New exploration on bifurcation for fractional-order quaternion-valued neural networks involving leakage delays

Abstract

During the past decades, many works on Hopf bifurcation of fractional-order neural networks are mainly concerned with real-valued and complex-valued cases. However, few publications involve the quaternion-valued neural networks which is a generalization of real-valued and complex-valued neural networks. In this present study, we explorate the Hopf bifurcation problem for fractional-order quaternion-valued neural networks involving leakage delays. Taking advantage of the Hamilton rule of quaternion algebra, we decompose the addressed fractional-order quaternion-valued delayed neural networks into the equivalent eight real valued networks. Then the delay-inspired bifurcation condition of the eight real valued networks are derived by making use of the stability criterion and bifurcation theory of fractional-order differential dynamical systems. The impact of leakage delay on the bifurcation behavior of the involved fractional-order quaternion-valued delayed neural networks has been revealed. Software simulations are implemented to support the effectiveness of the derived fruits of this study. The research supplements the work of Huang et al. (Neural Netw 117:67–93, 2019).

Introduction

It is common knowledge that neural networks have great application value in a lot of areas such as automatic control, biomedical science, image processing, associative memory, pattern recognition, etc. (Huang et al. 2019; Gu et al. 2020; Ouyang et al. 2020). Usually, time delay often appears in biological dynamical systems and artificial neural networks since there is the time lag of the response of different biological populations and signal propagation for different neurons. It is natural for us to introduce the time delay into neural networks and biological models. A great deal of scientific research proves that time delay often lead to various dynamical behavior such as periodic oscillation, disappearance of stability property, bifurcation, chaotic phenomenon, etc. (Zhang et al. 2017). Thus grasping the impact of time delay on the various dynamics of the involved delayed differential systems has become a key project in delayed differential equation area. In particular, revealing the effect of time delay on dynamics of all sorts of delayed neural networks is an important task in neural network field. Up to now, lots of researchers are dedicated to the investigation on delayed neural networks and plenty of outstanding achievements have been achieved. For instance, Aouiti et al. (2021) investigated the fixed-time stabilization and synchronization issue of delayed neural networks; In Xing et al. (2021), Xing et al. established a sufficient condition to ensure the stability behavior and the existence of bifurcation for an $(n+m)$-neuron double-ring neural networks involving multi-delays; In 2021, Nagamani et al. (2021) handled the problem of state estimation for BAM cellular neural networks concerning multi-proportional time delays. In 2015, Xu and Zhang (2015) focused on the existence and global exponential stability of anti-periodic solutions to delayed BAM neural networks. For more specific literature, we refer the readers to (Gan et al. 2020; Li et al. 2020; Kumar and Das 2020; Syed Ali et al. 2019; Xu and Zhang 2014; Xu et al. 2016; Xu and Zhang 2014; Njitacke et al. 2021; Doubla et al. 2021; Xu et al. 2021a, b; Pratap et al. 2020; Rajchakit et al. 2019; Iswarya et al. 2021; Saravanakumar et al. 2020, 2019).

Here we must point out that all publications above merely involve the real-valued neural networks which are one-dimensional valued case. In fact, there are still many other multi-dimensional valued neural networks which are often applied in realistic life. Complex-valued neural networks(CVNNs), which can be regarded as the extension of real-valued neural networks(RVNNs), play a key role in dealing with signal and innate information of neural networks. In particular, they can be usually utilized in all sorts of physical waves such as electronic wave, sound wave, elastic wave, optical wave, etc. In addition, there are another multi-dimensional valued neural networks which are called quaternion-valued neural networks(QVNNs). The QVNNs are firstly put up by Sudbery (1979) in 1843 and they can be regarded as the extension form of RVNNs and CVNNs. We formulate the quaternion number z as the form:$\mathcal{Q}:=\{z=z^R+i{z}^I+j{z}^J+k{z}^K\},$ where $z^R,z^I,z^J,z^K\in R$ and i, j, k comply with the following requirements:

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

The study about QVNNs has attracted much interest from a great many researchers since they own immense application value in many fields such as spatial rotation, image impression, color night vision, 3-D affine geometric transformation, etc. (Wang et al. 2021; Jiang and Wang 2021). Nowadays a great deal of publications on the dynamical peculiarities of QVNNs have been available. For instance, Wang et al. (2021) considered the synchronization problem for quaternion-valued fuzzy cellular neural networks involving leakage delays. Zhu and Sun (2019) investigated the stability for a class of quaternion-valued neural networks involving distributed delays and time-varying delays. Liu and Chen (2020) were concerned with the state estimation for quaternion-valued neural networks involving leakage delay and time-varying delays. For more related researches, one can see (Tu et al. 2019; Wei and Cao 2019; Popa 2018).

Notice that all the studies above about quaternion-valued neural networks are chiefly concerned with integer-order form and they are not concerned with fractional-order form. The reason is that the research on fractional-order differential systems has remained a slow development state because of the insufficiency of the basic theory of fractional-order dynamical system and the shortage of the real physical background. With the advancement of research on fractional-order dynamical system, it is found that fractional-order dynamical model is a more suitable tool to describe the real natural phenomenon in object work than the classical integer-order ones since they own the hereditary function and memory trait for numerous materials and changing process (Isokawa et al. 2003; Kusamichi et al. 2004). At present, fractional-order delayed quaternion-valued neural networks have already attracted much interest from scientific community and a large number of valuable achievements about fractional-order delayed quaternion-valued neural networks have been published. For example, Pratap et al. (2020) explored the finite-time Mittag-Leffler stability for fractional-order quaternion-valued neural networks involving impulsive effect. Wei et al. (2021) made a detailed discussion on stabilization and synchronization control issue for fractional-order quaternion-valued fuzzy memristive neural networks. Yang et al. (2018) talked about the stability and synchronization issues for a class of fractional-order quaternion-valued neural networks. For more concrete researches, one can see (Shu et al. 2021; Xu et al. 2021, 2020; Li and Chen 2020).

Hopf bifurcation is one of key dynamical properties of delayed neural networks. The so-called Hopf bifurcation points that the differential system which depends on a certain parameter owns the change in essential characteristics and a family of periodic solutions take place near the equilibrium point if the parameter has minor change. During the past decades, a lot of studies focused on the Hopf bifurcation of integer-order delayed neural networks (see Xu 2018; Wang et al. 2019). Recently, there are some works on the Hopf bifurcation of fractional-order delayed real-valued and complex-valued neural networks (see Li et al. 2018; Xu et al. 2021). However, the study on the fractional-order delayed quaternion-valued neural networks is very few. Inspired by the analysis above, in the present study, we are to carry out the investigation about the stability and the onset of Hopf bifurcation for fractional-order delayed quaternion-valued neural networks. To sum up, we are to tackle the following two problems: (1) Establish the sufficient condition ensuring the stability and the onset of Hopf bifurcation for the involved fractional-order quaternion-valued neural networks involving leakage delays; (2) Reveal the impact of the leakage delay on the stability and the onset of Hopf bifurcation for the involved fractional-order quaternion-valued neural networks involving leakage delays.

In this research, we are to deal with the following fractional-order quaternion-valued neural networks involving leakage delays:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{d^{\rho }{w}_1(t)}{d{t}^{\rho }}=-a{w}_1(t-\sigma )+b_1{g}_1(w_2(t)),}\hfill \\ {\displaystyle \frac{d^{\rho }{w}_2(t)}{d{t}^{\rho }}=-a{w}_2(t-\sigma )+b_2{g}_2(w_1(t)),}\hfill \end{array}\right)\end{array}$$ 1.1

where $0<\rho <1$, $w_i(t)\in \mathcal{Q}$ represents the state of the ith neuron at time t, $a>0$ stands for the self-feedback parameter, $\sigma \ge 0$ denotes the leakage delay, $b_i\in \mathcal{Q}(i=1,2)$ stands for the connection weight, $g_h\in \mathcal{Q}(h=1,2)$ stands for the activation function.

We plan this works as follows. Part 2 prepares some related knowledge about fractional-order differential systems and quaternion algebra. Part 3 displays the established sufficient condition guaranteeing the stability and the onset of Hopf bifurcation of model (1.1). Part 4 executes computer simulation to check the truthfulness of the established main theoretical results in the research. Part 5 completes this research.

Remark 1.1

Based on the following neural networks:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{d{w}_1(t)}{\mathit{dt}}=-a{w}_1(t-\sigma )+b_1{g}_1(w_2(t)),}\hfill \\ {\displaystyle \frac{d{w}_2(t)}{\mathit{dt}}=-a{w}_2(t-\sigma )+b_2{g}_2(w_1(t)),}\hfill \end{array}\right)\end{array}$$

where $a,b_1,b_2$ are real numbers. $w_i(t)\in R$ represents the state of the ith neuron at time t, $a>0$ stands for the self-feedback parameter, $\sigma \ge 0$ denotes the leakage delay, $b_i\in R(i=1,2)$ stands for the connection weight, $g_h\in R(h=1,2)$ stands for the activation function, and according to the discussion on fractional-order differential equation and quaternion-valued neural networks, we then construct the following quaternion-valued neural networks with leakage delay:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{d^{\rho }{w}_1(t)}{d{t}^{\rho }}=-a{w}_1(t-\sigma )+b_1{g}_1(w_2(t)),}\hfill \\ {\displaystyle \frac{d^{\rho }{w}_2(t)}{d{t}^{\rho }}=-a{w}_2(t-\sigma )+b_2{g}_2(w_1(t)),}\hfill \end{array}\right)\end{array}$$

where $0<\rho <1$, $w_i(t)\in \mathcal{Q}$ represents the state of the ith neuron at time t, $a>0$ stands for the self-feedback parameter, $\sigma \ge 0$ denotes the leakage delay, $b_i\in \mathcal{Q}(i=1,2)$ stands for the connection weight, $g_h\in \mathcal{Q}(h=1,2)$ stands for the activation function.

Preliminaries and assumptions

In this part, some related lemmas and necessary knowledge on fractional-order differential systems and quaternion algebra are prepared.

Definition 2.1

Podlubny (1999) Define the Caputo-type fractional-order derivative as follows:

$$\begin{array}{c}\hfill {\mathcal{D}}^{\rho }k(\delta )=\frac{1}{\mathrm{\Gamma }(h-\rho )}{\int }_{{\delta }_0}^{\delta }\frac{k^{(h)}(v)}{{(\delta -v)}^{\rho -h+1}}dv,\end{array}$$

where $k(\delta )\in ([{\delta }_0,\infty ),R),\mathrm{\Gamma }(s)={\int }_0^{\infty }{\delta }^{s-1}{e}^{-\delta }d\delta ,$ $\delta \ge {\delta }_0$, $h\in Z^{+},h-1\le \rho <h.$

Define the Laplace transform of Caputo-type fractional order derivative as follows:

$$\begin{array}{c}\hfill \mathcal{L}\{{\mathcal{D}}^{\rho }g(t);s\}=s^{\rho }\mathcal{G}(s)-\sum _{l=0}^{k-1}{s}^{\rho -l-1}{g}^{(l)}(0),k-1\le \rho <k\in Z^{+},\end{array}$$

where $\mathcal{G}(s)=\mathcal{L}\{g(t)\}.$ If $g^{(j)}(0)=0,j=1,2,\dots ,k,$ then $\mathcal{L}\{{\mathcal{D}}^{\rho }g(t);s\}=s^{\rho }\mathcal{G}(s).$

Lemma 2.1

Deng et al. (2007) Give this system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{d^{{\rho }_1}{\mathcal{H}}_1(t)}{d{t}^{{\rho }_1}}=q_{11}{\mathcal{H}}_1(t-{\sigma }_{11})+q_{12}{\mathcal{H}}_2(t-{\sigma }_{12})+\cdots +q_{1n}{\mathcal{H}}_n(t-{\sigma }_{1n}),}\hfill \\ {\displaystyle \frac{d^{{\rho }_2}{\mathcal{H}}_2(t)}{d{t}^{{\rho }_2}}=q_{21}{\mathcal{H}}_1(t-{\sigma }_{21})+q_{22}{\mathcal{H}}_2(t-{\sigma }_{22})+\cdots +q_{2n}{\mathcal{H}}_n(t-{\sigma }_{2n}),}\hfill \\ {\displaystyle ⋮}\hfill \\ {\displaystyle \frac{d^{{\rho }_n}{\mathcal{H}}_n(t)}{d{t}^{{\rho }_n}}=q_{n1}{\mathcal{H}}_1(t-{\sigma }_{n1})+q_{n2}{\mathcal{H}}_2(t-{\sigma }_{n2})+\cdots +q_{\mathit{nn}}{\mathcal{H}}_n(t-{\sigma }_{\mathit{nn}}),}\hfill \end{array}\right)\end{array}$$ 2.1

where ${\rho }_j\in (0,1)(j=1,2,\dots ,n)$. Set

$$\begin{array}{c}\hfill \mathrm{\Delta }(\xi )=\left[\begin{array}{cccc}{\xi }^{{\rho }_1}-q_{11}{e}^{-\xi {\sigma }_{11}}& -q_{12}{e}^{-\xi {\sigma }_{12}}& \cdots & -q_{1n}{e}^{-\xi {\sigma }_{1n}}\\ -q_{21}{e}^{-\xi {\sigma }_{12}}& {\xi }^{{\rho }_2}-q_{22}{e}^{-\xi {\sigma }_{22}}& \cdots & -q_{2n}{e}^{-\xi {\sigma }_{2n}}\\ ⋮& ⋮& \ddots & ⋮\\ -q_{n1}{e}^{-\xi {\sigma }_{n1}}& -q_{n2}{e}^{-\xi {\sigma }_{n2}}& \cdots & {\xi }^{{\rho }_n}-q_{\mathit{nn}}{e}^{-\xi {\sigma }_{\mathit{nn}}}\end{array}\right],\end{array}$$ 2.2

then we say that the equilibrium point of model (2.1) is asymptotically stable in Lyapunov sense if each root of $det(\mathrm{\Delta }(\xi ))=0$ possesses the negative real parts.

Let $a,b\in \mathcal{Q}$, then we can assume that $a=a^R+i{a}^I+j{a}^J+k{a}^K,b=b^R+i{b}^I+j{b}^J+k{b}^K.$ We have

$$\begin{array}{cc}\hfill a\pm b=& (a^R\pm b^R)+i(a^I\pm b^I)\hfill \\ \hfill & +j(a^J\pm b^J)+k(a^K\pm b^K),\hfill \end{array}$$ 2.3

and

$$\begin{array}{cc}\hfill ab=& (a^R{b}^R-a^I{b}^I-a^J{b}^J-a^K{b}^K)\hfill \\ \hfill & +i(a^R{b}^I+a^I{b}^R+a^J{b}^K-a^K{b}^J)\hfill \\ \hfill & +j(a^R{b}^J-a^I{b}^K+a^J{b}^R+a^K{b}^I)\hfill \\ \hfill & +k(a^R{b}^K+a^I{b}^J-a^J{b}^I+a^K{b}^R).\hfill \end{array}$$ 2.4

To establish the key conclusions of this work, we need the assumptions as follows:

• $(\mathcal{A}1)\text{Assume}\text{that}{w}_l=w_l^R+i{w}_l^I+j{w}_l^J+k{w}_l^K,w_l^R,w_l^I,w_l^J,w_l^K\in R,l=1,2.$

  $(\mathcal{A}2)\text{Assume}\text{that}{b}_l=b_l^R+i{b}_l^I+j{b}_l^J+k{b}_l^K,b_l^R,b_l^I,b_l^J,b_l^K\in R,l=1,2.$

  $(\mathcal{A}3)\text{Assume}\text{that}{g}_1=g_1^R(w_2^R)+i{g}_1^I(w_2^I)+j{g}_1^J(w_2^J)+k{g}_1^K(w_2^K)$ and $g_2=g_2^R(w_1^R)+i{g}_2^I(w_1^I)+j{g}_2^J(w_1^J)+k{g}_2^K(w_1^K).$

  $(\mathcal{A}4)\text{The}\text{partial}\text{derivatives}\text{of}{g}_1^R(w_2^R),g_1^I(w_2^I),g_1^J(w_2^J),g_1^K(w_2^K),g_2^R(w_1^R),g_2^I(w_1^I),g_2^J(w_1^J),g_2^K(w_1^K)$  exist and are continuous, $g_l^R(0)=0,g_l^I(0)=0,g_l^J(0)=0,g_l^K(0)=0,l=1,2.$

Stability and Hopf bifurcation

In the part, we are to study the stability and Hopf bifurcation for model (1.1). In view of $(\mathcal{A}1)-(\mathcal{A}3)$, one has

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{d^{\rho }{w}_1(t)}{d{t}^{\rho }}=-a\left[w_1^R,(t-\sigma ),+,i,w_1^I,(t-\sigma )\right)}\hfill \\ \left({\displaystyle +j{w}_1^J(t-\sigma )+k{w}_1^K(t-\sigma )}\right]\hfill \\ {\displaystyle +(b_1^R+i{b}_1^I+j{b}_1^J+k{b}_1^K)\left[g_1^R,(w_2^R(t))\right)}\hfill \\ \left({\displaystyle +i{g}_1^I(w_2^I(t))+j{g}_1^J(w_2^J(t))+k{g}_1^K(w_2^K(t))}\right],\hfill \\ {\displaystyle \frac{d^{\rho }{w}_2(t)}{d{t}^{\rho }}=-a\left[w_2^R,(t-\sigma ),+,i,w_2^I,(t-\sigma )\right)}\hfill \\ \left({\displaystyle +j{w}_2^J(t-\sigma )+k{w}_2^K(t-\sigma )}\right]\hfill \\ {\displaystyle +(b_2^R+i{b}_2^I+j{b}_2^J+k{b}_2^K)\left[g_2^R,(w_1^R(t))\right)}\hfill \\ \left({\displaystyle +i{g}_2^I(w_1^I(t))+j{g}_2^J(w_1^J(t))+k{g}_2^K(w_1^K(t))}\right],\hfill \end{array}\right)\end{array}$$ 3.1

which leads to

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{d^{\rho }{w}_1(t)}{d{t}^{\rho }}=-a{w}_1^R(t-\sigma )+b_1^R{g}_1^R(w_2^R(t))-b_1^I{g}_1^I(w_2^I(t))}\hfill \\ {\displaystyle -b_1^J{g}_1^J(w_2^J(t))-b_1^K{g}_1^K(w_2^K(t))}\hfill \\ {\displaystyle +i\left[-,a,w_1^I,(t-\sigma ),+,b_1^R,g_1^I,(w_2^I(t)),+,b_1^I,g_1^R,(w_2^R(t))\right)}\hfill \\ {\displaystyle \left(+,b_1^I,g_1^K,(w_2^K(t)),+,b_1^K,g_1^J,(w_2^J(t))\right]}\hfill \\ {\displaystyle +j\left[-,a,w_1^J,(t-\sigma ),+,b_1^R,g_1^J,(w_2^J(t)),+,b_1^I,g_1^K,(w_2^K(t))\right)}\hfill \\ {\displaystyle \left(+,b_1^J,g_1^R,(w_2^R(t)),+,b_1^K,g_1^I,(w_2^I(t))\right]}\hfill \\ {\displaystyle +k\left[-,a,w_1^K,(t-\sigma ),+,b_1^R,g_1^K,(w_2^K(t)),+,b_1^I,g_1^J,(w_2^J(t))\right)}\hfill \\ {\displaystyle \left(+,b_1^J,g_1^I,(w_2^I(t)),+,b_1^K,g_1^R,(w_2^R(t))\right],}\hfill \\ {\displaystyle \frac{d^{\rho }{w}_2(t)}{d{t}^{\rho }}=-a{w}_2^R(t-\sigma )+b_2^R{g}_2^R(w_1^R(t))-b_2^I{g}_2^I(w_1^I(t))}\hfill \\ {\displaystyle -b_2^J{g}_2^J(w_1^J(t))-b_2^K{g}_2^K(w_1^K(t))}\hfill \\ {\displaystyle +i\left[-,a,w_2^I,(t-\sigma ),+,b_2^R,g_2^I,(w_1^I(t)),+,b_2^I,g_2^R,(w_1^R(t))\right)}\hfill \\ {\displaystyle \left(+,b_2^I,g_2^K,(w_1^K(t)),+,b_2^K,g_2^J,(w_1^J(t))\right]}\hfill \\ {\displaystyle +j\left[-,a,w_2^J,(t-\sigma ),+,b_2^R,g_2^J,(w_1^J(t)),+,b_2^I,g_2^K,(w_1^K(t))\right)}\hfill \\ {\displaystyle \left(+,b_2^J,g_2^R,(w_1^R(t)),+,b_2^K,g_2^I,(w_1^I(t))\right]}\hfill \\ {\displaystyle +k\left[-,a,w_2^K,(t-\sigma ),+,b_2^R,g_2^K,(w_1^K(t)),+,b_2^I,g_2^J,(w_1^J(t))\right)}\hfill \\ {\displaystyle \left(+,b_2^J,g_2^I,(w_1^I(t)),+,b_2^K,g_2^R,(w_1^R(t))\right].}\hfill \end{array}\right)\end{array}$$ 3.2

It follows from (3.2) that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{d^{\rho }{w}_1^R(t)}{d{t}^{\rho }}=-a{w}_1^R(t-\sigma )+b_1^R{g}_1^R(w_2^R(t))-b_1^I{g}_1^I(w_2^I(t))}\hfill \\ -b_1^J{g}_1^J(w_2^J(t))-b_1^K{g}_1^K(w_2^K(t)),\hfill \\ {\displaystyle \frac{d^{\rho }{w}_1^I(t)}{d{t}^{\rho }}=-a{w}_1^I(t-\sigma )+b_1^R{g}_1^J(w_2^J(t))+b_1^I{g}_1^K(w_2^K(t))}\hfill \\ +b_1^J{g}_1^R(w_2^R(t))+b_1^K{g}_1^I(w_2^I(t)),\hfill \\ {\displaystyle \frac{d^{\rho }{w}_1^J(t)}{d{t}^{\rho }}=-a{w}_1^J(t-\sigma )+b_1^R{g}_1^K(w_2^K(t))+b_1^I{g}_1^J(w_2^J(t))}\hfill \\ +b_1^J{g}_1^I(w_2^I(t))+b_1^K{g}_1^R(w_2^R(t)),\hfill \\ {\displaystyle \frac{d^{\rho }{w}_1^K(t)}{d{t}^{\rho }}=-a{w}_1^K(t-\sigma )+b_1^R{g}_1^K(w_2^K(t))+b_1^I{g}_1^J(w_2^J(t))}\hfill \\ +b_1^J{g}_1^I(w_2^I(t))+b_1^K{g}_1^R(w_2^R(t)),\hfill \\ {\displaystyle \frac{d^{\rho }{w}_2^R(t)}{d{t}^{\rho }}=-a{w}_2^R(t-\sigma )+b_2^R{g}_2^R(w_1^R(t))-b_2^I{g}_2^I(w_1^I(t))}\hfill \\ -b_2^J{g}_2^J(w_1^J(t))-b_2^K{g}_2^K(w_1^K(t)),\hfill \\ {\displaystyle \frac{d^{\rho }{w}_2^I(t)}{d{t}^{\rho }}=-a{w}_2^I(t-\sigma )+b_2^R{g}_2^I(w_1^I(t))+b_2^I{g}_2^R(w_1^R(t))}\hfill \\ +b_2^I{g}_2^K(w_1^K(t))+b_2^K{g}_2^J(w_1^J(t)),\hfill \\ {\displaystyle \frac{d^{\rho }{w}_2^J(t)}{d{t}^{\rho }}=-a{w}_2^J(t-\sigma )+b_2^R{g}_2^J(w_1^J(t))+b_2^I{g}_2^K(w_1^K(t))}\hfill \\ +b_2^J{g}_2^R(w_1^R(t))+b_2^K{g}_2^I(w_1^I(t)),\hfill \\ {\displaystyle \frac{d^{\rho }{w}_2^K(t)}{d{t}^{\rho }}=-a{w}_2^K(t-\sigma )+b_2^R{g}_2^K(w_1^K(t))+b_2^I{g}_2^J(w_1^J(t))}\hfill \\ +b_2^J{g}_2^I(w_1^I(t))+b_2^K{g}_2^R(w_1^R(t)).\hfill \end{array}\right)\end{array}$$ 3.3

By $(\mathcal{A}4)$, it is easy to see that system (3.3) owns a unique zero equilibrium point. The linear system of (3.3) near the zero equilibrium point takes the form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{d^{\rho }{w}_1^R(t)}{d{t}^{\rho }}=-a{w}_1^R(t-\sigma )+c_{11}{w}_2^R(t)+c_{12}{w}_2^I(t)}\hfill \\ +c_{13}{w}_2^J(t)+c_{14}{w}_2^K(t),\hfill \\ {\displaystyle \frac{d^{\rho }{w}_1^I(t)}{d{t}^{\rho }}=-a{w}_1^I(t-\sigma )+c_{21}{w}_2^J(t)+c_{22}{w}_2^K(t)}\hfill \\ +c_{23}{w}_2^R(t)+c_{24}{w}_2^I(t),\hfill \\ {\displaystyle \frac{d^{\rho }{w}_1^J(t)}{d{t}^{\rho }}=-a{w}_1^J(t-\sigma )+c_{31}{w}_2^K(t)+c_{32}{w}_2^J(t)}\hfill \\ +c_{33}{w}_2^I(t)+c_{34}{w}_2^R(t),\hfill \\ {\displaystyle \frac{d^{\rho }{w}_1^K(t)}{d{t}^{\rho }}=-a{w}_1^K(t-\sigma )+c_{41}{w}_2^K(t)+c_{42}{w}_2^J(t)}\hfill \\ +c_{43}{w}_2^I(t)+c_{44}{w}_2^R(t),\hfill \\ {\displaystyle \frac{d^{\rho }{w}_2^R(t)}{d{t}^{\rho }}=-a{w}_2^R(t-\sigma )+c_{51}{w}_1^R(t)+c_{52}{w}_1^I(t)}\hfill \\ +c_{53}{w}_1^J(t)+c_{54}{w}_1^K(t),\hfill \\ {\displaystyle \frac{d^{\rho }{w}_2^I(t)}{d{t}^{\rho }}=-a{w}_2^I(t-\sigma )+c_{61}{w}_1^I(t)+c_{62}{w}_1^R(t)}\hfill \\ +c_{63}{w}_1^K(t)+c_{64}{w}_1^J(t),\hfill \\ {\displaystyle \frac{d^{\rho }{w}_2^J(t)}{d{t}^{\rho }}=-a{w}_2^J(t-\sigma )+c_{71}{w}_1^J(t)+c_{72}{w}_1^K(t)}\hfill \\ +c_{73}{w}_1^R(t)+c_{74}{w}_1^I(t),\hfill \\ {\displaystyle \frac{d^{\rho }{w}_2^K(t)}{d{t}^{\rho }}=-a{w}_2^K(t-\sigma )+c_{81}{w}_1^K(t)+c_{82}{w}_1^J(t)}\hfill \\ +c_{83}{w}_1^I(t)+c_{84}{w}_1^R(t),\hfill \end{array}\right)\end{array}$$ 3.4

where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle c_{11}=b_1^R{g}_1^{{}^{\prime }R}(0),c_{12}=-b_1^I{g}_1^{{}^{\prime }I}(0),c_{13}=-b_1^J{g}_1^{{}^{\prime }J}(0),c_{14}=-b_1^K{g}_1^{{}^{\prime }K}(0),}\hfill \\ {\displaystyle c_{21}=b_1^R{g}_1^{{}^{\prime }J}(0),c_{22}=b_1^I{g}_1^{{}^{\prime }K}(0),c_{23}=b_1^J{g}_1^{{}^{\prime }R}(0),c_{24}=b_1^K{g}_1^{{}^{\prime }I}(0),}\hfill \\ {\displaystyle c_{31}=b_1^R{g}_1^{{}^{\prime }K}(0),c_{32}=b_1^I{g}_1^{{}^{\prime }J}(0),c_{33}=b_1^J{g}_1^{{}^{\prime }I}(0),c_{34}=b_1^K{g}_1^{{}^{\prime }R}(0),}\hfill \\ {\displaystyle c_{41}=b_1^R{g}_1^{{}^{\prime }K}(0),c_{42}=b_1^I{g}_1^{{}^{\prime }J}(0),c_{43}=b_1^J{g}_1^{{}^{\prime }I}(0),c_{44}=b_1^K{g}_1^{{}^{\prime }R}(0),}\hfill \\ {\displaystyle c_{51}=b_2^R{g}_2^{{}^{\prime }R}(0),c_{52}=-b_2^I{g}_2^{{}^{\prime }I}(0),c_{53}=-b_2^J{g}_2^{{}^{\prime }J}(0),c_{54}=-b_2^K{g}_2^{{}^{\prime }K}(0),}\hfill \\ {\displaystyle c_{61}=b_2^R{g}_2^{{}^{\prime }I}(0),c_{62}=b_2^I{g}_2^{{}^{\prime }R}(0),c_{63}=b_2^I{g}_2^{{}^{\prime }K}(0),c_{64}=b_2^K{g}_2^{{}^{\prime }J}(0),}\hfill \\ {\displaystyle c_{71}=b_2^R{g}_2^{{}^{\prime }J}(0),c_{72}=b_2^I{g}_2^{{}^{\prime }K}(0),c_{73}=b_2^J{g}_2^{{}^{\prime }R}(0),c_{74}=b_2^K{g}_2^{{}^{\prime }I}(0),}\hfill \\ {\displaystyle c_{81}=b_2^R{g}_2^{{}^{\prime }K}(0),c_{82}=b_2^I{g}_2^{{}^{\prime }J}(0),c_{83}=b_2^J{g}_2^{{}^{\prime }I}(0),c_{84}=b_2^K{g}_2^{{}^{\prime }R}(0).}\hfill \end{array}\right)\end{array}$$ 3.5

The characteristic equation of (3.4) owns the form:

$$\begin{array}{c}\hfill det\left[\begin{array}{cc}{\mathcal{M}}_1& {\mathcal{M}}_2\\ {\mathcal{M}}_3& {\mathcal{M}}_1\end{array}\right]=0,\end{array}$$ 3.6

where

$$\begin{array}{cc}& {\mathcal{M}}_1=\left[\begin{array}{cccc}{s}^{\rho }+a{e}^{-s\sigma }& 0& 0& 0\\ 0& s^{\rho }+a{e}^{-s\sigma }& 0& 0\\ 0& 0& s^{\rho }+a{e}^{-s\sigma }& 0\\ 0& 0& 0& s^{\rho }+a{e}^{-s\sigma }\end{array}\right],\hfill \end{array}$$ 3.7

$$\begin{array}{cc}& {\mathcal{M}}_2=\left[\begin{array}{cccc}-c_{11}& -c_{12}& -c_{13}& -c_{14}\\ -c_{23}& -c_{24}& -c_{21}& -c_{22}\\ -c_{34}& -c_{33}& -c_{32}& -c_{31}\\ -c_{44}& -c_{43}& -c_{42}& -c_{41}\end{array}\right],\hfill \end{array}$$ 3.8

$$\begin{array}{cc}& {\mathcal{M}}_3=\left[\begin{array}{cccc}-c_{51}& -c_{52}& -c_{53}& -c_{54}\\ -c_{62}& -c_{61}& -c_{64}& -c_{63}\\ -c_{73}& -c_{74}& -c_{71}& -c_{72}\\ -c_{84}& -c_{83}& -c_{82}& -c_{81}\end{array}\right].\hfill \end{array}$$ 3.9

It follows from (3.6) that

$$\begin{array}{c}\hfill det\left[\begin{array}{cccc}{(s^{\rho }+a{e}^{-s\sigma })}^2-d_{11}& -d_{12}& -d_{13}& -d_{14}\\ -d_{21}& {(s^{\rho }+a{e}^{-s\sigma })}^2-d_{22}& -d_{23}& -d_{24}\\ -d_{31}& -d_{32}& {(s^{\rho }+a{e}^{-s\sigma })}^2-d_{33}& -d_{34}\\ -d_{41}& -d_{42}& -d_{43}& {(s^{\rho }+a{e}^{-s\sigma })}^2-d_{44}\end{array}\right]=0,\end{array}$$ 3.10

where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle d_{11}=c_{51}{c}_{11}+c_{52}{c}_{23}+c_{53}{c}_{34}+c_{54}{c}_{44},}\hfill \\ {\displaystyle d_{12}=c_{51}{c}_{12}+c_{52}{c}_{24}+c_{53}{c}_{33}+c_{54}{c}_{43},}\hfill \\ {\displaystyle d_{13}=c_{51}{c}_{13}+c_{52}{c}_{21}+c_{53}{c}_{32}+c_{54}{c}_{42},}\hfill \\ {\displaystyle d_{14}=c_{51}{c}_{14}+c_{52}{c}_{22}+c_{53}{c}_{31}+c_{54}{c}_{41},}\hfill \\ {\displaystyle d_{21}=c_{62}{c}_{11}+c_{61}{c}_{23}+c_{63}{c}_{34}+c_{63}{c}_{44},}\hfill \\ {\displaystyle d_{22}=c_{62}{c}_{12}+c_{61}{c}_{24}+c_{63}{c}_{33}+c_{63}{c}_{43},}\hfill \\ {\displaystyle d_{23}=c_{62}{c}_{13}+c_{61}{c}_{21}+c_{63}{c}_{32}+c_{63}{c}_{42},}\hfill \\ {\displaystyle d_{24}=c_{62}{c}_{14}+c_{61}{c}_{22}+c_{63}{c}_{31}+c_{63}{c}_{41},}\hfill \\ {\displaystyle d_{31}=c_{73}{c}_{11}+c_{74}{c}_{23}+c_{71}{c}_{34}+c_{72}{c}_{44},}\hfill \\ {\displaystyle d_{32}=c_{73}{c}_{12}+c_{74}{c}_{24}+c_{71}{c}_{33}+c_{72}{c}_{43},}\hfill \\ {\displaystyle d_{33}=c_{73}{c}_{13}+c_{74}{c}_{21}+c_{71}{c}_{32}+c_{72}{c}_{42},}\hfill \\ {\displaystyle d_{34}=c_{73}{c}_{14}+c_{74}{c}_{22}+c_{71}{c}_{31}+c_{72}{c}_{41},}\hfill \\ {\displaystyle d_{41}=c_{84}{c}_{11}+c_{83}{c}_{23}+c_{82}{c}_{34}+c_{81}{c}_{44},}\hfill \\ {\displaystyle d_{42}=c_{84}{c}_{12}+c_{83}{c}_{24}+c_{82}{c}_{33}+c_{81}{c}_{43},}\hfill \\ {\displaystyle d_{43}=c_{84}{c}_{13}+c_{83}{c}_{21}+c_{82}{c}_{32}+c_{81}{c}_{42},}\hfill \\ {\displaystyle d_{44}=c_{84}{c}_{14}+c_{83}{c}_{22}+c_{82}{c}_{31}+c_{81}{c}_{41}.}\hfill \end{array}\right)\end{array}$$ 3.11

By (3.10), one has

$$\begin{array}{c}\hfill {\varphi }^8+{\varrho }_3{\varphi }^6+{\varrho }_2{\varphi }^4+{\varrho }_1{\varphi }^2+{\varrho }_0=0,\end{array}$$ 3.12

where $\varphi =s^{\rho }+a{e}^{-s\sigma }$ and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {\varrho }_0=d_{11}{d}_{22}{d}_{33}{d}_{44}+d_{11}(d_{23}{d}_{34}{d}_{42}+d_{24}{d}_{32}{d}_{43})}\hfill \\ {\displaystyle -d_{11}{d}_{33}{d}_{24}{d}_{42}-d_{23}{d}_{32}{d}_{11}{d}_{44}-d_{34}{d}_{43}{d}_{11}{d}_{22}}\hfill \\ {\displaystyle -d_{12}{d}_{21}{d}_{33}{d}_{44}-d_{21}(d_{13}{d}_{34}{d}_{42}+d_{14}{d}_{32}{d}_{43})}\hfill \\ {\displaystyle +d_{14}{d}_{21}{d}_{43}{d}_{33}+d_{13}{d}_{21}{d}_{32}{d}_{44}+d_{12}{d}_{21}{d}_{34}{d}_{43}}\hfill \\ {\displaystyle +d_{12}{d}_{23}{d}_{31}{d}_{44}+d_{13}{d}_{24}{d}_{31}{d}_{42}+d_{14}{d}_{31}{d}_{43}{d}_{22}}\hfill \\ {\displaystyle -d_{14}{d}_{23}{d}_{31}{d}_{42}-d_{13}{d}_{31}{d}_{22}{d}_{44}-d_{12}{d}_{24}{d}_{31}{d}_{43}}\hfill \\ {\displaystyle -d_{41}(d_{12}{d}_{23}{d}_{34}+d_{13}{d}_{24}{d}_{32})-d_{14}{d}_{41}{d}_{22}{d}_{33}}\hfill \\ {\displaystyle +d_{14}{d}_{23}{d}_{32}{d}_{41}+d_{13}{d}_{34}{d}_{41}{d}_{22}+d_{12}{d}_{24}{d}_{41}{d}_{33},}\hfill \\ {\displaystyle {\varrho }_1=-[d_{33}{d}_{44}(d_{11}+d_{22})+d_{11}{d}_{22}(d_{11}+d_{44})]}\hfill \\ {\displaystyle -(d_{23}{d}_{34}{d}_{42}+d_{24}{d}_{32}{d}_{43})+d_{24}{d}_{42}(d_{11}+d_{33})}\hfill \\ {\displaystyle +d_{23}{d}_{32}(d_{11}+d_{44})+d_{34}{d}_{43}(d_{11}+d_{22})}\hfill \\ {\displaystyle +d_{12}{d}_{21}(d_{33}+d_{44})-d_{14}{d}_{21}{d}_{42}-d_{13}{d}_{21}{d}_{32}}\hfill \\ {\displaystyle +d_{13}{d}_{31}(d_{22}+d_{44})-d_{12}{d}_{23}{d}_{31}-d_{14}{d}_{31}{d}_{43}}\hfill \\ {\displaystyle +d_{14}{d}_{41}(d_{22}+d_{33})-d_{13}{d}_{34}{d}_{41}-d_{12}{d}_{24}{d}_{41},}\hfill \\ {\displaystyle {\varrho }_2=(d_{11}+d_{22})(d_{33}+d_{44})+d_{33}{d}_{44}+d_{11}{d}_{22}-d_{24}{d}_{42}}\hfill \\ {\displaystyle -d_{23}{d}_{32}-d_{34}{d}_{43}-d_{12}{d}_{21}-d_{13}{d}_{31}-d_{14}{d}_{41},}\hfill \\ {\displaystyle {\varrho }_3=-(d_{11}+d_{22}+d_{33}+d_{44}).}\hfill \end{array}\right)\end{array}$$ 3.13

By computer, we can obtain the roots of (3.12). Here we denote the root of (3.12) by ${\varphi }_j(j=1,2,\dots ,8)$ and assume that

$$\begin{array}{c}\hfill {\varphi }_j={\psi }_j+i{\omega }_j,j=1,2,\dots ,8,\end{array}$$ 3.14

where ${\psi }_j$ and ${\omega }_j$ stand for the real parts and imaginary parts of ${\varphi }_j$, respectively. Then

$$\begin{array}{c}\hfill s^{\rho }+a{e}^{-s\sigma }={\psi }_j+i{\omega }_j,j=1,2,\dots ,8,\end{array}$$ 3.15

Assume that $s=i\theta =\theta \left(cos,\frac{\pi }{2},+,i,sin,\frac{\pi }{2}\right)$ is the root of (3.15), then it follows from (3.15) that

$$\begin{array}{cc}& {\theta }^{\rho }\left(cos,\frac{\rho \pi }{2},+,i,sin,\frac{\rho \pi }{2}\right)+a(cos\theta \sigma -isin\theta \sigma )\hfill \\ \hfill & ={\psi }_j+i{\omega }_j,j=1,2,\dots ,8,\hfill \end{array}$$ 3.16

which leads to

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle cos\theta \sigma =\frac{1}{a}\left({\psi }_j,-,{\theta }^{\rho },cos,\frac{\rho \pi }{2}\right),}\hfill \\ {\displaystyle sin\theta \sigma =\frac{1}{a}\left(-,{\omega }_j,+,{\theta }^{\rho },sin,\frac{\rho \pi }{2}\right).}\hfill \end{array}\right)\end{array}$$ 3.17

In view of (3.17), one has

$$\begin{array}{c}\hfill {\left[\frac{1}{a},\left({\psi }_j,-,{\theta }^{\rho },cos,\frac{\rho \pi }{2}\right)\right]}^2+{\left[\frac{1}{a},\left(-,{\omega }_j,+,{\theta }^{\rho },sin,\frac{\rho \pi }{2}\right)\right]}^2=1,\end{array}$$ 3.18

which implies that

$$\begin{array}{cc}& {\theta }^{2\rho }-2\left[{\psi }_j,cos,\frac{\rho \pi }{2},+,{\omega }_j,sin,\frac{\rho \pi }{2}\right]{\theta }^{\rho }\hfill \\ \hfill & +{\psi }_j^2+{\omega }_j^2-a^2=0.\hfill \end{array}$$ 3.19

Now we obtain the following conclusion:

Lemma 3.1

For (3.19), if ${\psi }_j^2+{\omega }_j^2>a^2$, then (3.19) owns eight positive real roots:

$$\begin{array}{cc}\hfill {\theta }_j=& \left[{\psi }_j,cos,\frac{\rho \pi }{2},+,{\omega }_j,sin,\frac{\rho \pi }{2}\right)\hfill \\ \hfill & {\left(+,\sqrt{{\left({\psi }_j,cos,\frac{\rho \pi }{2},+,{\omega }_j,sin,\frac{\rho \pi }{2}\right)}^2-4({\psi }_j^2+{\omega }_j^2-a^2)}\right]}^{\frac{1}{\rho }},\hfill \end{array}$$ 3.20

where $j=1,2,\dots ,8.$

Proof

The conclusion of Lemma 3.1 is easily obtained by solving the common algebraic equation. Here we omit it.

By (3.17), we get

$$\begin{array}{cc}\hfill {\sigma }_j^{(l)}=& \frac{1}{{\theta }_j}\left[arccos,\frac{1}{a},\left({\psi }_j,-,{\theta }^{\rho },cos,\frac{\rho \pi }{2}\right),+,2,l,\pi \right],\hfill \\ \hfill & j=1,2,\dots ,8;l=0,1,2,\dots .\hfill \end{array}$$ 3.21

Let

$$\begin{array}{c}\hfill {\sigma }_0=min\{{\sigma }_j^{(l)}\},{\theta }_0{=\theta |}_{\sigma ={\sigma }_0},j=1,2,\dots ,8;l=0,1,2,\dots .\end{array}$$ 3.22

$\square$

Now we list the necessary assumption as follows:

($\mathcal{A}5)$   $sin{\theta }_0{\sigma }_{0}cos\frac{(\rho -1)\pi }{2}+cos{\theta }_0{\sigma }_{0}sin\frac{(\rho -1)\pi }{2}>0,$

Lemma 3.2

If $s(\sigma )={\delta }_1(\sigma )+i{\delta }_2(\sigma )$ is the root of Eq. (3.12) around $\sigma ={\sigma }_0$ which satisfies ${\delta }_1({\sigma }_0)=0,{\delta }_2({\sigma }_0)={\theta }_0$, then $Re\left[\frac{\mathit{ds}}{d\sigma }\right]{|}_{\sigma ={\sigma }_0,\theta ={\theta }_0}>0.$

Proof

Based on (3.12), one gets

$$\begin{array}{cc}& \left[8,{\left(s^{\rho },+,a,s^{-s\sigma }\right)}^7,+,6,{\varrho }_3,{\left(s^{\rho },+,a,s^{-s\sigma }\right)}^5\right)\hfill \\ \hfill & \left(,+,4,{\varrho }_2,{\left(s^{\rho },+,a,s^{-s\sigma }\right)}^3,+,2,{\varrho }_1,\left(s^{\rho },+,a,s^{-s\sigma }\right)\right]\hfill \\ \hfill & \times \left[\left(\rho ,s^{\rho -1},-,a,e^{-s\sigma },\sigma \right),\frac{\mathit{ds}}{d\sigma },-,a,e^{-s\sigma },s\right]=0.\hfill \end{array}$$ 3.23

Then

$$\begin{array}{c}\hfill {\left[\frac{\mathit{ds}}{d\sigma }\right]}^{-1}=\frac{\rho s^{\rho -1}-a{e}^{-s\sigma }\sigma }{a{e}^{-s\sigma }s}=\frac{\rho s^{\rho -1}}{a{e}^{-s\sigma }s}-\frac{\sigma }{s}.\end{array}$$ 3.24

Hence

$$\begin{array}{cc}& \text{Re}\left\{\frac{\mathit{ds}}{d\sigma }\right\}{|}_{\sigma ={\sigma }_0,\theta ={\theta }_0}\hfill \\ \hfill & =\text{Re}\left\{\frac{\rho s^{\rho -1}}{a{e}^{-s\sigma }s}\right\}{|}_{\sigma ={\sigma }_0,\theta ={\theta }_0}\hfill \\ \hfill & =\text{Re}\left\{\frac{\rho {\theta }_0^{\rho -1}\left[cos,\frac{(\rho -1)\pi }{2},+,i,sin,\frac{(\rho -1)\pi }{2}\right]}{ia{\theta }_0(cos{\theta }_0{\sigma }_0-isin{\theta }_0{\sigma }_0)}\right\}\hfill \\ \hfill & =\frac{\rho a{\theta }_0^{\rho }\left[sin,{\theta }_0,{\sigma }_0,cos,\frac{(\rho -1)\pi }{2},+,cos,{\theta }_0,{\sigma }_0,sin,\frac{(\rho -1)\pi }{2}\right]}{a^2{\theta }_0^2}.\hfill \end{array}$$ 3.25

In view of $(\mathcal{A}5)$, we have

$$\begin{array}{c}\hfill \text{Re}\left\{{\left[\frac{\mathit{ds}}{d\sigma }\right]}^{-1}\right\}{|}_{\sigma ={\sigma }_0,\theta ={\theta }_0}>0.\end{array}$$ 3.26

This ends the proof. $\square$

Now we give the hypothesis as follows:

($\mathcal{A}6)$   Assume that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {\mathcal{Z}}_1={\upsilon }_1>0,}\hfill \\ {\displaystyle {\mathcal{Z}}_2=det\left[\begin{array}{cc}{\upsilon }_1& 1\\ {\upsilon }_3& {\upsilon }_2\end{array}\right]>0,}\hfill \\ {\displaystyle {\mathcal{Z}}_3=det\left[\begin{array}{ccc}{\upsilon }_1& 1& 0\\ {\upsilon }_3& {\upsilon }_2& {\upsilon }_1\\ {\upsilon }_5& {\upsilon }_4& {\upsilon }_3\end{array}\right]>0,}\hfill \\ {\displaystyle {\mathcal{Z}}_4=det\left[\begin{array}{cccc}{\upsilon }_1& 1& 0& 0\\ {\upsilon }_3& {\upsilon }_2& {\upsilon }_1& 1\\ {\upsilon }_5& {\upsilon }_4& {\upsilon }_3& {\upsilon }_2\\ {\upsilon }_7& {\upsilon }_6& {\upsilon }_5& {\upsilon }_4\end{array}\right]>0,}\hfill \\ {\displaystyle {\mathcal{Z}}_5=det\left[\begin{array}{ccccc}{\upsilon }_1& 1& 0& 0& 0\\ {\upsilon }_3& {\upsilon }_2& {\upsilon }_1& 1& 0\\ {\upsilon }_5& {\upsilon }_4& {\upsilon }_3& {\upsilon }_2& {\upsilon }_1\\ {\upsilon }_7& {\upsilon }_6& {\upsilon }_5& {\upsilon }_4& {\upsilon }_3\\ 0& {\upsilon }_8& {\upsilon }_7& {\upsilon }_6& {\upsilon }_5\end{array}\right]>0,}\hfill \\ {\displaystyle {\mathcal{Z}}_6=det\left[\begin{array}{cccccc}{\upsilon }_1& 1& 0& 0& 0& 0\\ {\upsilon }_3& {\upsilon }_2& {\upsilon }_1& 1& 0& 0\\ {\upsilon }_5& {\upsilon }_4& {\upsilon }_3& {\upsilon }_2& {\upsilon }_1& 1\\ {\upsilon }_0& {\upsilon }_6& {\upsilon }_5& {\upsilon }_4& {\upsilon }_3& {\upsilon }_2\\ 0& {\upsilon }_8& {\upsilon }_7& {\upsilon }_6& {\upsilon }_5& {\upsilon }_4\\ 0& 0& 0& {\upsilon }_8& {\upsilon }_7& {\upsilon }_6\end{array}\right]>0,}\hfill \\ {\displaystyle {\mathcal{Z}}_7=det\left[\begin{array}{ccccccc}{\upsilon }_1& 1& 0& 0& 0& 0& 0\\ {\upsilon }_3& {\upsilon }_2& {\upsilon }_1& 1& 0& 0& 0\\ {\upsilon }_5& {\upsilon }_4& {\upsilon }_3& {\upsilon }_2& {\upsilon }_1& 1& 0\\ {\upsilon }_0& {\upsilon }_6& {\upsilon }_5& {\upsilon }_4& {\upsilon }_3& {\upsilon }_2& {\upsilon }_1\\ 0& {\upsilon }_8& {\upsilon }_7& {\upsilon }_6& {\upsilon }_5& {\upsilon }_4& {\upsilon }_3\\ 0& 0& 0& {\upsilon }_8& {\upsilon }_7& {\upsilon }_6& {\upsilon }_5\\ 0& 0& 0& 0& 0& {\upsilon }_8& {\upsilon }_7\end{array}\right]>0,}\hfill \\ {\displaystyle {\mathcal{Z}}_8={\upsilon }_8{\mathcal{Z}}_7>0.}\hfill \end{array}\right)\end{array}$$ 3.27

where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {\upsilon }_1=8a,}\hfill \\ {\displaystyle {\upsilon }_2=28a^2+{\varrho }_3,}\hfill \\ {\displaystyle {\upsilon }_3=56a^3+6{\varrho }_{3}a,}\hfill \\ {\displaystyle {\upsilon }_4=70a^4+15{\varrho }_3{a}^2+{\varrho }_2,}\hfill \\ {\displaystyle {\upsilon }_5=56a^5+20{\varrho }_3{a}^3+4{\varrho }_{2}a,}\hfill \\ {\displaystyle {\upsilon }_6=28a^6+15{\varrho }_3{a}^4+6{\varrho }_2{a}^2+{\varrho }_1,}\hfill \\ {\displaystyle {\upsilon }_7=8a^7+6{\varrho }_3{a}^5+4{\varrho }_2{a}^3+2{\varrho }_{1}a,}\hfill \\ {\displaystyle {\upsilon }_8=a^8+{\rho }_3{a}^6+{\rho }_2{a}^4+{\varrho }_1{a}^2+{\varrho }_0.}\hfill \end{array}\right)\end{array}$$ 3.28

Lemma 3.3

Assume that $\sigma =0$ and $(\mathcal{A}6)$ is fulfilled, then model (1.1) is locally asymptotically stable.

Proof

Let $\sigma =0$, then it follows from Eq. (3.12) that

$$\begin{array}{cc}& {(\lambda +a)}^8+{\varrho }_3{(\lambda +a)}^6+{\varrho }_2{(\lambda +a)}^4\hfill \\ \hfill & +{\varrho }_1{(\lambda +a)}^2+{\varrho }_0=0.\hfill \end{array}$$ 3.29

Namely,

$$\begin{array}{cc}& {\lambda }^8+{\upsilon }_1{\lambda }^7+{\upsilon }_2{\lambda }^6+{\upsilon }_3{\lambda }^5\hfill \\ \hfill & +{\upsilon }_4{\lambda }^4+{\upsilon }_5{\lambda }^3+{\upsilon }_6{\lambda }^2+{\upsilon }_7\lambda +{\upsilon }_8=0.\hfill \end{array}$$ 3.30

In term of $(\mathcal{A}6)$, one knows that each root ${\varsigma }_i$ of (3.30) satisfies $|\text{arg}({\varsigma }_i)|>\frac{\rho \pi }{2}(i=1,2,\dots ,8).$ Therefore, Lemma 3.3 is right. This completes the proof. $\square$

According to the investigation above, the following results are established.

Theorem 3.1

Under the assumptions $(\mathcal{A}1)$ -$(\mathcal{A}6)$. If $\sigma$ falls into in the interval $[0,{\sigma }_0)$ then the zero equilibrium point of model (1.1) is locally asymptotically stable and a Hopf bifurcation arises at the zero equilibrium point if $\sigma$ is equal to ${\sigma }_0.$

Remark 3.1

In 2019, although Huang et al. (2019) explored the stability behavior and Hopf bifurcation phenomenon for delayed fractional-order quaternion-valued neural networks, they are only focused on the impact of the transmission delay on stability and the existence of Hopf bifurcation of the considered networks. They are not concerned with the impact of leakage delays on the stability and the existence of Hopf bifurcation of the considered networks. In the current work, we have detailedly analyzed the role of leakage delay in stabilizing the neural networks. Up to now, few authors focus on this topic. In addition, some flexible variable substitution and analysis techniques show the bright spot of this study. This research supplements and perfects the previous studies (e.g., Huang et al. 2019; Xu et al. 2021).

Remark 3.2

Although we have dealt with the bifurcation issue of system (1.1) by virtue of the same approach as that in Xu et al. (2021a), some mathematical analysis techniques are very different. for example, variable substitution technique, characteristics equation and the analysis on the roots of the characteristics equation are different.

Software simulations

To check the correctness of the derived key results of this work, we employ the computer simulations for the following given neural network model:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{d^{\rho }{w}_1(t)}{d{t}^{\rho }}=-a{w}_1(t-\sigma )+b_1{g}_1(w_2(t)),}\hfill \\ {\displaystyle \frac{d^{\rho }{w}_2(t)}{d{t}^{\rho }}=-a{w}_2(t-\sigma )+b_2{g}_2(w_1(t)),}\hfill \end{array}\right)\end{array}$$ 4.1

where $w_1(t)=w_1^R(t)+i{w}_1^I(t)+j{w}_1^J(t)+k{w}_1^K(t),w_2(t)=w_2^R(t)+i{w}_2^I(t)+j{w}_2^J(t)+k{w}_2^K(t),a=0.2$ and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle b_1=-1.5+0.6i+1.8j+0.7k,}\hfill \\ {\displaystyle b_2=1.3+0.8i-0.9j+1.6k,}\hfill \\ {\displaystyle g_1(w_2)=tanh(w_2^R)+itanh(w_2^I)+jtanh(w_2^J)+ktanh(w_2^K),}\hfill \\ {\displaystyle g_2(w_1)=tanh(w_1^R)+itanh(w_1^I)+jtanh(w_1^J)+ktanh(w_1^K).}\hfill \end{array}\right)\end{array}$$

Apparently, the model (4.1) possesses the zero equilibrium point. In order to illustrate the impact of leakage delay on the stability behavior and bifurcation phenomenon, we select $\rho =0.93$. Taking advantage of Matlab software, we get ${\sigma }_0=0.15$ and ${\theta }_0=4.8803$. Meanwhile, we can verify that all the requirements in Theorem 3.1 hold true. Therefore one knows that the zero equilibrium point of system (4.1) is locally asymptotically stable for $\sigma \in [0,{\sigma }_0)$ and a family of Hopf bifurcation periodic solution arise around the zero equilibrium point. To illustrate this fact, we select $\sigma =0.12<{\sigma }_0=0.15$ and $\sigma =0.18>{\sigma }_0=0.15$ to carry out computer simulations. Figures 1, 2 and 3 stand for the simulation results for $\sigma =0.12<{\sigma }_0=0.15$ and Figs. 4, 5 and 6 stand for the simulation results for $\sigma =0.18>{\sigma }_0=0.15$. From Figs. 1, 2 and 3, one knows that the state of the neuron will gradually tend to zero. From Figs. 4, 5 and 6, one knows that system (4.1) loses its stability and the state of the neuron will keep a periodic oscillationary level in the vicinity of the zero equilibrium point. In addition, to explain this phenomenon distinctly, we also draw the related bifurcation plots which are shown in Figs. 7, 8, 9, 10, 11, 12, 13 and 14.

Fig. 1.

The state plots for neural networks (4.1) with $\sigma =0.12<{\sigma }_0=0.15$. The zero equilibrium point is locally asymptotically stable

Fig. 2.

The relation of two states for neural networks (4.1) with $\sigma =0.12<{\sigma }_0=0.15$. The zero equilibrium point is locally asymptotically stable

Fig. 3.

The phase trajectories for neural networks (4.1) with $\sigma =0.12<{\sigma }_0=0.15$. The zero equilibrium point is locally asymptotically stable

Fig. 4.

The state plots for neural networks (4.1) with $\sigma =0.18>{\sigma }_0=0.15$. A Hopf bifurcation takes place in the vicinity of the zero equilibrium point

Fig. 5.

The relation of two states for neural networks (4.1) with $\sigma =0.18>{\sigma }_0=0.15$. A Hopf bifurcation takes place in the vicinity of the zero equilibrium point

Fig. 6.

The phase trajectories for neural networks (4.1) with $\sigma =0.18>{\sigma }_0=0.15$. A Hopf bifurcation takes place in the vicinity of the zero equilibrium point

Fig. 7.

The bifurcation plot for neural networks (4.1): $\sigma$-$w_1^R$. The bifurcation value is approximately equal to 0.15

Fig. 8.

The bifurcation plot for neural networks (4.1): $\sigma$-$w_1^I$. The bifurcation value is approximately equal to 0.15

Fig. 9.

The bifurcation plot for neural networks (4.1): $\sigma$-$w_1^J$. The bifurcation value is approximately equal to 0.15

Fig. 10.

The bifurcation plot for neural networks (4.1): $\sigma$-$w_1^K$. The bifurcation value is approximately equal to 0.15

Fig. 11.

The bifurcation plot for neural networks (4.1): $\sigma$-$w_2^R$. The bifurcation value is approximately equal to 0.15

Fig. 12.

The bifurcation plot for neural networks (4.1): $\sigma$-$w_2^I$. The bifurcation value is approximately equal to 0.15

Fig. 13.

The bifurcation plot for neural networks (4.1): $\sigma$-$w_2^J$. The bifurcation value is approximately equal to 0.15

Fig. 14.

The bifurcation plot for neural networks (4.1): $\sigma$-$w_2^K$. The bifurcation value is approximately equal to 0.15

Conclusions

The leakage delay plays a key role in revealing the dynamical behavior of fractional-order quaternion-valued neural networks. However, so far, there are few works that deal with the impact of leakage delay on the stability and Hopf bifurcation for fractional-order quaternion-valued neural networks. On the basis of the previous studies, we set up a type of fractional-order quaternion-valued neural networks involving leakage delays. Making use of stability theory and the criterion of the onset of Hopf bifurcation of fractional-order differential dynamical system, we have derived a novel condition (See Theorem 3.1) to guarantee the stability and the onset of Hopf bifurcation for fractional-order quaternion-valued neural networks involving leakage delays. Necessary computer simulation results are clearly displayed to illustrate this fact. The derived results replenish and improve some previous works. The derived results can be applied in image processing, intelligent control, network optimization and so on. Also, the study thoughts can also be applied to deal with numerous other fractional-order quaternion-valued delayed neural networks. It is a pity that this work only involves a single leakage delay. For more leakage delays, the question will become more complex. We will deal with this aspect in near future.

Data availability

No data were used to support this study.

Declarations

Conflict of interests

The authors declare that they have no competing interests.