Exponential synchronization of quaternion-valued memristor-based Cohen–Grossberg neural networks with time-varying delays: norm method

Abstract

In this paper, the exponential synchronization of quaternion-valued memristor-based Cohen–Grossberg neural networks with time-varying delays is discussed. By using the differential inclusion theory and the set-valued map theory, the discontinuous quaternion-valued memristor-based Cohen–Grossberg neural networks are transformed into an uncertain system with interval parameters. A novel controller is designed to achieve the control goal. With some inequality techniques, several criteria of exponential synchronization for quaternion-valued memristor-based Cohen–Grossberg neural networks are given. Different from the existing results using decomposition techniques, a direct analytical approach is used to study the synchronization problem by introducing an improved one-norm method. Moreover, the activation function is less restricted and the Lyapunov analysis process is simpler. Finally, a numerical simulation is given to prove the validity of the main results.

Introduction

In 1971, Professor Leon O. Chua (1971) of the University of California, Berkeley proposed the memristor, which is a combination of the words “Memory” and “Resistor”. He believes that resistance, capacitance and inductance represent the relationship between the four important elements of voltage, current, charge and magnetic flux in electronics. But the element that represents the relationship between charge and magnetic flux doesn’t exist yet, so he named it the memristor. In 2008, researchers from HP Company (Strukov et al. 2008) made the first nano memristor device, which sparked a wave of in memristor research. Previous studies have shown that memristor has the synaptic function of neurons in the human brain (Pershin and Ventra 2010; Merrikh-Bayat and Shouraki 2011). Due to this special characteristic, memristor is used to create new neural networks to simulate the brain. So, the memristor-based neural networks become the focus of research (Wei et al. 2020; Chen et al. 2020; Zheng et al. 2018; Shi et al. 2017; Di Marco et al. 2018; Zheng et al. 2018; Cheng and Shi 2022).

Quaternion is a mathematical concept invented by Hamilton in 1843. Compared with complex-valued neural networks, the state, activation function and connection weight of quaternion-valued neural network are all quaternion. According to Hamilton’s rule, the multiplication of quaternion is not commutative. Obviously, many of the conclusions that hold for real and complex numbers no longer hold for quaternion, which also makes the study of quaternion more difficult and stagnated for quite a long time. In recent years, the application field of quaternion has been continuously expanded and achieved relatively ideal results. Currently, it mainly involves image processing (Hua et al. 2015), aerospace industry technology (Goodman 1977), artificial intelligence (Bhatti et al. 2020), quantum mechanics (Hasan and Mandal 2020) and other fields has been found, and some achievements have been made.

Especially, quaternion has broad application prospects and advantages in data modeling and processing of three-dimensional and four-dimensional space. For example, three imaginary parts of a pure quaternion are used to represent the three primary colors (RGB) in color space. In this way, the overall processing of color images can be realized without RGB separate processing, which can reflect the relevance between the three colors well. Research results show that the quaternion-valued neural networks exhibit better performance in image compression (Isokawa et al. 2003), 3D wind prediction (Ujang et al. 2011), color night vision (Luo et al. 2010), and other aspects.

In recent years, many scholars have studied the dynamics of quaternion-valued neural networks (Hu et al. 2022; Wei et al. 2023; Wang et al. 2023; Duan et al. 2020; Song et al. 2022; Zhang and Jian 2022; Wei and Cao 2019; Huang et al. 2021; Li et al. 2023; Xu et al. 2023, 2022). For example, the stability of fractional-order quaternion-valued neural networks was investigated in Hu et al. (2022); Wang et al. (2023). Wei et al. (2023) discussed the fixed-time control of the memristor-based quaternion-valued neural networks. Duan et al. (2020) studied the stability of a class of quaternion-valued neural networks with discrete and distributed delays. Song et al. (2022) investigated the stochastic quaternion-valued neural networks model with variable coefficients and neutral delays. In Zhang and Jian (2022), Liu et al. studied the exponential synchronization of quaternion-valued memristive delayed neural networks by quantized intermittent control. Wei and Cao (2019) researched the fixed-time synchronization program of quaternion-valued memristive neural networks. The authors Xu et al. discussed bifurcation mechanism for fractional-order three-triangle multi-delayed neural networks. Delay-induced periodic oscillation for fractional-order neural networks with mixed delays was studied in Xu et al. (2022).

As the existence of cellular neural networks, bidirectional associative memory neural networks and Hopfield neural networks promotion. Therefore, once the Cohen–Grossberg neural networks was proposed, researchers have paid more and more attention to it and obtained some interesting results. In Zhang et al. (2019), studied the adaptive synchronization problem of delayed Cohen–Grossberg inertial neural networks. Kong et al. investigated the fixed-time synchronization problem of Cohen–Grossberg drive-response neural networks with discontinuous neuron activations, in Kong et al. (2019). Shi and Cao (2020) studied the finite-time synchronization of memristor-based Cohen–Grossberg neural networks with time-varying delays. In 2022, Zhang et al. (2022) studied the global exponential stability of neutral-type Cohen–Grossberg neural networks with multiple time-varying discrete and neutral delays. Cheng and Shi (2023) discussed the exponential synchronization and asymptotic synchronization of Quaternion-Valued Memristor-Based Cohen-Grossberg neural networks.

At present, the research of Cohen–Grossberg neural networks mainly focuses on the real and complex domain, and the research of quaternion field is relatively few, which is a very challenging new subject for us. This is the motivation for our current study. Based on the above analysis, exponential synchronization of quaternion-valued memristor-based Cohen–Grossberg neural networks with time-varying delays is discussed in this paper. The main contributions can be summarized as follows:

• (i)Compared with real-valued neural networks and complex-valued neural networks, the significant advantages of quaternion-valued neural networks are their low-dimensionality and high efficiency in processing multi-dimensional data.
• (ii)Different from reference (Li et al. 2019), this paper doesn’t divide the quaternion-valued memristor-based Cohen–Grossberg neural networks into real and imaginary parts. We use a direct analytical approach to study the synchroniza-tion problem by introducing an improved one-norm, which simplifies the proof process.

This paper is organized as follows. In Sect. 2, we propose the quaternion-valued memristor-based Cohen–Grossberg neural networks with time-varying delays. In the third section, with some inequality techniques, the criterion of global exponential synchronization and global asymptotic synchronization for quaternion-valued memristor-based Cohen–Grossberg neural networks is given. In the fourth section, a numerical simulation experiment is given.

Notations: The notation used in this article is fairly standard. In this paper, let $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{Q}$ represent real numbers, complex numbers, and quaternion respectively. $m=m^R+m^{I}i+m^{J}j+m^{K}k\in \mathbb{Q}$, $m^R,m^I,m^J,m^K\in \mathbb{R}$. Let $m^T$ represent the transpose of m. The conjugate and the conjugate transpose of m are represented by $\overline{m}=m^R-m^{I}i-m^{J}j-m^{K}k$ and $m^{\ast }={(m^R-m^{I}i-m^{J}j-m^{K}k)}^T$. The modulus of m is written as

$$\begin{array}{c}\hfill |m|=\sqrt{m\overline{m}}=\sqrt{{\left(m^R\right)}^2+{\left(m^l\right)}^2+{\left(m^l\right)}^2+{\left(m^K\right)}^2}.\end{array}$$

Furthermore, for $m={(m_1,m_2,\dots ,m_n)}^T$, ${\Vert m\Vert }_1={\sum }_{i=1}^n|m_i|$ denotes the 1-norm of m. And co(A) represents the closure of the convex hull of A.

Problem description and preliminaries

The quaternion $m\in \mathbb{Q}$ can be described as:

$$\begin{array}{c}\hfill m=m^R+m^{I}i+m^{J}j+m^{K}k,\end{array}$$

where $m^R,m^I,m^J,m^K\in \mathbb{R}$, the imaginary parts i, j, k obey the Hamilton rule:

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

For the quaternion $m,n\in \mathbb{Q}$, where $n=n^R+n^{I}i+n^{J}j+n^{K}k$, then we can denote $m+n$ as follows

$$\begin{array}{c}\hfill m+n=m^R+n^R+\left(m^I,+,n^I\right)i+\left(m^J,+,n^J\right)j+\left(m^K,+,n^K\right)k.\end{array}$$

By Hamilton rule, we can denote mn as:

$$\begin{array}{cc}\hfill mn=& \left(m^R,n^R,-,m^I,n^I,-,m^J,n^J,-,m^K,n^K\right)\hfill \\ \hfill & +\left(m^R,n^I,+,m^I,n^R,+,m^J,n^K,-,m^K,n^J\right)i\hfill \\ \hfill & +\left(m^R,n^J,+,m^J,n^R,+,m^K,n^J,-,m^I,n^K\right)j\hfill \\ \hfill & +\left(m^R,n^K,+,m^K,n^R,+,m^I,n^J,-,m^J,n^I\right)k.\hfill \end{array}$$

In this paper, we consider a quaternion-valued memristor-based Cohen–Grossberg neural network with time-varying delays, which can be described by the following delayed differential equations.

$$\begin{array}{cc}\hfill {\dot{m}}_q(t)=& -a_q(m_q(t))(b_q(m_q(t))-\sum _{s=1}^n{c}_{\mathit{qs}}(m_q(t))f_s(m_s(t))\hfill \\ \hfill & -\sum _{s=1}^n{d}_{\mathit{qs}}(m_q(t))f_s(m_s(t-\tau (t)))-I_q),\hfill \end{array}$$ 1

for $q=1,2,\cdots ,n$, where $m_q(t)\in \mathbb{Q}$ stands for the state vector of the neuron. $a_q(m_q(t))\in \mathbb{Q}$ represent the amplification function, and $b_q(m_q(t))\in \mathbb{Q}$ stand for the behaved function. $f_s(·)$ denotes the activation function. $I_q$ is the external input. $\tau (t)$ is the transmission delay, which satisfies $0\le \tau (t)<\tau (\tau \ge 0)$ and $\dot{\tau }(t)\le \xi \le 1$. The initial condition of system (1) is chosen to be $m_q(s)={\psi }_q(s),-\tau \le s\le 0$, where ${\psi }_q(s)={({\psi }_1(s),{\psi }_2(s),\cdots ,{\psi }_n(s))}^T\in ([-\tau ,0],{\mathbb{Q}}^n)$. $c_{\mathit{qs}}(m_q(t))$ and $d_{\mathit{qs}}(m_q(t))$ represent the quaternion-valued memristor-based connection weights satisfying the following conditions:

$$\begin{array}{cc}\hfill & c_{\mathit{qs}}(m_q(t))\hfill \\ \hfill & =\left\{\begin{array}{c}{\acute{c}}_{\mathit{qs}}=c_{1qs}^R+c_{1qs}^{I}i+c_{1qs}^{J}j+c_{1qs}^{K}k,|m_q(t)|<T_q\hfill \\ {\stackrel{`}{c}}_{\mathit{qs}}=c_{2qs}^R+c_{2qs}^{I}i+c_{2qs}^{J}j+c_{2qs}^{K}k,|m_q(t)|\ge T_q\hfill \end{array}\right),\hfill \\ \hfill & d_{\mathit{qs}}(m_q(t))\hfill \\ \hfill & =\left\{\begin{array}{c}{\acute{d}}_{\mathit{qs}}=d_{1qs}^R+d_{1qs}^{I}i+d_{1qs}^{J}j+d_{1qs}^{K}k,|m_q(t)|<T_q\hfill \\ {\stackrel{`}{d}}_{\mathit{qs}}=d_{2qs}^R+d_{2qs}^{I}i+d_{2qs}^{J}j+d_{2qs}^{K}k,|m_q(t)|\ge T_q\hfill \end{array}\right),\hfill \end{array}$$

where $T_q>0$ is the switch.

The following assumptions are important for this paper.

$(H_1)$: For $q=1,2,\cdots ,n$, there exist positive constants ${\lambda }_s$ and $l_s$, such that

$$\begin{array}{cc}\hfill & \Vert f_s(n_s(t))-f_s(m_s(t)){\Vert }_1\le {\lambda }_s{\Vert n_s(t)-m_s(t)\Vert }_1,\hfill \\ \hfill & \Vert f_s(m_s(t)){\Vert }_1\le l_s.\hfill \end{array}$$

$(H_2)$: The amplification function $a_q(·)$ satisfies:

$$\begin{array}{cc}\hfill & \Vert a_q(n_q(t))-a_q(m_q(t)){\Vert }_1\le {\mu }_q{\Vert n_q(t)-m_q(t)\Vert }_1,\hfill \\ \hfill & \Vert a_q(n_q(t)){\Vert }_1\le {\rho }_q,\hfill \end{array}$$

where ${\mu }_q,{\varrho }_q$ are positive constants.

$(H_3)$:There exists positive constant ${\omega }_q$, such that:

$$\begin{array}{cc}\hfill & \Vert a_q(n_q(t))b_q(n_q(t))-a_q(m_q(t))b_q(m_q(t)){\Vert }_1\hfill \\ \hfill & \le {\omega }_q{\Vert n_q(t)-m_q(t)\Vert }_1,\hfill \end{array}$$

holds.

The sign function of $m\in \mathbb{Q}$ is denoted by $sgn(m)=sgn(m^R)+isgn(m^I)+jsgn(m^J)+ksgn(m^K).$ As we know, $\forall x\in \mathbb{R}$, the absolute value of x is denoted by $|x|=sgn(x)x$. $\forall x\in \mathbb{C}$, one-norm of $x=x^R+x^I\left(x^R,,,x^I,\in ,R^M\right)$ is written as ${\Vert x\Vert }_1=\Vert x^R{\Vert }_1+{\Vert x^I\Vert }_1=sgn{\left(x^R\right)}^T{x}^R+sgn{\left(x^I\right)}^T{x}^I=1/2(sgn{(x)}^{\ast }x+sgn(x)x^{\ast })$. Next, we will give the definition of the improved one-norm of quaternion.

Definition 2.1

The improved one-norm of quaternion m is defined by:

$$\begin{array}{cc}\hfill {\Vert m\Vert }_1=& \Vert m^R{\Vert }_1+\Vert m^I{\Vert }_1+\Vert m^J{\Vert }_1+{\Vert m^K\Vert }_1\hfill \\ \hfill & =sgn{\left(m^R\right)}^T{m}^R+sgn{\left(m^I\right)}^T{m}^I\hfill \\ \hfill & +sgn{\left(m^J\right)}^T{m}^J+sgn{\left(m^K\right)}^T{m}^K\hfill \\ \hfill & =1/2(sgn{(m)}^{\ast }m+sgn(m)m^{\ast }.\hfill \end{array}$$

Based on the theory of differential inclusion set valued map and the above analysis, system (1) can be written as:

$$\begin{array}{cc}\hfill {\dot{m}}_q(t)\in & -a_q(m_q(t))(b_q(m_q(t))-\sum _{s=1}^{n}co\left(c_{\mathit{qs}}^{-},,,c_{\mathit{qs}}^{+}\right)f_s(m_s(t))\hfill \\ \hfill & -\sum _{s=1}^{n}co\left(d_{\mathit{qs}}^{-},,,d_{\mathit{qs}}^{+}\right)f_s(m_s(t-\tau (t))-I_q),\hfill \end{array}$$ 2

where $c_{\mathit{qs}}^{\mathit{max}}=max\{|{\acute{c}}_{\mathit{qs}}|,|{\stackrel{`}{c}}_{\mathit{qs}}|\},d_{\mathit{qs}}^{\mathit{max}}=max\{|{\acute{d}}_{\mathit{qs}}|,|{\stackrel{`}{d}}_{\mathit{qs}}|\}$.

Differential inclusion means that there is $c_{\mathit{qs}}(t)\in co\left(b_{\mathit{qs}}^{-},,,b_{\mathit{qs}}^{+}\right),d_{\mathit{qs}}(t)\in co\left(d_{\mathit{qs}}^{-},,,d_{\mathit{qs}}^{+}\right)$ such that

$$\begin{array}{cc}\hfill & {\dot{m}}_q(t)=-a_q(m_q(t))(b_q(m_q(t))-\sum _{s=1}^n{c}_{\mathit{qs}}(t)f_s(m_s(t))\hfill \\ \hfill & -\sum _{s=1}^n{d}_{\mathit{qs}}(t)f_s(m_s(t-\tau (t))-I_q).\hfill \end{array}$$ 3

Correspondingly, the response system can be defined as:

$$\begin{array}{cc}\hfill {\dot{n}}_q(t)=& -a_q(n_q(t))(b_q(n_q(t))-\sum _{s=1}^n{c}_{\mathit{qs}}(n_q(t))f_s(n_s(t))\hfill \\ \hfill & -\sum _{s=1}^n{d}_{\mathit{qs}}(n_q(t))f_s(n_s(t-\tau (t)))-I_q)+u_q(t),\hfill \end{array}$$ 4

for $q=1,2,\cdots ,n$, where $n_q(t)\in \mathbb{Q}$ stand for the state vector of the neuron. $a_q(n_q(t))\in \mathbb{Q}$ represent the amplification function, and $b_q(n_q(t))\in \mathbb{Q}$ stand for the behaved function. $u_q(t)\in \mathbb{Q}$ is the controller. $n_q(s)={\varphi }_q(s),-\tau \le s\le 0$, where ${\varphi }_q(s)={({\varphi }_1(s),{\varphi }_2(s),\cdots ,{\varphi }_n(s))}^T$ $\in ([-\tau ,0],Q^n)$.

Similar to the analysis of (2) and (3), we have from (4) that

$$\begin{array}{cc}\hfill {\dot{n}}_q(t)=& -a_q(n_q(t))(b_q(n_q(t))-\sum _{s=1}^n{c}_{\mathit{qs}}^{\prime }(t)f_s(n_s(t))\hfill \\ \hfill & -\sum _{s=1}^n{d}_{\mathit{qs}}^{\prime }(t)f_s(n_s(t-\tau (t))-I_q)+u_q(t),\hfill \end{array}$$ 5

where $b_{\mathit{qs}}^{\prime }(t)\in co\left(b_{\mathit{qs}}^{-},,,b_{\mathit{qs}}^{+}\right)$, $d_{\mathit{qs}}^{\prime }(t)\in co\left(d_{\mathit{qs}}^{-},,,d_{\mathit{qs}}^{+}\right)$.

Remark 2

The assumptions in this paper are less restrictive to the activation functions and easier to verify. Recently, Wei and Cao (2019) studied fixed-time synchronization of the memristive quaternion-valued neural networks with the following conditions:

$$\begin{array}{cc}\hfill & co\left(c_{\mathit{qs}}^{-},,,c_{\mathit{qs}}^{+}\right)f_s(n_s(t))-co\left(c_{\mathit{qs}}^{-},,,c_{\mathit{qs}}^{+}\right)f_s(m_s(t))\hfill \\ \hfill & \subseteq co\left(c_{\mathit{qs}}^{-},,,c_{\mathit{qs}}^{+}\right)(f_s(n_s(t))-f_s(m_s(t)));\hfill \\ \hfill & co\left(d_{\mathit{qs}}^{-},,,d_{\mathit{qs}}^{+}\right)f_s(n_s(t))-co\left(d_{\mathit{qs}}^{-},,,d_{\mathit{qs}}^{+}\right)f_s(m_s(t))\hfill \\ \hfill & \subseteq co\left(d_{\mathit{qs}}^{-},,,d_{\mathit{qs}}^{+}\right)(f_s(n_s(t))-f_s(m_s(t))).\hfill \end{array}$$

While, according to Yang et al. (2014), we know that the above condition is not right. When ${\delta }_1=c_{\mathit{qs}}^{-}$ and ${\delta }_2=c_{\mathit{qs}}^{+}$, for any $m_s,n_s\in \mathbb{Q}$, there is no $\delta \in co(d_{\mathit{qs}}^{-},d_{\mathit{qs}}^{+})$ such that

$$\begin{array}{cc}\hfill {\delta }_1{f}_s(n_s)-{\delta }_2{f}_s(m_s)=& b_{\mathit{qs}}^{-}{f}_s(n_s)-b_{\mathit{qs}}^{+}{f}_s(m_s)\hfill \\ \hfill & =\delta (f_s(n_s)-f_s(m_s)).\hfill \end{array}$$ 6

Based on the above discussion, in the paper we take $b_{\mathit{qs}}(t)\in co\left(b_{\mathit{qs}}^{-},,,b_{\mathit{qs}}^{+}\right)$, $d_{\mathit{qs}}(t)$ $\in co\left(d_{\mathit{qs}}^{-},,,d_{\mathit{qs}}^{+}\right)$, $b_{\mathit{qs}}^{\prime }(t)\in co\left(b_{\mathit{qs}}^{-},,,b_{\mathit{qs}}^{+}\right)$, $d_{\mathit{qs}}^{\prime }(t)\in co\left(d_{\mathit{qs}}^{-},,,d_{\mathit{qs}}^{+}\right)$.

In this paper, we define the error system as $e_q(t)=n_q(t)-m_q(t)$, $(q=1,2,\cdots ,n)$.

$$\begin{array}{cc}\hfill {\dot{e}}_q(t)& =-(a_q(n_q(t))b_q(n_q(t))-a_q(m_q(t))b_q(m_q(t)))\hfill \\ \hfill & +a_q(n_q(t))(\sum _{s=1}^n{c}_{\mathit{qs}}^{\prime }(t){\tilde{f}}_s(e_s(t))+\sum _{s=1}^n{d}_{\mathit{qs}}^{\prime }(t){\tilde{f}}_s(e_s(t-\tau (t))))\hfill \\ \hfill & +(a_q(n_q(t))-a_q(m_q(t)))(\sum _{s=1}^n{c}_{\mathit{qs}}^{\prime }(t)f_s(m_s(t))\hfill \\ \hfill & +\sum _{s=1}^n{d}_{\mathit{qs}}^{\prime }(t)f_s(m_s(t-\tau (t))))\hfill \\ \hfill & +a_q(m_q(t))(\sum _{s=1}^n{c}_{\mathit{qs}}^{\prime }(t)-\sum _{s=1}^n{c}_{\mathit{qs}}(t))f_s(m_s(t))\hfill \\ \hfill & +(a_q(n_q(t))-a_q(m_q(t)))I_q\hfill \\ \hfill & +a_q(m_q(t))(\sum _{s=1}^n{d}_{\mathit{qs}}^{\prime }(t)-\sum _{s=1}^n{d}_{\mathit{qs}}(t))f_s(m_s(t-\tau (t)))+u_q(t),\hfill \end{array}$$ 7

where ${\tilde{f}}_s(e_s(t))=f_s(n_s(t))-f_s(m_s(t))$, ${\tilde{f}}_s(e_s(t-\tau (t)))=f_s(n_s(t-\tau (t)))-f_s(m_s(t-\tau (t)))$.

Lemma 2.1

Li and Cao (2015) When $t\in (-\tau ,+\infty )$, The function V(t) is non-negative and satisfies the following inequality:

$$\begin{array}{c}\hfill D^{+}V(t)\le -\eta V(t)+\zeta V(t-\tau (t)),\end{array}$$ 8

where $t>0$, and $\eta$, $\zeta$, are positive constants with $\eta >\zeta >0$, then

$$\begin{array}{c}\hfill V(t)\le \underset{-\tau \le \theta \le 0}{sup}V(\theta )e^{-rt},\end{array}$$

where r is the positive solution of the equation: $r=\alpha -\beta e^{-r\tau }$.

Lemma 2.2

Peng et al. (2022) If $n(t)=n^R(t)+i{n}^I(t)+j{n}^J(t)+k{n}^K(t)$, $h(t)=h^R(t)+i{h}^I(t)+j{h}^J(t)+k{h}^K(t)$ the following formula holds

$$\begin{array}{cc}\hfill & (1)n{(t)}^{\ast }sgn(h(t))+sgn{(h(t))}^{\ast }n(t)\le n{(t)}^{\ast }sgn(n(t))\hfill \\ \hfill & +sgn{(n(t))}^{\ast }n(t)=2{\Vert n(t)\Vert }_1;\hfill \\ \hfill & (2)D^{+}(n{(t)}^{\ast }sgn(n(t))+sgn{(n(t))}^{\ast }n(t))\hfill \\ \hfill & =sgn{(n(t))}^{\ast }\dot{n}(t)+\dot{n}(t)sgn(n(t));\hfill \\ \hfill & {(3)\Vert h(t)n(t)\Vert }_1\le {\Vert h(t)\Vert }_1{\Vert n(t)\Vert }_1;\hfill \\ \hfill & (4)sgn{(n(t))}^{\ast }sgn(n(t))={\Vert sgn(n(t))\Vert }_1.\hfill \end{array}$$

Definition 2.2

Yang et al. (2014) The drive system (1) and the response system (4) will reach global exponential synchronization, if there exist positive constants M and r such that

$$\begin{array}{c}\hfill \Vert n_q(t)-m_q(t)\Vert \le M\underset{-\tau \le s\le 0}{sup}\Vert {\varphi }_s(t)-{\psi }_s(t)\Vert e^{-rt},\end{array}$$ 9

where $q=1,2,\cdots ,n$, hold for $t\ge 0$.

Main results

In this section, we study the exponential synchronization and global asymptotic synchronization of quaternion-valued memristor-based Cohen–Grossberg neural networks with time-varying delays.

The controller is designed as the following:

$$\begin{array}{cc}\hfill u_q(t)& =-a_q(m_q(t))(\sum _{s=1}^n{c}_{\mathit{qs}}^{\prime }(t)\hfill \\ \hfill & -\sum _{s=1}^n{c}_{\mathit{qs}}(t))f_s(m_s(t))-(a_q(n_q(t))-a_q(m_q(t)))I_q\hfill \\ \hfill & -a_q(m_q(t))(\sum _{s=1}^n{d}_{\mathit{qs}}^{\prime }(t)\hfill \\ \hfill & -\sum _{s=1}^n{d}_{\mathit{qs}}(t))f_s(m_s(t-\tau (t)))-k_q(n_q(t)-m_q(t)).\hfill \end{array}$$ 10

The following notations will be used:

$$\begin{array}{c}\hfill {\tilde{c}}_{\mathit{qs}}=\underset{t\ge 0}{sup}|c_{\mathit{qs}}^{\prime }(t)|,{\tilde{d}}_{\mathit{qs}}=\underset{t\ge 0}{sup}|d_{\mathit{qs}}^{\prime }(t)|.\end{array}$$

Theorem 3.1

Under Assumption $H_1-H_3$ and the controller (10), the following conditions are true

$$\begin{array}{cc}\hfill \eta & =k_q-{\omega }_q-\sum _{s=1}^n{\rho }_q{\lambda }_s\Vert {\tilde{c}}_{\mathit{qs}}{\Vert }_1-\sum _{s=1}^n{\mu }_q{l}_s{\Vert {\tilde{c}}_{\mathit{qs}}\Vert }_1\hfill \\ \hfill & -\sum _{s=1}^n{\mu }_q{l}_s{\Vert {\tilde{d}}_{\mathit{qs}}\Vert }_1;\hfill \\ \hfill \zeta & =\sum _{s=1}^n{\omega }_q{l}_s{\Vert {\tilde{d}}_{\mathit{qs}}\Vert }_1;\hfill \\ \hfill \eta & >\zeta >0,\hfill \end{array}$$

where $q=1,2,\cdots ,n$, then the system (1) and the system (4) will reach exponential synchronization.

Proof

Consider the following Lyapunov functional

$$\begin{array}{c}\hfill V(t)=\frac{1}{2}\sum _{q=1}^n(sgn{(e_q(t))}^{\ast }{e}_q(t)+e_q{(t)}^{\ast }sgn(e_q(t))),\end{array}$$ 11

Taking the upper right Dini derivative of V(t):

$$\begin{array}{cc}\hfill D^{+}V(t)& =\frac{1}{2}\sum _{q=1}^n(sgn{(e_q(t))}^{\ast }{\dot{e}}_q(t)\hfill \\ \hfill & +{\dot{e}}_q{(t)}^{\ast }sgn(e_q(t)))\hfill \\ \hfill & =\frac{1}{2}\sum _{q=1}^{n}sgn{(e_q(t))}^{\ast }\hfill \\ \hfill & [-(a_q(n_q(t))b_q(n_q(t))-a_q(m_q(t))b_q(m_q(t)))\hfill \\ \hfill & +a_q(n_q(t))\sum _{s=1}^n{c}_{\mathit{qs}}^{\prime }(t){\tilde{f}}_s(e_s(t))\hfill \\ \hfill & +a_q(n_q(t))\sum _{s=1}^n{d}_{\mathit{qs}}^{\prime }(t){\tilde{f}}_s(e_s(t-\tau (t)))\hfill \\ \hfill & +(a_q(n_q(t))-a_q(m_q(t)))\sum _{s=1}^n(c_{\mathit{qs}}^{\prime }(t)f_s(m_s(t))\hfill \\ \hfill & +d_{\mathit{qs}}^{\prime }(t)f_s(m_s(t-(t))))\hfill \\ \hfill & -k_q{e}_q(t)]+\frac{1}{2}\sum _{q=1}^n[-(a_q(n_q(t))b_q(n_q(t))\hfill \\ \hfill & -a_q(m_q(t))b_q(m_q(t)))\hfill \\ \hfill & +a_q(n_q(t))\sum _{s=1}^n{c}_{\mathit{qs}}^{\prime }(t){\tilde{f}}_s(e_s(t))\hfill \\ \hfill & +a_q(n_q(t))\sum _{s=1}^n{d}_{\mathit{qs}}^{\prime }(t){\tilde{f}}_s(e_s(t-\tau (t)))\hfill \\ \hfill & +(a_q(n_q(t))-a_q(m_q(t)))\sum _{s=1}^n(c_{\mathit{qs}}^{\prime }(t)f_s(m_s(t))\hfill \\ \hfill & +d_{\mathit{qs}}^{\prime }(t)f_s(m_s(t-(t))))-k_q{e}_q(t){]}^{\ast }sgn(e_q(t))\hfill \\ \hfill & =-\frac{1}{2}\sum _{q=1}^n(sgn{(e_q(t))}^{\ast }(a_q(n_q(t))b_q(n_q(t))\hfill \\ \hfill & -a_q(m_q(t))b_q(m_q(t)))\hfill \\ \hfill & +(a_q(n_q(t))b_q(n_q(t))-a_q(m_q(t))b_q(m_q(t)){)}^{\ast }sgn(e_q(t)))\hfill \\ \hfill & +\frac{1}{2}\sum _{q=1}^n\sum _{s=1}^n\hfill \\ \hfill & (sgn{(e_q(t))}^{\ast }{a}_q(n_q(t))c_{\mathit{qs}}^{\prime }(t){\tilde{f}}_s(e_s(t))\hfill \\ \hfill & +{\tilde{f}}_s{(e_s(t))}^{\ast }{c}_{\mathit{qs}}^{\prime }{(t)}^{\ast }{a}_q{(n_q(t))}^{\ast }sgn(e_q(t))\hfill \\ \hfill & +\frac{1}{2}\sum _{q=1}^n\sum _{s=1}^n(sgn{(e_q(t))}^{\ast }{a}_q(n_q(t))d_{\mathit{qs}}^{\prime }(t)\hfill \\ \hfill & \times {\tilde{f}}_s(e_s(t-(t)))+{\tilde{f}}_s{(e_s(t-(t)))}^{\ast }{d}_{\mathit{qs}}^{\prime }{(t)}^{\ast }\times a_q{(n_q(t))}^{\ast }sgn(e_q(t)))\hfill \\ \hfill & +\frac{1}{2}\sum _{q=1}^n\sum _{s=1}^n(sgn{(e_q(t))}^{\ast }(a_q(n_q(t))-a_q(m_q(t)))\hfill \\ \hfill & \times c_{\mathit{qs}}^{\prime }(t)f_s(m_s(t))\hfill \\ \hfill & +f_s{(m_s(t))}^{\ast }{c}_{\mathit{qs}}^{\prime }{(t)}^{\ast }\times (a_q(n_q(t))\hfill \\ \hfill & -a_q(m_q(t)){)}^{\ast }sgn(e_q(t)))\hfill \\ \hfill & +\frac{1}{2}\sum _{q=1}^n\sum _{s=1}^n(sgn{(e_q(t))}^{\ast }(a_q(n_q(t))\hfill \\ \hfill & -a_q(m_q(t)))\times d_{\mathit{qs}}^{\prime }(t)f_s(m_s(t-\tau (t)))\hfill \\ \hfill & +f_s{(m_s(t-\tau (t)))}^{\ast }\hfill \\ \hfill & \times d_{\mathit{qs}}^{\prime }{(t)}^{\ast }(a_q(n_q(t))-a_q(m_q(t)){)}^{\ast }sgn(e_q(t)))\hfill \\ \hfill & -\frac{1}{2}\sum _{q=1}^n{k}_q(sgn{(e_q(t))}^{\ast }{e}_q(t)+e_q{(t)}^{\ast }sgn(e_q(t))).\hfill \end{array}$$ 12

It follows from Lemma 2.2, there exist five constants ${\lambda }_q,l_s,{\mu }_q,{\rho }_q,{\omega }_q>0$, such that

$$\begin{array}{cc}\hfill & -\frac{1}{2}\sum _{q=1}^n(sgn{(e_q(t))}^{\ast }(a_q(n_q(t))b_q(n_q(t))\hfill \\ \hfill & -a_q(m_q(t))b_q(m_q(t)))\hfill \\ \hfill & +(a_q(n_q(t))b_q(n_q(t))-a_q(m_q(t))b_q(m_q(t)){)}^{\ast }sgn(e_q(t)))\hfill \\ \hfill & \le \sum _{q=1}^n\Vert a_q(n_q(t))b_q(n_q(t))-a_q(m_q(t))b_q(m_q(t)){\Vert }_1\le {\omega }_q{\Vert e_q(t)\Vert }_1;\hfill \end{array}$$ 13

$$\begin{array}{cc}\hfill & \frac{1}{2}\sum _{q=1}^n\sum _{s=1}^n(sgn{(e_q(t))}^{\ast }{a}_q(n_q(t))c_{\mathit{qs}}^{\prime }(t){\tilde{f}}_s(e_s(t))\hfill \\ \hfill & +{\tilde{f}}_s{(e_s(t))}^{\ast }{c}_{\mathit{qs}}^{\prime }{(t)}^{\ast }{a}_q{(n_q(t))}^{\ast }sgn(e_q(t)))\hfill \\ \hfill & \le \sum _{q=1}^n\sum _{s=1}^n{\Vert a_q(n_q(t))c_{\mathit{qs}}^{\prime }(t){\tilde{f}}_s(e_s(t))\Vert }_1\hfill \\ \hfill & \le \sum _{q=1}^n\sum _{s=1}^n\Vert a_q(n_q(t)){\Vert }_1\Vert {\tilde{c}}_{\mathit{qs}}{\Vert }_1{\Vert {\tilde{f}}_s(e_s(t))\Vert }_1\hfill \\ \hfill & \le \sum _{q=1}^n\sum _{s=1}^n{\rho }_q{\lambda }_s\Vert {\tilde{c}}_{\mathit{qs}}{\Vert }_1{\Vert e_q(t)\Vert }_1;\hfill \end{array}$$ 14

$$\begin{array}{cc}\hfill & \frac{1}{2}\sum _{q=1}^n\sum _{s=1}^n(sgn{(e_q(t))}^{\ast }{a}_q(n_q(t))d_{\mathit{qs}}^{\prime }(t){\tilde{f}}_s(e_s(t-\tau (t)))\hfill \\ \hfill & +{\tilde{f}}_s{(e_s(t-\tau (t)))}^{\ast }{d}_{\mathit{qs}}^{\prime }{(t)}^{\ast }{a}_q{(n_q(t))}^{\ast }sgn(e_q(t)))\hfill \\ \hfill & \le \sum _{q=1}^n\sum _{s=1}^n{\Vert a_q(n_q(t))d_{\mathit{qs}}^{\prime }(t){\tilde{f}}_s(e_s(t-\tau (t)))\Vert }_1\hfill \\ \hfill & \le \sum _{q=1}^n\sum _{s=1}^n\Vert a_q(n_q(t)){\Vert }_1\Vert {\tilde{d}}_{\mathit{qs}}{\Vert }_1{\Vert {\tilde{f}}_s(e_s(t-\tau (t)))\Vert }_1\hfill \\ \hfill & \le \sum _{q=1}^n\sum _{s=1}^n{\rho }_q{\lambda }_s\Vert {\tilde{d}}_{\mathit{qs}}{\Vert }_1{\Vert e_q(t-\tau (t))\Vert }_1;\hfill \end{array}$$ 15

$$\begin{array}{cc}\hfill & \frac{1}{2}\sum _{q=1}^n\sum _{s=1}^n(sgn{(e_q(t))}^{\ast }(a_q(n_q(t))-a_q(m_q(t)))c_{\mathit{qs}}^{\prime }(t)\times f_s(m_s(t))\hfill \\ \hfill & +f_s{(m_s(t))}^{\ast }{c}_{\mathit{qs}}^{\prime }{(t)}^{\ast }(a_q(n_q(t))-a_q(m_q(t)){)}^{\ast }\times sgn(e_q(t)))\hfill \\ \hfill & \le \sum _{q=1}^n\sum _{s=1}^n{\Vert (a_q(n_q(t))-a_q(m_q(t)))c_{\mathit{qs}}^{\prime }(t)f_s(m_s(t))\Vert }_1\hfill \\ \hfill & \le \sum _{q=1}^n\sum _{s=1}^n\Vert a_q(n_q(t))-a_q(m_q(t)){\Vert }_1\Vert {\tilde{c}}_{\mathit{qs}}{\Vert }_1{\Vert f_s(m_s(t))\Vert }_1\hfill \\ \hfill & \le \sum _{q=1}^n\sum _{s=1}^n{\mu }_q{l}_s\Vert {\tilde{c}}_{\mathit{qs}}{\Vert }_1{\Vert e_q(t)\Vert }_1;\hfill \end{array}$$ 16

$$\begin{array}{cc}\hfill & \frac{1}{2}\sum _{q=1}^n\sum _{s=1}^n(sgn{(e_q(t))}^{\ast }(a_q(n_q(t))-a_q(m_q(t)))d_{\mathit{qs}}^{\prime }(t)\hfill \\ \hfill & \times f_s(m_s(t-\tau (t)))\hfill \\ \hfill & +f_s{(m_s(t-\tau (t)))}^{\ast }{d}_{\mathit{qs}}^{\prime }{(t)}^{\ast }(a_q(n_q(t))-a_q(m_q(t)){)}^{\ast }sgn(e_q(t)))\hfill \\ \hfill & \le \sum _{q=1}^n\sum _{s=1}^n{\Vert (a_q(n_q(t))-a_q(m_q(t)))d_{\mathit{qs}}^{\prime }(t)f_s(m_s(t-\tau (t)))\Vert }_1\hfill \\ \hfill & \le \sum _{q=1}^n\sum _{s=1}^n\Vert a_q(n_q(t))-a_q(m_q(t)){\Vert }_1\Vert {\tilde{d}}_{\mathit{qs}}{\Vert }_1{\Vert f_s(m_s(t-\tau (t)))\Vert }_1\hfill \\ \hfill & \le \sum _{q=1}^n\sum _{s=1}^n{\mu }_q{l}_s\Vert {\tilde{d}}_{\mathit{qs}}{\Vert }_1{\Vert e_q(t)\Vert }_1;\hfill \end{array}$$ 17

$$\begin{array}{cc}\hfill -& \frac{1}{2}\sum _{q=1}^n{k}_q(sgn{(e_q(t))}^{\ast }{e}_q(t)+e_q{(t)}^{\ast }sgn(e_q(t)))\hfill \\ \hfill & =-k_q{\Vert e_q(t)\Vert }_1.\hfill \end{array}$$ 18

Based on formulas (13)–(18), it can be obtained that

$$\begin{array}{cc}\hfill & D^{+}V(t)\le -\sum _{q=1}^n(k_q-{\omega }_q-\sum _{s=1}^n{\rho }_q{\lambda }_s{\Vert {\tilde{c}}_{\mathit{qs}}\Vert }_1\hfill \\ \hfill & -\sum _{s=1}^n{\mu }_q{l}_s\Vert {\tilde{c}}_{\mathit{qs}}{\Vert }_1-\sum _{s=1}^n{\mu }_q{l}_s\Vert {\tilde{d}}_{\mathit{qs}}{\Vert }_1){\Vert e_q(t)\Vert }_1\hfill \\ \hfill & +\sum _{q=1}^n\sum _{s=1}^n{\omega }_q{l}_s\Vert {\tilde{d}}_{\mathit{qs}}{\Vert }_1{\Vert e_q(t-\tau (t))\Vert }_1.\hfill \end{array}$$ 19

According to Theorem 3.1, we can obtain

$$\begin{array}{cc}\hfill D^{+}V(t)\le & -\alpha V(t)+\beta V(t-\tau (t)).\hfill \end{array}$$ 20

Based on Lemma 2.2, it can be obtained that

$$\begin{array}{c}\hfill V(t)\le \underset{-\tau \le \theta \le 0}{sup}V(\theta )e^{-rt},\end{array}$$ 21

where r is the positive solution of the equation: $r=\alpha -\beta e^{-r\tau }$.

It is obvious that

$$\begin{array}{c}\hfill \Vert e_q(t)\Vert \le M\underset{-\tau \le s\le 0}{sup}\Vert {\varphi }_s(t)-{\psi }_s(t)\Vert e^{-rt},\end{array}$$ 22

therefore, under the designed controller (10), The drive system (1) and the response system (4) reach global exponential synchronization. Thus completing the proof. $\square$

Corollary 3.1

For three given assumptions $(H_3)-(H_3)$, The drive system (1) and the response system (4) reach global asymptotic synchronization under the following controller:

$$\begin{array}{cc}\hfill u_q(t)& =-a_q(m_q(t))(\sum _{s=1}^n{c}_{\mathit{qs}}^{\prime }(t)\hfill \\ \hfill & -\sum _{s=1}^n{c}_{\mathit{qs}}(t))f_s(m_s(t))\hfill \\ \hfill & -a_q(m_q(t))(\sum _{s=1}^n{d}_{\mathit{qs}}^{\prime }(t)\hfill \\ \hfill & -\sum _{s=1}^n{d}_{\mathit{qs}}(t))f_s(m_s(t-\tau (t)))\hfill \\ \hfill & -(a_q(n_q(t))-a_q(m_q(t)))I_q-k_q(n_q(t)-m_q(t)),\hfill \end{array}$$ 23

where

$$\begin{array}{cc}\hfill & k_q>{\omega }_q+\sum _{s=1}^n{\rho }_q{\lambda }_s{\Vert {\tilde{c}}_{\mathit{qs}}\Vert }_1\hfill \\ \hfill & +\sum _{s=1}^n{\mu }_q{l}_s{\Vert {\tilde{c}}_{\mathit{qs}}\Vert }_1\hfill \\ \hfill & +\sum _{s=1}^n{\mu }_q{l}_s{\Vert {\tilde{d}}_{\mathit{qs}}\Vert }_1+1,\hfill \\ \hfill & \sum _{s=1}^n{\omega }_q{l}_s{\Vert {\tilde{d}}_{\mathit{qs}}\Vert }_1<1-\xi ,\hfill \end{array}$$

in which ${\omega }_q,{\mu }_q,{\lambda }_q,l_s,{\rho }_q$ are all constants.

Proof

Consider the following auxiliary function

$$\begin{array}{cc}\hfill V_q(t)& =\frac{1}{2}\sum _{q=1}^n(sgn{(e_q(t))}^{\ast }{e}_q(t)+e_q{(t)}^{\ast }sgn(e_q(t)))\hfill \\ \hfill & +\frac{1}{2}\sum _{q=1}^n\sum _{s=1}^n{\int }_{t-\tau (t)}^{t}sgn{(e_q(t))}^{\ast }{e}_q(t)+e_q{(t)}^{\ast }sgn(e_q(t))ds.\hfill \end{array}$$ 24

Then, taking the upper right Dini derivative of V(t):

$$\begin{array}{cc}\hfill D^{+}V(t)& \le -\sum _{q=1}^n\{k_q-{\omega }_q-{\rho }_q{\lambda }_s\Vert {\tilde{c}}_{\mathit{qs}}{\Vert }_1-{\mu }_q{l}_s{\Vert {\tilde{c}}_{\mathit{qs}}\Vert }_1\hfill \\ \hfill & -{\mu }_q{l}_s\Vert {\tilde{d}}_{\mathit{qs}}{\Vert }_1-1\}{\Vert e_q(t)\Vert }_1\hfill \\ \hfill & +\sum _{q=1}^n\sum _{s=1}^n({\omega }_q{l}_s\Vert {\tilde{d}}_{\mathit{qs}}{\Vert }_1-(1-\xi ))\times {\Vert e_q(t-\tau (t))\Vert }_1<0.\hfill \end{array}$$ 25

Based on the above discussion, we can conclude that the response system (4) and the drive system (1) achieve global asymptotic synchronization. So the proof is finished. $\square$

Remark 3

When discussing the dynamical behavior of quaternion-valued Cohen–Grossberg neural networks, scholars have proposed many methods to overcome the difficulty of non-commutative quaternion multiplication. Such as decomposition (Li et al. 2019) and lexicographical order method (Wei et al. 2020), etc.

Numerical examples

Considering the following two dimensional quaternion-valued memristor-based Cohen–Grossberg neural networks model:

$$\begin{array}{cc}\hfill {\dot{m}}_1(t)& =-a_1(m_1(t))(b_1(m_1(t))\hfill \\ \hfill & -c_{11}(m_1(t))f(m_1(t))-c_{12}(m_1(t))f(m_2(t))\hfill \\ \hfill & -d_{11}(m_1(t))f(m_1(t-\sigma (t)))\hfill \\ \hfill & -d_{12}(m_1(t))f(m_2(t-\sigma (t)))-I_1),\hfill \\ \hfill {\dot{m}}_2(t)& =-a_2{m}_2(t)(b_2(m_2(t))\hfill \\ \hfill & -c_{21}(m_1(t))f(m_1(t))-c_{22}(m_2(t))f(m_2(t))\hfill \\ \hfill & -d_{21}(m_2(t))f(m_1(t-\sigma (t)))\hfill \\ \hfill & -d_{22}(m_2(t))f(m_2(t-\sigma (t)))-I_2).\hfill \end{array}$$ 26

The memristive connection weights are

$$\begin{array}{cc}\hfill c_{11}\left(m_1\left(t\right)\right)& =\left\{\begin{array}{cc}0.3-0.6i+1.3j-0.6k,\hfill & |m_1(t)|<1,\\ 0.8-0.7i+1.0j-0.5k,\hfill & |m_1(t)|\ge 1,\end{array}\right)\hfill \\ \hfill c_{12}\left(m_1\left(t\right)\right)& =\left\{\begin{array}{cc}-0.1-0.4i-0.1j-0.3k,\hfill & |m_2(t)|\ge 1,\\ -0.5-0.9i-0.5j-0.7k,\hfill & |m_2(t)|<1,\end{array}\right)\hfill \\ \hfill c_{21}\left(m_2\left(t\right)\right)& =\left\{\begin{array}{cc}0.6-0.3i+0.8j-0.4k,\hfill & |m_2(t)|\ge 1,\\ 1.1+0.7i-0.7j+0.6k,\hfill & |m_2(t)|<1,\end{array}\right)\hfill \\ \hfill c_{22}\left(m_2\left(t\right)\right)& =\left\{\begin{array}{cc}-1.2-0.1i-1.3j-0.2k,\hfill & |m_2(t)|\ge 1,\\ -0.8-0.3i-0.7j-0.3k,\hfill & |m_2(t)|<1,\end{array}\right)\hfill \\ \hfill d_{11}\left(m_1\left(t\right)\right)& =\left\{\begin{array}{cc}-1.5+0.6i-0.5j+0.3k,\hfill & |m_1(t)|\ge 1,\\ -1.4+0.8i-0.4j+0.6k,\hfill & |m_1(t)|<1,\end{array}\right)\hfill \\ \hfill d_{12}\left(m_1\left(t\right)\right)& =\left\{\begin{array}{cc}-0.1-0.9i-0.1j-0.6k,\hfill & |m_1(t)|\ge 1,\\ -0.5-0.5i-0.5j-0.7k,\hfill & |m_1(t)|<1,\end{array}\right)\hfill \\ \hfill d_{21}\left(m_2\left(t\right)\right)& =\left\{\begin{array}{cc}-1.2-1.1i-1.3j-1.3k,\hfill & |m_2(t)|\ge 1,\\ -0.8-0.2i-0.6j-0.1k,\hfill & |m_2(t)|<1,\end{array}\right)\hfill \\ \hfill d_{22}\left(m_2\left(t\right)\right)& =\left\{\begin{array}{cc}0.3-0.5i+0.2j7-0.4k,\hfill & |m_2(t)|\ge 1,\\ 0.5-0.8i+0.4j-0.7k,\hfill & |m_2(t)|<1,\end{array}\right)\hfill \\ \hfill \end{array}$$

The response system is:

$$\begin{array}{cc}\hfill {\dot{n}}_1(t)& =-a_1{n}_1(t)(b_1(n_1(t))\hfill \\ \hfill & -c_{11}(n_1(t))f(n_1(t))-c_{12}(n_1(t))f(n_2(t))\hfill \\ \hfill & -d_{11}(n_1(t))f(n_1(t-\sigma (t)))\hfill \\ \hfill & -d_{12}(n_1(t))f(n_2(t-\sigma (t)))-I_1)+u_1(t),\hfill \\ \hfill {\dot{n}}_2(t)& =-a_2{n}_2(t)(b_2(n_2(t))\hfill \\ \hfill & -c_{21}(n_1(t))f(n_1(t))-c_{22}(n_2(t))f(n_2(t))\hfill \\ \hfill & -d_{21}(n_2(t))f(n_1(t-\sigma (t)))\hfill \\ \hfill & -d_{22}(n_2(t))f(n_2(t-\sigma (t)))-I_2)+u_2(t).\hfill \end{array}$$ 27

Meanwhile, set the time delay $\tau (t)=0.75-0.25cos(t)$, such that $\tau =1$, activation function $f(m_q)=0.1tanh(m_q)$ $(q=1,2)$, $a_1{m}_1(t)=(0.3+sin(m_1^R(t)))+(0.3+sin(m_1^R(t)))i+(0.3+sin(m_1^R(t)))j+(0.3+sin(m_1^R(t)))k$, $a_2{m}_2(t)=(0.3+cos(m_2^R(t)))+(0.3+cos(m_2^R(t)))i+(0.3+cos(m_2^R(t)))j+(0.3+cos(m_2^R(t)))k$, $b_q(m_q(t))=2m_q(t)$, $k_1=k_2=10$ and external inputs $I_1$ = $I_2$ = 0. Through the delay function $\dot{\tau }(t)\le \xi \le 1$, we can get that $\xi =0.25$. It can be easily calculated that ${\lambda }_1={\lambda }_2=0.1$, $l_1=l_2=0.1$, ${\mu }_1={\mu }_2=1$ ${\rho }_1=0.5$,${\rho }_2=1.2$,${\omega }_1={\omega }_2=2.8$, then one has:

$$\begin{array}{cc}\hfill & k_1-{\omega }_1-\sum _{s=1}^n{\rho }_1{\lambda }_s{\Vert {\tilde{c}}_{1s}\Vert }_1\hfill \\ \hfill & -\sum _{s=1}^n{\mu }_1{l}_s\Vert {\tilde{c}}_{1s}{\Vert }_1-\sum _{s=1}^n{\mu }_1{l}_s{\Vert {\tilde{d}}_{1s}\Vert }_1\hfill \\ \hfill =& 5.82>\sum _{s=1}^n{\omega }_1{l}_s{\Vert {\tilde{d}}_{1s}\Vert }_1=1.1512;\hfill \\ \hfill & k_2-{\omega }_1-\sum _{s=1}^n{\rho }_2{\lambda }_s{\Vert {\tilde{c}}_{2s}\Vert }_1\hfill \\ \hfill & -\sum _{s=1}^n{\mu }_2{l}_s\Vert {\tilde{c}}_{2s}{\Vert }_1-\sum _{s=1}^n{\mu }_2{l}_s{\Vert {\tilde{d}}_{2s}\Vert }_1\hfill \\ \hfill =& 5.172>\sum _{s=1}^n{\omega }_2{l}_s{\Vert {\tilde{d}}_{2s}\Vert }_1=2.044.\hfill \end{array}$$

According to the above analysis, all the conditions deduced in Theorem 3.1 are satisfied. The response system and the drive system achieve global exponential synchronization under the controller (10).

Now, all the conditions derived in Theorem 3.1 are satisfied. Figures 1,  2,  3 and 4 depict the trajectories of error variables $e_q^R(t)$, $e_q^I(t)$, $e_q^J(t)$ and $e_q^K(t)$ $(q=1,2)$ without controller. Figures 5 and 6 depict the error of state variables $e_q^R(t)$, $e_q^I(t)$, $e_q^J(t)$ and $e_q^K(t)$ $(q=1,2)$ with the controller (10). According to the simulation results, the drive system (26) and response system (27) can reach synchronization under controller (10), which verify our main results.

Fig. 1.

The trajectories of error variables $e_q^R(t)$ without controller, $(q=1,2)$

Fig. 2.

The trajectories of error variables $e_q^I(t)$ without controller, $(q=1,2)$

Fig. 3.

The trajectories of error variables $e_q^J(t)$ without controller, $(q=1,2)$

Fig. 4.

The trajectories of error variablesr $e_q^K(t)$ without controller, $(q=1,2)$

Fig. 5.

The trajectories of error variables $e_q^R(t),e_q^I(t)$ $(q=1,2)$

Fig. 6.

The trajectories of error variables $e_q^J(t),e_q^K(t)$ $(q=1,2)$

Conclusions

We introduced synchronization control for a general kind of delayed quaternion-valued memristor-based Cohen–Grossberg neural networks. Through constructing the Lyapunov functions, using analytical techniques, and building a now controller, several novel control strategies were presented to investigate the synchronization of delayed quaternion-valued memristor-based systems. Some conditions that can be easily verified were established to achieve synchronization of the considered neural networks. Furthermore, a direct analytical approach was introduced to study the synchronization problem by introducing an improved one-norm method. Finally, the effectiveness of our main results was verified by numerical simulation.

Funding

Funding were provided by the National Natural Science Foundation of China under Grant 61703354; Key Laboratory of Numerical Simulation of Sichuan Provincial Universities KLNS-2023SZFZ001; Natural Science Foundation of Sichuan Province 2022NSFSC0529; the CUIT (KYQN202324; KYTD202243); the Scientific Research Foundation of Chengdu University of Information Technology KYTZ202184.

Data availability

Data on the results of the study may be obtained from the corresponding author upon reasonable request.

Declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.