Dynamic Behavior of a Class of Six-Neuron Fractional BAM Neural Networks

Abstract

In this paper, the stability and Hopf bifurcation of a six-neuron fractional BAM neural network model with multiple delays are considered. By transforming the multiple-delays model into a fractional-order neural network model with a delay through the variable substitution, we prove the conditions for the existence of Hopf bifurcation at the equilibrium point. Finally, our results are verified by numerical simulations.

1. Introduction

The research content of neural networks is quite extensive, reflecting the characteristics of this interdisciplinary technological field. The study of neural network dynamic behavior is a prerequisite for its successful application to practical problems. Due to the nonlinear characteristics of neural networks, many neural network models have dynamic properties such as equilibrium points, periodic solutions, bifurcation, chaos, traveling wave solutions, etc. In recent years, many outstanding scholars have conducted stability analyses on neural networks. For example, Mao and Wang [1] studied the local and global stability of a class of coupled ring network systems with an arbitrary number of neurons and provided sufficient conditions for the existence of branches under different types of delay. Fan et al. [2] analyzed the stability, bifurcation, and chaos dynamics of equilibrium points in a simplified memristor-based fractional-order neural network model. In [3], Xu studied the stability of a class of simplified BAM neural networks at the zero equilibrium point, analyzed the conditions for the generation of Hopf bifurcation, and established the global existence of periodic orbits using related theories.

With the deepening of research on neural networks, many scholars have found that in the practical application of neural networks, due to the influence of various objective factors, there is a time delay in the signal propagation of different neurons. Thus, they have chosen to introduce time delays into the neural network model, which indicates that the propagation of neuronal signals depends on both the current state and on a certain time or period in the past. The introduction of time delays can affect the stability of neural network models, shifting them from stable operation to bifurcation, chaos, oscillation, and other phenomena at the equilibrium points, which is of great significance in studying and solving practical problems [4,5,6,7,8,9]. At present, the research on time-delay neural network models has attracted the attention of many scholars and has led to many excellent results. In [10], Cheng and Xie introduced time delay into a class of triangular neural network models, studied the stability of the zero equilibrium using the characteristic equation, and provided the critical value for Hopf bifurcation. Lin and Xu [11] presented a class of reaction–diffusion neural network models with leakage delays and obtained sufficient conditions for the model to generate Hopf bifurcation with time delays. Mao and Wang established a class of multi-delay four-coupling neural network models and studied the dynamic behavior changes generated at the equilibrium point by introducing different types of time delays from the literature into the model [12]. All of these studies indicate that the existence of time delays can affect the stability of the model. Selecting time delays as parameters in the model allows researchers to more accurately study the dynamic behavior of the system, and has a positive role in promoting the solution of practical problems. For more references on this subject, interested readers may refer to [13,14,15,16,17].

In recent years, with the in-depth study of fractional-order correlation theory and application it has been found that, as compared with integer order differential equations, fractional-order differential equations can better describe complex processes such as heritability and memory in practical problems. As such, this approach has become widely used in physics, biology, engineering, and other fields. It is well known that fractional-order calculus is not merely an extension of integer-order differential and integral calculus; rather, it possesses unique properties that allow for a more truthful and accurate reflection of the system dynamics behavior. At present, fractional calculus has been widely used in neural network systems, especially in neural network models with time delay [18,19,20]. In [21], Xu and Mu discussed the integer order case and the fractional order case of a BAM neural network model with time delay. Through this comparison, it was found that the time delay has an important effect on the stability of the equilibrium points and the generation of Hopf bifurcation in the two kinds of BAM neural network systems with time delay. However, by adjusting the time delay value, the stability region of the fractional BAM neural network can be changed and the time of bifurcation appearance can be prolonged. Xu et al. [22] proposed a new fractional 4D neural network model with double delay. By establishing a series of new stability theories and bifurcation conditions, they fully verified the different time delay effects on fractional order neural network system dynamics. In addition, Li and Liu proposed new methods to determine the finite-time stability of fractional BAM neural network systems in the literature [23]. When time delay is introduced into a fractional-order neural network system, the stability of the system is changed, leading to bifurcation, chaos, periodic solutions, and other phenomena [24,25,26,27]. The research on fractional neural networks has attracted many excellent scholars, and the changes in the dynamic behavior of such networks is the main content of the present study. To date, many excellent results have been achieved; see [28,29,30,31].

In [31], Liu and Li studied the high-dimensional bifurcation analysis of the following integer-order BAM neural networks model with two delays, studying the stability of the systems at the equilibrium points and the conditions for the existence of Hopf bifurcation using the sum of delays as parameters:

$$\left\{\begin{array}{cc}\hfill {\dot{x}}_1\left(t\right)=& -{\mu }_1{x}_1\left(t\right)+c_{21}{f}_1\left(x_2(t-{\tau }_2)\right)+c_{31}{f}_1\left(x_3(t-{\tau }_2)\right)+c_{41}{f}_1\left(x_4(t-{\tau }_2)\right)\hfill \\ & +c_{51}{f}_1\left(x_5(t-{\tau }_2)\right)+c_{61}{f}_1\left(x_6(t-{\tau }_2)\right),\hfill \\ \hfill {\dot{x}}_2\left(t\right)=& -{\mu }_2{x}_2\left(t\right)+c_{12}{f}_2\left(x_1(t-{\tau }_1)\right),\hfill \\ \hfill {\dot{x}}_3\left(t\right)=& -{\mu }_3{x}_3\left(t\right)+c_{13}{f}_3\left(x_1(t-{\tau }_1)\right),\hfill \\ \hfill {\dot{x}}_4\left(t\right)=& -{\mu }_4{x}_4\left(t\right)+c_{14}{f}_4\left(x_1(t-{\tau }_1)\right),\hfill \\ \hfill {\dot{x}}_5\left(t\right)=& -{\mu }_5{x}_5\left(t\right)+c_{15}{f}_5\left(x_1(t-{\tau }_1)\right),\hfill \\ \hfill {\dot{x}}_6\left(t\right)=& -{\mu }_6{x}_6\left(t\right)+c_{16}{f}_6\left(x_1(t-{\tau }_1)\right),\hfill \end{array}\right.$$

(1)

where ${\mu }_i$$(i=1,2,\cdots ,6)$ is the decay rates of the internal neurons, the real constants $c_{i1}$ $(i=1,2,\cdots ,6)$ and $c_{1i}$ show the connection weight between the I-layer and the J-layer, $f_i$ is an activation function, $x_i\left(t\right)$$(i=1,2,\cdots ,6)$ represents the state of the i-th neuron, and ${\tau }_1$ and ${\tau }_2$ represent the time delay of signal transmission between I-layer neurons and J-layer neurons.

A neural network is a kind of complex dynamic system. Many scholars often ignore the delay caused by information transmission between neurons. Actually, the time delay in dynamic systems is universal, and it is more practical to study the delay differential equation model. In addition, fractional calculus is the general case of integral calculus, which can more truly and accurately describe the dynamic behavior changes of the system. Based on the above analysis, and on the basis of previous system (1), we propose the following fractional-order BAM neural network model with multiple delays:

$$\left\{\begin{array}{cc}\hfill D^q{x}_1\left(t\right)=& -{\mu }_1{x}_1\left(t\right)+c_{21}{f}_1\left(x_2(t-{\tau }_2)\right)+c_{31}{f}_1\left(x_3(t-{\tau }_2)\right)+c_{41}{f}_1\left(x_4(t-{\tau }_2)\right)\hfill \\ & +c_{51}{f}_1\left(x_5(t-{\tau }_2)\right)+c_{61}{f}_1\left(x_6(t-{\tau }_2)\right),\hfill \\ \hfill D^q{x}_2\left(t\right)=& -{\mu }_2{x}_2\left(t\right)+c_{12}{f}_2\left(x_1(t-{\tau }_1)\right),\hfill \\ \hfill D^q{x}_3\left(t\right)=& -{\mu }_3{x}_3\left(t\right)+c_{13}{f}_3\left(x_1(t-{\tau }_1)\right),\hfill \\ \hfill D^q{x}_4\left(t\right)=& -{\mu }_4{x}_4\left(t\right)+c_{14}{f}_4\left(x_1(t-{\tau }_1)\right),\hfill \\ \hfill D^q{x}_5\left(t\right)=& -{\mu }_5{x}_5\left(t\right)+c_{15}{f}_5\left(x_1(t-{\tau }_1)\right),\hfill \\ \hfill D^q{x}_6\left(t\right)=& -{\mu }_6{x}_6\left(t\right)+c_{16}{f}_6\left(x_1(t-{\tau }_1)\right),\hfill \end{array}\right.$$

(2)

where $q\in (0,1]$, $x_i$ stands for the state of the i-th neuron, $f_i$ stands for the activation function for the i-th neuron, ${\tau }_i$ stands for the time delay, and ${\mu }_i,{\beta }_i$ are constants.

In this paper, we extend a class of six neuron BAM neural network model with time delay from integer order to fractional order based on previous studies, then study the stability of the system at the zero equilibrium point and the condition for the existence of Hopf bifurcation. The characteristic equation is obtained by linearizing the fractional six-neuron BAM neural network system with time delay. The distribution of the roots of the characteristic equation is studied to calculate the time delay when the system loses stability and Hopf bifurcation occurs at the zero equilibrium point.

The main contribution of studying six-dimensional neural networks in this paper is to study the needs of more complex high-dimensional neural networks, which are more in line with objective reality. The key technique in studying the stability and bifurcation of fractional order time-delay systems is to consider the influence of fractional order in the characteristic equation, as well as how to linearize the system by Laplace transform.

The rest of this paper is arranged as follows. In Section 2, the basics of fractional differential equations are provided. In Section 3, the stability of the system at the equilibrium point and the existence of Hopf branches are analyzed. In Section 4, the validity of the conclusion is verified by numerical simulation. Finally, in Section 5 the conclusions of this paper are presented.

2. Preliminaries

There are three classical definitions of fractional derivatives, namely, Riemann–Liouville fractional derivatives, Caputo fractional derivatives, and Grunwald–Letnikov fractional derivatives. Among them, Caputo fractional derivatives are a generalization of integer derivatives, which can be calculated numerically using the initial value of the integer order. Therefore, this section provides the definition of Caputo fractional derivatives, along with Lemma 1, which is needed later.

 Definition 1. 

The definition of a Caputo fractional derivative [32] is 

$${}_a^C{D}_t^{p}f\left(t\right)=\frac{1}{\Gamma (k-p)}{\int }_a^t\frac{f^{\left(k\right)}ds}{{(t-s)}^{p+1-k}},$$

where $k-1<p<k\in Z^{+}$, $\Gamma \left(s\right)={\int }_0^{\infty }{t}^{s-1}{e}^{-t}dt$.

By Laplace transformation of this Caputo fractional derivative, we can obtain the following form: 

$$L\{D^{p}f\left(t\right);s\}=s^{p}F\left(s\right)-\sum _{l=0}^{n-1}{s}^{p-l-1}{f}^{\left(l\right)}\left(0\right),n-1<p<n\in Z^{+},$$

where $f^{\left(l\right)}\left(0\right)=0,l=1,2,\dots ,n$, $L\{D^{p}f\left(t\right);s\}=s^{p}F\left(s\right)$. 

 Lemma 1 

([33]). Consider the following multi-variable fractional order system:

$$\left\{\begin{array}{c}{D}^{p_1}{u}_1\left(t\right)=l_{11}{u}_1\left(t\right)+l_{12}{u}_2\left(t\right)+\cdots +l_{1(n-1)}{u}_{n-1}\left(t\right)+l_{1n}{u}_n\left(t\right),\\ D^{p_2}{u}_2\left(t\right)=l_{21}{u}_1\left(t\right)+l_{22}{u}_2\left(t\right)+\cdots +l_{2(n-1)}{u}_{n-1}\left(t\right)+l_{2n}{u}_n\left(t\right),\\ \cdots \cdots \\ D^{p_n}{u}_n\left(t\right)=l_{n1}{u}_1\left(t\right)+l_{n2}{u}_2\left(t\right)+\cdots +l_{n(n-1)}{u}_{n-1}\left(t\right)+l_{nn}{u}_n\left(t\right),\end{array}\right.$$

(3)

where p_i ∈ (0, 1] (i = 1, 2,…, n) , σ $is equal to the inverse of the lowest common multiple of the denominator$μ_i $of$ p_i, p_i = $\frac{v_i}{{\mu }_i}$, (v_i, μ_i) = 1, v_i ≤ μ_i ,$and$ v_i, μ_i ∈ Z⁺ $for$ i = 1, 2,…, n. $Furthermore, we define$

$$▵\left(\lambda \right)=\left[\begin{array}{cccc}{\lambda }^{\gamma p_1}-l_{11}& {-l}_{12}& \cdots & {-l}_{1n}\\ {-l}_{21}& {\lambda }^{\gamma p_2}-l_{22}& \cdots & {-l}_{2n}\\ \cdots & \cdots & \ddots & ⋮\\ {-l}_{n1}& {-l}_{n2}& \cdots & {\lambda }^{\gamma p_n}-l_{nn}\end{array}\right].$$

It is clear that the zero solution is the equilibrium point of system (2). When all the roots of the equation $det\left(▵\left(\lambda \right)\right)=0$ satisfy $\mid arg\left(\lambda \right)\mid >\frac{\sigma \pi }{2}$, the equilibrium point of system (2) is globally and asymptotically Lyapunov stable. 

3. Stability and Hopf Bifurcation

First, we carry out the following transformation:

$$\left\{\begin{array}{c}\tau ={\tau }_1+{\tau }_2,\hfill \\ u_1\left(t\right)=x_1(t-{\tau }_1),\hfill \\ u_2\left(t\right)=x_2\left(t\right),\hfill \\ u_3\left(t\right)=x_3\left(t\right),\hfill \\ u_4\left(t\right)=x_4\left(t\right),\hfill \\ u_5\left(t\right)=x_5\left(t\right),\hfill \\ u_6\left(t\right)=x_6\left(t\right).\hfill \end{array}\right.$$

(4)

In view of (4), system (2) can be transformed into

$$\left\{\begin{array}{cc}\hfill D^q{\mu }_1\left(t\right)=& -{\mu }_1{\mu }_1\left(t\right)+c_{21}{f}_1\left({\mu }_2(t-\tau )\right)+c_{31}{f}_1\left({\mu }_3(t-\tau )\right)+c_{41}{f}_1\left({\mu }_4(t-\tau )\right)\hfill \\ & +c_{51}{f}_1\left({\mu }_5(t-\tau )\right)+c_{61}{f}_1\left({\mu }_6(t-\tau )\right),\hfill \\ \hfill D^q{\mu }_2\left(t\right)=& -{\mu }_2{\mu }_2\left(t\right)+c_{12}{f}_2\left({\mu }_1\left(t\right)\right),\hfill \\ \hfill D^q{\mu }_3\left(t\right)=& -{\mu }_3{\mu }_3\left(t\right)+c_{13}{f}_3\left({\mu }_1\left(t\right)\right),\hfill \\ \hfill D^q{\mu }_4\left(t\right)=& -{\mu }_4{\mu }_4\left(t\right)+c_{14}{f}_4\left({\mu }_1\left(t\right)\right),\hfill \\ \hfill D^q{\mu }_5\left(t\right)=& -{\mu }_5{\mu }_5\left(t\right)+c_{15}{f}_5\left({\mu }_1\left(t\right)\right),\hfill \\ \hfill D^q{\mu }_6\left(t\right)=& -{\mu }_6{\mu }_6\left(t\right)+c_{16}{f}_6\left({\mu }_1\left(t\right)\right).\hfill \end{array}\right.$$

(5)

$$\begin{array}{c}\left(\mathbf{H}\mathbf{1}\right)f_i\in C^1,f_i\left(0\right)=0,(i=1,2,\cdots 6).\hfill \end{array}$$

From (H1), the linearization transformation of system (5) is

$$\left\{\begin{array}{cc}\hfill D^q{\mu }_1\left(t\right)=& -{\mu }_1{\mu }_1\left(t\right)+m_{21}{\mu }_2(t-\tau )+m_{31}{\mu }_3(t-\tau )+m_{41}{\mu }_4(t-\tau )\hfill \\ & +m_{51}{\mu }_5(t-\tau )+m_{61}{\mu }_6(t-\tau ),\hfill \\ \hfill D^q{\mu }_2\left(t\right)=& -{\mu }_2{\mu }_2\left(t\right)+n_{12}{\mu }_1\left(t\right),\hfill \\ \hfill D^q{\mu }_3\left(t\right)=& -{\mu }_3{\mu }_3\left(t\right)+n_{13}{\mu }_1\left(t\right),\hfill \\ \hfill D^q{\mu }_4\left(t\right)=& -{\mu }_4{\mu }_4\left(t\right)+n_{14}{\mu }_1\left(t\right),\hfill \\ \hfill D^q{\mu }_5\left(t\right)=& -{\mu }_5{\mu }_5\left(t\right)+n_{15}{\mu }_1\left(t\right),\hfill \\ \hfill D^q{\mu }_6\left(t\right)=& -{\mu }_6{\mu }_6\left(t\right)+n_{16}{\mu }_1\left(t\right),\hfill \end{array}\right.$$

(6)

where $m_{i1}=c_{i1}{f}_i^{\prime }\left(0\right)$ and $n_{1i}={c_{1i}f}_i^{\prime }\left(0\right)$$(i=2,3,4,5,6)$. Then, we calculate the characteristic equation of system (6) as follows:

$$det\left[\begin{array}{cccccc}{s}^q+{\mu }_1& {-m}_{21}{e}^{-s\tau }& {-m}_{31}{e}^{-s\tau }& {-m}_{41}{e}^{-s\tau }& {-m}_{51}{e}^{-s\tau }& {-m}_{61}{e}^{-s\tau }\\ {-n}_{12}& s^q+{\mu }_2& 0& 0& 0& 0\\ {-n}_{13}& 0& s^q+{\mu }_3& 0& 0& 0\\ {-n}_{14}& 0& 0& s^q+{\mu }_4& 0& 0\\ {-n}_{15}& 0& 0& 0& s^q+{\mu }_5& 0\\ {-n}_{16}& 0& 0& 0& 0& s^q+{\mu }_6\end{array}\right]=0.$$

Further, we can obtain

$$s^{6q}+a_5{s}^{5q}+a_4{s}^{4q}+a_3{s}^{3q}+a_2{s}^{2q}+a_1{s}^q+a_0+(b_4{s}^{4q}+b_3{s}^{3q}+b_2{s}^{2q}+b_1{s}^q+b_0)e^{-s\tau }=0,$$

(7)

where

$$\begin{array}{cc}\hfill a_5=& \sum _{i=1}^6{\mu }_i,\hfill \\ \hfill a_4=& \sum _{i=1}^6\sum _{j=1,j>i}^6{\mu }_i{\mu }_j,\hfill \\ \hfill a_3=& \sum _{i=1}^6\sum _{j=1,j>i}^6\sum _{k=1,k>j}^6{\mu }_i{\mu }_j{\mu }_k,\hfill \\ \hfill a_2=& \sum _{i=1}^6\sum _{j=1,j>i}^6\sum _{k=1,k>j}^6\sum _{l=1,l>k}^6{\mu }_i{\mu }_j{\mu }_k{\mu }_l,\hfill \\ \hfill a_1=& \sum _{i=1}^6\sum _{j=1,j>i}^6\sum _{k=1,k>j}^6\sum _{l=1,l>k}^6\sum _{p=1,p>l}^6{\mu }_i{\mu }_j{\mu }_k{\mu }_l{\mu }_p,\hfill \\ \hfill a_0=& {\mu }_1{\mu }_2{\mu }_3{\mu }_4{\mu }_5{\mu }_6,\hfill \end{array}$$

$$\begin{array}{cc}\hfill b_4=& -\sum _{i=2}^6{n}_{1i}{m}_{i1},\hfill \\ \hfill b_3=& -\sum _{i=2}^6\sum _{j=2,j\ne i}^6{n}_{1i}{m}_{i1}{\mu }_j,\hfill \\ \hfill b_2=& -\sum _{i=2}^6\sum _{2\le j<k\le 6,j,k\ne i}{n}_{li}{m}_{i1}{\mu }_j{\mu }_k,\hfill \\ \hfill b_1=& -\sum _{i=2}^6\sum _{2\le j<k<l\le 6,j,k,l\ne i}^6{n}_{1i}{m}_{i1}{\mu }_j{\mu }_k{\mu }_l,\hfill \\ \hfill b_0=& -\sum _{i=2}^6\sum _{2\le j<k<l<p\le 6,j,k,l,p\ne i}^6{n}_{1i}{m}_{i1}{\mu }_j{\mu }_k{\mu }_l{\mu }_p.\hfill \end{array}$$

Subsequently, in order to more accurately analyze the stability at the equilibrium point of system (5), we make the following assumptions.

(H2) Each of the following expressions holds:

$$\begin{array}{c}{D}_1=a_5>0,\hfill \\ D_2=\left|\begin{array}{cc}{a}_5& 1\\ a_3+b_3& a_4+b_4\end{array}\right|>0,\hfill \\ D_3=\left|\begin{array}{ccc}{a}_5& 1& 0\\ a_3+b_3& a_4+b_4& a_5\\ a_1+b_1& a_2+b_2& a_3+b_3\end{array}\right|>0,\hfill \\ D_4=\left|\begin{array}{cccc}{a}_5& 1& 0& 0\\ a_3+b_3& a_4+b_4& a_5& 1\\ a_1+b_1& a_2+b_2& a_3+b_3& a_4+b_4\\ 0& a_0+b_0& a_1+b_1& a_2+b_2\end{array}\right|>0,\hfill \\ D_5=\left|\begin{array}{ccccc}{a}_5& 1& 0& 0& 0\\ a_3+b_3& a_4+b_4& a_5& 1& 0\\ a_1+b_1& a_2+b_2& a_3+b_3& a_4+b_4& a_5\\ 0& a_0+b_0& a_1+b_1& a_2+b_2& a_3+b_3\\ 0& 0& 0& a_0+b_0& a_1+b_1\end{array}\right|>0,\hfill \\ D_6=\left|\begin{array}{cccccc}{a}_5& 1& 0& 0& 0& 0\\ a_3+b_3& a_4+b_4& a_5& 1& 0& 0\\ a_1+b_1& a_2+b_2& a_3+b_3& a_4+b_4& a_5& 1\\ 0& a_0+b_0& a_1+b_1& a_2+b_2& a_3+b_3& a_4+b_4\\ 0& 0& 0& a_0+b_0& a_1+b_1& a_2+b_2\\ 0& 0& 0& 0& 0& a_0+b_0\end{array}\right|>0.\hfill \end{array}$$

(8)

 Lemma 2. 

When $\tau =0$ and (H2) holds, system (5) is locally asymptotically stable.

 Proof. 

If $\tau =0$, Equation (7) is equal to

$$s^{6q}+a_5{s}^{5q}+(a_4+b_4)s^{4q}+(a_3+b_3)s^{3q}+(a_2+b_2)s^{2q}+(a_1+b_1)s^q+(a_0+b_0)=0.$$

(9)

Let $\lambda =s^q$; Equation (9) then becomes a univariate equation of higher degree with respect to $\lambda$, as shown below:

$$\begin{array}{c}{\lambda }^6+a_5{\lambda }^5+(a_4+b_4){\lambda }^4+(a_3+b_3){\lambda }^3+(a_2+b_2){\lambda }^2+(a_1+b_1)\lambda +(a_0+b_0)=0.\end{array}$$

(10)

The Routh–Hurwitz criterion shows that all roots of Equation (10) have strictly negative real parts. We know that every root ${\lambda }_i$ satisfies $|arg\left({\lambda }_j\right)|>\frac{\pi }{2q}>\frac{\pi }{2},(q\in (0,1)),$$j=(1,2,\cdots ,6)$, which shows that the real part of all roots of the characteristic Equation (10) is less than zero. Per Lemma 2, the zero equilibrium point of system (5) is asymptotically stable.

Next, we discuss the sufficient conditions for system (5) to generate Hopf bifurcation near the equilibrium point.

When $\tau >0$, we can now make the following necessary assumptions and lemmas in order to move forward with the work.

(H3) $a_0^2-b_0^2<0$.

□

 Lemma 3. 

If (H3) holds, there are at least a pair of pure imaginary roots in Equation (7).

 Proof. 

Let $s=iw$$(w>0)$ be a pure imaginary root of the characteristic Equation (7); then, the characteristic Equation (7) becomes

$$\begin{array}{cc}& (w^{6q}(cos3q\pi +isin3q\pi )+a_5{w}^{5q}(cos\frac{5q\pi }{2}+isin\frac{5q\pi }{2})+a_4{w}^{4q}(cos2q\pi +isin2q\pi )\hfill \\ & +a_3{w}^{3q}(cos\frac{3q\pi }{2}+isin\frac{3q\pi }{2})+a_2{w}^{2q}(cosq\pi +isinq\pi )+a_1{w}^q(cos\frac{q\pi }{2}+isin\frac{q\pi }{2})+a_0)\hfill \\ & +(b_4{w}^{4q}(cos2q\pi +isin2q\pi )+b_3{w}^{3q}(cos\frac{3q\pi }{2}+isin\frac{3q\pi }{2})+b_2{w}^{2q}(cosq\pi +isinq\pi )\hfill \\ & +b_1{w}^q(cos\frac{q\pi }{2}+isin\frac{q\pi }{2})+b_0)(cosw\tau -isinw\tau )=0.\hfill \end{array}$$

(11)

By separating the real and imaginary parts of Equation (11), we have

$$\begin{array}{cc}& (b_4{w}^{4q}cos2q\pi +b_3{w}^{3q}cos\frac{3q\pi }{2}+b_2{w}^{2q}cosq\pi +b_1{w}^{q}cos\frac{q\pi }{2}+b_0)cosw\tau \hfill \\ & +(b_4{w}^{4q}sin2q\pi +b_3{w}^{3q}sin\frac{3q\pi }{2}+b_2{w}^{2q}sinq\pi +b_1{w}^{q}sin\frac{q\pi }{2})sinw\tau \hfill \\ & =-(w^{6q}cos3q\pi +a_5{w}^{5q}cos\frac{5q\pi }{2}+(a_4{w}^{4q}cos2q\pi +a_3{w}^{3q}cos\frac{3q\pi }{2}\hfill \\ & +a_2{w}^{2q}cosq\pi +a_1{w}^{q}cos\frac{q\pi }{2}+d_0)\hfill \end{array}$$

(12)

and

$$\begin{array}{cc}& (b_4{w}^{4q}sin2q\pi +b_3{w}^{3q}sin\frac{3q\pi }{2}+b_2{w}^{2q}sinq\pi +b_1{w}^{q}sin\frac{q\pi }{2})cosw\tau \hfill \\ & -(b_4{w}^{4q}cos2q\pi +b_3{w}^{3q}cos\frac{3q\pi }{2}+b_2{w}^{2q}cosq\pi +b_1{w}^{q}cos\frac{q\pi }{2}+b_0)sinw\tau \hfill \\ & =-(w^{6q}sin3q\pi +a_5{w}^{5q}sin\frac{5q\pi }{2}+(a_4{w}^{4q}sin2q\pi +a_3{w}^{3q}sin\frac{3q\pi }{2}\hfill \\ & +a_2{w}^{2q}sinq\pi +a_1{w}^{q}sin\frac{q\pi }{2}).\hfill \end{array}$$

(13)

Let

$$\begin{array}{cc}\hfill E_1=& b_4{w}^{4q}cos2q\pi +b_3{w}^{3q}cos\frac{3q\pi }{2}+b_2{w}^{2q}cosq\pi +b_1{w}^{q}cos\frac{q\pi }{2}+b_0,\hfill \\ \hfill E_2=& b_4{w}^{4q}sin2q\pi +b_3{w}^{3q}sin\frac{3q\pi }{2}+b_2{w}^{2q}sinq\pi +b_1{w}^{q}sin\frac{q\pi }{2},\hfill \\ \hfill E_3=& w^{6q}cos3q\pi +a_5{w}^{5q}cos\frac{5q\pi }{2}+a_4{w}^{4q}cos2q\pi +a_3{w}^{3q}cos\frac{3q\pi }{2}\hfill \\ & +a_2{w}^{2q}cosq\pi +a_1{w}^{q}cos\frac{q\pi }{2}+a_0,\hfill \\ \hfill E_4=& w^{6q}sin3q\pi +a_5{w}^{5q}sin\frac{5q\pi }{2}+a_4{w}^{4q}sin2q\pi +a_3{w}^{3q}sin\frac{3q\pi }{2}\hfill \\ & +a_2{w}^{2q}sinq\pi +a_1{w}^{q}sin\frac{q\pi }{2}.\hfill \end{array}$$

(14)

Then, we have

$$\begin{array}{c}{E}_{1}cosw\tau +E_{2}sinw\tau =-E_3,\hfill \end{array}$$

(15)

$$\begin{array}{c}{E}_{2}cosw\tau -E_{1}sinw\tau =-E_4.\hfill \end{array}$$

(16)

Further, we can obtain

$$\begin{array}{c}cosw\tau =\frac{-(E_1{E}_3+E_2{E}_4)}{E_1^2+E_2^2},\hfill \end{array}$$

(17)

$$\begin{array}{c}sinw\tau =\frac{E_1{E}_4-E_2{E}_3}{E_1^2+E_2^2}.\hfill \end{array}$$

(18)

We can now square the two sides of Equations (15) and (16) and add them together to obtain the following formula:

$$\begin{array}{c}{E}_1^2+E_2^2=E_3^2+E_4^2.\hfill \end{array}$$

(19)

From the Equations (14) and (19), we have

$$\begin{array}{c}{w}^{12q}+C_{11}{w}^{11q}+C_{10}{w}^{10q}+C_9{w}^{9q}+C_8{w}^{8q}+C_7{w}^{7q}+C_6{w}^{6q}+C_5{w}^{5q}\hfill \\ +C_4{w}^{4q}+C_3{w}^{3q}+C_2{w}^{2q}+C_1{w}^q+C_0=0,\hfill \end{array}$$

(20)

where

$$\begin{array}{cc}\hfill C_{11}=& 2a_5(cos\frac{5q\pi }{2}cos3q\pi +sin\frac{5q\pi }{2}sin3q\pi ),\hfill \\ \hfill C_{10}=& a_5^2+2a_4(cos3q\pi cos2q\pi +sin3q\pi sin2q\pi ),\hfill \\ \hfill C_9=& 2a_3(cos3q\pi cos\frac{3q\pi }{2}+sin3q\pi sin\frac{3q\pi }{2})+2a_4{a}_5(cos2q\pi cos\frac{5q\pi }{2}+sin2q\pi sin\frac{5q\pi }{2}),\hfill \\ \hfill C_8=& a_4^2-b_4^2+2a_2(cos3q\pi cosq\pi +sin3q\pi sinq\pi )+2a_3{a}_5(cos\frac{5q\pi }{2}cos\frac{3q\pi }{2}\hfill \\ & +sin\frac{5q\pi }{2}sin\frac{3q\pi }{2}),\hfill \\ \hfill C_7=& 2a_1(cos3q\pi cos\frac{q\pi }{2}+sin3q\pi sin\frac{q\pi }{2})+2a_2{a}_5(cosq\pi cos\frac{5q\pi }{2}+sinq\pi sin\frac{5q\pi }{2})\hfill \\ & +2a_3{a}_4(cos2q\pi cos\frac{3q\pi }{2}+sin2q\pi sin\frac{3q\pi }{2})-2b_3{b}_4(cos2q\pi cos\frac{3q\pi }{2}+sin2q\pi sin\frac{3q\pi }{2}),\hfill \\ \hfill C_6=& a_3^2-b_3^2+2a_{0}cos3q\pi +2a_1{a}_5(cos\frac{q\pi }{2}cos\frac{5q\pi }{2}+sin\frac{q\pi }{2}sin\frac{5q\pi }{2})+2a_2{a}_4(cosq\pi cos2q\pi \hfill \\ & +sinq\pi sin2q\pi )-2b_2{b}_4(cosq\pi cos2q\pi +sinq\pi sin2q\pi ),\hfill \\ \hfill C_5=& 2a_0{a}_{5}cos\frac{5q\pi }{2}+2a_1{a}_4(cos2q\pi cos\frac{q\pi }{2}+sin2q\pi sin\frac{q\pi }{2})+2a_2{a}_3(cosq\pi cos\frac{3q\pi }{2}\hfill \\ & +sinq\pi sin\frac{3q\pi }{2})-2b_1{b}_4(cos2q\pi cos\frac{q\pi }{2}+sin2q\pi sin\frac{q\pi }{2})-2b_3{b}_4(cos2q\pi cos\frac{3q\pi }{2}\hfill \\ & +sin2q\pi sin\frac{3q\pi }{2}),\hfill \\ \hfill C_4=& a_2^2-b_2^2+2a_0{a}_{4}cos2q\pi +2a_1{a}_3(cos\frac{3q\pi }{2}cos\frac{q\pi }{2}+sin\frac{3q\pi }{2}sin\frac{q\pi }{2})-2b_0{b}_{4}cos2q\pi \hfill \\ & -2b_1{b}_3(cos\frac{3q\pi }{2}cos\frac{q\pi }{2}+sin\frac{3q\pi }{2}sin\frac{q\pi }{2}),\hfill \\ \hfill C_3=& 2a_0{a}_{3}cos\frac{3q\pi }{2}+2a_1{a}_2(cosq\pi cos\frac{q\pi }{2}+sinq\pi sin\frac{q\pi }{2})-2b_0{b}_{3}cos\frac{3q\pi }{2}\hfill \\ & -2b_1{b}_2(cosq\pi cos\frac{q\pi }{2}+sinq\pi sin\frac{q\pi }{2}),\hfill \\ \hfill C_2=& a_1^2-b_1^2+2a_0{a}_{2}cosq\pi -2b_0{b}_{2}cosq\pi ,\hfill \\ \hfill C_1=& 2a_0{a}_{1}cos\frac{q\pi }{2}-2b_0{b}_{1}cos\frac{q\pi }{2},\hfill \\ \hfill C_0=& a_0^2-b_0^2.\hfill \end{array}$$

(21)

Let

$$\begin{array}{cc}\hfill f\left(w\right)=& w^{12q}+C_{11}{w}^{11q}+C_{10}{w}^{10q}+C_9{w}^{9q}+C_8{w}^{8q}+C_7{w}^{7q}+C_6{w}^{6q}\hfill \\ & +C_5{w}^{5q}+C_4{w}^{4q}+C_3{w}^{3q}+C_2{w}^{2q}+C_1{w}^q+C_0.\hfill \end{array}$$

(22)

Obviously, $\underset{w\to +\infty }{lim}f\left(w\right)=+\infty$, as (H3) $a_0^2-b_0^2<0$; then, Equation (20) has at least one positive real root. The real root of Equation (20) can be expressed as $w_k,k=1,2,\cdots ,N$. Then, by substituting it into Equation (17), we have

$$\begin{array}{c}{\tau }_k^{\left(j\right)}=\frac{1}{w_k}arccos\left(\frac{-(E_1{E}_3+E_2{E}_4)}{(E_1^2+E_2^2)}+2j\pi \right),(k=1,2,\dots ,N;j=0,1,2,\dots ).\hfill \end{array}$$

(23)

In this case, $w_k$ is a pair of pure virtual roots of Equation (7). Now, we define ${\tau }_0=min\left\{{\tau }_k^{\left(0\right)}\right\}$ and $w_0$ for Equation (7) corresponding to the ${\tau }_0$ root, that is, the characteristic Equation (7) has at least one pair of pure virtual roots. This completes the proof. □

Next, we verify the transversality condition of system (2) to judge the existence of Hopf bifurcation. The following are assumed:

$$\begin{array}{c}\left(\mathbf{H}\mathbf{4}\right)\frac{K_R{L}_R+K_I{L}_I}{L_R^2+L_I^2}\ne 0.\hfill \end{array}$$

 Lemma 4. 

If (H4) is satisfified, the transversality condition holds.

 Proof. 

Let $s\left(\tau \right)=\alpha \left(\tau \right)+iw\left(\tau \right)$ be the root of Equation (6) near $\tau ={\tau }_0$ that satisfies $\alpha \left({\tau }_0\right)=0,w\left({\tau }_0\right)=w_0$ and find the transversality condition. To verify the transversality condition, that is, the derivative of Equation (6) with respect to $\tau$, we have

$$\begin{array}{c}(6q{s}^{6q-1}+5q{a}_5{s}^{5q-1}+4q{a}_4{s}^{4q-1}+3q{a}_3{s}^{3q-1}+2q{a}_2{s}^{2q-1}+q{a}_1{s}^{q-1})\frac{ds}{d\tau }\hfill \\ +(4q{b}_4{s}^{4q-1}+3q{b}_3{s}^{3q-1}+2q{b}_2{s}^{2q-1}+q{b}_1{s}^{q-1})e^{-s\tau }\frac{ds}{d\tau }\hfill \\ -(b_4{s}^{4q}+b_3{s}^{3q}+b_2{s}^{2q}+b_1{s}^q+b_0)(\tau e^{-s\tau }\frac{ds}{d\tau }+s{e}^{-s\tau })=0.\hfill \end{array}$$

(24)

By simplifying the above Equation (24), we can obtain the following:

$${\left(\frac{d\lambda }{d\tau }\right)}^{-1}=\frac{K\left(s\right)}{L\left(s\right)}-\frac{\tau }{s},$$

(25)

where

$$\begin{array}{cc}\hfill K\left(s\right)=& (6q{s}^{6q-1}+5q{a}_5{s}^{5q-1}+4q{a}_4{s}^{4q-1}+3q{a}_3{s}^{3q-1}+2q{a}_2{s}^{2q-1}+q{a}_1{s}^{q-1})\hfill \\ & +(4q{b}_4{s}^{4q-1}+3q{b}_3{s}^{3q-1}+2q{b}_2{s}^{2q-1}+q{b}_1{s}^{q-1})e^{-s\tau },\hfill \\ \hfill L\left(s\right)=& (b_4{s}^{4q}+b_3{s}^{3q}+b_2{s}^{2q}+b_1{s}^q+b_0)\left(s{e}^{-s\tau }\right).\hfill \end{array}$$

(26)

The real numbers $K_R,K_I,L_R,L_I$ are defined as follows:

$$\begin{array}{c}K\left(iw\right)=K_R+i{K}_I,\hfill \\ L\left(iw\right)=L_R+i{L}_I,\hfill \end{array}$$

where $K_R,L_R$ and $K_I,L_I$, respectively $K\left(s\right)$ and $L\left(s\right)$ in $\tau ={\tau }_0$ ($s=i{w}_0$), are at this time the real part and imaginary part. From (H4), we have

$$\begin{array}{c}Re\left({\left(\frac{ds}{d\tau }\right)}^{-1}{|}_{\tau ={\tau }_0}\right)=\frac{K_R{L}_R+K_I{L}_I}{L_R^2+L_R^2}\ne 0.\hfill \end{array}$$

(27)

The transversal condition is satisfied. This completes the proof. □

It is obvious from Lemmas 2 and 3 that when (H3) and (H4) hold, system (5) loses stability at the zero equilibrium point and a Hopf bifurcation appears.

 Theorem 1. 

If (H1)–(H4) hold, then:

 (1) 

When $\tau \in [0,{\tau }_0)$, the zero equilibrium point of system (5) has global asymptotic stability, while when $\tau >{\tau }_0$, the zero solution of fractional order system (5) is unstable.

 (2) 

When $\tau ={\tau }_0$, System (5) loses stability at the zero equilibrium point and Hopf bifurcation occurs.

4. Numerical Simulations

In this section, we provide a numerical simulation example to support the validity of the above conclusions. Consider the following system:

$$\left\{\begin{array}{cc}\hfill D^{0.8}{x}_1\left(t\right)=& -0.5x_1\left(t\right)+tanh\left(x_2(t-{\tau }_2)\right)-1.6tanh\left(x_3(t-{\tau }_2)\right)+2tanh\left(x_4(t-{\tau }_2)\right)\hfill \\ & +tanh\left(x_5(t-{\tau }_2)\right)-3tanh\left(x_6(t-{\tau }_2)\right),\hfill \\ \hfill D^{0.8}{x}_2\left(t\right)=& -0.6x_2\left(t\right)+tanh\left(x_1(t-{\tau }_1)\right),\hfill \\ \hfill D^{0.8}{x}_3\left(t\right)=& -0.2x_3\left(t\right)+tanh\left(x_1(t-{\tau }_1)\right),\hfill \\ \hfill D^{0.8}{x}_4\left(t\right)=& -0.6x_4\left(t\right)-tanh\left(x_1(t-{\tau }_1)\right),\hfill \\ \hfill D^{0.8}{x}_5\left(t\right)=& -0.5x_5\left(t\right)+2tanh\left(x_1(t-{\tau }_1)\right),\hfill \\ \hfill D^{0.8}{x}_6\left(t\right)=& -0.4x_6\left(t\right)+tanh\left(x_1(t-{\tau }_1)\right).\hfill \end{array}\right.$$

(28)

Obviously, system (28) has an equilibrium point (0,0,0,0,0,0). By numerical calculation, we have $D_1=2.8000>0,D_2=9.9860>0,D_3=57.0115>0,D_4=92.9994>0,D_5=332.6698>0,D_6=65.4694>0$, that is, (H1) and (H2) hold such that the Equation (20) has two positive real roots. In this circumstance, $w_0=1.9547,{\tau }_0=0.4840,\frac{K_R{L}_R+K_I{L}_I}{L_R^2+L_I^2}=0.3875\ne 0$, (H3), and (H4) all hold as well. Take ${\tau }_1=0.2,{\tau }_2=0.25$, ${\tau }_0={\tau }_1+{\tau }_2=0.45<0.4840$; then, Figure 1 and Figure 2 can be obtained through numerical simulation. We can conclude that the zero equilibrium point of system (28) is locally asymptotically stable. When taking ${\tau }_1=0.3,{\tau }_2=0.25$, that is, $\tau =0.55>0.4840$, it can be seen from Figure 3 and Figure 4 that the system loses stability and Hopf bifurcation appears at the zero equilibrium point.

5. Conclusions

Time-delay neural networks have very rich dynamic behaviors, among which stability and bifurcation phenomena are important dynamic behaviors. Therefore, it is very effective and applicable to study the dynamic behavior of such systems in solving both theoretical and practical problems. This paper analyzes a fractional-order BAM neural network model with double delays for a class of six neurons to investigate the existence of stability and Hopf bifurcation. By taking the sum of the two delays $\tau ={\tau }_1+{\tau }_2$ as the bifurcation parameter, we study the stability of the system and the sufficient conditions for the generation of Hopf bifurcation. The analysis shows that time delay affects the stability of the system; when $\tau$ exceeds the critical value ${\tau }_0$, the equilibrium point of the system loses stability and Hopf bifurcation occurs. These results are verified by a numerical simulation example.