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

Authors: Bingnan Tang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

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 delayϑ1=ϑ2=ϑ. Secondly, the sufficient criterion to guarantee the stability and the emergence of Hopf bifurcation for given neural networks with the delay ϑ2=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 ϑ1=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.


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## Body

## 1. 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 [1–6]. 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 reaction-diffusion 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 [12–23].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 [24–27]. The 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 [24–27]. 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 fractional-order neural networks are available (see [28–31]).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 fractional-order 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 fractional-order PD control of Hopf bifurcations for delayed fractional-order small-world networks. For more detailed publications, one can see [35–38]. Our concern arises: what influence do different delays have on the stability peculiarity and bifurcation phenomenon of the involved neural networks? The 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:(1)du1tdt=−α1u1tβ1u1t−γ1h2u2t−ϑ2,du2tdt=−α2u2tβ2u2t−γ2h1u1t−ϑ1,where ujt stands for the state of the jth neuron; αj⋅ stands for the amplification function; βj⋅ denotes the behaved function; γj denotes the connection weight; hj⋅ represents the signal function of the jth neuron at time t; and ϑj 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):(2)dψu1tdtψ=−α1u1tβ1u1t−γ1h2u2t−ϑ2,dψu2tdtψ=−α2u2tβ2u2t−γ2h1u1t−ϑ1,where ujt stands for the state of the jth neuron; αj⋅ stands for the amplification function; βj⋅ denotes the behaved function; γj denotes the connection weight; hj⋅ represents the signal function of the jth neuron at time t; ϑj 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:(i)
Q1βj0=0,hj0=0,j=1,2.(ii)
Q2∃α¯jandα¯jsuch that0<α¯j≤αj≤α¯j,j=1,2.This paper is structured as follows. Section2 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.


## 2. Basic Theory

In this part, some necessary knowledge on fractional calculus is prepared.Definition 1.
(see [40]). The Caputo fractional-order derivative can be defined in the following:(3)Dψhϱ=1Γn−ψ∫ϱ0ϱhlνϱ−νψ−l+1dν,where hϱ∈ϱ0,∞,R,Γν=∫0∞ϱν−1e−ϱdϱ, ϱ≥ϱ0, and l∈Z+,l−1≤ψ<l.
The Laplace transform ofDψ is given by(4)LDψht;s=sψℋs−∑k=0i−1sψ−k−1wk0,i−1≤ψ<i∈Z+,where ℋs=Lht. If hk0=0,k=1,2,…,i, then LDψht;s=sψℋs.Definition 2.
(see [41]). We say that u10,u20 is an equilibrium point of system (2) provided that(5)−α1u10β1u10−γ1h2u20=0,−α2u20β2u20−γ2h1u10=0.Lemma 1.
(see [42, 43]). Let u∗ be the equilibrium point of the following system:(6)dψutdtψ=ht,ut,u0=u0,where ψ∈0,1 and ht,ut:R+×Rm⟶Rm. u∗ is locally asymptotically stable provided that total eigenvalues ϑ of ∂ht,u/∂u|u=u∗ satisfy argϑ>ψπ/2.Lemma 2.
(see [44]). Consider the system(7)dψ1W1tdtψ1=k11W1t−ϑ11+k12W2t−ϑ12+⋯+k1nWmt−ϑ1n,dψ2W2tdtψ2=k21W1t−ϑ21+k22W2t−ϑ22+⋯+k2nWmt−ϑ2n,⋮dψnWntdtψn=kn1W1t−ϑn1+kn2W2t−ϑn2+⋯+knnWnt−ϑnn,where 0<ψj<1j=1,2,…,n, the initial conditions Wjt=ϕjt∈C−maxj,hϑjh,0, and t∈−maxj,hϑjh,0, j,h=1,2,…,n. Set(8)Δσ=σψ1−k11e−σϑ11−k12e−σϑ12⋯−k1ne−σϑ1n−k21e−σϑ12σψ2−k22e−σϑ22⋯−k2ne−σϑ2n⋮⋮⋱⋮−kn1e−σϑn1−kn2e−σϑn2⋯σψn−knne−σϑnn.The zero solution of system (8) is Lyapunov asymptotically stable provided that total roots of detΔσ=0 have negative real parts.


## 3. Bifurcation for System (2) with ϑ1=ϑ2=ϑ

In this part, we will investigate the stability and the appearance of Hopf bifurcation for system (2) with ϑ1=ϑ2=ϑ. In terms of Q1 and Q2, it is easy to see that system (2) has a unique equilibrium point ℰ0,0. Under the conditions Q1 and Q2, system (2) becomes(9)dψu1tdtψ=−α1u1tβ1u1t−γ1h2u2t−ϑ,dψu2tdtψ=−α2u2tβ2u2t−γ2h1u1t−ϑ.The linear system of model (9) around ℰ0,0 takes the following form: (10)dψu1tdtψ=−α10β1′0u1t+α10γ1h2′0u2t−ϑ,dψu2tdtψ=−α20β2′0u2t+α20γ2h1′0u1t−ϑ.Let(11)a1=−α10β1′0,a2=α10γ1h2′0,b1=−α20β2′0,b2=α20γ2h1′0.Then, system (10) becomes(12)dψu1tdtψ=a1u1t+a2u2t−ϑ,dψu2tdtψ=b1u2t+b2u1t−ϑ.Thus, we obtain the following associated characteristic equation for (12):(13)detsψ−a1−a2e−sϑ−b2e−sϑsψ−b1=0.According to (13), we get(14)s2ψ−a1+b1sψ+a1b1−a2b2e−2sϑ=0.Assume thatQ3a1+b1<0,a1b1>a2b2.Lemma 3.
Ifϑ=0 and Q3 is fulfilled, then model (9) is locally asymptotically stable.Proof.
Whenϑ=0, then (14) becomes(15)λ2−a1+b1λ+a1b1−a2b2=0.
ByQ3, one knows that all roots λl of (15) satisfy argλl>ψπ/2l=1,2. So, Lemma 3 holds. The proof ends.Suppose thats=iξ=ξcosπ/2+isinπ/2 is a root of (14). Then, one has(16)a2b2cos2ξϑ=ξ2ψcosψπ−a1+b1ξψcosψπ2+a1b1,a2b2sin2ξϑ=ξ2ψsinψπ−a1+b1ξψsinψπ2.By (16), one has(17)a22b22=ξ2ψcosψπ−a1+b1ξψcosψπ2+a1b12+ξ2ψsinψπ−a1+b1ξψsinψπ22,which can be written in the following form:(18)ξ4ψ+c3ξ3ψ+c2ξ2ψ+c1ξψ+c0=0,where(19)c0=a12b12−a22b22,c1=−2a1b1a1+b1cosψπ2,c2=a1+b12,c3=−2a1+b12cosψπcosψπ2−sinψπsinψπ2.Denote(20)ℳ1ξ=ξ4ψ+c3ξ3ψ+c2ξ2ψ+c1ξψ+c0,ℳ2η=η4+c3η3+c2η2+c1η+c0.In view of Lemma 3.7 of [45], we have the following conclusion.Lemma 4.
(i) Ifci>0i=0,1,2,3 and Q3 is true, then equation (14) has no root with zero real parts. (ii) If c0<0, then equation (14) has at least a pair of purely imaginary roots. (iii) If c0>0 and ∃v0>0 which satisfies ℳ1v0<0, then equation (14) possesses at least two pairs of purely imaginary roots.For the proof of this lemma, one can see [45]. Here we omit it.Assume that equation (14) has four positive real roots ξhh=1,2,3,4. By (16), one has(21)ϑfl=12ξfarcsinξf2ψsinψπ−a1+b1ξfψsinψπ/2a2b2+2lπ,where l=0,1,2,…,f=1,2,3,4. Set(22)ϑ0=minf=1,2,3,4ϑf0,ξ0=ξ|ϑ=ϑ0.Next, the following assumption is prepared.Q4A11A21+A12A22>0, where(23)A11=a1+b1ψξ0ψ−1cosψ−1π2−2ψξ02ψ−1cos2ψ−1π2,A12=a1+b1ψξ0ψ−1sinψ−1π2−2ψξ02ψ−1sin2ψ−1π2,A21=2a2b2ξ0sin2ξ0ϑ0,A22=2a2b2ξ0cos2ξ0ϑ0.Lemma 5.
Ifsϑ=χϑ+iξϑ is the root of equation (14) around ϑ=ϑ0 such that χϑ0=0,ξϑ0=ψ0, then we get Reds/dϑ|ϑ=ϑ0,ξ=ξ0>0.Proof.
In terms of equation (14), one has(24)2ψs2ψ−1dsdϑ−a1+b1ψsψ−1dsdϑ+2a2b2e−2sϑdsdϑϑ+s=0.
Then,(25)dsdϑ−1=a1+b1ψsψ−1−2ψs2ψ−12a2b2se−2sϑ−ϑs.
Thus,(26)Redsdϑϑ=ϑ0,ξ=ξ0−1=Rea1+b1ψsψ−1−2ψs2ψ−12a2b2se−2sϑϑ=ϑ0,ξ=ξ0=A11A21+A12A22A212+A222.
According toQ4, one gets(27)Redsdϑϑ=ϑ0,ξ=ξ0−1>0.
The proof ends.
On the basis of above analysis, it is easy to establish the following result.Theorem 1.
IfQ1−Q4 are true, then the equilibrium point ℰ0,0 of system (2) is locally asymptotically stable when ϑ∈0,ϑ0 and a Hopf bifurcation will happen around the equilibrium point ℰ0,0 if ϑ=ϑ0.


## 4. Bifurcation for System (2) with ϑ2=0

In this part, we will investigate the stability and the appearance of Hopf bifurcation for system (2) with ϑ2=0. In terms of Q1 and Q2, it is easy to see that system (2) has a unique equilibrium point ℰ0,0. Under the conditions Q1 and Q2, system (2) becomes(28)dψu1tdtψ=−α1u1tβ1u1t−γ1h2u2t,dψu2tdtψ=−α2u2tβ2u2t−γ2h1u1t−ϑ1.The linear system of model (28) around ℰ0,0 takes the form: (29)dψu1tdtψ=−α10β1′0u1t+α10γ1h2′0u2t,dψu2tdtψ=−α20β2′0u2t+α20γ2h1′0u1t−ϑ1.That is,(30)dψu1tdtψ=a1u1t+a2u2t,dψu2tdtψ=b1u2t+b2u1t−ϑ1,where a1,a2,b1,b1 are defined by (11). Thus, we obtain the following associated characteristic equation for (30):(31)detsψ−a1−a2−b2e−sϑ1sψ−b1=0.According to (31), we get(32)s2ψ−a1+b1sψ+a1b1−a2b2e−sϑ1=0.Lemma 6.
Ifϑ1=0 and Q3 is fulfilled, then model (28) is locally asymptotically stable.Proof.
Whenϑ1=0, then (32) becomes(33)λ2−a1+b1λ+a1b1−a2b2=0.
ByQ3, one knows that all roots λl of (33) satisfy argλl>ψπ/2l=1,2. So, Lemma 6 holds. The proof ends.
Suppose thats=iϱ=ϱcosπ/2+isinπ/2 is a root of (32). Then, one has(34)a2b2cosϱϑ1=ϱ2ψcosψπ−a1+b1ϱψcosψπ2+a1b1,a2b2sinϱϑ1=ϱ2ψsinψπ−a1+b1ϱψsinψπ2.
By (34), one has (35)a22b22=ϱ2ψcosψπ−a1+b1ϱψcosψπ2+a1b12+ϱ2ψsinψπ−a1+b1ϱψsinψπ22,which can be written in the following form:(36)ϱ4ψ+c3ϱ3ψ+c2ϱ2ψ+c1ϱψ+c0=0,where cii=0,1,2,3 is defined by (19). Denote(37)N1ϱ=ϱ4ψ+c3ϱ3ψ+c2ϱ2ψ+c1ϱψ+c0,N2ζ=ζ4+c3ζ3+c2ζ2+c1ζ+c0.
In view of Lemma 3.7 of [45], we have the following conclusions.Lemma 7.
(i) Ifci>0i=0,1,2,3 and Q3 is true, then equation (32) has no root with zero real parts. (ii) If c0<0, then equation (32) has at least a pair of purely imaginary roots. (iii) If c0>0 and ∃u0>0 which satisfies N1u0<0, then equation (32) possesses at least two pairs of purely imaginary roots.For the proof of this lemma, one can see [45]. Here we omit it.Assume that equation (32) has four positive real roots ϱ1hh=1,2,3,4. By (34), one has(38)ϑ1fl=1ϱfarcsinϱf2ψsinψπ−a1+b1ϱfψsinψπ/2a2b2+2lπ,where l=0,1,2,…,f=1,2,3,4. Set(39)ϑ10=minf=1,2,3,4ϑ1f0,ϱ0=ϱ|ϑ1=ϑ10.Next, the following assumption is prepared.Q5ℬ11ℬ21+ℬ12ℬ22>0, where(40)ℬ11=a1+b1ψϱ0ψ−1cosψ−1π2−2ψϱ02ψ−1cos2ψ−1π2,ℬ12=a1+b1ψϱ0ψ−1sinψ−1π2−2ψϱ02ψ−1sin2ψ−1π2,ℬ21=a2b2ϱ0sinϱ0ϑ10,ℬ22=a2b2ϱ0cosϱ0ϑ10.Lemma 8.
Ifsϑ1=αϑ1+iϱϑ1 is the root of equation (32) around ϑ1=ϑ10 such that αϑ10=0,ϱϑ10=ϱ0, then we get Reds/dϑ|ϑ1=ϑ10,ϱ=ϱ0>0.Proof.
In terms of equation (32), one has(41)2ψs2ψ−1dsdϑ1−a1+b1ψsψ−1dsdϑ1+a2b2e−sϑ1dsdϑ1ϑ1+s=0.
Then,(42)dsdϑ1−1=a1+b1ψsψ−1−2ψs2ψ−1a2b2se−sϑ1−ϑ1s.
Thus,(43)Redsdϑ1ϑ1=ϑ10,ϱ=ϱ0−1=Rea1+b1ψsψ−1−2ψs2ψ−1a2b2se−sϑ1ϑ1=ϑ10,ϱ=ϱ0=ℬ11ℬ21+ℬ12ℬ22ℬ212+ℬ222.
According toQ5, one gets(44)Redsdϑ1ϑ1=ϑ10,ϱ=ϱ0−1>0.
The proof ends.
On the basis of above analysis, it is easy to establish the following result.Theorem 2.
IfQ1,Q2,Q3, and Q5 are true, then the equilibrium point ℰ0,0 of system (28) is locally asymptotically stable when ϑ1∈0,ϑ10 and a Hopf bifurcation will happen around the equilibrium point ℰ0,0 if ϑ1=ϑ10.


## 5. Bifurcation for System (2) with ϑ1=0

In this part, we will investigate the stability and the appearance of Hopf bifurcation for system (2) with ϑ1=0. In terms of Q1 and Q2, it is easy to see that system (2) has a unique equilibrium point ℰ0,0. Under the conditions Q1 and Q2, system (2) becomes(45)dψu1tdtψ=−α1u1tβ1u1t−γ1h2u2t−ϑ2,dψu2tdtψ=−α2u2tβ2u2t−γ2h1u1t.The linear system of model (45) around ℰ0,0 takes the form (46)dψu1tdtψ=−α10β1′0u1t+α10γ1h2′0u2t−ϑ2,dψu2tdtψ=−α20β2′0u2t+α20γ2h1′0u1t.That is,(47)dψu1tdtψ=a1u1t+a2u2t−ϑ2,dψu2tdtψ=b1u2t+b2u1t,where a1,a2,b1,b1 are defined by (11). Thus, we obtain the following associated characteristic equation for (47):(48)detsψ−a1−a2e−sϑ2−b2sψ−b1=0.According to (48), we get(49)s2ψ−a1+b1sψ+a1b1−a2b2e−sϑ2=0.Lemma 9.
Ifϑ2=0 and Q3 is fulfilled, then model (45) is locally asymptotically stable.Proof.
Whenϑ2=0, then (49) becomes(50)λ2−a1+b1λ+a1b1−a2b2=0.
ByQ3, one knows that all roots λj of (50) satisfy argλj>ψπ/2j=1,2. So, Lemma 9 holds. The proof ends.
Suppose thats=iσ=σcosπ/2+isinπ/2 is a root of (49). Then, one has(51)a2b2cosσϑ2=σ2ψcosψπ−a1+b1σψcosψπ/2+a1b1,a2b2sinσϑ2=σ2ψsinψπ−a1+b1σψsinψπ/2.
By (51), one has(52)a22b22=σ2ψcosψπ−a1+b1σψcosψπ/2+a1b12+σ2ψsinψπ−a1+b1σψsinψπ/22,which can be written in the following form:(53)σ4ψ+c3σ3ψ+c2σ2ψ+c1σψ+c0=0,where cii=0,1,2,3 is defined by (19). Denote(54)K1σ=σ4ψ+c3σ3ψ+c2σ2ψ+c1σψ+c0,K2ς=ς4+c3ς3+c2ς2+c1ς+c0.
In view of Lemma 3.7 of [45], we have the following conclusions.Lemma 10.
(i) Ifci>0i=0,1,2,3 and Q3 is true, then equation (49) has no root with zero real parts. (ii) If c0<0, then equation (49) has at least a pair of purely imaginary roots. (iii) If c0>0 and ∃w0>0 which satisfies K1w0<0, then equation (49) possesses at least two pairs of purely imaginary roots.For the proof of this lemma, one can see [45]. Here we omit it.Assume that equation (49) has four positive real roots ϱ2hh=1,2,3,4. By (51), one has(55)ϑ2fl=1σfarcsinσf2ψsinψπ−a1+b1σfψsinψπ/2a2b2+2lπ,where l=0,1,2,…,f=1,2,3,4. Set(56)ϑ20=minf=1,2,3,4ϑ2f0,σ0=σ|ϑ2=ϑ20.Next, the following assumption is prepared.Q6C11C21+C12C22>0, where(57)C11=a1+b1ψσ0ψ−1cosψ−1π2−2ψσ02ψ−1cos2ψ−1π2,C12=a1+b1ψσ0ψ−1sinψ−1π2−2ψσ02ψ−1sin2ψ−1π2,C21=a2b2σ0sinσ0ϑ10,C22=a2b2σ0cosσ0ϑ10.Lemma 11.
Ifsϑ2=βϑ2+iσϑ2 is the root of equation (32) around ϑ2=ϑ20 such that βϑ20=0,σϑ20=ϱ0, then we get Reds/dϑ2|ϑ2=ϑ20,σ=σ0>0.Proof.
In terms of equation (49), one has(58)2ψs2ψ−1dsdϑ2−a1+b1ψsψ−1dsdϑ2+a2b2e−sϑ2dsdϑ2ϑ2+s=0.
Then,(59)dsdϑ2−1=a1+b1ψsψ−1−2ψs2ψ−1a2b2se−sϑ2−ϑ2s.
Thus,(60)Redsdϑ2ϑ2=ϑ20,σ=σ0−1=Rea1+b1ψsψ−1−2ψs2ψ−1a2b2se−sϑ2ϑ2=ϑ20,σ=σ0=C11C21+C12C22C212+C222.
According toQ6, one gets(61)Redsdϑ2ϑ2=ϑ20,σ=σ0−1>0.
The proof ends.
On the basis of above analysis, it is easy to establish the following result.Theorem 3.
IfQ1,Q2,Q3, and Q6 are true, then the equilibrium point ℰ0,0 of system (45) is locally asymptotically stable when ϑ2∈0,ϑ20 and a Hopf bifurcation will happen around the equilibrium point ℰ0,0 if ϑ1=ϑ20.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. They 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 fractional-order 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) ϑ1=ϑ2=ϑ, (2) ϑ1, and (3) ϑ2) 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.


## 6. Three Examples

Example 1.
Given the following system:(62)dψu1tdtψ=−4.25+sinu1t1.22u1t−1.23tanhu2t−ϑ,dψu2tdtψ=−6.13+sinu2t2.13u2t−2.05tanhu1t−ϑ.
Clearly, system (62) possesses the equilibrium point ℰ0,0. Using the computer software, one derives φ0=0.0499 and ϑ0=0.142 and the hypotheses Q1−Q4 in Theorem 1 are fulfilled. So, one knows that ℰ0,0 of model (62) maintains locally asymptotically stable state provided that 0≤ϑ<0.142. Choosing ϑ=0.12<ϑ0=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>ϑ0=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 ℰ0,0. We give bifurcation plots (see Figures 3 and 4) to display the bifurcation point. Besides, the quantity change for ξ0 and ϑ0 is shown in Table 1.Figure 1
Simulation plots of model (62) with ϑ=0.12<ϑ0=0.1420.
(a)(b)(c)(d)Figure 2
Simulation plots of model (62) with ϑ=0.23>ϑ0=0.1420.
(a)(b)(c)(d)Figure 3
Bifurcation plot of model (62): ϑ−u1.Figure 4
Bifurcation plot of model (62): ϑ−u2.Table 1
The change plot ofξ0 and ϑ0 for model (62).
ξ0ϑ00.00880.02510.03950.11240.04990.14200.12140.34550.14710.41880.17640.50220.25060.7135Example 2.
Given the following system:(63)dψu1tdtψ=−4.25+sinu1t1.22u1t−1.23tanhu2t,dψu2tdtψ=−6.13+sinu2t2.13u2t−2.05tanhu1t−ϑ1.
Clearly, system (63) possesses the equilibrium point ℰ0,0. Using the computer software, one derives ϱ0=0.0955 and ϑ1=0.2380 and the hypotheses Q1−Q3 and Q5 in Theorem 2 are fulfilled. So, one knows that ℰ0,0 of model (63) maintains locally asymptotically stable state provided that 0≤ϑ1<0.2380. Choosing ϑ1=0.16<ϑ10=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 ϑ1≥0.142. Selecting ϑ1=0.28>ϑ10=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 ℰ0,0. We give bifurcation plots (see Figures 7 and 8) to display the bifurcation point. Besides, the quantity change for ϱ0 and ϑ0 is shown in Table 2.Figure 5
Simulation plots of model (63) with ϑ1=0.16<ϑ10=0.2380.
(a)(b)(c)(d)Figure 6
Simulation plots of model (63) with ϑ1=0.28>ϑ10=0.2380.
(a)(b)(c)(d)Figure 7
Bifurcation plot of model (63): ϑ1−u1.Figure 8
Bifurcation plot of model (63): ϑ1−u2.Table 2
The change plot ofϱ0 and ϑ10 of model (63).
ϱ0ϑ100.03620.09020.05260.13110.09550.23800.12430.30980.22180.55310.24560.61230.28470.7099Example 3.
Given the following system:(64)dψu1tdtψ=−4.25+sinu1t1.22u1t−1.23tanhu2t−ϑ2,dψu2tdtψ=−6.13+sinu2t2.13u2t−2.05tanhu1t.
Clearly, system (64) possesses the equilibrium point ℰ0,0. Using the computer software, one derives σ0=0.0973 and ϑ20=0.2380 and the hypotheses Q1−Q3 and Q6 in Theorem 3 are fulfilled. So, one knows that ℰ0,0 of model (64) maintains locally asymptotically stable state provided that 0≤ϑ2<0.2380. Choosing ϑ2=0.18<ϑ20=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 ϑ2≥0.2380. Selecting ϑ2=0.31>ϑ20=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 ℰ0,0. We give bifurcation plots (see Figures 11 and 12) to display the bifurcation point. Besides, the quantity change for σ0 and ϑ20 is shown in Table 3.Figure 9
Simulation plots of model (64) with ϑ2=0.18<ϑ20=0.2380.
(a)(b)(c)(d)Figure 10
Simulation plots of model (64) with ϑ2=0.31>ϑ20=0.2380.
(a)(b)(c)(d)Figure 11
Bifurcation plot of model (64): ϑ2versusu1.Figure 12
Bifurcation plot of model (64): ϑ2versusu2.Table 3
The change plot ofσ0 and ϑ20 of model (64).
σ0ϑ200.03220.07880.04230.10340.09730.23800.12530.30650.25850.63220.28840.70550.33580.8213


## 7. 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)ϑ1=ϑ2=ϑ; (2) ϑ2=0; and (3) ϑ1=0. The investigation shows that the delay plays a crucial role in stabilizing and generating Hopf bifurcation for the addressed network model. Three differential sufficient criteria are set up to guarantee the stability and the emergence of Hopf bifurcation of the considered neural networks under different cases. The software simulations verify the established theoretical findings. The derived conclusions replenish and consummate the earlier publications. The 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.



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