Bifurcation Study on Fractional-Order Cohen–Grossberg Neural Networks Involving Delays

This work is chiefly concerned with the stability behavior and the appearance of Hopf bifurcation of fractional-order delayed Cohen–Grossberg neural networks. Firstly, we study the stability and the appearance of Hopf bifurcation of the involved neural networks with identical delay 
 
 
 
 ϑ
 
 
 1
 
 
 =
 
 
 ϑ
 
 
 2
 
 
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 ϑ
 
 . Secondly, the sufficient criterion to guarantee the stability and the emergence of Hopf bifurcation for given neural networks with the delay 
 
 
 
 ϑ
 
 
 2
 
 
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 0
 
 is set up. Thirdly, we derive the sufficient condition ensuring the stability and the appearance of Hopf bifurcation for given neural networks with the delay 
 
 
 
 ϑ
 
 
 1
 
 
 =
 0
 
 . The investigation manifests that the delay plays a momentous role in stabilizing networks and controlling the Hopf bifurcation of the addressed fractional-order delayed neural networks. At last, software simulation results successfully verified the rationality of the analytical results. The theoretical findings of this work can be applied to design, control, and optimize neural networks.


Introduction
It is common knowledge that neural networks have broad application perspectives in a number of subjects such as data science, biology, artificial intelligence, control engineering, biomedical treatment, and so on [1][2][3][4][5][6]. During a long period, in order to further grasp the inherent laws and dynamic characteristics of neural networks, a great deal of scholars have made great efforts to study neural networks and fruitful results have constantly been emerging. For instance, Hajihosseini et al. [7] investigated the Hopf bifurcation for a class of delayed neural networks by applying the frequency domain approach. In [8], Lin et al. [8] dealt with Turing instability and Turing-Hopf bifurcation for a kind of reactiondiffusion neural networks with leakage term. In 2019, Aouiti et al. [9] established the sufficient condition to ensure the existence and exponential stability of piecewise pseudo almost periodic solution for delayed inertial neural networks with impulse. In 2020, Xu et al. [10] obtained new results on pseudo almost periodic solutions for quaternion-valued delayed fuzzy cellular neural networks. Kumar and Das [11] made a detailed discussion on exponential stability of delayed BAM neural network by applying the matrix measure method. For more related knowledge on these dynamics of neural networks, we refer the readers to [12][13][14][15][16][17][18][19][20][21][22][23].
It is worth pointing out that all the mentioned works are concerned with the integer-order neural networks. For a long time in the past, the development of fractional calculus has been kept in a very low state due to the lack of practical background and the imperfection of the theoretical system. In recent decades, fractional calculus has been found to have wide applications in numerous fields such as electromagnetic waves, biological engineering, control theory, network science, and so on [24][25][26][27].
e fractional calculus has greater advantages than integer-order ones since it can better portray memory nature and hereditary information for the dynamical systems in natural world [24][25][26][27]. Nowadays, many scholars pay much attention to the fractional-order dynamical models. Especially, the study on fractional-order neural networks has aroused the close attention of many scholars. Many outstanding achievements on fractionalorder neural networks are available (see [28][29][30][31]).
Hopf bifurcation, which acts as a vital dynamical phenomenon of neural networks, naturally arouses the extensive research interest from plenty of scholars. During the past several decades, lots of researchers only deal with the Hopf bifurcation for integer-order neural networks. However, the investigation on Hopf bifurcation for fractional-order ones is very scarce. At present, some works on Hopf bifurcation for fractional-order neural networks have been reported. For example, Eshaghi et al. [32] discussed the Hopf bifurcation, chaos control, and synchronization for a chaotic fractionalorder model. Yuan et al. [33] obtained the new results on bifurcation for fractional-order complex-valued delayed neural networks. Xiao et al. [34] dealt with the fractionalorder PD control of Hopf bifurcations for delayed fractionalorder small-world networks. For more detailed publications, one can see [35][36][37][38]. Our concern arises: what influence do different delays have on the stability peculiarity and bifurcation phenomenon of the involved neural networks? e answer will play a key role in controlling neural networks. Inspired by this viewpoint, in this present work, we shall deal with the Hopf bifurcation of delayed neural networks by selecting different delays as bifurcation parameters. Detailed works include the following points: (a) probe into the influence of delay on stability and Hopf bifurcation of fractional-order Cohen-Grossberg neural networks with delays and (b) establish some new sufficient conditions to ensure the stability and the appearance of Hopf bifurcation for fractional-order Cohen-Grossberg neural networks by regarding different delays as bifurcation parameters.
In [39], Liu and Xu studied the following integer-order delayed Cohen-Grossberg neural networks: where u j (t) stands for the state of the jth neuron; α j (·) stands for the amplification function; β j (·) denotes the behaved function; c j denotes the connection weight; h j (·) represents the signal function of the jth neuron at time t; and ϑ j is a delay that corresponds to the finite speed of the axonal signal transmission at time t, j � 1, 2 [39]. By selecting the sum of two delays as bifurcation parameter, the authors established the sufficient conditions to guarantee the stability and the emergence of Hopf bifurcation of model (1). In addition, with the aid of the center manifold theory and the normal form knowledge, the direction and stability of bifurcating periodic solutions are analyzed. Inspired by the analysis above, on the basis of model (1), we establish the following fractional-order version of model (1): where u j (t) stands for the state of the jth neuron; α j (·) stands for the amplification function; β j (·) denotes the behaved function; c j denotes the connection weight; h j (·) represents the signal function of the jth neuron at time t; ϑ j is a delay that corresponds to the finite speed of the axonal signal transmission at time t, j � 1, 2 [39]; and ψ ∈ (0, 1) is a constant. In order to find the key results of this work, we first give the assumptions as follows: is paper is structured as follows. Section 2 gives some related knowledge on fractional differential systems. In Sections 3-5, the stability behavior and the emergence of Hopf bifurcation for neural networks (2) with three different cases are investigated. In Section 6, three examples are prepared to indicate the rationality of the established key conclusions and bifurcation diagrams are clearly displayed. Section 7 concludes the paper.

Basic Theory
In this part, some necessary knowledge on fractional calculus is prepared.

□
Suppose that s � iξ � ξ(cos(π/2) + i sin(π/2)) is a root of (14). en, one has By (16), one has which can be written in the following form: where Mathematical Problems in Engineering Denote In view of Lemma 3.7 of [45], we have the following conclusion. (14) has no root with zero real parts. (ii) If c 0 < 0, then equation (14) has at least a pair of purely imaginary (14) possesses at least two pairs of purely imaginary roots.
For the proof of this lemma, one can see [45]. Here we omit it.
Suppose that s � iϱ � ϱ(cos(π/2) + i sin(π/2)) is a root of (32). en, one has By (34), one has which can be written in the following form: where c i (i � 0, 1, 2, 3) is defined by (19). Denote In view of Lemma 3.7 of [45], we have the following conclusions. For the proof of this lemma, one can see [45]. Here we omit it.
Proof. In terms of equation (32), one has en, e proof ends. On the basis of above analysis, it is easy to establish the following result.  (28) is locally asymptotically stable when ϑ 1 ∈ 0, ϑ 10 and a Hopf bifurcation will happen around the equilibrium point E(0, 0) if ϑ 1 � ϑ 10 .

Bifurcation for System (2) with ϑ 1 = � 0
In this part, we will investigate the stability and the appearance of Hopf bifurcation for system (2) with ϑ 1 � 0. In terms of (Q1) and (Q2), it is easy to see that system (2) has a unique equilibrium point E(0, 0). Under the conditions (Q1) and (Q2), system (2) becomes e linear system of model (45) around E(0, 0) takes the form at is, where a 1 , a 2 , b 1 , b 1 are defined by (11). us, we obtain the following associated characteristic equation for (47): According to (48), we get Lemma 9. If ϑ 2 � 0 and (Q3) is fulfilled, then model (45) is locally asymptotically stable.

⎧ ⎨ ⎩ (51)
By (51), one has (52) which can be written in the following form: where c i (i � 0, 1, 2, 3) is defined by (19). Denote In view of Lemma 3.7 of [45], we have the following conclusions. For the proof of this lemma, one can see [45]. Here we omit it.
Assume that equation (49) has four positive real roots ϱ 2h (h � 1, 2, 3, 4). By (51), one has where l � 0, 1, 2, . . . , f � 1, 2, 3, 4. Set Next, the following assumption is prepared.(Q6) Proof. In terms of equation (49), one has (58) en, us, According to (Q6), one gets e proof ends. On the basis of above analysis, it is easy to establish the following result.  (45) is locally asymptotically stable when ϑ 2 ∈ 0, ϑ 20 and a Hopf bifurcation will happen around the equilibrium point E(0, 0) if ϑ 1 � ϑ 20 . Remark 1. In [39], Liu and Xu have dealt with the bifurcation of integer-order delayed Cohen-Grossberg neural networks (1) by selecting the sum of two delays as bifurcation parameter. ey did not consider the impact of two different delays on the stability and Hopf bifurcation of the neural networks (1). In this paper, we set up new fractionalorder delayed Cohen-Grossberg neural networks (2). We have focused on the impact of two different delays on the stability and Hopf bifurcation of the established neural networks (2) by regarding the three different delays ((1) ϑ 1 � ϑ 2 � ϑ, (2) ϑ 1 , and (3) ϑ 2 ) as bifurcation parameters. Based on this viewpoint, we think that our work is a good complement and improvement of Liu and Xu [39].

Remark 2.
In [2], the authors studied the Hopf bifurcation of integer-order delayed neural networks by the frequency domain approach. Now we cannot investigate the Hopf bifurcation of fractional-order delayed Cohen-Grossberg neural networks by the frequency domain approach.
Remark 3. Now we cannot investigate the direction and stability of Hopf bifurcation of the involved neural networks due to the lack of theoretical tools.

Three Examples
Example 1. Given the following system: Clearly, system (62) possesses the equilibrium point E(0, 0). Using the computer software, one derives φ 0 � 0.0499 and ϑ 0 � 0.142 and the hypotheses (Q1) − (Q4) in eorem 1 are fulfilled. So, one knows that E(0, 0) of model (62) maintains locally asymptotically stable state provided that 0 ≤ ϑ < 0.142. Choosing ϑ � 0.12 < ϑ 0 � 0.142, we get the simulation plot ( Figure 1) which manifests that the states of two neurons will tend to be zero. Model (62) will be unstable, and the Hopf bifurcation phenomenon takes place provided that ϑ ≥ 0.142. Selecting ϑ � 0.23 > ϑ 0 � 0.142, one gets the simulation plot ( Figure 2) which shows that the states of two neurons will keep periodic vibration in the vicinity of E(0, 0). We give bifurcation plots (see Figures 3 and 4) to display the bifurcation point. Besides, the quantity change for ξ 0 and ϑ 0 is shown in Table 1.
Example 2. Given the following system:    asymptotically stable state provided that 0 ≤ ϑ 1 < 0.2380. Choosing ϑ 1 � 0.16 < ϑ 10 � 0.2380, we get the simulation plot ( Figure 5) which manifests that the states of two neurons will tend to be zero. Model (63) will be unstable, and the Hopf bifurcation phenomenon takes place provided that ϑ 1 ≥ 0.142. Selecting ϑ 1 � 0.28 > ϑ 10 � 0.2380, one gets the simulation plot ( Figure 6) which shows that the states of two neurons will keep periodic vibration in the vicinity of E(0, 0). We give bifurcation plots (see Figures 7 and 8) to display the bifurcation point. Besides, the quantity change for ϱ 0 and ϑ 0 is shown in Table 2.
Example 3. Given the following system:     provided that 0 ≤ ϑ 2 < 0.2380. Choosing ϑ 2 � 0.18 < ϑ 20 � 0.2380, we get the simulation plot (Figure 9) which manifests that the states of two neurons will tend to be zero. Model (63) will be unstable, and the Hopf bifurcation phenomenon takes place provided that ϑ 2 ≥ 0.2380. Selecting ϑ 2 � 0.31 > ϑ 20 � 0.2380, one gets the simulation plot ( Figure 10) which shows that the states of two neurons will keep periodic vibration in the vicinity of E(0, 0). We give bifurcation plots (see Figures 11 and 12) to display the bifurcation point. Besides, the quantity change for σ 0 and ϑ 20 is shown in Table 3.

Conclusions
As everyone knows, the stability nature and Hopf bifurcation of fractional-order models have become a focus problem in dynamical systems. In this study, we handle the stability and the emergence of Hopf bifurcation of a class of fractional-order delayed Cohen-Grossberg neural networks under three delay cases: (1) ϑ 1 � ϑ 2 � ϑ; (2) ϑ 2 � 0;  and (3) ϑ 1 � 0. e investigation shows that the delay plays a crucial role in stabilizing and generating Hopf bifurcation for the addressed network model. ree differential sufficient criteria are set up to guarantee the stability and the emergence of Hopf bifurcation of the considered neural networks under different cases. e software simulations verify the established theoretical findings. e derived conclusions replenish and consummate the earlier publications. e derived results can also help us manage the operation of the neural networks. In the near future, we will study the Hopf bifurcation with multiple delays and consider the Hopf bifurcation by choosing other parameters as bifurcation parameters. Also, we will deal with the Hopf bifurcation of fractional-order neural networks with different orders.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there are no conflicts of interest.