# Further Exploration on Bifurcation for Fractional-Order Bidirectional Associative Memory (BAM) Neural Networks concerning Time Delay∗

Authors: Nengfa Wang; Changjin Xu; Zixin Liu
Journal: Complexity
 (2021)

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## Abstract

This work principally considers the stability issue and the emergence of Hopf bifurcation for a class of fractional-order BAM neural network models concerning time delays. Through the detailed analysis on the distribution of the roots of the characteristic equation of the involved fractional-order delayed BAM neural network systems, we set up a new delay-independent condition to guarantee the stability and the emergence of Hopf bifurcation for the investigated fractional-order delayed BAM neural network systems. The work indicates that delay is a significant element that has a vital impact on the stability and the emergence of Hopf bifurcation in fractional-order delayed BAM neural network systems. The simulation figures and bifurcation plots are clearly presented to verify the derived key research results. The established conclusions of this work have significant guiding value in regulating and optimizing neural networks.


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## Body

## 1. Introduction

Neural networks have been found to have immense application prospect in a lot of subject areas such as modeling human brain, remote sensing, biological science, pattern recognition, artificial intelligence, and control technique [1, 2]. Usually, time delay often occurs in neural network systems due to the lag of the response of signal transmission of the neurons in neural networks. Thus, it is necessary for us to establish the delayed neural networks to describe the real situation of neural networks. Generally speaking, time delay often gives rise to the disappearance of stability, periodic oscillation, chaotic behavior, and so on [3, 4]. In order to grasp the effect of time delay on various dynamical properties of neural networks, miscellaneous delayed neural networks have been built and studied. Up to now, a great deal of valuable publications has been achieved. For instance, Aouiti et al. [5] investigated the existence and global exponential stability of pseudo almost periodic solution to delayed BAM neural networks involving leakage delays by virtue of fixed point theory and mathematical inequality skills. Yang et al. [6] studied the almost automorphic solution to high-order delayed BAM neural networks by means of the exponential dichotomy theory, Banach contraction mapping law, and differential inequality strategy. Maharajan et al. [7] set up a new global robust exponential stability condition for a class of uncertain BAM neural network systems involving mixed time delays. Popa [8] focused on the global μ-stability for impulsive complex-valued BAM neural networks concerning mixed delays. Sowmiya et al. [9] made a detailed analysis on mean-square asymptotic stability for impulsive discrete-time stochastic BAM neural networks involving Markovian jumping and multiple delays. For details, we refer the readers to [10, 11].The general BAM networks are given by(1)p˙it=−αipit+∑j=1majihiqjt−ηji+Pi,q˙jt=−βjqjt+∑i=1nbijkjpit−ζij+Qj,where i=1,2,...,n;j=1,2,...,m, αi and βj describe the stability of internal neuron processes on P-layer and Q-layer, respectively; aji,bij represent the connection weights; pit and qjt denote the states of the neurons on P-layer and Q-layer, respectively; hi and kj are activation functions; Pi,Qj denote the inputs; ηji and ζij are time delays. The model (1) describes the change law of different neurons which lie in two layers. For details, please see [12, 13].System (1) is a large-scale nonlinear dynamical model. It owns very complicated dynamical properties. In order to have a good command of the internal law of network system (1), many researchers pay much attention to some simplified versions of delayed neural network models. By the investigation on various dynamical peculiarities of the simplified neural network systems, we are able to grasp the potential dynamical properties for large-scale delayed neural network systems. During the past several years, a lot of works on the simplified neural network models have been published. For example, Hajihosseini et al. [14] discussed the bifurcation problem for recurrent neural networks involving three neurons. Kaslik and Balint [15] investigated the Neimark–Sacker bifurcation of a discrete-time delayed neural network system involving two neurons. Ge and Xu [16] obtained the sufficient condition to ensure the stability and the onset of Hopf bifurcation for delayed neural networks involving four neurons. Yang and Ye [17] dealt with the stability and bifurcation behavior for delayed BAM neural network involving five neurons. As to more concrete literatures on this theme, one can see [4, 18].All above publications are only restricted to the integer-order dynamical equations. In recent years, fractional calculus has displayed wide application value in a lot of fields such as heat and mass transfer, electromagnetic and electrodynamics, control science, population systems, biophysics, and neural networks [19–27]. The study shows that fractional calculus can be regarded as a very useful tool to describe the object issues in the real world because it owns the memory property and hereditary function during the dynamic change process [28, 29]. Recently, fractional calculus has become the biggest concern of the present day world. In particular, fractional-order neural networks have also become one of the key hot issues in neural network area. Delay-induced Hopf bifurcation is a significant dynamical property in delayed dynamical models. However, it is a pity that a great deal of works is only concerned with delay-induced Hopf bifurcation for integer-order dynamical system concerning delays and few publications focus on the fractional-order case (see [30, 31]). In fractional-order neural networks, what is the effect of time delay and fractional-order on the stability and bifurcation? The solution of this problem is beneficial to the design of neural networks. Up to now, there are many bifurcation problems that are expected to be solved. This viewpoint stimulates us to deal with the delay-induced Hopf bifurcation of delayed neural networks involving multiple neurons.Based on the neural networks (1), we consider the following fractional-order simplified delayed neural networks:(2)dξu1tdtξ=−ku1t+a11hu4t−ζ+a12lu5t−ζ+a13lu6t−ζ,dξu2tdtξ=−ku2t+a21hu5t−ζ+a22lu6t−ζ+a23lu4t−ζ,dξu3tdtξ=−ku3t+a31hu6t−ζ+a32lu4t−ζ+a33lu5t−ζ,dξu4tdtξ=−ku4t+a41hu1t−ζ+a42lu2t−ζ+a43lu3t−ζ,dξu5tdtξ=−ku5t+a51hu2t−ζ+a52lu3t−ζ+a53lu1t−ζ,dξu6tdtξ=−ku6t+a61hu3t−ζ+a62lu1t−ζ+a63lu2t−ζ,where ξ∈0,1 is a real number; k describes the stability of internal neuron processes on P-layer and Q-layer; aiji=1,2,3,4,5,6;j=1,2,3 represents the connection weights; uiti=1,2,3 denotes the state of the i-neuron on P-layer; ujtj=4,5,6 denotes the state of the j-neuron on Q-layer; h and l are activation functions. In order to establish the key results of this work, we make the following hypothesis:(3)H1h,l∈C1,h0=l0=0.The remainder of this article is planned as follows. Section2 presents the key theories on fractional calculus. Section 3 displays the main conclusions on stability and Hopf bifurcation for neural networks (2). Section 4 executes software simulations to illustrate the key conclusions of this article. Section 5 ends this work with a simple conclusion.


## 2. Indispensable Definitions and Lemmas

In this part, we give several necessary definitions and lemmas about fractional calculus which will be used in the next part.Definition 1 (see [32]).
Define Caputo fractional-order derivative as follows:(4)Dξuϱ=1Γl−ξ∫ϱ0ϱulsϱ−sξ−l+1ds,where uϱ∈ϱ0,∞,R,Γs=∫0∞ϱs−1e−ϱdϱ, ϱ≥ϱ0, andl∈Z+,ξ∈l−1,l.Lemma 1 (see [33, 34]).
Consider the following model:(5)dξutdtξ=wt,ut,u0=u0,where ξ∈0,1 and wt,ut:R+×Rn⟶Rn,n∈Z+. Let u∗ be the equilibrium point of system (5). If every eigenvalue (denoted by ς) of ∂wt,u/∂u|u=u∗ obeys argς>ξπ/2, then we say that u∗ is locally asymptotically stable.Lemma 2 (see [35]).
Consider the following model:(6)dξ1ℋ1tdtξ1=e11ℋ1t−ζ11+e12ℋ2t−ζ12+⋯+e1lℋlt−ζ1l,dξ2ℋ2tdtξ2=e21ℋ1t−ζ21+e22ℋ2t−ζ22+⋯+e2lℋlt−ζ2l,⋮dξlℋtdtξl=el1ℋ1t−ζl1+el2ℋ2t−ζl2+⋯+ellℋlt−ζll,where ξi∈0,1i=1,2,...,l, the initial value ℋit=ωit∈C−maxi,hζih,0, t∈−maxi,hζil,0, and i,h=1,2,...,l. Denote(7)Δη=ηξ1−e11e−ηζ11−e12e−ηζ12⋯−e1le−ηζ1l−e21e−ηζ12ηξ2−e22e−ηζ22⋯−e2le−ηζ2l⋮⋮⋱⋮−el1e−ηζl1−el2e−ηζl2⋯ηξl−elle−ηζll.
Then, the zero solution of system (6) is said to be asymptotically stable in Lyapunov sense provided that every root of detΔη=0 owns negative real parts.


## 3. Exploration on Delay-Induced Hopf Bifurcation

In this part, by discussing the characteristic equation of system (2) and setting the time delay as bifurcation parameter, we will establish the delay-independent sufficient condition to guarantee the stability and the onset of Hopf bifurcation for system (2).In view ofH1, one can easily know that system (2) owns the unique equilibrium U0,0,0,0,0,0. The linear system of system (2) at the zero equilibrium U0,0,0,0,0,0 owns the expression:(8)dξu1tdtξ=−ku1t+b11u4t−ζ+b12u5t−ζ+b13u6t−ζ,dξu2tdtξ=−ku2t+b21u5t−ζ+b22u6t−ζ+b23u4t−ζ,dξu3tdtξ=−ku3t+b31u6t−ζ+b32u4t−ζ+b33u5t−ζ,dξu4tdtξ=−ku4t+b41u1t−ζ+b42u2t−ζ+b43u3t−ζ,dξu5tdtξ=−ku5t+b51u2t−ζ+b52u3t−ζ+b53u1t−ζ,dξu6tdtξ=−ku6t+b61u3t−ζ+b62u1t−ζ+b63u2t−ζ,where bi1=ai1hi1′0,bi2=ai2l′0,bi3=ai3l′0,i=1,2,3,4,5,6. The characteristic equation for equation (8) owns the expression:(9)detsξ+k00−b11e−sζ−b12e−sζ−b13e−sζ0sξ+k0−b23e−sη−b21e−sζ−b22e−sζ00sξ+k−b32e−sη−b33e−sζ−b31e−sζ−b41e−sζ−b42e−sζ−b43e−sζsξ+k00−b53e−sζ−b51e−sζ−b52e−sζ0sξ+k0−b62e−sζ−b63e−sζ−b61e−sζ00sξ+k=0.By equation (9), we get(10)U1s+U2se−2sζ+U3se−4sζ+U4se−6sζ=0,where(11)U1s=s6ξ+μ5s5ξ+μ4s4ξ+μ3s3ξ+μ2s2ξ+μ1sξ+μ0,U2s=ν4s4ξ+ν3s3ξ+ν2s2ξ+ν1sξ+ν0,U3s=χ2s2ξ+χ1sξ+χ0,U4s=ρ0=−c11c22c33,where(12)μ0=k6−k4c13c31+c12c21+c23c32−k3c11c23c31+c13c21c32,μ1=6k2−4k3c13c31+c12c21+c23c32−3k2c11c23c31+c13c21c32+2kc33c11+c22+c11c22,μ2=9k4−6k2c13c31+c12c21+c23c32−3kc11c23c31+c13c21c32+c33c11+c22+c11c22,μ3=26k3−4kc13c31+c12c21+c23c32−c11c23c31+c13c21c32,μ4=15k2−c13c31+c12c21+c23c32,μ5=6k,ν0=k2c11c31c22+c12c21c33+c11c23c32−k4c11+c22+c33,ν1=2kc11c31c22+c12c21c33+c11c23c32−4k3c11+c22+c33,ν2=c11c31c22+c12c21c33+c11c23c32−6k2,ν3=−4kc11+c22+c33,ν4=−c11+c22+c33,χ0=k2c33c11+c22+c11c22,χ1=2kc33c11+c22+c11c22,χ2=c33c11+c22+c11c22,where(13)c11=b41b11+b42b23+b43b32,c12=b41b12+b42b21+b43b33,c13=b41b13+b42b22+b43b31,c21=b53b11+b51b23+b52b32,c22=b53b12+b51b21+b52b33,c23=b53b13+b51b22+b52b31,c31=b62b11+b63b23+b61b32,c32=b62b12+b63b21+b61b33,c33=b62b13+b63b22+b61b31.By virtue of (10), we get(14)U1se4sζ+U2se2sζ+U3s+U4se−2sζ=0.Assume thats=iϑ=ϑcosπ/2+isinπ/2 is the root of (14) and denote the real parts and imaginary parts of Ujsj=1,2,3,4 by UjRs and UjIsj=1,2,3,4, respectively. It follows from (14) that(15)U1Rϑcos4ϑζ−U1Iϑsin4ϑζ+U2Rϑ+U4Rϑcos2ϑζ+U4Iϑ−U2Iϑsin2ϑζ=−U3Rϑ,U1Iϑcos4ϑζ+U1Rϑsin4ϑζ+U2Iϑ+U4Iϑcos2ϑζ+U2Rϑ−U4Rϑsin2ϑζ=−U3Iϑ,where(16)U1Rϑ=ϑ6ξcos3ξπ+μ5ϑ5ξcos5ξπ2+μ4ϑ4ξcos2ξπ+μ3ϑ3ξcos3ξπ2+μ2ϑ2ξcosξπ+μ1ϑξcosξπ2+μ0,U1Iϑ=ϑ6ξsin3ξπ+μ5ϑ5ξsin5ξπ2+μ4ϑ4ξsin2ξπ+μ3ϑ3ξsin3ξπ2+μ2ϑ2ξsinξπ+μ1ϑξsinξπ2,U2Rϑ=ν4ϑ4ξcos2ξπ+ν3ϑ3ξcos3ξπ2+ν2ϑ2ξcosξπ+ν1ϑξcosξπ2+ν0,U2Iϑ=ν4ϑ4ξsin2ξπ+ν3ϑ3ξsin3ξπ2+ν2ϑ2ξsinξπ+ν1ϑξsinξπ2,U3Rϑ=χ2ϑ2ξcosξπ+χ1ϑξcosξπ2+χ0,U3Iϑ=χ2ϑ2ξsinξπ+χ1ϑξsinξπ2,U4Rϑ=ρ0,U4Iϑ=0.In view of (16), we can rewrite (15) as(17)U1Rϑcos4ϑζ−U1Iϑsin4ϑζ+U2Rϑ+U4Rϑcos2ϑζ−U2Iϑsin2ϑζ=−U3Rϑ,U1Iϑcos4ϑζ+U1Rϑsin4ϑζ+U2Iϑ+cos2ϑζ+U2Rϑ−U4Rϑsin2ϑζ=−U3Iϑ,According tosin2ϑζ=±1−cos22ϑζ, we are to deal with two cases.(i)
Ifsin2ϑζ=1−cos22ϑζ, it follows from the first equation of (17) that(18)2U1Rϑ2cos22ϑζ−1−2U1Iϑcos2ϑζ1−cos22ϑζ+U2Rϑ+U4Rϑcos2ϑζ−U2Iϑ1−cos22ϑζ=−U3Rϑ,which leads to(19)2U1Rϑ2cos22ϑζ−1+U2Rϑ+U4Rϑcos2ϑζ+U3Rϑ2−2U1Iϑcos2ϑζ−U2Iϑ21−cos22ϑζ.Then, one gets(20)ρ1cos42ϑζ+ρ2cos32ϑζ+ρ3cos22ϑζ+ρ4cos2ϑζ+ρ5=0,where(21)ρ1=16U1R2ϑ+4U1I2ϑ,ρ2=8U1RϑU2Rϑ+U4Rϑ+4U1IϑU2Iϑ,ρ3=U2Rϑ+U4Rϑ2+8U1RϑU3Rϑ−2U1Rϑ+U2I2ϑ−4U1R2ϑ,ρ4=2U2Rϑ+U4RϑU3Rϑ−U1Rϑ−4U1IϑU2Iϑ,ρ5=U3Rϑ−2U1Rϑ2−U2I2ϑ.Suppose thatcos2ϑζ=η and set(22)hη=η4+ρ2ρ1η3+ρ3ρ1η2+ρ4ρ1η+ρ5ρ1,then(23)dhηdη=4η3+3ρ2ρ1η2+2ρ3ρ1η+ρ4ρ1.Let(24)4η3+3ρ2ρ1η2+2ρ3ρ1η+ρ4ρ1=0.Assume thaty=η+ρ2/4ρ1, then (24) can be expressed as(25)y3+r1y+r2=0,where(26)r1=ρ32ρ1−3ρ2216ρ12,r2=ρ2332ρ13−ρ2ρ38ρ12+ρ44ρ1.Denote(27)δ1=r222+r133,δ2=−1+i32.By (25), one gets(28)y1=−r22+δ13+−r22−δ13,y2=−r22+δ13δ2+−r22−δ13δ22,y3=−r22+δ13δ22+−r22−δ13δ2.According to the analysis above, one can obtain the expression ofcos2ϑζ. Then, one can derive the expression of sin2ϑζ. Here, we suppose that(29)cos2ϑζ=φ1ϑ,sin2ϑζ=φ2ϑ.Hence,(30)φ12ϑ+φ22ϑ=1.By virtue of computer software, one can easily derive the root (sayϑ) of (30). Thus, one has(31)ζ1l=12ϑarccosφ1ϑ+2lπ,l=0,1,2,….(ii)
Ifsin2ϑζ=−1−cos22ϑζ, by means of the same method, one can also derive(32)cos2ϑζ=ψ1ϑ,sin2ϑζ=ψ2ϑ.Then,(33)ψ12ϑ+ψ22ϑ=1.By virtue of computer software, we can derive the root (sayϑ) of (33). Then,(34)ζ2k=12ϑarccosψ1ϑ+2kπ,k=0,1,2,….Define(35)ζ0=minζ1l,ζ2l,l=0,1,2,….In the sequel, we are to verify the transversality condition to ensure the onset of Hopf bifurcation. The following hypothesis is needed. (H2) AR∗ℬR∗+AI∗ℬI∗>0, where(36)AR∗=6ξϑ06ξ−1cos6ξ−1π2+5ξμ5ϑ05ξ−1cos5ξ−1π2+4ξμ4ϑ04ξ−1×cos4ξ−1π2+3ξμ3ϑ03ξ−1cos3ξ−1π2+2ξμ2ϑ02ξ−1cos2ξ−1π2+ξμ1ϑ0ξ−1cosξ−1π2+4ξν4ϑ04ξ−1cos4ξ−1π2+3ξν3ϑ03ξ−1cos3ξ−1π2+2ξν2ϑ02ξ−1cos2ξ−1π2+ξν1ϑ0ξ−1cosξ−1π2cos2ϑ0ζ0+4ξν4ϑ04ξ−1sin4ξ−1π2+3ξν3ϑ03ξ−1sin3ξ−1π2+2ξν2ϑ02ξ−1sin2ξ−1π2+ξν1ϑ0ξ−1sinξ−1π2sin2ϑ0ζ0+2ξχ2ϑ02ξ−1cos2ξ−1π2+ξχ1ϑ0ξ−1cosξ−1π2cos4ϑ0ζ0+2ξχ2ϑ02ξ−1sin2ξ−1π2+ξχ1ϑ0ξ−1sinξ−1π2sin4ϑ0ζ0,AI∗=6ξϑ06ξ−1sin6ξ−1π2+5ξμ5ϑ05ξ−1sin5ξ−1π2+4ξμ4ϑ04ξ−1×sin4ξ−1π2+3ξμ3ϑ03ξ−1sin3ξ−1π2+2ξμ2ϑ02ξ−1sin2ξ−1π2+ξμ1ϑ0ξ−1sinξ−1π2+4ξν4ϑ04ξ−1cos4ξ−1π2+3ξν3ϑ03ξ−1cos3ξ−1π2+2ξν2ϑ02ξ−1cos2ξ−1π2+ξν1ϑ0ξ−1cosξ−1π2sin2ϑ0ζ0−4ξν4ϑ04ξ−1sin4ξ−1π2+3ξν3ϑ03ξ−1sin3ξ−1π2+2ξν2ϑ02ξ−1sin2ξ−1π2+ξν1ϑ0ξ−1sinξ−1π2cos2ϑ0ζ0−2ξχ2ϑ02ξ−1cos2ξ−1π2+ξχ1ϑ0ξ−1cosξ−1π2sin4ϑ0ζ0+2ξχ2ϑ02ξ−1sin2ξ−1π2+ξχ1ϑ0ξ−1sinξ−1π2cos4ϑ0ζ0,ℬR∗=2ϑ0ν4ϑ04ξcos2ξπ+ν3ϑ03ξcos3ξπ2+ν2ϑ02ξcosξπ+ν1ϑ0ξcosξπ2+ν0sin2ϑ0ζ0+2ϑ0ν4ϑ04ξsin2ξπ+ν3ϑ03ξsin3ξπ2+ν2ϑ02ξsinξπ+ν1ϑ0ξsinξπ2+ν0cos2ϑ0ζ0,ℬI∗=2ϑ0ν4ϑ04ξcos2ξπ+ν3ϑ03ξcos3ξπ2+ν2ϑ02ξcosξπ+ν1ϑ0ξcosξπ2+ν0cos2ϑ0ζ0+2ϑ0ν4ϑ04ξsin2ξπ+ν3ϑ03ξsin3ξπ2+ν2ϑ02ξsinξπ+ν1ϑ0ξsinξπ2+ν0sin2ϑ0ζ0.Lemma 3.
Assume thatsζ=ϕ1ζ+iϕ2ζ is the root of (10) at ζ=ζ0 and ϕ1ζ0=0,ϕ2ζ0=ϑ0, then Reds/dζζ=ζ0,ϑ=ϑ0>0.Proof.
By virtue of (10), we get(37)dU1sdζ+dU2sdζe−2sζ−2e−2sζdsdζζ+sU2s+dU3sdζe−4sζ,−4e−4sζdsdζζ+sU3s+dU4sdζe−6sζ−6e−6sζdsdζζ+sU4s=0.
Since(38)dU1sdζ=6ξs6ξ−1+5ξμ5s5ξ−1+4ξμ4s4ξ−1+3ξμ3s3ξ−1+2ξμ2s2ξ−1+ξμ1sξ−1dsdζ,dU2sdζ=4ξν4s4ξ−1+3ξν3s3ξ−1+2ξν2s2ξ−1+ξν1sξ−1dsdξ,dU3sdζ=2ξχ2s2ξ−1+ξχ1sξ−1dsdζ,dU4sdζ=0,then by (37) and (38), one gets(39)dsdζ−1=Asℬs−ζs,where(40)As=6ξs6ξ−1+5ξμ5s5ξ−1+4ξμ4s4ξ−1+3ξμ3s3ξ−1+2ξμ2s2ξ−1+ξμ1sξ−1+4ξν4s4ξ−1+3ξν3s3ξ−1+2ξν2s2ξ−1+ξν1sξ−1e−2sζ+2ξχ2s2ξ−1+ξχ1sξ−1e−4sζ,ℬs=2se−2sζν4s4ξ+ν3s3ξ+ν2s2ξ+ν1sξ+ν0+4sχ2s2ξ+χ1sξ+χ0e−4sζ+6sρ0e−6sζ.
It follows from (H2) that(41)Redsdζ−1|ζ=ζ0,ϑ=ϑ0=AR∗ℬR∗+AI∗ℬI∗AR∗2+ℬI∗2>0.
This completes the proof.
Let(42)τ1=μ5,τ2=μ4+ν4,τ3=μ3+ν3,τ4=μ2+ν2+χ2,τ5=μ1+ν1+χ1,τ6=μ0+ν0+χ0+ρ0.
Next, the following assumption is needed:
(H3) the following inequalities are true:(43)G1=τ1>0,G2=detτ11τ3τ2>0,G3=detτ110τ3τ2τ1τ5τ4τ3>0,G4=detτ1100τ3τ2τ11τ5τ4τ3τ20τ6τ5τ4>0,G5=detτ11000τ3τ2τ110τ5τ4τ3τ2τ10τ6τ5τ4τ3000τ6τ5>0,G6=τ6>0.Lemma 4.
Ifζ=0 and (ℋ3) is fulfilled, then system (2) is locally asymptotically stable.Proof.
Obviously, (10) with ζ=0 owns the following expression:(44)U1s+U2s+U3s+U4s=0.
Namely,(45)λ6+τ1λ5+τ2λ4+τ3λ3+τ4λ2+τ5λ+τ6=0.
By means of (H3), one knows that every root λi of (45) satisfies argλi>ξπ/2i=1,2,...,6. So, we can obtain that Lemma 3 holds. This ends the proof.
According to the study above, the following result is built.Theorem 1.
If (H1)–(H3) hold true, then the equilibrium point U0,0,0,0,0,0 of system (2) is locally asymptotically stable proved that ζ∈0,ζ0 and a Hopf bifurcation is to arise around U0,0,0,0,0,0 if ζ=ζ0.Remark 1.
Theorem1 shows that ζ0 is a critical value which determines whether system (2) is stable or unstable. If ζ<ζ0, then system (2) is stable, and if ζ>ζ0, then system (2) becomes unstable and a family of periodic solutions will appear near U0,0,0,0,0,0.Remark 2.
In [3, 4], Cheng et al. studied the stability and Hopf bifurcation of integer-order delayed neural networks. They obtain the characteristic equation by applying integer-order differential equation theory and determinant knowledge. In this work, we investigate the stability and Hopf bifurcation of fractional-order delayed neural networks. We obtain the characteristic equation by applying fractional-order differential equation theory, Laplace transform, and determinant knowledge. The investigation on the distribution of the characteristic roots for characteristic equation of fractional-order neural networks is more difficult that of integer-order case. From this viewpoint, we think that our work replenishes and improves the earlier works of Cheng et al. [3, 4].


## 4. Software Simulation Plots

Give the fractional-order neural network system:(46)dξu1tdtξ=−u1t+0.8tanhu4t−ζ+0.8tanhu5t−ζ−0.9tanhu6t−ζ,dξu2tdtξ=−u2t+0.5tanhu5t−ζ−0.2tanhu6t−ζ+0.5tanhu4t−ζ,dξu3tdtξ=−u3t−0.5tanhu6t−ζ−0.5tanhu4t−ζ+0.8tanhu5t−ζ,dξu4tdtξ=−u4t+0.2tanhu1t−ζ+0.6tanhu2t−ζ−0.8tanhu3t−ζ,dξu5tdtξ=−u5t−0.9tanhu2t−ζ−0.5tanhu3t−ζ+0.7tanhu1t−ζ,dξu6tdtξ=−u6t+0.2tanhu3t−ζ−1.2tanhu1t−ζ−0.9tanhu2t−ζ.Apparently, neural network system (46) owns the unique zero equilibrium point U0,0,0,0,0,0. Let ξ=0.94. By means of computer software, one can derive ζ0=0.75 and ϑ0=2.0922. By virtue of algebraic computation with computer, one can verify that the assumptions H1–H3 of Theorem 1 hold. Then, one can conclude that the zero equilibrium point U0,0,0,0,0,0 of neural network system (46) is locally asymptotically stable provided that ζ∈0,0.75. To illustrate this fact, we carry out computer simulations. We carry out numerical discretizations of model (46) by Adams–Bashforth–Moulton numerical algorithm. The integration algorithm starts with the solutions of system (46) in terms of R-L integral. The implicit discretization approach is applied to construct the tactics. We select ζ=0.67<ζ0=0.75. The computer simulation figures are presented in Figure 1 which shows the locally asymptotically stable behavior of the neural network system (46). When ζ passes through the critical value ζ0=0.75, then the delay-induced Hopf bifurcation of neural network system (46) will arise in the vicinity of U0,0,0,0,0,0. To explain this fact, we select ζ=0.9>ζ0=0.75. The computer simulation figures are presented in Figure 2 which shows the Hopf bifurcation phenomenon of neural network system (46). The initial conditions are 0.09,0.038,0.02,0.02,−0.039,0.08 and the time step is 0.0035 and the time of simulation is 250 seconds. To display the Hopf bifurcation phenomenon of neural network system (46) intuitively, we also draw the bifurcation plots which can be seen in Figures 3–8. From Figures 3–8, one can easily know that the bifurcation value of neural network system is 0.75.Figure 1
Stability property for neural network model (46) involving ζ=0.65<ζ0=0.75.
(a)(b)(c)(d)Figure 2
Hopf bifurcation for neural network model (46) involving ζ=0.9>ζ0=0.75.
(a)(b)(c)(d)Figure 3
Bifurcation figure of neural network model (46): ζ-u1.Figure 4
Bifurcation figure of neural network model (46): ζ-u2.Figure 5
Bifurcation figure of neural network model (4.2):ζ-u3.Figure 6
Bifurcation figure of neural network model (4.2):ζ-u4.Figure 7
Bifurcation figure of neural network model (46): ζ-u5.Figure 8
Bifurcation figure of neural network model (46): ζ-u6.Remark 3.
In Figure1, the subfigures 1–10 stand for the relation of the variable in horizontal axis and vertical axis. The subfigures 11–26 stand for the relation of the variable in horizontal axis, vertical axis, and vertical axis. In Figure 2, the subfigures 1–10 stand for the relation of the variable in horizontal axis and vertical axis. The subfigures 11–26 stand for the relation of the variable in horizontal axis, vertical axis, and vertical axis.


## 5. Conclusions

Delay-induced Hopf bifurcation phenomenon is a significant dynamical behavior in delayed dynamical models. In particular, delay-induced Hopf bifurcation in neural network area has attracted much attention from good many scholars in the whole world. During the past decades, some researchers have investigated the Hopf bifurcation problem of fractional-order delayed neural networks. However, the major works are only concerned with the low-dimensional delayed fractional-order delayed neural networks; few works focus on the high-dimensional fractional-order ones. In this work, we mainly focus on the stability problem and the appearance of Hopf bifurcation of high-dimensional fractional-order delayed BAM neural network systems. The study shows that when the time delay keeps in a suitable range, the neural network systems will remain a stable state and if the time delay passes the critical value, then a Hopf bifurcation will take place around the equilibrium point of the involved neural networks. Thus, the time delay is a momentous factor that affects the stability and Hopf bifurcation for the investigated neural networks. In the end, the software simulation results and bifurcation diagrams efficaciously illustrate the effectiveness of the crucial analytical conclusions.



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*Source: 9096727-2021-10-22.xml*