Stability and Hopf bifurcation analysis of a fractional-order ring-hub structure neural network with delays under parameters delay feedback control

1.   Introduction

The construction of neural networks in biological system is complex and multifaceted. Inspired by the structure and function of biological neural networks, artificial neural networks have attracted the attention of many scholars and have been applied in many fields, such as disease diagnosis, signal and image processing, associative memory, combinatorial optimization, artificial intelligence and pattern recognition ^[1,2,3,4]. Through further research, scholars have proposed adaptive neural networks ^[5], feedback neural networks ^[6], recurrent neural networks ^[7], cellular neural networks ^[8] and so on. In addition, due to the different capacities of transmitting, receiving and processing information among different neurons, time delay cannot be ignored for neural networks. Time delay can lead to dynamic behaviors of neural networks with poor performance, oscillation, bifurcation and chaos ^[9,10]. It is known that Hopf bifurcation phenomenon is universal in the neural networks ^[11,12]. Considering that modeling work involving biological or physical processes takes time to complete, multiple time delays occur naturally ^[13,14].

In recent years, fractional calculus has been widely used in engineering mathematics, physics, biology and economics because of its genetic and wireless storage properties ^{[15,16,17,18]}. Based on this property, fractional integration operator is introduced into neural network systems. Researches have shown that fractional-order systems can more accurately represent memory features compared to integer-order systems, thereby enabling the ability of neurons to effectively transmit and process information. Therefore, fractional-order neural network systems have gained widespread attention. Huang et al. ^[19] discussed fractional order-induced bifurcations in a delayed neural network with three neurons. Xu et al. ^[20] investigated Hopf bifurcation on simplified BAM neural networks with multiple delays. In ^[21], a fractional-order recurrent neural network is proposed and the stability and Hopf bifurcation are investigated.

Although there have been many studies on bifurcation of fractional-order neural networks, most of them only consider neural networks with a single structure. In general, the structure of neural networks can be divided into ring structure, star structure, linked structure, and hub structure. In view of the diversity and complexity of real neural networks, the impact of network topology on the dynamic behavior of the network cannot be ignored. Tao et al. ^[22] put forward a bidirectional super-ring-shaped delayed neural network polymer with $n$ neurons and investigated the influence of time delay on the dynamics of the network. In ^[23], a class of $4n$-dimensional delayed neural networks with radial-ring configuration and bidirectional coupling is proposed. Further research shows that neural networks with hub structure are beneficial to information integration. In addition, the ring structure of the network can effectively solve the congestion problem caused by the increase of information traffic. Based on the characteristics of the two structures, Chen et al. ^[24] studied the stability and Hopf bifurcation of a high-dimensional fractional delayed neural network containing a ring-hub structure. In order to achieve better performance of neural networks, inspired by ^[24], this paper considers the following a fractional-order $4$-neuron ring-hub structure neural network with two time delays:

where $\theta\in(0, 1]$ is the fractional order, $D^{\theta}$ denotes the Caputo fractional derivative; $x_{i}(t)(i = 1, 2, 3, 4)$ denotes the state variable of the $i$th neuron at time $t$; $\sigma > 0$ describes the internal decay rate of neurons; $c_{ij}(i, j = 1, 2, 3, 4)$ stands for the connection weight from the $i$th neuron to the $j$th neuron; $f$ represents the activation function; $\tau_{2}\geq0$ is the time delay of signal transmission from central neuron $x_{4}(t)$ to peripheral neuron $x_{i}(t)(i = 1, 2, 3)$; conversely, the time delay of signal transmission from peripheral neuron to central neuron is $\tau_{1}\geq0$. The initial conditions of system (1.1) are $x_{i}(\xi) = \phi_{i}(\xi), \phi_{i}(\xi)\geq0, i = 1, 2, 3, 4, -\sigma \leq \xi \leq0$, $\sigma = max\{\tau _{1}, \tau _{2}\}$. Throughout this paper, we make the following assumption:

$(H1)\quad f\in C^{1}, f(0) = 0, f'(0)\neq0$.

In the fields of biology, economy, fluid dynamics and communication security, the transmission delay of a signal often affects the overall response of system. Considering that bifurcation caused by time delay may be harmful, a controller is introduced to control the bifurcation phenomena. Bifurcation control generally involves delaying the occurrence of inherent bifurcation, modifying the critical value of existing bifurcation point, and stabilizing the bifurcation solutions or branches. There are various methods of bifurcation control. For instance, Ma et al. ^[25] investigated a parametric delay feedback controller on van der Pol–Duffing oscillator, which makes the coefficients of the system dependent on time delay. In ^[26], a hybrid control strategy is proposed and the existence of Hopf bifurcation with time-delays Hopfield neural network is analyzed. A delayed feedback controller ^[27] is used for Hopf bifurcation control of small-world network model. In order to achieve optimal dynamic behaviors, the controller proposed in ^[27] is implemented in this paper, and the following parameters delay feedback control of ring-hub structure neural network is considered

where $ke^{-P(\tau_{1})}$ is parameters delay feedback controller. Let $P(\tau_{1}) = p\tau_{1}$, $p$ is called the decay rate. Note that the feedback controller $ke^{-p\tau_{1}}$ is a function that decreases exponentially with time delay. This means that the feedback effect of past states diminishes with time $t$.

For systems with two delays and delay dependent parameters, the method of stability switching curves is proposed to study the dynamic behaviors of system. In ^[28], a class of integer-order two delays models with delay dependent parameters is considered. The corresponding characteristic equation has the following form

where $P_{i}(\lambda, \tau), i = 0, 1, 2$ are polynomials in $\lambda$ and $P_{i}$ only depend on $\tau$. Authors proposed a geometric method to study Eq (1.3), which obtains the stability switching curves in the whole two time delays parameter plane. The direction of bifurcation in stability switching curves is determined according to the direction of the characteristic roots crossing the imaginary axis. The stability condition of system at the equilibrium point is obtained. This method is applied to study the effect of two delays on stability of an HIV infection model ^[29] and a planktonic resource-consumer system ^[30]. In addition, the geometric method ^[28] is also applicable to the characteristic equation with the form of

There are few papers discussing the stability of neural networks with multiple time delays changing simultaneously and delay dependent parameters. In this paper, the method of stability switching curves is first used to discuss the stability and the existence of Hopf bifurcation of a fractional-order neural network with a composite ring-hub structure with two time delays changing simultaneously and delay dependent parameters. The main contribution of this paper are as follows:

$(\text{i})$ Considering the complex topology of neural networks, a fractional-order two time delays neural network with a composite ring-hub structure is considered.

$(\text{ii})$ By taking the time delay as the bifurcation parameter, sufficient conditions for the stability and the existence of Hopf bifurcation are established. The research results show that delays and the fractional-order can affect the stability of system.

$(\text{iii})$ A parameters delay feedback controller is introduced into the fractional-order neural network system with a composite ring-hub structure, and controls successfully Hopf bifurcation.

$(\text{iv})$ It is the first time to apply the method of stability switching curves to a fractional-order neural network system. The influence of time delays changing simultaneously on the stability of fractional-order controlled system (1.2) is analyzed.

The rest of this paper is organized as follows: In Section 2, we consider the local stability and the existence of Hopf bifurcation of system (1.1). In Section 3, the parameters delay feedback controller is introduced into system (1.1). The stability and existence of Hopf bifurcation of system (1.2) are discussed by using the method of stability switching curves. In Section 4, numerical simulation is adopted to verify the correctness of theoretical results. Finally, the conclusions are given in Section 5.

2.   Stability and Hopf bifurcation analysis of system (1.1)

In this section, $\tau = \tau_{1}+\tau_{2}$ is selected as the bifurcation parameter to study the local stability of the equilibrium and the existence of Hopf bifurcation of system (1.1). Based on assumption $(H1)$, it can be concluded that the equilibrium of system (1.1) is the origin $O(0, 0, 0, 0)$. The corresponding linear system of system (1.1) at $O$ is

where $\varphi_{ij} = c_{ij}f'(0)(i, j = 1, 2, 3, 4)$.

By applying Laplace transformation, the characteristic equation of system (2.1) is

that is

where

The stability of the equilibrium $O$ is discussed in two scenarios.

Case Ⅰ: $\tau_{1} = \tau_{2} = 0$.

If $\tau_{1} = \tau_{2} = 0$, Equation (2.3) becomes

where

Lemma 2.1. ^[31] For the following fractional-order system: $D^{\theta}x(t) = Ax(t), A\in R^{n\times n}$, the equilibrium of system is locally asymptotically stable if and only if all the eigenvalues $s_{i}(i = 1, 2, \ldots, n)$ of $A$ satisfy $\lvert arg(s _{i}) \rvert > \theta \pi/2$, where $\theta\in(0, 1]$.

According to Lemma 2.1 and Routh-Hurwitz criterion, we have the following theorem.

Theorem 2.1. System (1.1) is locally asymptotically stable if and only if $D_{i} > 0 (i = 1, 2, 3, 4)$ holds, where $D_{i}$ is defined as follows

Proof. If $D_{i} > 0(i = 1, 2, 3, 4)$ holds, we can conclude that all the roots $s_{i}$ satisfy $|arg(s _{i}) | > \theta \pi/2(i = 1, 2, 3, 4)$. It follows from Lemma 2.1 that system $(1.1)$ with $\tau_{1} = \tau_{2} = 0$ is locally asymptotically stable.

Case Ⅱ: $\tau_{1} > 0, \tau_{2} > 0.$

Let $\tau = \tau_{1}+\tau_{2}$, Equation (2.3) becomes

where

Assuming that the characteristic equation (2.5) has a purely imaginary root $s = \omega(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2})(\omega > 0)$. Substituting $s$ into Eq (2.5) and separating real and imaginary parts, we get

where

By calculation, one obtains

Since $\cos^{2}\omega\tau+\sin^{2}\omega\tau = 1$, we can obtain

where

Let

To obtain the main results, we make the following assumption:

then $\lim\limits_{\omega \rightarrow \infty }F(\omega) = +\infty$ and Eq (2.8) has at least one positive root $\omega_{i}$. According to Eq (2.7), we have

Denote

There exists a simple pair of purely imaginary roots for Eq (2.5) when $\tau = \tau_{0}$, and all roots of Eq (2.5) for $\tau\in(0, \tau_{0})$ have strictly negative real parts. Let $s(\tau) = \mu(\tau) +i\omega(\tau) (\omega > 0)$ be the root of Eq (2.5) near $\tau = \tau_{i}^{(k)}$ complying with $\mu(\tau_{i}^{(k)}) = 0, \omega (\tau_{i}^{(k)}) = \omega_{i}$. Substituting $s(\tau)$ into Eq (2.5) and taking the derivative of $s$ with respect to $\tau$, one gets

where

It follows from Eq (2.11)

where

Due to $\Psi_{1} ^{2}+\Psi_{2} ^{2} > 0$, we have

If assumption

is holds, then the transversality condition $Re\left [\frac{\mathrm{d} s}{\mathrm{d} \tau } \right]\vert_{{\tau = \tau _{i}^{(k)}}}\neq 0$ holds.

Based on the above analysis, the following theorem can be obtained.

Theorem 2.2. For system (1.1), assumptions $(H1)-(H3)$ hold.

$({i})$ If $\tau\in[0, \tau_{0})$, then the equilibrium $O$ is locally asymptotically stable;

$({ii})$ If $\tau > \tau_{0}$, then system (1.1) undergoes Hopf bifurcation at $O$ when $\tau = \tau_{0}$.

3.   Stability and Hopf bifurcation analysis of system (1.2)

In this section, we introduce the parameters delay feedback controller into system (1.1). The stability and the existence of Hopf bifurcation of system (1.2) are studied by applying the method of stability switching curves.

The linearized system (1.2) at equilibrium $O(0, 0, 0, 0)$ is given by

The characteristic equation corresponding to system (3.1) can be obtained

where

When $\tau_{1} = \tau_{2} = 0$, the characteristic equation (3.2) reduces to

where

In order to obtain the main result, we assume

According the Routh-Hurwitz criterion and Lemma 2.1, we present the following theorem.

Theorem 3.1. For $\tau_{1} = \tau_{2} = 0$, if $(H1)$ and $(H4)$ hold, then the equilibrium $O$ of system (1.2) is locally asymptotically stable.

3.1.   Stability switching curves

In this subsection, when $\tau _{1} > 0, \tau _{2} > 0$, and $\tau _{1}\neq \tau _{2}$, the characteristic equation (3.2) with two time delays and one-delay dependent coefficients is discussed using the method in ^[28]. We firstly give the following basic assumptions:

$(\text{i})$ Existence of a principal term: $deg(P_{0}(s, \tau_{1}))\geq max\left \{ deg(P_{1}(s, \tau_{1})), deg(P_{2}(s, \tau_{1})) \right \}$.

$(\text{ii})$ No zero frequency: $s = 0$ is not a characteristic root for any $\tau_{1}\in I$, i.e.,

$(\text{iii})$ The polynomials $P_{l}(s, \tau_{1}), l = 0, 1, 2$ have no common factor.

$(\text{iv})$ No large oscillation: $\lim\limits_{\underset{\left | s \right |\rightarrow \infty }{Res\geq 0}}\underset{\tau_{1}\in I}{\sup}(\left | \frac{P_{1}(s, \tau_{1})}{P_{0}(s, \tau_{1})} \right |+\left | \frac{P_{2}(s, \tau_{1})}{P_{0}(s, \tau_{1})} \right |) < 1$.

$(\text{v})$ $P_{l}(i\omega, \tau_{1})\neq0, l = 0, 1, 2$ for any $\tau_{1}\in I$ and $\omega\in\mathbb{R}_{+}$.

$(\text{vi})$ For any $\omega\in\mathbb{R}_{+}$, at least one of $\left|P_{l}(i\omega, \tau_{1})\right|, l = 0, 1, 2$ tends to $\infty$ as $\tau_{1}\rightarrow -\infty$. If there are multiple such $P_{l}$, then these functions tend to infinity at different rates.

Next, we verify that the above assumptions $(\hbox{i})$–$(\hbox{vi})$ hold for Eq (3.2).

For $P_{0}(s, \tau_{1}), P_{1}(s, \tau_{1}), P_{2}(s, \tau_{1})$, $(\hbox{i})$ is automatically satisfied.

Due to $P_{0}(0, \tau_{1})+P_{1}(0, \tau_{1})+P_{2}(0, \tau_{1}) = A_{2}-e^{-p\tau_{1}}A_{4}+A_{7}\neq0$, $(\hbox{ii})$ is true.

If $(\hbox{iii})$ does not hold, then there exists a common factor $c(s, \tau_{1})$ for $P_{l}(s, \tau_{1}), l = 0, 1, 2$, which causes Eq (3.2) to be decomposed into the product of another transcendental equation satisfying $(\hbox{iii})$ and $c(s, \tau_{1})$.

Since

the condition $(\hbox{iv})$ holds.

According to the expression for $P_{l}(s, \tau_{1}), l = 0, 1, 2$, $(\hbox{v})$ is naturally true. The presentation of assumption $(\hbox{vi})$ helps to reduce the cases of graph for $S_{n}^{\pm }(\omega, \tau_{1})$.

Assume that $s = \omega(\cos\frac{\pi }{2}+i\sin\frac{\pi }{2})(\omega > 0)$ is a pure imaginary root of $D(s, \tau_{1}, \tau_{2}) = 0$. Substituting it into Eq (3.2), we get

where

We know that $s = \omega(\cos\frac{\pi }{2}+i\sin\frac{\pi }{2})(\omega > 0)$ is the root of Eq (3.2) if and only if the right side of Eq (3.4) must form a triangle on the complex plane. Based on the relationship between the three sides of the triangle, for some $(\tau_{1}, \tau_{2})\in I\times \mathbb{R}_{+}$, then $(\omega, \tau_{1})$ satisfies

Inequalities (3.6) are equivalent to

The feasible region $\Omega = \{(\omega, \tau_{1})\in I\times\mathbb{R}_{+}:F_{i}(\omega, \tau_{1})\geq0, i = 1, 2, 3\}$, defined by inequalities (3.6) or (3.7) for $(\omega, \tau_{1})$, is such that the characteristic equation (3.2) may have solutions for $\tau_{2}\in\mathbb{R}_{+}$. As discussed in ^[28], for each connected region $\Omega_{k}$ of $\Omega$, we define the admissible range for $\omega$ as $I_{k} = [\omega_{k}^{l}, \omega_{k}^{r}], k = 1, 2, \cdots, N$. For each $\omega\in I_{k}$, there exists $\tau_{1}$-intervals $I_{\omega}^{k} = [\tau_{1, \omega}^{k, l}, \tau_{1, \omega}^{k, r}]$ such that Eqs (3.6) or (3.7) holds.

Let $\theta _{1}(\omega, \tau_{1})$, $\theta _{2}(\omega, \tau_{1})$ be the angles formed by $1$ and $a_{1}(\omega, \tau_{1})e^{-s\tau_{1}}$, $1$ and $a_{2}(\omega, \tau_{1})e^{-s(\tau_{1}+\tau_{2})}$, respectively. According to the law of cosine, we obtain

In the following, two possible cases are considered.

$1)$ If $Im(a_{1}(\omega, \tau_{1})e^{-i\omega \tau_{1}}) > 0$, we obtain

therefore

and

It follows from Eqs (3.10) and (3.11) that

and

$2)$ If $Im(a_{1}(\omega, \tau_{1})e^{-i\omega \tau_{1}}) < 0$, then

therefore

and

Similarly, according to Eqs (3.14) and (3.15), we get

and

Once the values of $(\omega, \tau_{1})$ satisfying Eqs (3.12) or (3.16) are determined, then the critical values of $\tau_{2}$ are obtained according to Eqs (3.13) or (3.17). Now we define the function $S_{n}^{\pm}:\Omega\rightarrow \mathbb{R}$

In $\Omega$, the roots of $S_{n}^{\pm }(\omega, \tau_{1})$ are denoted as $\tau_{1}^{i\pm }, i = 1, 2, \cdots$, and the corresponding $\tau_{2}$ values are obtained

When $\omega$ takes all the values in the whole interval $I_{k}$, we have the following curve on $\Omega$

and the stability switching curves

3.2.   Crossing direction

In this subsection, we consider the direction of $s(\tau_{1}, \tau_{2})$ crossing the imaginary axis when $(\tau_{1}, \tau_{2})$ deviates from a crossing curve. Define the direction in which $\omega$ increases as the positive direction of the curve. As moving along the positive direction of the curve, the region on the left-hand (right-hand) side of the curve is called the region on the left (right).

Assume that $(\tau _{1}^{\ast }, \tau _{2}^{\ast })\in \mathcal{T}$, then there is an $\omega ^{\ast } > 0$ such that $(\pm i\omega^{\ast}, \tau _{1}^{\ast }, \tau _{2}^{\ast })$ is a pair of pure imaginary roots of Eq (3.2). If $\frac{\partial D}{\partial s}(iw^{\ast }, \tau_{1}^{\ast }, \tau _{2}^{\ast })\neq 0$, then $s(\tau _{1}, \tau _{2}) = \mu (\tau _{1}, \tau _{2})\pm i\omega (\tau _{1}, \tau _{2})$ is a pair of conjugate complex roots of Eq (3.2), which satisfies $\mu (\tau _{1}^{\ast }, \tau _{2}^{\ast }) = 0$ and $\omega (\tau _{1}^{\ast }, \tau _{2}^{\ast }) = \omega ^{\ast }$ in the neighborhood of $(\tau _{1}^{\ast }, \tau _{2}^{\ast })$. Define the tangent vector of $\mathcal{T}$ along the positive direction as $\mathit{\boldsymbol{n}}_{\bf{1}} = (\frac{\partial \tau _{1}}{\partial \omega }, \frac{\partial \tau _{2}}{\partial \omega })$, and the normal vector to $\mathcal{T}$ pointing to the region on the right as $\mathit{\boldsymbol{n}}_{\bf{2}} = (\frac{\partial \tau _{2}}{\partial \omega }, -\frac{\partial \tau _{1}}{\partial \omega })$. When $(\tau_{1}, \tau_{2})$ moves along the direction $\mathit{\boldsymbol{n}}_{\bf{3}} = (\frac{\partial \tau _{1}}{\partial\mu }, \frac{\partial \tau _{2}}{\partial\mu })$, the direction in which the characteristic roots crossing the imaginary axis is determined by the sign of the inner product of $\mathit{\boldsymbol{n}}_{\bf{2}}$ and $\mathit{\boldsymbol{n}}_{\bf{3}}$. Let

Regarding $\tau_{1}$ and $\tau_{2}$ as a function of $\mu$ and $\omega$ at a neighborhood of $(0, \omega^{\ast})$, and giving the following assumption

by the implicit function theorem and $(H5)$, we have

where

Let

Since $R_{0}^{2}+I_{0}^{2}\geq0$, we have

If $sign(\delta(\tau_{1}^{\ast }, \tau_{2}^{\ast })) > 0(<0)$, then the pair of eigenvalues $\mu(\tau_{1}, \tau_{2})\pm i\omega(\tau_{1}, \tau_{2})$ of Eq (3.2) crosses the imaginary axis to the right (left)-hand region of the curves $\mathcal{T}$, when $(\tau_{1}, \tau_{2})$ moves along the positive direction of the crossing curves $\mathcal{T}$.

Based on the above discussion, the following theorem is given

Theorem 3.2. If $sign(\delta(\tau_{1}^{\ast }, \tau_{2}^{\ast })) > 0(<0)$, then a pair of pure imaginary roots of the characteristic equation $D(s, \tau_{1}, \tau_{2}) = 0$ crosses the imaginary axis from left to right when $(\tau_{1}, \tau_{2})$ passes through the crossing curve to the right (left) region, where

where $P_{i}^{\ast } = P_{i}(i\omega ^{\ast }, \tau_{1} ^{\ast })$ and $P_{i\tau_{1}}^{\ast } = \frac{\partial P_{i}}{\partial \tau_{1}}(i\omega ^{\ast }, \tau_{1} ^{\ast }), i = 0, 1, 2$.

Proof. Since

By direct calculation there is

which completes the proof.

In the rest of this subsection, we further assume that

$(H6)\quad\frac{\partial S_{n}^{\pm }(\omega, \tau_{1})}{\partial \tau_{1}} \ne 0,$ for any ${(\tau_{1}, \tau_{2})}\in\mathcal{T}$.

To sum up, we have the the following theorem.

Theorem 3.3. Assumptions $(H1)$ and $(H4)-(H6)$ hold.

$(i)$ If $(\tau_{1}, \tau_{2})\in \mathcal{T}$, then system (1.2) has a locally asymptotically stable equilibrium $O$;

$(ii)$ If $(\tau_{1}, \tau_{2})$ crosses the stability switching curves $\mathcal{T}$, then system (1.2) undergoes Hopf bifurcation at $O$.

4.   Numerical simulation

In this section, we consider the following system

where $f(\cdot) = \tanh(\cdot), \theta = 0.96$ and $P(\tau_{1}) = p\tau_{1}$. System (4.1) can be discussed in the following two cases.

Case Ⅰ: $k = 0$. Obviously, system (4.1) becomes uncontrolled system (1.1) when $k = 0$. The characteristic equation corresponding to system (1.1) can be obtained

The relevant discussions are as follows.

$1)$ When $\tau_{1} = 0$ and $\tau_{2} = 0$, it is easy to calculate that $D_{1} = 4.4 > 0$, $D_{2} = 30.275 > 0$ and $D_{3} = {276.031} > 0$, $D_{4} = 773.3284 > 0$. Therefore, it follows from Lemma 2.1 and Theorem 2.1 that the equilibrium $O$ of system (4.1) is locally asymptotically stable (see Figure 1).

$2)$ When $\tau_{1} > 0$ and $\tau_{2} > 0$, from the definition of $F(\omega)$, we have

$c_{8} = -1.648$, then $(H2)$ is satisfied. $\omega_{0} = 1.079$ and $\tau_{0} = 1.5985$ can be calculated from Eqs (2.8) and (2.9). According to the conditions of Theorem 2.2, it is also easy to validate that the equilibrium $O$ of system (4.1) is locally asymptotically stable when $\tau = 1.4 < \tau_{0}$ (see Figure 2). System (4.1) exhibits Hopf bifurcation at the equilibrium $O$ when $\tau = 1.6 > \tau_{0}$ (see Figure 3).

Case Ⅱ: $k\neq0$. System (4.1) corresponds to controlled system (1.2). Let $k = -0.5, p = 0.05$, the characteristic equation of system (1.2) can be obtained

where

System (4.1) can be discussed in the following two cases.

$1)$ When $\tau_{1} = 0$ and $\tau_{2} = 0$, we can calculate $B_{4}(B_{1}B_{2}B_{3}-B_{1}^{2}B_{4}-B_{3}^{2}) = 1398.964 > 0$, then $(H4)$ is satisfied. So the original equilibrium $O$ is locally asymptotically stable from Theorem 3.1 (see Figure 4).

$2)$ When $\tau_{1} > 0, \tau_{2} > 0$ and $\tau_{1}\neq\tau_{2}$, the curves $\mathcal{C}$ and the feasible region $\Omega$ are shown in Figure 5. The curves $\mathcal{C}$ on $\Omega$ form stability switching curves $\mathcal{T}$ on $(\tau_{1}, \tau_{2})$ plane as in Figure 6a. $sign(\delta(\tau_{1}^{\ast }, \tau_{2}^{\ast })) = 1 > 0$ can be calculated, then the crossing direction of stability changes from left to right according to Theorem 3.2 (see Figure 6a). A partial enlargement is shown by the green area of Figure 6a (see Figure 6b). Choosing $(\tau_{1}, \tau_{2}) = (0.6, 1.5)$, we find that $O$ is locally asymptotically stable (see Figure 7). When $(\tau_{1}, \tau_{2})$ passes through stability switching curves from left to right along the arrow direction, by choosing $(\tau_{1}, \tau_{2}) = (0.65, 1.8)$, system (4.1) exhibits Hopf bifurcation at $O$ (see Figure 8). When $(\tau_{1}, \tau_{2}) = (0.6, 1)$, the original equilibrium $O$ of system (4.1) is locally asymptotically stable (see Figure 9). Compared with Figures 3 and 9, Hopf bifurcation of system with parameters delay feedback control could be delayed.

In order to highlight the control effect, we fix $p = 0.05, \theta = 0.96$ and take different parameter values for $k$ to discuss. With the different values of $k = -0.5, -0.6, -0.7$, the corresponding stability switching curves are shown in Figure 10. In order to observe Figure 10 clearly, we give a partial enlargement of Figure 10. In particular, the region of the locally asymptotically stability of the equilibrium increases as $k$ decreases when $p = 0.05, \theta = 0.96$ (see Figure 11). In addition, the influence of different fractional order values on the stability of system (4.1) is discussed as in Figure 12 when fixed $k = -0.5, p = 0.05$. When $\theta = 0.9, 0.96, 0.98$, its corresponding stability switching curves are shown in Figure 12, respectively. As can be seen from the partial enlargement of Figure 12, the region of the locally asymptotically stability of the equilibrium $O$ decreases as $\theta$ increases when $k = -0.5, p = 0.05$ (see Figure 13). The results show that fractional order also can affect the stability of system.

5.   Conclusions

In this paper, Hopf bifurcation control for a fractional-order two delays neural network with ring-hub structure has been studied. By selecting time delay as bifurcation parameters, the conditions for the local asymptotic stability of the equilibrium and the existence of Hopf bifurcation for uncontrolled system and controlled system are obtained respectively. In particular, the stability of system is discussed by the method of stability switching curves, and the region of the locally asymptotically stability of the equilibrium in $(\tau_{1}, \tau_{2})$ plane is obtained. The research results show that the introduction of parameters delay feedback controller can effectively control Hopf bifurcation of system. In addition, the influence of the change of control parameters on the stability of system are discussed. It is found that the region of the locally asymptotically stability at the equilibrium increases with the decrease of feedback control parameter $k$. When the other parameters remain unchanged, the region of the locally asymptotically stability at the equilibrium decreases as the increase of the order $\theta$. Therefore, the stability of the equilibrium can be effectively improved by adjusting the feedback control parameters $k$ and fractional-order $\theta$. Our work has important significance for regulating the stability of network system performance. Especially in fields such as artificial intelligence, mechanical control, and image processing, our research can be applied in controlling Hopf bifurcation caused by time delays.

The characteristic equation of the following form is studied by using the method of stability switching curves in this paper:

For fractional-order models with two delays and delay dependent parameters, the method of stability switching curves can also be applied to the following more general the characteristic equation

However, since we need to consider a quadrilateral constructed by the four terms on the left hand side of Eq (5.1), the analysis of Eq (5.1) is more complicated and difficult.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No.11761040). We greatly appreciate the editors' and the anonymous referees' careful reading and helpful suggestions which have made the manuscript a real significant improvement.

Conflict of interest

All authors declare no conflicts of interest in this paper.