Stability and Hopf Bifurcation of a Class of Six-Neuron Fractional BAM Neural Networks with Multiple Delays

Abstract

In this paper, we study the stability and Hopf bifurcation of a class of six-neuron fractional BAM neural networks with multiple delays. Firstly, the model is transformed into a fractional neural network model with two nonidentical delays by using variable substitution. Then, by assigning a value to one of the time delays and selecting the remaining time delays as parameters, the critical value of Hopf bifurcation for different time delays is calculated. The study shows that when the time lag exceeds its critical value, the equilibrium point of the system will lose its stability and generate Hopf bifurcation. Finally, the correctness of theoretical analysis is verified by simulation.

1. Introduction

Artificial neural networks (ANNs), also referred to as neural networks (NNs), is a distributed parallel information processing algorithm mathematical model that imitates the behavior characteristics of biological neural networks. Because of its strong adaptability, high fault tolerance, fast computing speed, and other capabilities, it has important applications in many fields, such as pattern recognition, image processing, intelligent signal sensing analysis, target tracking, and identification. In references [1,2] Hopfield proposed single-layer and layered feedback neural networks, respectively. After that, a large number of scholars and researchers further studied Hopfield’s method, thus forming the research upsurge of artificial neural networks in recent years. Associative memory network is an important branch of neural networks which obtains the original pattern by simulating human brain and storing sample patterns with the weight of neural networks and performing large-scale parallel computation. Among all kinds of associative memory network models, the bidirectional associative memory (BAM) neural network proposed by B. Kosko [3] in 1988 is the most widely used. The BAM neural network is a kind of feedback system with bidirectional stability, which has the characteristics of large-scale parallel data processing and strong self-organization. Research on the BAM neural network has been one of the hot topics in recent years. Song studied the stability and bifurcation problems of a class of BAM neural networks with three neurons in the literature [4]. Cao [5] gave a sufficient criterion for the stability and Hopf bifurcation generation of a class of BAM neural networks with four neurons. In reference [6], Gopalsamy put forward the meaning of leakage delay of BAM neural networks. Cheng [7] presented a class of three-layer neural networks, and discussed the direction of generating Hopf bifurcation and the stability of bifurcation periodic solution.

Fractional calculus originated in 1695, almost at the same time as classical calculus, and has a history of more than 300 years. However, due to the limitations of technology and environment, fractional calculus was only discussed and studied in the pure mathematics field at first. After 1974, the theory and application of fractional calculus developed rapidly, and it was applied in biological systems, control systems, chaos and turbulence, electrochemistry, viscoelastic mechanics, and many other fields. Matignon [8] and Deng [9] both studied the stability of finite dimensional fractional order systems and give a method to judge stability of fractional order linear systems. Li proposed the definition of Mittag–Leffler stability in the literature [10], and further proposed the definition of generalized Mittag–Leffler stability [11], which enriched the stability theory of fractional calculus. Huang discussed the bifurcation control problem for a class of predator–prey systems with incommensurate fractional order delays [12]. Compared with the integer differential system, the fractional differential dynamic model can more accurately describe some non-classical phenomena in natural science and engineering applications, but the current fractional calculus theory is not perfect and in many aspects needs in-depth research, such as bifurcation direction. Fractional calculus still has broad development prospects and will have important research significance in the future.

Time delays exist in mechanical, physical, biological, and economic system models. The existence of a time delay, on the one hand, will destroy the dynamic performance of the system and make the system unstable. On the other hand, in some control systems, people can make the system reach the ideal state by changing the time delay. The use of time delays can better solve practical problems, so the study of dynamic behavior of all kinds of systems with time delays has important theoretical research significance and practical application value. In recent years, because the fractional order delay differential equation can describe the dynamic behavior of the system more accurately and has better control of the system, more and more scholars have become committed to studying the dynamic behavior of the fractional order BAM neural network model with multiple delays. Xu discussed the global existence of periodic solutions of BAM neural networks with six neurons in document [13]. In reference [14], the stability and Hopf bifurcation direction of integer order BAM neural networks with different distribution of six neurons are discussed, respectively. Then, in references [15,16,17], for the fractional BAM neural network with six neurons, a sufficient criterion for the stability of the model solution and bifurcation generation are given. In [18,19], Huang studied the stability and bifurcation problems of integer order BAM neural networks with 6 and ($n+2$) neurons, respectively, and discussed the fractional order BAM neural networks with four neurons in [20,21], and gave the sufficient conditions for the stability of system solutions and Hopf bifurcation generation. The model in the literature [18] is as follows.

$$\left\{\begin{array}{cc}\hfill {\dot{x}}_1\left(t\right)=& -{\mu }_1{x}_1\left(t\right)+c_{11}{f}_1\left(y_1(t-{\tau }_3)\right)+c_{21}{f}_1\left(y_2(t-{\tau }_3)\right)\hfill \\ \hfill & +c_{31}{f}_1\left(y_3(t-{\tau }_3)\right)+c_{41}{f}_1\left(y_4(t-{\tau }_3)\right),\hfill \\ \hfill {\dot{x}}_2\left(t\right)=& -{\mu }_2{x}_2\left(t\right)+c_{12}{f}_2\left(y_1(t-{\tau }_4)\right)+c_{22}{f}_2\left(y_2(t-{\tau }_4)\right)\hfill \\ \hfill & +c_{32}{f}_2\left(y_3(t-{\tau }_4)\right)+c_{42}{f}_2\left(y_4(t-{\tau }_4)\right),\hfill \\ \hfill {\dot{y}}_1\left(t\right)=& -{\mu }_3{y}_1\left(t\right)+d_{11}{g}_1\left(x_1(t-{\tau }_1)\right)+d_{21}{g}_1\left(x_2(t-{\tau }_2)\right),\hfill \\ \hfill {\dot{y}}_2\left(t\right)=& -{\mu }_4{y}_2\left(t\right)+d_{12}{g}_2\left(x_1(t-{\tau }_1)\right)+d_{22}{g}_2\left(x_2(t-{\tau }_2)\right),\hfill \\ \hfill {\dot{y}}_3\left(t\right)=& -{\mu }_5{y}_3\left(t\right)+d_{13}{g}_3\left(x_1(t-{\tau }_1)\right)+d_{23}{g}_3\left(x_2(t-{\tau }_2)\right),\hfill \\ \hfill {\dot{y}}_4\left(t\right)=& -{\mu }_6{y}_4\left(t\right)+d_{14}{g}_4\left(x_1(t-{\tau }_1)\right)+d_{24}{g}_4\left(x_2(t-{\tau }_2)\right),\hfill \end{array}\right.$$

(1)

where $c_{ij},d_{ji}(i=1,2,3,4;j=1,2)$ represents the connection weight between the neuron of I-layer and the neuron of J-layer. ${\mu }_k(k=1,2,\dots ,6)$ is the decay rates of internal neurons, $f_j,g_i(i=1,2,3,4;j=1,2)$ are activation functions, ${\tau }_i(1=1,2,3,4)$ represents the transmission delay between I-layer neurons and J-layer neurons. See reference [18] for parameter details.

In this paper, the stability of fractional BAM neural networks with multiple delays is discussed. Different from the model established in reference [18], the model established in this paper introduces a fractional derivative and assumes that ${\eta }_1\not\equiv {\eta }_2$ is more suitable for practical application. Similar to the method of reference [20], this paper also converts the multi-delay model into double-delay model, but the number of neurons is increased in this paper when establishing the model, which increases the complexity of the model in some respects.

Compared with the integer order BAM neural network with multiple delays ([22,23]), the dynamic behavior of the network will become more complex due to the existence of the fractional order. The description of the dynamic behavior of the system will depend on the flexible application of the fractional derivative knowledge and the stability and bifurcation theory of the fractional system.

It is worth noting that previous studies generally studied the stability and bifurcation of fractional order systems with the same order. In this paper, six different fractional orders are considered in the modeling process. With the increase of dimension of neural networks and different fractional orders, then the characteristic equation of model will become a more complex higher-order transcendental equation. It will be challenging to analyze the distribution of the roots of the characteristic equation of the model.

This article is organized as follows. In Section 2, the definition of the Caputo fractional derivative and some lemmas are given. In Section 3, sufficient conditions for asymptotic stability of equilibrium point and Hopf bifurcation generation of the system (3) are obtained by a linearizing system (3) and we analyze its characteristic equation with time delay as a parameter. The two delays in system (4) are discussed as parameters, respectively. In Section 4, two examples are given to verify the correctness of the theoretical results. Finally, the main conclusions are given in Section 5.

2. Preliminaries

The commonly used fractional derivatives are the Riemann–Liouville derivative, the Caputo derivative, and the Grunwald–Letnikov derivative. Considering that the Caputo derivative makes it easier to select a initial value, this paper adopts a Caputo-type fractional derivative to calculate. In order to obtain the main results of this paper, the following definitions and lemmas are given.

Definition 1.

Caputo fractional derivative [24] is defined as

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

where $\mathsf{\Gamma }(•)$ is the Gamma function.

The Laplace transform of the fractional derivative of Caputo is

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

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

Lemma 1

([9]). Consider the following fractional linear system with multiple state variables.

$$\left\{\begin{array}{c}{D}^{q_1}{y}_1\left(t\right)={\overline{I}}_{11}{y}_1\left(t\right)+{\overline{I}}_{12}{y}_2\left(t\right)+\cdots +{\overline{I}}_{1(n-1)}{y}_{n-1}\left(t\right)+{\overline{I}}_{1n}{y}_n\left(t\right),\hfill \\ D^{q_2}{y}_2\left(t\right)={\overline{I}}_{21}{y}_1\left(t\right)+{\overline{I}}_{22}{y}_2\left(t\right)+\cdots +{\overline{I}}_{2(n-1)}{y}_{n-1}\left(t\right)+{\overline{I}}_{2n}{y}_n\left(t\right),\hfill \\ \cdots \cdots \hfill \\ D^{q_n}{y}_n\left(t\right)={\overline{I}}_{n1}{y}_1\left(t\right)+{\overline{I}}_{n2}{y}_2\left(t\right)+\cdots +{\overline{I}}_{n(n-1)}{y}_{n-1}\left(t\right)+{\overline{I}}_{nn}{y}_n\left(t\right),\hfill \end{array}\right.$$

(2)

where $q_i\in (0,1](i=1,2,\dots ,n)$. The characteristic equation $▵\left(s\right)$ of the system (2) is as follows.

$$▵\left(s\right)=\left[\begin{array}{cccc}{s}^{q_1}-{\overline{I}}_{11}& {\overline{I}}_{12}& \cdots & {\overline{I}}_{1n}\\ {\overline{I}}_{21}& s^{q_2}-{\overline{I}}_{22}& \cdots & {\overline{I}}_{2n}\\ \cdots & \cdots & \ddots & ⋮\\ {\overline{I}}_{n1}& {\overline{I}}_{n2}& \cdots & s^{q_n}-{\overline{I}}_{nn}\end{array}\right].$$

If all the real parts of the roots of equation $\left|▵\left(s\right)\right|=0$ are negative, then the zero solution of system (2) is Lyapunov globally asymptotically stable.

Based on the above discussion, this paper improves system (1) and establishes the following fractional order BAM neural network model with multiple delays.

$$\left\{\begin{array}{cc}\hfill D^{q_1}{u}_1\left(t\right)=& -{\mu }_1{u}_1\left(t\right)+c_{11}{f}_1\left(v_1(t-{\tau }_3)\right)+c_{21}{f}_1\left(v_2(t-{\tau }_3)\right)\hfill \\ \hfill & +c_{31}{f}_1\left(v_3(t-{\tau }_3)\right)+c_{41}{f}_1\left(v_4(t-{\tau }_3)\right),\hfill \\ \hfill D^{q_2}{u}_2\left(t\right)=& -{\mu }_2{u}_2\left(t\right)+c_{12}{f}_2\left(v_1(t-{\tau }_4)\right)+c_{22}{f}_2\left(v_2(t-{\tau }_4)\right)\hfill \\ \hfill & +c_{32}{f}_2\left(v_3(t-{\tau }_4)\right)+c_{42}{f}_2\left(v_4(t-{\tau }_4)\right),\hfill \\ \hfill D^{q_3}{v}_1\left(t\right)=& -{\mu }_3{v}_1\left(t\right)+d_{11}{g}_1\left(u_1(t-{\tau }_1)\right)+d_{21}{g}_1\left(u_2(t-{\tau }_2)\right),\hfill \\ \hfill D^{q_4}{v}_2\left(t\right)=& -{\mu }_4{v}_2\left(t\right)+d_{12}{g}_2\left(u_1(t-{\tau }_1)\right)+d_{22}{g}_2\left(u_2(t-{\tau }_2)\right),\hfill \\ \hfill D^{q_5}{v}_3\left(t\right)=& -{\mu }_5{v}_3\left(t\right)+d_{13}{g}_3\left(u_1(t-{\tau }_1)\right)+d_{23}{g}_3\left(u_2(t-{\tau }_2)\right),\hfill \\ \hfill D^{q_6}{v}_4\left(t\right)=& -{\mu }_6{v}_4\left(t\right)+d_{14}{g}_4\left(u_1(t-{\tau }_1)\right)+d_{24}{g}_4\left(u_2(t-{\tau }_2)\right).\hfill \end{array}\right.$$

(3)

In the above model, $D^{q_i}$ is the Caputo fractional derivative and $0<q_i\le 1,$ $i=1,2,\dots ,6$.

Now, let us do the following transformation

$$\left\{\begin{array}{c}{\eta }_1={\tau }_1+{\tau }_3,\hfill \\ {\eta }_2={\tau }_2+{\tau }_4,\hfill \\ y_1\left(t\right)=u_1(t-{\tau }_1),\hfill \\ y_2\left(t\right)=u_2(t-{\tau }_2),\hfill \\ y_3\left(t\right)=v_1\left(t\right),\hfill \\ y_4\left(t\right)=v_2\left(t\right),\hfill \\ y_5\left(t\right)=v_3\left(t\right),\hfill \\ y_6\left(t\right)=v_4\left(t\right).\hfill \end{array}\right.$$

Then the equivalent form of the system (3) can be obtained, that is,

$$\left\{\begin{array}{cc}\hfill D^{q_1}{y}_1\left(t\right)=& -{\mu }_1{y}_1\left(t\right)+c_{11}{f}_1\left(y_3(t-{\eta }_1)\right)+c_{21}{f}_1\left(y_4(t-{\eta }_1)\right)\hfill \\ \hfill & +c_{31}{f}_1\left(y_5(t-{\eta }_1)\right)+c_{41}{f}_1\left(y_6(t-{\eta }_1)\right),\hfill \\ \hfill D^{q_2}{y}_2\left(t\right)=& -{\mu }_2{y}_2\left(t\right)+c_{12}{f}_2\left(y_3(t-{\eta }_2)\right)+c_{22}{f}_2\left(y_4(t-{\eta }_2)\right)\hfill \\ \hfill & +c_{32}{f}_2\left(y_5(t-{\eta }_2)\right)+c_{42}{f}_2\left(y_6(t-{\eta }_2)\right),\hfill \\ \hfill D^{q_3}{y}_3\left(t\right)=& -{\mu }_3{y}_3\left(t\right)+d_{11}{g}_1\left(y_1\left(t\right)\right)+d_{21}{g}_1\left(y_2\left(t\right)\right),\hfill \\ \hfill D^{q_4}{y}_4\left(t\right)=& -{\mu }_4{y}_4\left(t\right)+d_{12}{g}_2\left(y_1\left(t\right)\right)+d_{22}{g}_2\left(y_2\left(t\right)\right),\hfill \\ \hfill D^{q_5}{y}_5\left(t\right)=& -{\mu }_5{y}_5\left(t\right)+d_{13}{g}_3\left(y_1\left(t\right)\right)+d_{23}{g}_3\left(y_2\left(t\right)\right),\hfill \\ \hfill D^{q_6}{y}_6\left(t\right)=& -{\mu }_6{y}_6\left(t\right)+d_{14}{g}_4\left(y_1\left(t\right)\right)+d_{24}{g}_4\left(y_2\left(t\right)\right).\hfill \end{array}\right.$$

(4)

In order to carry out the follow-up work smoothly, the following hypothesis is given.

Hypothesis 1 (H1).

$f_i,g_j\in C^1,f_i\left(0\right)=g_j\left(0\right)=0,i=1,2;j=1,2,3,4$.

3. Local Stability of Equilibrium Point and Existence of Hopf Bifurcation

From (H1), we know that the zero point of the system (4) is its equilibrium point. The linearized form of system (4) is obtained as follows.

$$\left\{\begin{array}{cc}\hfill D^{q_1}{y}_1\left(t\right)=& -{\mu }_1{y}_1\left(t\right)+a_{11}{y}_3(t-{\eta }_1)+a_{21}{y}_4(t-{\eta }_1)+a_{31}{y}_5(t-{\eta }_1)+a_{41}{y}_6(t-{\eta }_1),\hfill \\ \hfill D^{q_2}{y}_2\left(t\right)=& -{\mu }_2{y}_2\left(t\right)+a_{12}{y}_3(t-{\eta }_2)+a_{22}{y}_4(t-{\eta }_2)+a_{32}{y}_5(t-{\eta }_2)+a_{42}{y}_6(t-{\eta }_2),\hfill \\ \hfill D^{q_3}{y}_3\left(t\right)=& -{\mu }_3{y}_3\left(t\right)+b_{11}{y}_1\left(t\right)+b_{21}{y}_2\left(t\right),\hfill \\ \hfill D^{q_4}{y}_4\left(t\right)=& -{\mu }_4{y}_4\left(t\right)+b_{12}{y}_1\left(t\right)+b_{22}{y}_2\left(t\right),\hfill \\ \hfill D^{q_5}{y}_5\left(t\right)=& -{\mu }_5{y}_5\left(t\right)+b_{13}{y}_1\left(t\right)+b_{23}{y}_2\left(t\right),\hfill \\ \hfill D^{q_6}{y}_6\left(t\right)=& -{\mu }_6{y}_6\left(t\right)+b_{14}{y}_1\left(t\right)+b_{24}{y}_2\left(t\right),\hfill \end{array}\right.$$

(5)

where $a_{ij}=c_{ij}{f}_j^{\prime }\left(0\right),b_{ji}=d_{ji}{g}_i^{\prime }\left(0\right),i=1,2,3,4;j=1,2.$ It is easy to find its characteristic equation

$$\left|\begin{array}{cccccc}{\lambda }^{q_1}+{\mu }_1& 0& -a_{11}{e}^{-\lambda {\eta }_1}& -a_{21}{e}^{-\lambda {\eta }_1}& -a_{31}{e}^{-\lambda {\eta }_1}& -a_{41}{e}^{-\lambda {\eta }_1}\\ 0& {\lambda }^{q_2}+{\mu }_2& -a_{12}{e}^{-\lambda {\eta }_2}& -a_{22}{e}^{-\lambda {\eta }_2}& -a_{32}{e}^{-\lambda {\eta }_2}& -a_{42}{e}^{-\lambda {\eta }_2}\\ -b_{11}& -b_{21}& {\lambda }^{q_3}+{\mu }_3& 0& 0& 0\\ -b_{12}& -b_{22}& 0& {\lambda }^{q_4}+{\mu }_4& 0& 0\\ -b_{13}& -b_{23}& 0& 0& {\lambda }^{q_5}+{\mu }_5& 0\\ -b_{14}& -b_{24}& 0& 0& 0& {\lambda }^{q_6}+{\mu }_6\end{array}\right|=0,$$

where

$$M_1\left(\lambda \right)+M_2\left(\lambda \right)e^{-\lambda {\eta }_1}+M_3\left(\lambda \right)e^{-\lambda {\eta }_2}+M_4\left(\lambda \right)e^{-\lambda ({\eta }_1+{\eta }_2)}=0,$$

(6)

where

$$\begin{array}{cc}\hfill M_1\left(\lambda \right)=& \prod _{i=1}^6{\lambda }^{q_i}+\sum _{i=1}^6{\mu }_i\prod _{\genfrac{}{}{0pt}{}{j=1}{j\ne i}}^6{\lambda }^{q_j}+\sum _{i=1}^6\sum _{\genfrac{}{}{0pt}{}{j=1}{j>i}}^6{\mu }_i{\mu }_j\prod _{\genfrac{}{}{0pt}{}{k=1}{k\ne i,j}}^6{\lambda }^{q_k}+\sum _{i=1}^6\sum _{\genfrac{}{}{0pt}{}{j=1}{j>i}}^6\sum _{\genfrac{}{}{0pt}{}{k=1}{k>j}}^6{\lambda }^{q_i}{\lambda }^{q_j}{\lambda }^{q_k}\prod _{\genfrac{}{}{0pt}{}{l=1}{l\ne i,j,k}}^6{\mu }_l\hfill \\ & +\sum _{i=1}^6\sum _{\genfrac{}{}{0pt}{}{j=1}{j>i}}^6{\lambda }^{q_i}{\lambda }^{q_j}\prod _{\genfrac{}{}{0pt}{}{k=1}{k\ne i,j}}^6{\mu }_k+\sum _{i=1}^6{\lambda }^{q_i}\prod _{\genfrac{}{}{0pt}{}{j=1}{j\ne i}}^6{\mu }_j+\prod _{i=1}^6{\mu }_i,\hfill \\ \hfill M_2\left(\lambda \right)=& -\sum _{i=1}^4{a}_{i1}{b}_{1i}\left(\prod _{\genfrac{}{}{0pt}{}{j=2}{j\ne i+2}}^6{\lambda }^{q_j}+\sum _{\genfrac{}{}{0pt}{}{j=2}{j\ne i+2}}^6{\mu }_j\prod _{\genfrac{}{}{0pt}{}{k=2}{k\ne i+2,j}}^6{\lambda }^{q_k}+\sum _{\genfrac{}{}{0pt}{}{j=2}{j\ne i+2}}^6\sum _{\genfrac{}{}{0pt}{}{k=2,k>j}{k\ne i+2}}^6{\mu }_j{\mu }_k\prod _{\genfrac{}{}{0pt}{}{l=2}{l\ne i+2,j,k}}^6{\lambda }^{q_l}\right)\hfill \\ & -\sum _{i=1}^4{a}_{i1}{b}_{1i}\left(\sum _{\genfrac{}{}{0pt}{}{j=2}{j\ne i+2}}^6{\lambda }^{q_j}\prod _{\genfrac{}{}{0pt}{}{k=2}{k\ne i+2,j}}^6{\mu }_k+\prod _{\genfrac{}{}{0pt}{}{j=2}{j\ne i+2}}^6{\mu }_j\right),\hfill \\ \hfill M_3\left(\lambda \right)=& -\sum _{i=1}^4{a}_{i1}{b}_{1i}\left(\prod _{\genfrac{}{}{0pt}{}{j=1}{j\ne 2,i+2}}^6{\lambda }^{q_j}+\sum _{\genfrac{}{}{0pt}{}{j=1}{j\ne 2,i+2}}^6{\mu }_j\prod _{\genfrac{}{}{0pt}{}{k=1}{k\ne 2,i+2,j}}^6{\lambda }^{q_k}+\sum _{\genfrac{}{}{0pt}{}{j=1}{j\ne 2,i+2}}^6\sum _{\genfrac{}{}{0pt}{}{k=1,k>j}{k\ne 2,i+2}}^6{\mu }_j{\mu }_k\prod _{\genfrac{}{}{0pt}{}{l=1}{l\ne 2,i+2,j,k}}^6{\lambda }^{q_l}\right)\hfill \\ & -\sum _{i=1}^4{a}_{i1}{b}_{1i}\left(\sum _{\genfrac{}{}{0pt}{}{j=1}{j\ne 2,i+2}}^6{\lambda }^{q_j}\prod _{\genfrac{}{}{0pt}{}{k=1}{k\ne 2,i+2,j}}^6{\mu }_k+\prod _{\genfrac{}{}{0pt}{}{j=1}{j\ne 2,i+2}}^6{\mu }_j\right),\hfill \\ \hfill M_4\left(\lambda \right)=& \sum _{i=1}^4\sum _{\genfrac{}{}{0pt}{}{j=1}{j\ne i}}^4{b}_{1i}{b}_{2j}(a_{i1}{a}_{j2}-a_{i2}{a}_{j1})\left(\prod _{\genfrac{}{}{0pt}{}{k=3}{k\ne i+2,j+2}}^6{\lambda }^{q_k}+\sum _{\genfrac{}{}{0pt}{}{k=3}{k\ne i+2,j+2}}^6\sum _{\genfrac{}{}{0pt}{}{l=3}{l\ne k,i+2,j+2}}^6{\mu }_k{\lambda }^{q_l}+\prod _{\genfrac{}{}{0pt}{}{k=3}{k\ne i+2,j+2}}^6{\mu }_k\right).\hfill \end{array}$$

3.1. The Hopf Bifurcation of a System (3) with Time Delay ${\eta }_2$ as Parameter

When ${\eta }_1=0$, Equation (6) is equal to

$$M_1\left(\lambda \right)+M_2\left(\lambda \right)+\left(M_3\left(\lambda \right)+M_4\left(\lambda \right)\right)e^{-\lambda {\eta }_2}=0.$$

(7)

Suppose $\lambda =\overline{w}(\cos \frac{\pi }{2}+isin\frac{\pi }{2})(\overline{w}>0)$ is a pure imaginary root of Equation (7), and ${\overline{M}}_i^R,{\overline{M}}_i^I$ represents the real and imaginary parts of $M_i\left(i\overline{w}\right)(i=1,2,3,4)$, respectively. There are

$$\left\{\begin{array}{c}\hfill ({\overline{M}}_3^R+{\overline{M}}_4^R)cos\overline{w}{\eta }_2+({\overline{M}}_3^I+{\overline{M}}_4^I)sin\overline{w}{\eta }_2=-({\overline{M}}_1^R+{\overline{M}}_2^R),\\ \hfill -({\overline{M}}_3^R+{\overline{M}}_4^R)sin\overline{w}{\eta }_2+({\overline{M}}_3^I+{\overline{M}}_4^I)cos\overline{w}{\eta }_2=-({\overline{M}}_1^I+{\overline{M}}_2^I).\end{array}\right.$$

(8)

By transforming Equations (8)–(10), the following can be deduced.

$$\begin{array}{c}cos\overline{w}{\eta }_2=-\frac{({\overline{M}}_1^R+{\overline{M}}_2^R)({\overline{M}}_3^R+{\overline{M}}_4^R)+({\overline{M}}_1^I+{\overline{M}}_2^I)({\overline{M}}_3^I+{\overline{M}}_4^I)}{{({\overline{M}}_3^R+{\overline{M}}_4^R)}^2+{({\overline{M}}_3^I+{\overline{M}}_4^I)}^2},\hfill \\ sin\overline{w}{\eta }_2=\frac{({\overline{M}}_1^I+{\overline{M}}_2^I)({\overline{M}}_3^R+{\overline{M}}_4^R)-({\overline{M}}_1^R+{\overline{M}}_2^R)({\overline{M}}_3^I+{\overline{M}}_4^I)}{{({\overline{M}}_3^R+{\overline{M}}_4^R)}^2+{({\overline{M}}_3^I+{\overline{M}}_4^I)}^2},\hfill \end{array}$$

(9)

and

$$\begin{array}{c}H\left(\overline{w}\right)={({\overline{M}}_3^R+{\overline{M}}_4^R)}^2+{({\overline{M}}_3^I+{\overline{M}}_4^I)}^2-{({\overline{M}}_1^R+{\overline{M}}_2^R)}^2-{({\overline{M}}_1^I+{\overline{M}}_2^I)}^2=0.\hfill \end{array}$$

(10)

In order to carry out the follow-up work smoothly, the following hypothesis is given.

Hypothesis 2 (H2).

The Equation (10) has at least one positive real root.

Let $\overline{w}$ represent all positive real roots of Equation (10). The bifurcation point ${\overline{\eta }}_{20}$ of Equation (7) at ${\eta }_1=0$ is obtained from Equation (9), and its expression is as follows.

$$\begin{array}{c}{\overline{\eta }}_2^{\left(k\right)}=\frac{1}{\overline{w}}arccos\left(-\frac{({\overline{M}}_1^R+{\overline{M}}_2^R)({\overline{M}}_3^R+{\overline{M}}_4^R)+({\overline{M}}_1^I+{\overline{M}}_2^I)({\overline{M}}_3^I+{\overline{M}}_4^I)}{{({\overline{M}}_3^R+{\overline{M}}_4^R)}^2+{({\overline{M}}_3^I+{\overline{M}}_4^I)}^2}+2k\pi \right),k=0,1,2,3,\cdots .\hfill \\ {\overline{\eta }}_{20}=min\left\{{\overline{\eta }}_2^{\left(0\right)}\right\}.\hfill \end{array}$$

(11)

Lemma 2.

Take ${\eta }_2\in (0,{\overline{\eta }}_{20})$, when (H3) was established, the Equation (12) has at least one pair of pure imaginary roots.

Proof. 

When ${\eta }_1>0$, Equation (6) becomes

$$M_1\left(\lambda \right)+M_3\left(\lambda \right)e^{-\lambda {\eta }_2}+\left(M_2\left(\lambda \right)+M_4\left(\lambda \right)e^{-\lambda {\eta }_2}\right)e^{-\lambda {\eta }_1}=0.$$

(12)

Let $\lambda =w(cos\frac{\pi }{2}+isin\frac{\pi }{2})(w>0)$ be a pure imaginary root of Equation (12), and use $M_i^R,M_i^I$ to represent the real and imaginary parts of $M_i\left(iw\right)(i=1,2,3,4)$, respectively. There are

$$\left\{\begin{array}{c}\left(M_2^R+M_4^{R}cosw{\eta }_2+M_4^{I}sinw{\eta }_2\right)cosw{\eta }_1+\left(M_2^I+M_4^{I}cosw{\eta }_2-M_4^{R}sinw{\eta }_2\right)sinw{\eta }_1\hfill \\ =-\left(M_1^R+M_3^{R}cosw{\eta }_2+M_3^{I}sinw{\eta }_2\right),\hfill \\ \left(M_2^R+M_4^{R}cosw{\eta }_2+M_4^{I}sinw{\eta }_2\right)sinw{\eta }_1-\left(M_2^I+M_4^{I}cosw{\eta }_2-M_4^{R}sinw{\eta }_2\right)cosw{\eta }_1\hfill \\ =\left(M_1^I+M_3^{I}cosw{\eta }_2-M_3^{R}sinw{\eta }_2\right).\hfill \end{array}\right.$$

(13)

Then the following two equations can be obtained by the transformation derivation of the Equation (13).

$$\begin{array}{c}{\left(M_4^R\right)}^2+{\left(M_4^I\right)}^2+{\left(M_2^R\right)}^2+{\left(M_2^I\right)}^2-{\left(M_3^R\right)}^2-{\left(M_3^I\right)}^2-{\left(M_1^R\right)}^2-{\left(M_1^I\right)}^2\hfill \\ +2(M_2^I{M}_4^I+M_2^R{M}_4^R-M_1^I{M}_3^I-M_1^R{M}_3^R)cosw{\eta }_2\hfill \\ +2(M_2^R{M}_4^I-M_2^I{M}_4^R-M_1^R{M}_3^I+M_1^I{M}_3^R)sinw{\eta }_2=0,\hfill \end{array}$$

(14)

and

$$\begin{array}{c}cosw{\eta }_1=-\frac{E_1{E}_3+E_2{E}_4}{{\left(E_1\right)}^2+{\left(E_2\right)}^2},\hfill \\ sinw{\eta }_1=\frac{E_2{E}_3-E_1{E}_4}{{\left(E_1\right)}^2+{\left(E_2\right)}^2},\hfill \end{array}$$

(15)

where

$$\begin{array}{c}{E}_1=M_2^R+M_4^{R}cosw{\eta }_2+M_4^{I}sinw{\eta }_2,\hfill \\ E_2=M_2^I+M_4^{I}cosw{\eta }_2-M_4^{R}sinw{\eta }_2,\hfill \\ E_3=M_1^R+M_3^{R}cosw{\eta }_2+M_3^{I}sinw{\eta }_2,\hfill \\ E_4=M_1^I+M_3^{I}cosw{\eta }_2-M_3^{R}sinw{\eta }_2.\hfill \end{array}$$

□

In order to carry out the follow-up work smoothly, the following hypothesis is given.

Hypothesis 3 (H3).

Equation(14) has at least one positive real root.

Let Equation (14) exist m positive real roots, denoted as $w_j(j=1,2,\dots ,m)$, and then we can use Equation (15) to obtain

$$\begin{array}{c}{\eta }_{1j}^{\left(k\right)}=\frac{1}{w_j}arccos\left(-\frac{E_1{E}_3+E_2{E}_4}{{\left(E_1\right)}^2+{\left(E_2\right)}^2}+2k\pi \right),k=0,1,2,3,\dots ,j=1,2,\dots ,m.\hfill \end{array}$$

(16)

Let ${\eta }_{10}=\underset{j=1,2,\dots ,m}{min}\left\{{\eta }_{1j}^{\left(0\right)}\right\}$, and $w_0$ is the value corresponding to ${\eta }_{10}$. Then Equation (10) has at least one pair of pure imaginary roots. The proof ends.

Next we consider the stability of system (4) at equilibrium. For the smooth follow-up work, we propose the following assumptions.

Hypothesis 4 (H4).

$|arg\left(s^{*}\right)|>\frac{\pi }{2{\zeta }_i},i=1,2,\dots ,6$. (See below for the meanings of $s^{*}$ and ${\zeta }_i$.)

Lemma 3.

If the (H4) is true, the system (4) is asymptotically stable.

Proof. 

When ${\eta }_1={\eta }_2=0$, Equation (6) is equal to

$$M_1\left(\lambda \right)+M_2\left(\lambda \right)+M_3\left(\lambda \right)+M_4\left(\lambda \right)=0.$$

(17)

Now mark $q_i=\frac{{\gamma }_i}{{\zeta }_i}$, where ${\gamma }_i$ and ${\zeta }_i$($i=1,2,\dots ,6$) are mutual prime positive integers, and M is the least common multiple of ${\zeta }_i$. Let $\lambda =s^M$, then Equation (17) become unary n-degree polynomial about s, and $n=\sum _{i=1}^6\left(M{q}_i\right)$. According to the fundamental theorem of algebra, this equation has n complex roots. For convenience, expressed as $s^{*}$. Then, we obtain $|arg\left(s^{*}\right)|>\frac{\pi }{2M}$ from (H4). Therefore, all the roots of Equation (17) satisfy $|arg\left({\lambda }^{*}\right)|=\left|arg\left({\left(s^{*}\right)}^M\right)\right|>\frac{\pi }{2}$.

From Lemma 1, the system (6) is asymptotically stable at ${\eta }_1={\eta }_2=0$ if (H4) is satisfied. The proof ends.

□

Hypothesis 5 (H5).

$\frac{K_1{L}_1+K_2{L}_2}{L_1^2+L_2^2}\ne 0.$ (See below for the meanings of $K_i$ and $L_i(i=1,2).$)

Lemma 4.

If (H5) is satisfied, the transversality condition is valid.

Proof. 

Let $\lambda \left({\eta }_1\right)=\alpha \left({\eta }_1\right)+iw\left({\eta }_1\right)$ be the root of Equation (6) near ${\eta }_1={\eta }_{10}$ that satisfies $\alpha \left({\eta }_{10}\right)=0,w\left({\eta }_{10}\right)=w_0$ and now find the transversal condition. Let us take the derivative of (6) with respect to ${\eta }_1$, and obtain

$$\begin{array}{c}{M}_1^{\prime }\left(\lambda \right)\frac{d\lambda }{d{\eta }_1}+M_2^{\prime }\left(\lambda \right)e^{-\lambda {\eta }_1}\frac{d\lambda }{d{\eta }_1}-\left(\lambda +{\eta }_1\frac{d\lambda }{d{\eta }_1}\right)M_2\left(\lambda \right)e^{-\lambda {\eta }_1}+M_3^{\prime }\left(\lambda \right)e^{-\lambda {\eta }_2}\frac{d\lambda }{d{\eta }_1}-{\eta }_2{M}_3\left(\lambda \right)e^{-\lambda {\eta }_2}\frac{d\lambda }{d{\eta }_1}\hfill \\ +M_4^{\prime }\left(\lambda \right)e^{-\lambda ({\eta }_1+{\eta }_2)}\frac{d\lambda }{d{\eta }_1}-\left(\lambda +({\eta }_1+{\eta }_2)\frac{d\lambda }{d{\eta }_1}\right)M_4\left(\lambda \right)e^{-\lambda ({\eta }_1+{\eta }_2)}=0,\hfill \end{array}$$

(18)

where $M_i^{\prime }\left(\lambda \right)(i=1,2,3,4)$ is the first derivative of $M_i\left(\lambda \right)$ with respect to $\lambda$. Simplify the above equation to obtain

$$\frac{d\lambda }{d{\eta }_1}=\frac{K\left(\lambda \right)}{L\left(\lambda \right)},$$

where

$$\begin{array}{cc}\hfill K\left(\lambda \right)=& \lambda M_2\left(\lambda \right)e^{-\lambda {\eta }_1}+\lambda M_4\left(\lambda \right)e^{-\lambda ({\eta }_1+{\eta }_2)},\hfill \\ \hfill L\left(\lambda \right)=& M_1^{\prime }\left(\lambda \right)+M_2^{\prime }\left(\lambda \right)e^{-\lambda {\eta }_1}-{\eta }_1{M}_2\left(\lambda \right)e^{-\lambda {\eta }_1}+M_3^{\prime }\left(\lambda \right)e^{-\lambda {\eta }_2}-{\eta }_2{M}_3\left(\lambda \right)e^{-\lambda {\eta }_2}\hfill \\ & +M_4^{\prime }\left(\lambda \right)e^{-\lambda ({\eta }_1+{\eta }_2)}-({\eta }_1+{\eta }_2)M_4\left(\lambda \right)e^{-\lambda ({\eta }_1+{\eta }_2)}.\hfill \end{array}$$

Let us define the real numbers $K_1,K_2,L_1$ and $L_2$ as follows.

$$\begin{array}{c}K\left(iw\right)=K_1+i{K}_2,\hfill \\ L\left(iw\right)=L_1+i{L}_2,\hfill \end{array}$$

that is, $K_1,L_1$ and $K_2,L_2$ represent the real and imaginary parts of $K\left(\lambda \right)$ and $L\left(\lambda \right)$ at ${\eta }_1={\eta }_{10}$($\lambda =i{w}_0$), respectively.

The following equation can be obtained by assuming (H5).

$$\begin{array}{c}Re\left(\frac{d\lambda }{d{\eta }_1}{\mid }_{{\eta }_1={\eta }_{10}}\right)=\frac{K_1{L}_1+K_2{L}_2}{L_1^2+L_2^2}\ne 0.\hfill \end{array}$$

(19)

The transversal condition is satisfied. The proof ends.

□

To sum up, from Lemmas 2–4, the following theorem can be obtained.

Theorem 1.

If (H1)–(H5) hold, then we can obtain the following conclusion for system (3).

(1)

If ${\eta }_2\in [0,{\overline{\eta }}_{20})$, then when ${\eta }_1\in [0,{\eta }_{10})$, the zero equilibrium point of the fractional order system is asymptotically stable.

(2)

If ${\eta }_2\in [0,{\overline{\eta }}_{20})$, then when ${\eta }_1={\eta }_{10}$, the zero equilibrium point of fractional order system loses stability and produces Hopf bifurcation.

3.2. The Hopf Bifurcation of a System (3) with Time Delay ${\eta }_1$ as Parameter

When ${\eta }_2=0$, the Equation (6) becomes

$$M_1\left(\lambda \right)+M_3\left(\lambda \right)+\left(M_2\left(\lambda \right)+M_4\left(\lambda \right)\right)e^{-\lambda {\eta }_1}=0.$$

(20)

Assume that $\lambda =\overline{\varpi }(cos\frac{\pi }{2}+isin\frac{\pi }{2})(\overline{\varpi }>0)$ is a pure virtual root Equation (20), and use ${\overline{\mathcal{M}}}_i^R,{\overline{\mathcal{M}}}_i^I$ to represent the real and imaginary parts of $M_i\left(i\overline{\varpi }\right)(i=1,2,3,4)$, respectively. There are

$$\left\{\begin{array}{c}\hfill ({\overline{\mathcal{M}}}_2^R+{\overline{\mathcal{M}}}_4^R)cos\overline{\varpi }{\eta }_1+({\overline{\mathcal{M}}}_2^I+{\overline{\mathcal{M}}}_4^I)sin\overline{\varpi }{\eta }_1=-({\overline{\mathcal{M}}}_1^R+{\overline{\mathcal{M}}}_3^R),\\ \hfill -({\overline{\mathcal{M}}}_2^R+{\overline{\mathcal{M}}}_4^R)sin\overline{\varpi }{\eta }_1+({\overline{\mathcal{M}}}_2^I+{\overline{\mathcal{M}}}_4^I)cos\overline{\varpi }{\eta }_1=-({\overline{\mathcal{M}}}_1^I+{\overline{\mathcal{M}}}_3^I).\end{array}\right.$$

(21)

By transforming the Equation (21), it is easy to obtain the following formula.

$$\begin{array}{c}cos\overline{\varpi }{\eta }_1=-\frac{({\overline{\mathcal{M}}}_1^R+{\overline{\mathcal{M}}}_3^R)({\overline{\mathcal{M}}}_2^R+{\overline{\mathcal{M}}}_4^R)+({\overline{\mathcal{M}}}_1^I+{\overline{\mathcal{M}}}_3^I)({\overline{\mathcal{M}}}_2^I+{\overline{\mathcal{M}}}_4^I)}{{({\overline{\mathcal{M}}}_2^R+{\overline{\mathcal{M}}}_4^R)}^2+{({\overline{\mathcal{M}}}_2^I+{\overline{\mathcal{M}}}_4^I)}^2},\hfill \\ sin\overline{\varpi }{\eta }_1=\frac{({\overline{\mathcal{M}}}_1^I+{\overline{\mathcal{M}}}_3^I)({\overline{\mathcal{M}}}_2^R+{\overline{\mathcal{M}}}_4^R)-({\overline{\mathcal{M}}}_1^R+{\overline{\mathcal{M}}}_3^R)({\overline{\mathcal{M}}}_2^I+{\overline{\mathcal{M}}}_4^I)}{{({\overline{\mathcal{M}}}_2^R+{\overline{\mathcal{M}}}_4^R)}^2+{({\overline{\mathcal{M}}}_2^I+{\overline{\mathcal{M}}}_4^I)}^2},\hfill \end{array}$$

(22)

and

$$\begin{array}{c}G\left(\overline{\varpi }\right)={({\overline{\mathcal{M}}}_2^R+{\overline{\mathcal{M}}}_4^R)}^2+{({\overline{\mathcal{M}}}_2^I+{\overline{\mathcal{M}}}_4^I)}^2-{({\overline{\mathcal{M}}}_1^R+{\overline{\mathcal{M}}}_3^R)}^2-{({\overline{\mathcal{M}}}_1^I+{\overline{\mathcal{M}}}_3^I)}^2=0.\hfill \end{array}$$

(23)

Hypothesis 6 (H6).

Equation (23) has at least one positive real root.

Without loss of generality, if all positive real roots of Equation (23) are expressed as $\overline{\varpi }$, then the bifurcation point ${\overline{\eta }}_{10}$ of Equation (20) at ${\eta }_2=0$ can be deduced from Equation (22). Namely,

$$\begin{array}{c}{\overline{\eta }}_1^{\left(k\right)}=\frac{1}{\overline{\varpi }}arccos\left(-\frac{({\overline{\mathcal{M}}}_1^R+{\overline{\mathcal{M}}}_3^R)({\overline{\mathcal{M}}}_2^R+{\overline{\mathcal{M}}}_4^R)+({\overline{\mathcal{M}}}_1^I+{\overline{\mathcal{M}}}_3^I)({\overline{\mathcal{M}}}_2^I+{\overline{\mathcal{M}}}_4^I)}{{({\overline{\mathcal{M}}}_2^R+{\overline{\mathcal{M}}}_4^R)}^2+{({\overline{\mathcal{M}}}_2^I+{\overline{\mathcal{M}}}_4^I)}^2}+2k\pi \right),k=0,1,2,3,\cdots ,\hfill \\ {\overline{\eta }}_{10}=min\left\{{\overline{\eta }}_1^{\left(0\right)}\right\}.\hfill \end{array}$$

(24)

Lemma 5.

Take ${\eta }_1\in (0,{\overline{\eta }}_{10})$, when (H7) was established, Equation (25) has at least one pair of pure imaginary roots.

Proof. 

When ${\eta }_2>0$, Equation (6) can be written as

$$M_1\left(\lambda \right)+M_2\left(\lambda \right)e^{-\lambda {\eta }_1}+\left(M_3\left(\lambda \right)+M_4\left(\lambda \right)e^{-\lambda {\eta }_1}\right)e^{-\lambda {\eta }_2}=0.$$

(25)

Let $\lambda =\varpi (\cos \frac{\pi }{2}+isin\frac{\pi }{2})(\varpi >0)$ is a pure imaginary root of the Equation (25), and use ${\mathcal{M}}_i^R,{\mathcal{M}}_i^I$ to represent the real and imaginary parts of $M_i\left(i\varpi \right)(i=1,2,3,4)$, respectively. Then we can obtain

$$\left\{\begin{array}{c}\left({\mathcal{M}}_3^R+{\mathcal{M}}_4^{R}cos\varpi {\eta }_1+{\mathcal{M}}_4^{I}sin\varpi {\eta }_1\right)cos\varpi {\eta }_2+\left({\mathcal{M}}_3^I+{\mathcal{M}}_4^{I}cos\varpi {\eta }_1-{\mathcal{M}}_4^{R}sin\varpi {\eta }_1\right)sin\varpi {\eta }_2\hfill \\ =-\left({\mathcal{M}}_1^R+{\mathcal{M}}_2^{R}cos\varpi {\eta }_1+{\mathcal{M}}_2^{I}sin\varpi {\eta }_1\right),\hfill \\ \left({\mathcal{M}}_3^R+{\mathcal{M}}_4^{R}cos\varpi {\eta }_1+{\mathcal{M}}_4^{I}sin\varpi {\eta }_1\right)sin\varpi {\eta }_2-\left({\mathcal{M}}_3^I+{\mathcal{M}}_4^{I}cos\varpi {\eta }_1-{\mathcal{M}}_4^{R}sin\varpi {\eta }_1\right)cos\varpi {\eta }_2\hfill \\ =\left({\mathcal{M}}_1^I+{\mathcal{M}}_2^{I}cos\varpi {\eta }_1-{\mathcal{M}}_2^{R}sin\varpi {\eta }_1\right).\hfill \end{array}\right.$$

(26)

Equation (27) is obtained by summing and squaring both sides of Equation (26).

$$\begin{array}{c}{\left({\mathcal{M}}_1^R\right)}^2+{\left({\mathcal{M}}_1^I\right)}^2+{\left({\mathcal{M}}_2^R\right)}^2+{\left({\mathcal{M}}_2^I\right)}^2-{\left({\mathcal{M}}_3^R\right)}^2-{\left({\mathcal{M}}_3^I\right)}^2-{\left({\mathcal{M}}_4^R\right)}^2-{\left({\mathcal{M}}_4^I\right)}^2\hfill \\ +2({\mathcal{M}}_1^I{\mathcal{M}}_2^I+{\mathcal{M}}_1^R{\mathcal{M}}_2^R-{\mathcal{M}}_3^I{\mathcal{M}}_4^I-{\mathcal{M}}_3^R{\mathcal{M}}_4^R)cos\varpi {\eta }_1\hfill \\ +2({\mathcal{M}}_1^R{\mathcal{M}}_2^I-{\mathcal{M}}_1^I{\mathcal{M}}_2^R-{\mathcal{M}}_3^R{\mathcal{M}}_4^I+{\mathcal{M}}_3^I{\mathcal{M}}_4^R)sin\varpi {\eta }_1=0.\hfill \end{array}$$

(27)

Equation (28) is obtained by transformation as follows.

$$\begin{array}{c}cos\varpi {\eta }_2=-\frac{{\mathcal{E}}_1{\mathcal{E}}_3+{\mathcal{E}}_2{\mathcal{E}}_4}{{\left({\mathcal{E}}_3\right)}^2+{\left({\mathcal{E}}_4\right)}^2},\hfill \\ sin\varpi {\eta }_2=\frac{{\mathcal{E}}_2{\mathcal{E}}_3-{\mathcal{E}}_1{\mathcal{E}}_4}{{\left({\mathcal{E}}_3\right)}^2+{\left({\mathcal{E}}_4\right)}^2},\hfill \end{array}$$

(28)

where

$$\begin{array}{c}{\mathcal{E}}_1={\mathcal{M}}_1^R+{\mathcal{M}}_2^{R}cos\varpi {\eta }_1+{\mathcal{M}}_2^{I}sin\varpi {\eta }_1,\hfill \\ {\mathcal{E}}_2={\mathcal{M}}_1^I+{\mathcal{M}}_2^{I}cos\varpi {\eta }_1-{\mathcal{M}}_2^{R}sin\varpi {\eta }_1,\hfill \\ {\mathcal{E}}_3={\mathcal{M}}_3^R+{\mathcal{M}}_4^{R}cos\varpi {\eta }_1+{\mathcal{M}}_4^{I}sin\varpi {\eta }_1,\hfill \\ {\mathcal{E}}_4={\mathcal{M}}_3^I+{\mathcal{M}}_4^{I}cos\varpi {\eta }_1-{\mathcal{M}}_4^{R}sin\varpi {\eta }_1.\hfill \end{array}$$

□

Hypothesis 7(H7).

Equation (27) has at least one positive real root.

Let Equation (27) have n positive real roots, denoted as ${\varpi }_j(j=1,2,\dots ,n)$, and from Equation (28) we can easily obtain

$$\begin{array}{c}{\eta }_{2j}^{\left(k\right)}=\frac{1}{{\varpi }_j}arccos\left(-\frac{{\mathcal{E}}_1{\mathcal{E}}_3+{\mathcal{E}}_2{\mathcal{E}}_4}{{\left({\mathcal{E}}_3\right)}^2+{\left({\mathcal{E}}_4\right)}^2}+2k\pi \right),k=0,1,2,3,\dots ,j=1,2,\dots ,n.\hfill \end{array}$$

(29)

We have ${\eta }_{20}=\underset{j=1,2,\dots ,n}{min}\left\{{\eta }_{2j}^{\left(0\right)}\right\}$, where ${\varpi }_0$ is the value corresponding to ${\eta }_{20}$. At least one pair of pure imaginary roots exists in Equation (25). The proof ends.

Hypothesis 8 (H8).

$\frac{P_1{Q}_1+P_2{Q}_2}{Q_1^2+Q_2^2}\ne 0.$ (See below for the meanings of $P_i$ and $Q_i(i=1,2)$).

Lemma 6.

If (H8) is satisfied, then the transversality condition is true.

Proof. 

Let $\lambda \left({\eta }_2\right)=\alpha \left({\eta }_2\right)+i\varpi \left({\eta }_2\right)$ be the root of Equation (6) near ${\eta }_2={\eta }_{20}$ that satisfies $\alpha \left({\eta }_{20}\right)=0,\varpi \left({\eta }_{20}\right)={\varpi }_0$, and then find the transversal condition. If take the derivative of Equation (6) with respect to ${\eta }_2$, we obtain

$$\begin{array}{c}{\mathcal{M}}_1^{\prime }\left(\lambda \right)\frac{d\lambda }{d{\eta }_2}+{\mathcal{M}}_2^{\prime }\left(\lambda \right)e^{-\lambda {\eta }_1}\frac{d\lambda }{d{\eta }_2}-{\eta }_1{\mathcal{M}}_2\left(\lambda \right)e^{-\lambda {\eta }_1}\frac{d\lambda }{d{\eta }_2}+{\mathcal{M}}_3^{\prime }\left(\lambda \right)e^{-\lambda {\eta }_2}\frac{d\lambda }{d{\eta }_2}+{\mathcal{M}}_4^{\prime }\left(\lambda \right)e^{-\lambda ({\eta }_1+{\eta }_2)}\frac{d\lambda }{d{\eta }_2}\hfill \\ -\left(\lambda +{\eta }_2\frac{d\lambda }{d{\eta }_2}\right){\mathcal{M}}_3\left(\lambda \right)e^{-\lambda {\eta }_2}-\left(\lambda +({\eta }_1+{\eta }_2)\frac{d\lambda }{d{\eta }_2}\right){\mathcal{M}}_4\left(\lambda \right)e^{-\lambda ({\eta }_1+{\eta }_2)}=0,\hfill \end{array}$$

(30)

where ${\mathcal{M}}_i^{\prime }\left(\lambda \right)(i=1,2,3,4)$ is the first derivative of ${\mathcal{M}}_i\left(\lambda \right)$ with respect to $\lambda$. The following formula is obtained by simplifying the above formula.

$$\frac{d\lambda }{d{\eta }_2}=\frac{P\left(\lambda \right)}{Q\left(\lambda \right)}$$

and

$$\begin{array}{cc}\hfill P\left(\lambda \right)=& \lambda {\mathcal{M}}_3\left(\lambda \right)e^{-\lambda {\eta }_2}+\lambda {\mathcal{M}}_4\left(\lambda \right)e^{-\lambda ({\eta }_1+{\eta }_2)},\hfill \\ \hfill Q\left(\lambda \right)=& {\mathcal{M}}_1^{\prime }\left(\lambda \right)+{\mathcal{M}}_2^{\prime }\left(\lambda \right)e^{-\lambda {\eta }_1}-{\eta }_1{\mathcal{M}}_2\left(\lambda \right)e^{-\lambda {\eta }_1}+{\mathcal{M}}_3^{\prime }\left(\lambda \right)e^{-\lambda {\eta }_2}-{\eta }_2{\mathcal{M}}_3\left(\lambda \right)e^{-\lambda {\eta }_2}\hfill \\ & +{\mathcal{M}}_4^{\prime }\left(\lambda \right)e^{-\lambda ({\eta }_1+{\eta }_2)}-({\eta }_1+{\eta }_2){\mathcal{M}}_4\left(\lambda \right)e^{-\lambda ({\eta }_1+{\eta }_2)}.\hfill \end{array}$$

The real numbers $P_1,P_2,Q_1$ and $Q_2$ are defined as follows.

$$\begin{array}{c}P\left(i\varpi \right)=P_1+i{P}_2,\hfill \\ Q\left(i\varpi \right)=Q_1+i{Q}_2,\hfill \end{array}$$

that is, $P_1,Q_1$ and $P_2,Q_2$ represent the real and imaginary parts of $P\left(\lambda \right)$ and $Q\left(\lambda \right)$ at ${\eta }_2={\eta }_{20}$, respectively.

Then the following formula can be easily obtained from (H8),

$$\begin{array}{c}Re\left(\frac{d\lambda }{d{\eta }_2}{\mid }_{{\eta }_2={\eta }_{20}}\right)=\frac{P_1{Q}_1+P_2{Q}_2}{Q_1^2+Q_2^2}\ne 0.\hfill \end{array}$$

(31)

The transversal condition is satisfied. The proof ends.

□

In summary, from Lemma 3, Lemma 5, and Lemma 6, we can obtain Theorem 2 as follows.

Theorem 2.

If the (H1),(H4) and (H6)–(H8) hold, then we can obtain the following conclusion for the system (3).

(1)

If ${\eta }_1\in [0,{\overline{\eta }}_{10})$, then when ${\eta }_2\in [0,{\eta }_{20})$, the zero equilibrium point of the fractional order system is asymptotically stable.

(2)

If ${\eta }_1\in [0,{\overline{\eta }}_{10})$, then when ${\eta }_2={\eta }_{20}$, the zero equilibrium point of fractional order system loses stability and produces Hopf bifurcation.

4. Numerical Examples

In this section, we will give two numerical examples to support the theoretical results. Example 1 and Example 2 verify the validity and correctness of Theorem (1) and Theorem (2) respectively. It is obvious that the system (3) is equivalent to the system (4) and the system (3) form.

4.1. Example 1

Take $f_i\left(t\right)=g_j\left(t\right)=tanh\left(t\right),i=1,2;j=1,2,3,4.$ Consider the following system

$$\left\{\begin{array}{cc}\hfill D^{0.95}{u}_1\left(t\right)=& -0.5u_1\left(t\right)+tanh\left(v_1(t-{\tau }_3)\right)+tanh\left(v_2(t-{\tau }_3)\right)\hfill \\ \hfill & +tanh\left(v_3(t-{\tau }_3)\right)-tanh\left(v_4(t-{\tau }_3)\right),\hfill \\ \hfill D^{0.98}{u}_2\left(t\right)=& -0.6u_2\left(t\right)-tanh\left(v_1(t-{\tau }_4)\right)-tanh\left(v_2(t-{\tau }_4)\right)\hfill \\ \hfill & -2tanh\left(v_3(t-{\tau }_4)\right)+tanh\left(v_4(t-{\tau }_4)\right),\hfill \\ \hfill D^{0.96}{v}_1\left(t\right)=& -0.6v_1\left(t\right)-tanh\left(u_1(t-{\tau }_1)\right)-tanh\left(u_2(t-{\tau }_2)\right),\hfill \\ \hfill D^{0.93}{v}_2\left(t\right)=& -0.4v_2\left(t\right)+tanh\left(u_1(t-{\tau }_1)\right)+tanh\left(u_2(t-{\tau }_2)\right),\hfill \\ \hfill D^{0.94}{v}_3\left(t\right)=& -0.8v_3\left(t\right)+tanh\left(u_1(t-{\tau }_1)\right)+tanh\left(u_2(t-{\tau }_2)\right),\hfill \\ \hfill D^{0.92}{v}_4\left(t\right)=& -0.5v_4\left(t\right)+2tanh\left(u_1(t-{\tau }_1)\right)+tanh\left(u_2(t-{\tau }_2)\right).\hfill \end{array}\right.$$

(32)

It is clear that the system (32) has an equilibrium point (0, 0, 0, 0, 0, 0). ${\overline{\eta }}_{20}=2.3695$ is obtained by numerical calculation. Take ${\tau }_2=0.3,{\tau }_4=0.5$, which is ${\eta }_2={\tau }_2+{\tau }_4=0.8\in (0,{\overline{\eta }}_{20})$. Then we have ${\eta }_{10}=1.5147$, and $\frac{K_1{L}_1+K_2{L}_2}{L_1^2+L_2^2}=0.1480\ne 0$. That is, the (H1)–(H5) are satisfied. When taking ${\tau }_1=0.5,{\tau }_3=0.9$, that is, ${\eta }_1={\tau }_1+{\tau }_3=1.4<{\eta }_{10}$, the Figure 1 can be obtained through Matlab simulation. It is obvious that the zero equilibrium point of the system (32) is locally asymptotically stable. And take ${\tau }_1=0.5,{\tau }_3=1.1$, ${\eta }_1={\tau }_1+{\tau }_3=1.6>{\eta }_{10}$, it can be seen from Figure 2 that the zero-equilibrium point of the system loses its stability and produces Hopf bifurcation.

4.2. Example 2

Take $f_i\left(t\right)=g_j\left(t\right)=tanh\left(t\right),i=1,2;j=1,2,3,4.$ Consider the following system

$$\left\{\begin{array}{cc}\hfill D^{0.89}{u}_1\left(t\right)=& -0.6u_1\left(t\right)-tanh\left(v_1(t-{\tau }_3)\right)-2tanh\left(v_2(t-{\tau }_3)\right)\hfill \\ \hfill & -2tanh\left(v_3(t-{\tau }_3)\right)+2tanh\left(v_4(t-{\tau }_3)\right),\hfill \\ \hfill D^{0.94}{u}_2\left(t\right)=& -0.4u_2\left(t\right)-tanh\left(v_1(t-{\tau }_4)\right)+2tanh\left(v_2(t-{\tau }_4)\right)\hfill \\ \hfill & +2tanh\left(v_3(t-{\tau }_4)\right)-3tanh\left(v_4(t-{\tau }_4)\right),\hfill \\ \hfill D^{0.98}{v}_1\left(t\right)=& -0.5v_1\left(t\right)-tanh\left(u_1(t-{\tau }_1)\right)-tanh\left(u_2(t-{\tau }_2)\right),\hfill \\ \hfill D^{0.92}{v}_2\left(t\right)=& -0.3v_2\left(t\right)+tanh\left(u_1(t-{\tau }_1)\right)+tanh\left(u_2(t-{\tau }_2)\right),\hfill \\ \hfill D^{0.87}{v}_3\left(t\right)=& -0.6v_3\left(t\right)+tanh\left(u_1(t-{\tau }_1)\right)+tanh\left(u_2(t-{\tau }_2)\right),\hfill \\ \hfill D^{0.95}{v}_4\left(t\right)=& -0.3v_4\left(t\right)+2tanh\left(u_1(t-{\tau }_1)\right)+tanh\left(u_2(t-{\tau }_2)\right).\hfill \end{array}\right.$$

(33)

It is clear that the system (33) has an equilibrium point (0,0,0,0,0,0). ${\overline{\eta }}_{10}=0.5563$ is obtained by numerical calculation. Take ${\tau }_1=0.1,{\tau }_3=0.2$, which is ${\eta }_1={\tau }_1+{\tau }_3=0.3\in (0,{\overline{\eta }}_{10})$. We have ${\eta }_{20}=0.4574$, and $\frac{P_1{Q}_1+P_2{Q}_2}{Q_1^2+Q_2^2}=1.2745\ne 0$. That is, (H1), (H4), and (H6)–(H8) are satisfied. When taking ${\tau }_2=0.2,{\tau }_4=0.24$, that is, ${\eta }_2={\tau }_2+{\tau }_4=0.44<{\eta }_{20}$, the Figure 3 can be obtained through Matlab simulation. It is obvious that the zero equilibrium point of the system (33) is locally asymptotically stable. And take ${\tau }_2=0.23,{\tau }_4=0.24$, ${\eta }_2={\tau }_2+{\tau }_4=0.47>{\eta }_{20}$, it can be seen from Figure 4 that the zero-equilibrium point of the system loses its stability and produces Hopf bifurcation.

In [25,26], the unified program code can easily be edited for the numerical simulation. However, because there are six different fractional orders in this model, it is difficult to write Matlab code. Accordingly, we mainly use the Simulink tool in Matlab software to draw phase diagram and trajectory diagram of the model, which has achieved good results.

5. Conclusions

In this paper, we study the dynamic behavior of a class of fractional order simplified BAM neural networks with multiple delays.Firstly, the system with four delays is transformed into a system with two delays. Then, by fixing one delay ${\eta }_1={\tau }_1+{\tau }_3$ (${\eta }_2={\tau }_2+{\tau }_4$) and taking the other delay ${\eta }_2={\tau }_2+{\tau }_4$ (${\eta }_1={\tau }_1+{\tau }_3$) as a bifurcation parameter, the dynamic behavior of the fractional order system with two delays is analyzed, and sufficient conditions for the local stability of the zero solution and Hopf bifurcation are obtained. Through analysis, it is found that the fractional order delay system remains stable when the time delay is small, while when the time delay ${\eta }_2\left({\eta }_1\right)$ exceeds the critical value ${\eta }_{20}\left({\eta }_{10}\right)$, the zero solution of the nonlinear fractional order system will lose stability and produce Hopf bifurcation. Finally, two simulation examples are given to illustrate the effectiveness and feasibility of the main results. The fractional BAM neural network model with six neurons studied in this paper is relatively simple, and there are still many problems to be studied, such as the study of bifurcation control and the establishment of higher dimensional neural network model, which need to be further discussed.