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

: In this paper, a fractional-order two delays neural network with ring-hub structure is investigated. Firstly, the stability and the existence of Hopf bifurcation of proposed system are obtained by taking the sum of two delays as the bifurcation parameter. Furthermore, a parameters delay feedback controller is introduced to control successfully Hopf bifurcation. The novelty of this paper is that the characteristic equation corresponding to system has two time delays and the parameters depend on one of them. Selecting two time delays as the bifurcation parameters simultaneously, stability switching curves in ( τ 1 , τ 2 ) plane and crossing direction are obtained. Su ffi cient criteria for the stability and the existence of Hopf bifurcation of controlled system are given. Ultimately, numerical simulation shows that parameters delay feedback controller can e ff ectively control Hopf bifurcation of system.


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 θ ∈ (0, 1] is the fractional order, D θ denotes the Caputo fractional derivative; x i (t)(i = 1, 2, 3, 4) denotes the state variable of the ith neuron at time t; σ > 0 describes the internal decay rate of neurons; c i j (i, j = 1, 2, 3, 4) stands for the connection weight from the ith neuron to the jth neuron; f represents the activation function; τ 2 ≥ 0 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 τ 1 ≥ 0. The initial conditions of system (1.1) Throughout this paper, we make the following assumption: 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(τ 1 ) is parameters delay feedback controller.Let P(τ 1 ) = pτ 1 , p is called the decay rate.Note that the feedback controller ke −pτ 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 P 0 (λ, τ) + P 1 (λ, τ)e −λτ + P 2 (λ, τ)e −λτ 1 = 0, ( where P i (λ, τ), i = 0, 1, 2 are polynomials in λ and P i only depend on τ.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 P 0 (λ, τ) + P 1 (λ, τ)e −λτ + P 2 (λ, τ)e −λ(τ+τ 1 ) = 0. (1.4) 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: (i) Considering the complex topology of neural networks, a fractional-order two time delays neural network with a composite ring-hub structure is considered.
(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.
(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.
(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 fractionalorder 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.

Stability and Hopf bifurcation analysis of system (1.1)
In this section, τ = τ 1 + τ 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 φ i j = c i j f ′ (0)(i, j = 1, 2, 3, 4).
By applying Laplace transformation, the characteristic equation of system (2.1) is that is s 4θ + q 1 s 3θ + q 2 s 2θ + q 3 s θ + q 4 + (q 5 s 2θ + q 6 s θ + q 7 )e −s(τ where The stability of the equilibrium O is discussed in two scenarios.
Case I: where Lemma 2.1.[31] For the following fractional-order system: D θ x(t) = Ax(t), A ∈ R n×n , the equilibrium of system is locally asymptotically stable if and only if all the eigenvalues s i According to Lemma 2.1 and Routh-Hurwitz criterion, we have the following theorem.

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.
Next, we verify that the above assumptions (i)-(vi) hold for Eq (3.2).

Crossing direction
In this subsection, we consider the direction of s(τ 1 , τ 2 ) crossing the imaginary axis when (τ 1 , τ 2 ) deviates from a crossing curve.Define the direction in which ω 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).
Based on the above discussion, the following theorem is given )) > 0(< 0), then a pair of pure imaginary roots of the characteristic equation D(s, τ 1 , τ 2 ) = 0 crosses the imaginary axis from left to right when (τ 1 , τ 2 ) passes through the crossing curve to the right (left) region, where where P * i = P i (iω * , τ * 1 ) and P By direct calculation there is which completes the proof.
2) When τ 1 > 0, τ 2 > 0 and τ 1 τ 2 , the curves C and the feasible region Ω are shown in Figure 5.The curves C on Ω form stability switching curves T on (τ 1 , τ 2 ) plane as in Figure 6(a).sign(δ(τ * 1 , τ * 2 )) = 1 > 0 can be calculated, then the crossing direction of stability changes from left to right according to Theorem 3.2 (see Figure 6(a)).A partial enlargement is shown by the green area of Figure 6(a) (see Figure 6(b)).Choosing (τ 1 , τ 2 ) = (0.6, 1.5), we find that O is locally asymptotically stable (see Figure 7).When (τ 1 , τ 2 ) passes through stability switching curves from left to right along the arrow direction, by choosing (τ 1 , τ 2 ) = (0.65, 1.8), system (4.1)exhibits Hopf bifurcation at O (see Figure 8).When (τ 1 , τ 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, θ = 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, θ = 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 θ = 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 θ increases when k = −0.5, p = 0.05 (see Figure 13).The results show that fractional order also can affect the stability of system.

Conclusions
In this paper, Hopf bifurcation control for a fractional-order two delays neural network with ringhub 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 (τ 1 , τ 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 θ.Therefore, the stability of the equilibrium can be effectively improved by adjusting the feedback control parameters k and fractional-order θ.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.
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 P 0 (s, τ 1 ) + P 1 (s, τ 1 )e −sτ 1 + P 2 (s, τ 1 )e −sτ 2 + P 3 (s, τ 1 )e −s(τ 1 +τ 2 ) = 0. (5.1) 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.