# Periodic Oscillatory Phenomenon in Fractional-Order Neural Networks Involving Different Types of Delays

Authors: Nengfa Wang; Changjin Xu; Zixin Liu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This research is chiefly concerned with the stability and Hopf bifurcation for newly established fractional-order neural networks involving different types of delays. By means of an appropriate variable substitution, equivalent fractional-order neural network systems involving one delay are built. By discussing the distribution of roots of the characteristic equation of the established fractional-order neural network systems and selecting the delay as bifurcation parameter, a novel delay-independent bifurcation condition is derived. The investigation verifies that the delay is a significant parameter which has an important influence on stability nature and Hopf bifurcation behavior of neural network systems. The computer simulation plots and bifurcation graphs effectively illustrate the reasonableness of the theoretical fruits.


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## Body

## 1. Introduction

It is widely known that neural network systems own great application value in plenty of areas such as disease treatment, pattern recognition, intelligent control, biology, information processing, control technique, and so on [1–6]. In real world, time delay usually occurs in lots of neural networks and biological systems since there is delay of information transmission of distinct neurons and the response of diverse biotic populations. Therefore, grasping the effect of delay on the dynamic law for many dynamical models is a key topic in contemporary science. Generally speaking, the delay frequently leads to the disappearance of stability and the appearance of bifurcation, chaotic behavior, and a lot of other dynamic characteristics. So, a number of researchers pay much attention to various neural networks, and lots of excellent results have been reported. For instance, Lin and Zhang [7] considered the global asymptotic synchronization of delayed BAM neural networks by means of the LMI method and matrix measure method. Ding et al. [8] established a sufficient criterion guaranteeing the global robust exponential stability of BAM neural networks involving discrete delay and the distributed delay under uncertainty by applying an appropriate Lyapunov–Krasovskii function. In the work of Hou et al. [9], the authors dealt with the stability issue of discrete-time uncertain models concerning Markov jump and delay. In 2021, Zhou [10] investigated the global exponential dissipativity for recurrent neural networks involving proportional delays. For more works on this subject, one can see [11–13].It should be noted that all the involved publications mentioned above only focus on integer-order neural networks concerning delays and are not concerned with the fractional-order form. The research on fractional-order differential equation has remained a relatively slow development situation because of the lack of theoretical body of knowledge and realistic background. Until recently, fractional-order differential dynamical system has aroused widespread interest from many scholars since it owns great application prospect in a lot of fields such as control technique, various physical waves, neural network models, biological systems, and so on [14, 15]. Many scholars argue that that fractional-order differential model can more efficaciously characterize the actual situation in object world than the traditional integer-order differential model since it has preferable memory trait and hereditary property [16]. Recently, a lot of scholars have dedicated themselves to the study on dynamical behavior for fractional-order differential models. In particular, the research on fractional-order neural networks concerning delays has aroused widespread interest of many researchers and a lot of results on fractional-order neural networks concerning delays have been published (see [17–19]).Delay-induced bifurcation is an important topic in neural network area. Over the past few decades, a number of works on delay-induced bifurcation of integer-order neural networks have been reported and the basic bifurcation theory is sound. It is a pity that the study on Hopf bifurcation for fractional-order neural networks is relatively rare. In recent years, some researchers pay much attention to Hopf bifurcation of fractional-order delayed neural networks and some good results have been achieved. For instance, Xu et al. [20] made a detailed analysis on the stability and the onset of Hopf bifurcation for fractional-order BAM neural networks involving time delays. Djilali et al. [21] discussed the Turing–Hopf bifurcation for a fractional-order diffusive mussel-algae model. Hu et al. [22] investigated the Hopf bifurcation and chaos for a fractional-order memristor-based chaotic circuit system involving delays. Xu et al. [23] dealt with Hopf bifurcation of fractional-order BAM neural network models involving multiple delays. For more studies on this aspect, one can see [24, 25].It is worth noticing that in many cases, the study on bifurcation of fractional-order delayed neural networks is only concerned with discrete delay. Considering that in neural network systems, there are numerous parallel paths involving many diverse axon sizes and lengths [26], it is very important for us to introduce continuous distributed delays into neural networks to depict the signal transmission of neurons. In order to grasp the impact of the distributed delay on Hopf bifurcation for neural networks concerning time delay, plenty of authors have dedicated themselves to all kinds of neural networks involving distributed delays (see [27, 28]). In 2016, Karaoğlu et al. [29] analyzed the following neural networks concerning mixed delays:(1)dv1tdt=−v1t+γ11h11∫−∞tHt−sv1sds+γ12h12v2t−η2,dv2tdt=−v2t+γ21h21v1t−η1+γ22h22∫−∞tHt−sv2sds,where vlt,l=1,2, stands for the state of the lth neuron at time t, γij,i,j=1,2, denotes the connection weight and is real constant, ηl>0,l=1,2, denotes the delay, and H. denotes nonnegative bounded delay kernel defined on 0,∞, which reflects the impact of the past states on the present dynamics [29]. Usually, the kernel function takes the following two forms:(1)
∫−∞tHμdμ=1,Hμ=βe−βt−μ,β>0.(2)
∫−∞tHt−μdμ=1,Hμ=βe−βμ,β>0.In [29], Karaoğlu et al. took kernel function as case (2). By means of stability criterion and Hopf bifurcation theory of delayed differential equation, Karaoğlu et al. obtained a sufficient condition to guarantee the stability and the onset of Hopf bifurcation of system (1). In addition, with the aid of center manifold theorem and normal form theory, the concrete formula to determine bifurcation nature is given.Motivated by the discussion above, our concern is the stability and Hopf bifurcation for fractional-order neural networks involving discrete delays and distributed delays. On the basis of the work of Karaoğlu et al. [29], in this work, we modify system (1) as the following fractional-order version:(2)dv1ρtdtρ=−v1t+γ11h11∫−∞tHt−sv1sds+γ12h12v2t−η2,dv2ρtdtρ=−v2t+γ21h21v1t−η1+γ22h22∫−∞tHt−sv2sds,where 0<ρ<1 is a constant, vlt,l=1,2, stands for the state of the lth neuron at time t, γij,i,j=1,2, denotes the connection weight and is real constant, ηl>0,l=1,2, denotes the delay, and H. denotes nonnegative bounded delay kernel defined on 0,∞, which reflects the impact of the past states on the present dynamics [29], and the kernel function H. takes form (2).The novelty of this paper lies in constructing novel fractional-order neural networks including fractional-order and integer-order equations by applying an appropriate variable substitution. Up to now, there are very few publications on this topic.In order to establish the prime results of this work, the following assumptions are needed:A1hkl∈C,hkl0=0,k,l=1,2.A2η1+η2=η.This paper is organized as follows. Section2 gives some essential theories on fractional-order differential equation. Section 3 gives the bifurcation results for system (2) with kernel function (2). Section 4 carries out computer simulation to sustain the reasonableness of the derived main results. This research ends in Section 5.


## 2. Basic Theory on Fractional-Order Differential Equation

In this segment, we will make some preparations on some necessary theories on fractional-order differential equation.Definition 1 (see [30]).
Define Caputo-type fractional-order derivative as follows:(3)Dρwζ=1Γκ−ρ∫ζ0ζwκvζ−vρ−κ+1dv,where wζ∈ζ0,∞,R,Γv=∫0∞ζv−1e−ζdζ, ζ≥ζ0,κ∈Z+,κ−1≤ρ<κ.Lemma 1 (see [31, 32]).
Consider the following system:(4)dρutdtρ=gt,ut,u0=u0,where ρ∈0,1 and gt,ut:R+×Rn⟶Rn. Denote u0 as the equilibrium point of (4). Then, u0 is locally asymptotically stable if every eigenvalue μ of ∂gt,u/∂u|u=u0 satisfies argμ>ρπ/2.Lemma 2 (see [33]).
Consider the following system:(5)dρ1ℬ1tdtρ1=q11ℬ1t−η11+q12ℬ2t−η12+,⋯,+q1kℬkt−η1k,dρ2ℬ2tdtρ2=q21ℬ1t−η21+q22ℬ2t−η22+,⋯,+q2kℬkt−η2k,⋮dρkℬktdtρk=qk1ℬ1t−ηk1+qk2ℬ2t−ηk2+,⋯,+qkkℬkt−ηkk,where ρh∈0,1,h=1,2,…,k. Set(6)Δs=sρ1−q11e−sη11−q12e−sη12⋯−q1ke−sη1k−q21e−sη12sρ2−q22e−sη22⋯−q2ke−sη2k⋮⋮⋱⋮−pk1e−sηk1−qk2e−sηk2⋯sρk−qkke−sηkk.
The zero solution of system (5) is said to be asymptotically stable if every root of detΔs=0 owns negative real parts.
Give the system:(7)dρ1ℬ1tdtρ1=q11ℬ1t+q12ℬ2t+,⋯,+q1kℬkt,dρ2ℬ2tdtρ2=q21ℬ1t+q22ℬ2t+,⋯,+q2kℬkt,⋮dρkℬktdtρk=qk1ℬ1t+qk2ℬ2t+,⋯,+qkkℬkt,where ϱh∈0,1,h=1,2,…,k. The characteristic equation of system (7) takes the following form:(8)detsρ1−q11−q12⋯−q1k−q21sρ2−q22⋯−q2k⋮⋮⋱⋮−qk1−qk2⋯sρk−qkk=0.
Suppose thatψj=αj/βj,αj,βj∈Z+,αj,βj=1 and β is the lowest common multiple of βj of ψj,j=1,2,…,k.Lemma 3 (see [33]).
We say that the zero solution of system (7) is locally asymptotically stable if every root λs of the following equation:(9)detλβψ1−q11−q12e−sη12⋯−q1k−q21λβψ2−q22⋯−q2k⋮⋮⋱⋮−qk1−qk2⋯λβψk−qkk=0,obeys argλ>π/2β.


## 3. Bifurcation Study for Neural Networks (2)

In this segment, we will discuss the stability behavior and the emergence of Hopf bifurcation for neural networks (2). Let(10)v3t=∫−∞tHt−sv1sds=∫−∞tβeβt−sv1sds,v4t=∫−∞tHt−sv2sds=∫−∞tβeβt−sv2sds,and then(11)dv3tdt=∫−∞tβeβt−sv1sds′=−βv3t+βv1t,dv4tdt=∫−∞tβeβt−sv2sds′=−βv4t+βv2t.Thus, system (2) becomes the following equivalent form:(12)dv1ρtdtρ=−v1t+γ11h11v3t+γ12h12v2t−η2,dv2ρtdtρ=−v2t+γ21h21v1t−η1+γ22h22v4t,dv3tdt=−βv3t+βv1t,dv4tdt=−βv4t+βv2t.By virtue ofA1, one easily knows that system (12) has a unique zero equilibrium point. The linear system of equation (12) near the zero equilibrium point takes the following expression:(13)dv1ρtdtρ=−v1t+ϱ11v3t+ϱ12v2t−η2,dv2ρtdtρ=−v2t+ϱ21v1t−η1+ϱ22v4t,dv3tdt=−βv3t+βv1t,dv4tdt=−βv4t+βv2t,where ϱkl=γklhkl′0,k,l=1,2. The characteristic equation for (13) is given by(14)detsρ+1−ϱ12e−sη2−ϱ110−ϱ21e−sη1sρ+10−ϱ22−β0s+β00−β0s+β=0.Letρ=ζ/ξ, where ζ,ξ∈Z+ and ζ,ξ=1. Denote λ=s1/ξ. If η=0, which implies that η1=η2=0. Then, equation (14) takes the following form:(15)detsρ+1−ϱ12−ϱ110−ϱ21sρ+10−ϱ22−β0s+β00−β0s+β=0.Lemma 4.
For system (12), if η=0 and each root λ of equation (15) satisfies argλ>π/2ξ, then the zero equilibrium point of system (12) is locally asymptotically stable in Lyapunov sense.Proof.
It is easy to see that ifη=0, then the characteristic equation (14) can be changed into equation (15). Applying Lemma 3, we can conclude that Lemma 4 is true.
By equation (14), we obtain(16)s2ρ+2+ϵ1s2ρ+1+ϵ2s2ρ+2sρ+2+ϵ3sρ+1+ϵ4sρ+s2+ϵ5s+ϵ6+ϵ7s2+ϵ8s+ϵ9e−2sη=0,where(17)ϵ1=2β,ϵ2=β,ϵ3=βϱ11−ϱ22+4,ϵ4=2β+β2ϱ11−β2ϱ22,ϵ5=βϱ11−ϱ22+2,ϵ6=β+β2ϱ11−ϱ22,ϵ7=−ϱ12ϱ21,ϵ8=−2βϱ12ϱ21,ϵ9=−β2ϱ12ϱ21.
Suppose thats=iσ=σcosπ/2+isinπ/2 is the root of equation (16); then, we have(18)σ2ρ+2cos2ρ+2π2+isin2ρ+2π2+ϵ1σ2ρ+1cos2ρ+1π2+isin2ρ+1π2+ϵ2σ2ρcosρπ+isinρπ+2σρ+2cosρ+2π2+isinρ+2π2+ϵ3σρ+1cosρ+1π2+isinρ+1π2+ϵ4σρcosρπ2+isinρπ2−σ2+iϵ5σ+ϵ6+−ϵ7σ2+iϵ8σ+ϵ9cos2ση−isin2ση=0.
By virtue of equation (18), one gets(19)Q1cos2ση+Q2sin2ση=Q3,Q2cos2ση−Q1sin2ση=Q4,where(20)Q1=ϵ9−ϵ7σ2,Q2=ϵ8σ,Q3=−σ2ρ+2cos2ρ+2π2−ϵ1σ2ρ+1cos2ρ+1π2−ϵ2σ2ρcosρπ−2σρ+2cosρ+2π2−ϵ3σρ+1cosρ+1π2−ϵ4σρcosρπ2+σ2−ϵ6,Q4=−σ2ρ+2sin2ρ+2π2−ϵ1σ2ρ+1sin2ρ+1π2−ϵ2σ2ρsinρπ−2σρ+2sinρ+2π2−ϵ3σρ+1sinρ+1π2−ϵ4σρsinρπ2−ϵ5σ.
By (19), we have(21)cos2ση=Q1Q3+Q2Q4Q12+Q22,and(22)Q12+Q22=Q32+Q42.
In equation (20), denote(23)α1=−cos2ρ+2π2,α2=−ϵ1cos2ρ+1π2,α3=−ϵ2cosρπ,α4=−2cosρ+2π2,α5=−ϵ3cosρ+1π2,α6=−ϵ4cosρπ2,α7=−sin2ρ+2π2,α8=−ϵ1sin2ρ+1π2,α9=−ϵ2sinρπ,α10=−2sinρ+2π2,α11=−ϵ3sinρ+1π2,α12=−ϵ4sinρπ2,and then (20) becomes(24)Q1=ϵ9−ϵ7σ2,Q2=ϵ8σ,Q3=α1σ2ρ+2+α2σ2ρ+1+α3σ2ρ+α4σρ+2+α5σρ+1+α6σρ+σ2−ϵ6,Q4=α7σ2ρ+2+α8σ2ρ+1+α9σ2ρ+α10σρ+2+α11σρ+1+α12σρ−ϵ5σ.
According to (22) and (24), we get(25)ς1σ4ρ+4+ς2σ4ρ+3+ς3σ4ρ+2+ς4σ4ρ+1+ς5σ4ρ+ς6σ3ρ+4+ς7σ3ρ+3+ς8σ3ρ+2+ς9σ3ρ+1+ς10σ3ρ+ς11σ2ρ+4+ς12σ2ρ+3+ς13σ2ρ+2+ς14σ2ρ+1+ς15σ2ρ+ς16σρ+4+ς17σρ+3+ς18σρ+2+ς19σρ+1+ς20σρ+ς21σ4+ς22σ2+ς23=0,where(26)ς1=α12+α72,ς2=2α1α2+α7α8,ς3=α22+α82+2α1α3+α7α9,ς4=2α2α3+α8α9,ς5=α32+α92,ς6=2α1α4+α7α10,ς7=2α1α5+α2α4+α7α11+α8α10,ς8=2α1α6+α2α5+α3α4+α7α12+α8α11+α9α10,ς9=2α2α6+α3α5+α8α12+α9α11,ς10=2α3α6+α9α12,ς11=α42+α102+2α1,ς12=2α2+α4α5+α10α11−α7ϵ5,ς13=α52+α112+2α3−α1α6−α8ϵ5+α10α12+α4α6,ς14=2α5α6+α2ϵ6+α11α12−α9ϵ5,ς15=α62−2α3ϵ6+α12,ς16=2α4,ς17=2α5−α10ϵ5,ς18=2α6−α4ϵ6−α11ϵ5,ς19=−2α5ϵ6+α12ϵ5,ς20=−2α6ϵ6,ς21=1−ϵ72,ς22=−2ϵ7+ϵ52−ϵ82+2ϵ7ϵ9,ς23=ϵ62−ϵ92.
Denote(27)Sσ=ς1σ4ρ+4+ς2σ4ρ+3+ς3σ4ρ+2+ς4σ4ρ+1+ς5σ4ρ+ς6σ3ρ+4+ς7σ3ρ+3+ς8σ3ρ+2+ς9σ3ρ+1+ς10σ3ρ+ς11σ2ρ+4+ς12σ2ρ+3+ς13σ2ρ+2+ς14σ2ρ+1+ς15σ2ρ+ς16σρ+4+ς17σρ+3+ς18σρ+2+ς19σρ+1+ς20σρ+ς21σ4+ς22σ2+ς23.
Now the following assumption is given.A3ς23<0, where ς23 is defined by (26).Lemma 5.
IfA3 holds, then equation (16) possesses at least a pair of purely imaginary roots.Proof.
Clearly,S0=ς23<0 and limσ⟶∞Sσ=+∞; then, equation (25) possesses at least one positive root. That is to say, equation (16) possesses at least a couple of purely imaginary roots.
We cannot easily derive the solution of equation (25) since the powers of equation (25) are fractional number. Thus, we will change the powers of equation (25) to integer powers. Let y=σ1/ξ; then, σ=yξ. In view of equation (25), one has(28)ς1y4ζ+4ξ+ς2y4ζ+3ξ+ς3y4ζ+2ξ+ς4y4ζ+ξ+ς5y4ζ+ς6y3ζ+4ξ+ς7y3ζ+3ξ+ς8y3ζ+2ξ+ς9y3ζ+ξ+ς10y3ζ+ς11y2ζ+4ξ+ς12y2ζ+3ξ+ς13y2ζ+2ξ+ς14y2ζ+ξ+ς15y2ζ+ς16yζ+4ξ+ς17yζ+3ξ+ς18yζ+2ξ+ς19yζ+ξ+ς20yζ+ς21y4ξ+ς22y2ξ+ς23=0.
With the aid of computer, we can derive the roots of equation (28). Assume that equation (28) possesses the positive root (say yj); then, equation (25) possesses the positive root σj=yjξ. Assume that equation (28) owns k positive roots yj,j=1,2,…,k. By (21), one gets(29)ηjn=12σjarccosQ1Q3+Q2Q4Q12+Q22+2nπ,j=1,2,…,k;n=0,1,2,⋯.
Let(30)η0=ηj00minj=1,2,…,kηj0,σ0=σ|η=η0.
Next the following assumption is given:A4V11V21+V12V22>0, where(31)V11=2ρ+2σ02ρ+1cos2ρ+1π2+ϵ12ρ+1σ02ρcosρπ+2ρϵ2σ02ρ−1cos2ρ−1π2+2ρ+2σ0ρ+1cosρ+1π2+ϵ3ρ+1σ0ρcosρπ2+ϵ4ρσ0ρ−1cosρ−1π2+ϵ5+ϵ8cos2σ0η0+2ϵ7sin2σ0η0,V12=2ρ+2σ02ρ+1sin2ρ+1π2+ϵ12ρ+1σ02ρsinρπ+2ρϵ2σ02ρ−1sin2ρ−1π2+2ρ+2σ0ρ+1sinρ+1π2+ϵ3ρ+1σ0ρsinρπ2+ϵ4ρσ0ρ−1sinρ−1π2+2σ0−ϵ8sin2σ0η0+2ϵ7cos2σ0η0,V21=2σ0ϵ9−ϵ7σ02cos2σ0η0+2σ02ϵ8sin2σ0η0,V22=2σ02ϵ8cos2σ0η0−2σ0ϵ9−ϵ7σ02sin2σ0η0.Lemma 6.
Suppose thatsη=χ1η+iχ2η is the root of equation (16) near η=η0 that satisfies χ1η0=0,χ2η0=σ0; then, one obtains Reds/dηη=η0,σ=σ0>0.Proof.
By means of equation (16), one derives(32)2ρ+2s2ρ+1+ϵ12ρ+1s2ρ+2ρϵ2s2ρ−1+2ρ+2sρ+1+ϵ3ρ+1sρ+ϵ4ρsρ−1+2s+ϵ5dsdη+2ϵ7s+ϵ8e−2sηdsdη−2e−2sηdsdηη+sϵ7s2+ϵ8s+ϵ9=0,which implies(33)dsdη−1=V1sV2s−ηs,where(34)V1s=2ρ+2s2ρ+1+ϵ12ρ+1s2ρ+2ρϵ2s2ρ−1+2ρ+2sρ+1+ϵ3ρ+1sρ+ϵ4ρsρ−1+2s+ϵ5+2ϵ7s+ϵ8e−2sη,V2s=2se−2sηϵ7s2+ϵ8s+ϵ9.
Therefore,(35)Redsdηη=η0,σ=σ0−1=ReV1sV2sη=η0,σ=σ0=V11V21+V12V22V212+V222.
By virtue ofA4, we get(36)Redsdηη=η0,σ=σ0−1>0.
The proof finishes.
Based on the exploration above, the following results can be easily established.Theorem 1.
If assumptionsA1–A4 are satisfied and all the roots λ of equation (15) satisfy argλ>π/2ξ, then the zero equilibrium point of system (2) is locally asymptotically stable when η lies in 0,η0 and a Hopf bifurcation arises near the zero equilibrium point when η=η0.Remark 1.
Karaoğlu et al. [29] studied the stability behavior and Hopf bifurcation for integer-order neural network models concerning distributed delays and discrete delays. In this paper, we mainly deal with the stability behavior and Hopf bifurcation for fractional-order neural network models concerning distributed delays and discrete delays. The research method of Karaoğlu et al. [29] cannot be directly utilized to handle the fractional-order one. Based on this idea, we think that the study complements the research of Karaoğlu et al. [29].Remark 2.
Although many authors have studied the Hopf bifurcation issue of delayed fractional-order neural networks, all the equations of delayed fractional-order neural networks are fractional-order cases. In our study, neural network system (2) involves integer-order operator and fractional-order operator. So, there are many differences in research method. Furthermore, so far, there are only very few publications concerning this topic. Based on this idea, we believe that our research enriches the bifurcation theory of fractional-order differential equation.


## 4. Simulation Figures

Consider the system as follows:(37)dv1ρtdtρ=−v1t−0.5tanhv3t−1.8tanhv2t−η2,dv2ρtdtρ=−v2t+1.5tanhv1t−η1+1.7tanhv4t,dv3tdt=−0.5v3t+0.5v1t,dv4tdt=−0.5v4t+0.5v2t.Obviously system (37) possesses a unique zero equilibrium point. Choose ρ=0.78=39/50 which implies that ζ=39,ξ=50. By virtue of Matlab software, we derive σ0=2.113,η0=0.41,ϵ23=−0.4317,V11=0.3368,V12=0.4251,V21=−0.3128,V22=0.5035. Then, V11V21+V12V22=0.1087>0. In addition, all the roots (λ) of equation (15) satisfy argλ>π/2ξ. Thus, all conditions in Theorem 1 are fulfilled. So, the zero equilibrium point of system (37) is locally asymptotically stable state when η lies in 0,0.41. To verify this fact, we choose η=0.38<η0=0.41. The software simulation figures are given in Figure 1. It follows from Figure 1 that four neuron states v1, v2, v3, and v4 are to be closed to 0. Figure 1 includes 14 subfigures (Figures 1(a)–1(n)). Different subfigures show the relation of corresponding variables. If η crosses η0=0.41, then the delayed-induced Hopf bifurcation arises near (0,0,0,0). To verify this fact, we choose η=0.5>η0=0.41. The software simulation figures are given in Figure 2. Figure 2 includes 14 subfigures (Figures 2(a)–2(n)). Different subfigures show the relation of corresponding variables. It follows from Figure 2 that four states of four neurons will display the periodic oscillatory behavior. Furthermore, the bifurcation graphs are given in Figures 3–6, which show that the bifurcation value is 0.41 intuitively.Figure 1
The stable behavior of system (37) when η=0.38<η0=0.41.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)Figure 2
Hopf bifurcation phenomenon of system (37) when η=0.5>η0=0.41.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)Figure 3
Bifurcation graph of system (37): η-v1.Figure 4
Bifurcation graph of system (37): η-v2.Figure 5
Bifurcation graph of system (37): η-v3.Figure 6
Bifurcation graph of system (37): η-v4.


## 5. Conclusions

As is known to us, the stability behavior and Hopf bifurcation nature of many different neural networks are important topics in delayed dynamical systems. In this present study, the stability behavior and the emergence of Hopf bifurcation for fractional-order neural networks involving different types of delays have been investigated in detail. Making use of a suitable variable substitution, novel equivalent fractional-order forms of the involved delayed neural network models are built. Taking advantage of stability theory and bifurcation knowledge of fractional-order differential equation, a novel delay-independent bifurcation criterion for the fractional-order neural networks involving different types of delays is presented. The investigation indicates that time delay plays a significant role in stabilizing and controlling Hopf bifurcation for the involved fractional-order delayed neural networks. The research method can help us deal with the Hopf bifurcation issue of many other neural networks involving distributed delays in the near future. In this paper, we only consider the effect of the sum of two delaysη1 and η2 on bifurcation of the involved system. In the coming work, we will deal with the impact of two different delays η1 and η2 on bifurcation of the involved system.



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