frac{1}{1+e^{-{\xi }_d^{(i)}}}i\hfill \\ \hfill & +\frac{1}{1+e^{-{\xi }_d^{(j)}}}j+\frac{1}{1+e^{-{\xi }_d^{(k)}}}k,d=1,2.\hfill \end{array}$$

Choosing ${\lambda }_1={\lambda }_2={\lambda }_3={\lambda }_4={\lambda }_5=1$, $\alpha =0.1,\iota =0.3$. Obviously, ${\theta }_1=0.45,{\theta }_2=0.15$, $l_1^2=l_2^2=0.01$, $k_1^2=k_2^2=4$. According to Theorem 1, QVNN(2) is a GEDS,

$$\begin{array}{c}\hfill {\mathrm{\Xi }}_1=\left\{\xi ,\in ,{\mathbb{Q}}^2,\mid ,\Vert \xi \Vert ,\le ,6.4332\right\}\end{array}$$

is a GEAS of (2). Figures 1 and 2 show the simulation results of QVNN(2).

Figure 1.

Transient behavior trajectories of QVNN (2).

Figure 2.

Phase trajectories of QVNN (2).

Example 2

Considering the networks model:

$$\begin{array}{cc}\hfill \frac{\text{d}{\xi }_c(t)}{\text{d}t}=& -p_c{\xi }_c(t-\iota )+\sum _{d=1}^3(a_{\mathit{cd}}{f}_d({\xi }_d(t))+b_{\mathit{cd}}{f}_d({\xi }_d(t-\mu (t)))+m_{\mathit{cd}}{g}_d(\dot{{\xi }_d}(t-\omega (t))))+I_c(t),c=1,2,3,\hfill \end{array}$$ 4

$$\begin{array}{cc}& \text{where}{p}_1=15,p_2=15,p_3=15,a_{11}=0.3-0.1i+0.1j+0.3k,a_{12}\hfill \\ \hfill & =0.1-1.03i+1.05j+0.1k,a_{13}=0.3+0.1i+1.15j\hfill \\ \hfill & +0.1k,a_{21}=0.2+0.2i-1.03j+0.2k,a_{22}=0.4+0.4i+1.03j+1.25k,a_{23}\hfill \\ \hfill & =0.4-0.3i-1.05j+0.4k,a_{31}=0.2-0.3i\hfill \\ \hfill & +1.1j+0.2k,a_{32}=0.2+0.2i-1.03j+1.15k,a_{33}=0.2+0.3i-0.4j+1.3k,b_{11}\hfill \\ \hfill & =0.2-0.3i-0.2j-1.01k,b_{12}=0.8\hfill \\ \hfill & +0.3i+1.1j+1.1k,b_{13}=0.3-0.2i-1.15j+0.1k,b_{21}=0.3-0.6i-0.3j-1.2k,b_{22}\hfill \\ \hfill & =0.2-0.4i+1.1j+0.2k,b_{23}=0.3\hfill \\ \hfill & +1.01i+1.15j+1.02k,b_{31}=0.3-0.2i+1.01j+0.2k,b_{32}=0.5+1.2i+1.13j+0.5k,b_{33}\hfill \\ \hfill & =0.4-0.3i+1.1j+0.3k,m_{11}\hfill \\ \hfill & =0.1+1.5i+1.02j+0.2k,m_{12}=-1.02-0.5i+1.4j+1.1k,m_{13}=0.3-1.1i+1.15j+0.3k,m_{21}\hfill \\ \hfill & =-1.2+0.5i+1.04j\hfill \\ \hfill & +0.6k,m_{22}=0.1+1.1i+1.02j+0.3k,m_{23}=1.3+0.2i+1.15j+0.1k,m_{31}\hfill \\ \hfill & =1.2-0.4i+1.1j-1.2k,m_{32}=0.1+1.2i\hfill \\ \hfill & +0.3j+1.05k,m_{33}=-0.4+0.3i+0.05j+0.6k,\mu (t)=0.15cost+0.85,\omega (t)=0.25cost+0.75,I_1\hfill \\ \hfill & ={(1.08;0.5;0.4;0.5)}^T,I_2={(0.6;0.4;-0.5;0.5)}^T,I_3={(0.5;0.2;-0.1;0.5)}^T,f_d({\xi }_d)=g_d\left({\xi }_d\right)\hfill \\ \hfill & =0.1(|{\xi }^{(r)}|+|{\xi }^{(i)}|i+|{\xi }^{(j)}|j+|{\xi }^{(k)}|k),d=1,2,3.\hfill \end{array}$$

Choosing ${\lambda }_1={\lambda }_2={\lambda }_3={\lambda }_4={\lambda }_5=1,\iota =0.5$. Obviously, ${\theta }_1=0.15,{\theta }_2=0.25$, $l_1^2=l_2^2=k_1^2=k_2^2=0.01$. Figures 3 and 4show the simulation results of QVNN(3). According to Theorem 2, the QVNN(3) is a GDS, and

$$\begin{array}{c}\hfill {\mathrm{\Xi }}_2=\left\{\xi ,\in ,{\mathbb{Q}}^3,\mid ,\Vert \xi \Vert ,\le ,0.8342\right\}\end{array}$$

is a GAS of (3).

Figure 3.

Transient behavior trajectories of QVNN (3).

Figure 4.

Phase trajectories of QVNN (3).

Remark 2

Via the theorems and simulation results, systems can reach stability under different initial value conditions, time-varying delay, external input, and exist a GAS.

Conclusions

This paper researched the GAS of the neutral type QVNNs with time-varying and leakage delay. The QVNNs are studied by direct methods, rather than decomposition into real or complex numbers. It makes the calculation more convenient. Traditional assumptions on neutral type activation function are removed, which makes the results more pragmatic. Four new classes of Lyapunov functions are constructed by extended inequalities and Lyapunov theory to find the GAS of QVNNs. Systems can reach stability under leakage delay and additive delay. The concrete GAS estimation is given by simulations.

The estimation of GAS and GEAS for QVNNs with different time delays will be further considered in the following work. This examination holds significant importance in understanding the dynamical behavior of QVNNs and how their stability properties are influenced by temporal factors. By thoroughly analyzing QVNNs with differing time delays, develop a comprehensive framework for estimating stability metrics, which can be utilized in designing more robust and efficient QVNNs for a wide range of applications. This work will not only enhance our theoretical understanding of QVNNs but also contribute to the practical implementation of these networks in real-world scenarios.

Author contributions

Xili Wu conceived the manuscript. All authors contributed to manuscript preparation, and revised the manuscript.

Funding

This study was supported by by the Science and Technology Project of Chongqing Education Commission (Nos.KJQN202301214, KJQN202101212, KJQN202101220, KJZD-M202001201, KJZD-M202301202), the Natural Science Foundation Project of Chongqing(Nos.cstc2021jcyj-msxmX0051,CSTB2022NSCQ-MSX0393).

Data availability

In this paper, numerical simulation is used to carry out the experiment, and all the data have been presented in the manuscript.

Competing interests

The authors declare no competing interests.