# Bifurcations of a Fractional-Order Four-Neuron Recurrent Neural Network with Multiple Delays

Authors: Yu Fei; Rongli Li; Xiaofang Meng; Zhouhong Li
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

This paper investigates the bifurcation issue of fractional-order four-neuron recurrent neural network with multiple delays. First, the stability and Hopf bifurcation of the system are studied by analyzing the associated characteristic equations. It is shown that the dynamics of delayed fractional-order neural networks not only depend heavily on the communication delay but also significantly affects the applications with different delays. Second, we numerically demonstrate the effect of the order on the Hopf bifurcation. Two numerical examples illustrate the validity of the theoretical results at the end.


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## Body

## 1. Introduction

Recurrent neural network (RNN) is a type of recursive neural network that takes sequence data as input, recurses in the evolution direction of the sequence, and all nodes (recurrent units) are connected in a chain. Till now, several recurrent neural networks (RNNs) have been widely considered in various fields such as signal processing, optimizations control, image processing, robotics, pattern recognitions, and automatic control, so they have attracted extensive attention of researchers in recent years [1–7]. Since the applications of RNNs depend more heavily on dynamical neural networks, quite a few efforts have been undertaken to study their dynamical properties and a large number of useful results have been investigated, including oscillation, stability, bifurcation, synchronization, and chaos of various RNNs [8–14].As the matter of fact, for some applications of nonlinear dynamical models, time delay has a significant impact, and in addition to affecting stability, it causes oscillations and other unstable phenomena, such as chaos [15]. Communication delays and the response times of neurons are considered key factors in the performance of neural networks, and this is caused by the finite switching speed of amplifiers and the noninstantaneous signal transmission between neurons [16]. In recently years, many scholars have been interested in studying the dynamics of neural networks with such time delays [17–19]. It must be pointed out that exponential stabilization of memristor-based RNNs with disturbance and mixed time delays by periodically intermittent control has been considered by Wang et al. [20]. Using the appropriate Lyapunov–Krasovski functionals and applying matrix inequality approach methods, Zhou [21] discussed the passivity of a class of recurrent neural networks with impulse and multiproportional delays. Zhou and Zhao [22] investigated the exponential synchronization and polynomial synchronization of recurrent neural networks with and without proportional delays. Robust stability analysis of recurrent neural networks is studied in Refs. [23, 24]. Furthermore, time delays are ubiquitous and unavoidable in the real world. Due to the existence of delays, the system can become unstable, and the dynamic behavior of nonlinear systems becomes more difficult. Moreover, since the solution space of the delay dynamical is infinite, it makes the systems more complex and bifurcation occurs. Hence, it is necessary to consider the properties and dynamics of neural networks via delays, such as time delay [25, 26], multiple delays [27, 28] time-varying delays [29, 30], and so on. In 2013, Zhang and Yang [31] studied a four-neuron recurrent neural network with multiple delays, described as follows:(1)x1.t=−x1t+fx2t−τ1,x2.t=−x2t+fx3t−τ1,x3.t=−x3t+fx4t−τ1,x4.t=−x4t+ω1fx1t−τ2+ω2fx2t−τ2+ω3fx3t−τ2,C5α1β1+α2β2/α12+α22≠0where xiti=1,2,3,4 stand for state of the ith neuron at time t, ωk∈Rk=1,2,3 are the network parameters or weight, f⋅ is the connection function between neurons, and τj≥0j=1,2 are the communication time delay. By using the distribution of the solutions of the associated characteristic equation, the Hopf bifurcation and local stability of the four-dimensional RNNs with two delays are studied. For more recurrent neural network research results, see references [5, 10, 12, 20].In more than three centuries, fractional calculus has developed into a classical mathematical concept. Nonlinear dynamics systems have shown that it has an exceptionally important role in generalizing ordinary differentiation and integration to arbitrary noninteger order. Therefore, if we study the effects of the memory and genetics factors, fractional neural network is sometimes more realistic and more general than integer neural networks. In recent years, the application of fractional order neural networks has developed rapidly, and the complex dynamical behaviors of fractional neural networks has become a very important research hot points, such as stability or multistability, Hopf bifurcation, synchronization, chaos, and so on. For instance, in Ref. [32], the multistability of a fractional-order competitive neural networks with delay is investigated by using the fractional calculus and partitioning of state space. In Lu and Xue [33] study, adaptive synchronization is investigated for fractional delayed stochastic neural networks. Yuan and Huang [34] considered the quantitative analysis of fractional-order neural networks with time delay. Udhayakumar and Rajan [35] discussed Hopf bifurcation of a delayed fractional-order octonion-valued neural networks.We also know that Hopf bifurcations, which include subcritical and supercritical ones, can be used to efficiently design biochemical oscillators. Furthermore, fractional order neural networks with same delay cannot accurately describe the dynamical properties of real world neural networks compared with the ones with different delays. In recent years, some researchers have considered the dynamical behavior of fractional models with time delay [36–48]. In 2019 [49], we also investigated the existence of Hopf bifurcation for four-neuron fractional neural networks with leakage delays. To the best of our knowledge, so far there are few results on the Hopf bifurcation of four-dimensional fractional-order recurrent neural network with multiple delays are reported, and therefore, the study of Hopf bifurcation of fractional-order dynamical systems with multiple delays remains an open problem.Based on the above motivations, we are dedicated to presenting a theoretical exploration of stability and Hopf bifurcation for a four-neuron fractional-order recurrent neural network with multiple delays in this work. The main contributions can be highlighted as follows:(i)
A novel delayed fractional-order recurrent neural network with four-neuron and two different delays is studied(ii)
Double main dynamical properties of the fractional-order recurrent neural network with two delays are investigated: stability and oscillation(iii)
The Hopf bifurcation is discussed in terms of delays and orderIn the article, we shall give some some lemmas and definitions of fractional-order calculus in Section2, and models description in Section 3. In Section 4, the local stability of the trivial steady state of delayed fractional-order RNNs is examined by applying the associated characteristic equation. In addition, the authors will care about the Hopf bifurcation of fractional-order RNNs with multiple delays. In Section 5, two numerical examples are provided to demonstrate the theoretical results. The last section gives some conclusions.


## 2. Preliminaries

This section we will give some Caputo definitions and lemma for fractional calculus as a basis for the theoretical analysis and simulation proofs.Definition 1.
(see [50]). The fractional integral of order ϕ for a function fx is defined as follows:(2)Iϕfx=1Γϕ∫x0xx−sϕ−1fsds,where ϕ>0, and Γ⋅ is the Gamma function satisfying Γs=∫0∞xs−1e−xdx.Definition 2.
(see [50]). Caputo fractional derivative of order ϕ for a function ψx∈Ckx0,∞,R is defined by(3)Dϕψx=1Γn−ϕ∫x0xψksx−sϕ−k+1ds,where x≥x0 and k−1≤ϕ<k, k∈N+.
Moreover, whenϕ∈0,1, then(4)Dϕψx=1Γ1−ϕ∫x0xψ′sx−sϕds.Lemma 1 (see [51]).
Consider the following fractional order autonomous model.(5)Dϕu=Ju,u0=u0,in which 0<ϕ≤1, u∈Rk, and J∈Rk×k. Then the zero solution of the system (5) is asymptotically stable in the Lyapunov sense if all roots λi are the system (5) of character equation satisfy argλi>ϕπ/2i=1,2,…,k, and then each component of the states decays towards 0 like t−ϕ. In addition, this model is stable if and only if argλi≥ϕπ/2 and those critical eigenvalues that satisfy argλi=ϕπ/2 have geometric multiplicity one.


## 3. Mathematics Model Elaboration

This article considers the following four-neuron fractional-order recurrent neural network with two delays:(6)Dϕx1t=−x1t+fx2t−τ1,Dϕx2t=−x2t+fx3t−τ1,Dϕx3t=−x3t+fx4t−τ1,Dϕx4t=−x4t+ω1fx1t−τ2+ω2fx2t−τ2+ω3fx3t−τ2,where ϕ∈0,1 are fractional order; xiti=1,2,3,4 stand for state variables; ωii=1,2,3 denote the connection weights; the function of connecting neurons is denoted by fx⋅; and τ1 and τ2 are the communication time delays.Remark 1.
In fact, ifϕ=1, the fractional delayed neural networks (6) changes into the general neural network (1).
Accordingly, the main purpose of this article is to investigate the stability and the application of Hopf bifurcations of the neural networks (6) taking different time delays τ1 and τ2 as the bifurcation parameters by the method of stability analysis [52]. In addition, the effects of the order on the creation of the Hopf bifurcation for the proposed fractional order neural network with multiple delays are also numerically discussed.
Throughout of this paper, assume that the following condition holds true:
C1f⋅∈CR,R, f0=0, xfx>0, for x≠0.


## 4. Main Results

This section choosesτ1 or τ2 as a bifurcation parameter to study the stability analysis and Hopf bifurcation for the fractional order RNNs (6) and to study the bifurcation points accurately.


### 4.1. Bifurcation Depending onτ1 in Equation (6)

In this subsection, we first study the effects ofτ1 on bifurcations of system (6) by establishing τ2.Applying Taylor series formula, the following form of equation (6) at the origin is(7)Dϕx1t=−x1t+m1x2t−τ1,Dϕx2t=−x2t+m2x3t−τ1,Dϕx3t=−x3t+m3x4t−τ1,Dϕx4t=−x4t+m4x1t−τ2+m5x2t−τ2+m6x3t−τ2.By applying Laplace transformation, its characteristic equation is given as(8)detsϕ+1−m1esτ1000sϕ+1−m2e−sτ1000sϕ+1−m3e−sτ1−m4e−sτ2−m5e−sτ2−m6e−sτ2sϕ+1=0,where mk=f′0k=1,2,3,mk=ωjf′0j=1,2,3,k=4,5,6.From (8), we have(9)K1s+K2se−sτ1+K3se−2sτ1+K4se−3sτ1=0,where(10)K1s=s4ϕ+4s3ϕ+6s2ϕ+4sϕ+1,K2s=−m3m6s2ϕ+2sϕ+1e−sτ2,K3s=−m2m3m5sϕ+1e−sτ2,K4s=−m1m2m3m4e−sτ2.Multiplyingesτ1 and e2sτ1 on both sides of equation (9), respectively, we can obtain(11)K1se2sτ1+K2sesτ1+K3s+K4se−sτ1=0,K1sesτ1+K2s+K3se−sτ1+K4se−2sτ1=0.LetK1s=A1+iB1, K2s=A2+iB2, K3s=A3+iB3, K4s=A4+iB4, and from equation (9), we have(12)A1+iB1e2sτ1+A2+iB2esτ1+A3+iB3+A4+iB4e−sτ1=0,A1+iB1esτ1+A2+iB2+A3+iB3e−sτ1+A4+iB4e−2sτ1=0.Takes=iw=wcosπ/2+isinπ/2ω>0 be a purely imaginary root of equation (11). Apply inserting s into equation (11) and separating the imaginary and real parts yields the following equations:(13)A1cos2ωτ1−B1sin2ωτ1+A2+A4cosωτ1+B4−B2sinωτ1=−A3,B1cos2ωτ1+A1sin2ωτ1+B2+B4cosωτ1+A2−A4sinωτ1=−B3,A1cos2ωτ1−B1sin2ωτ1+A2+A4cosωτ1+B4−B2sinωτ1=−A3,B1cos2ωτ1+A1sin2ωτ1+B2+B4cosωτ1+A2−A4sinωτ1=−B3.Evidently,(14)cosωτ1=F12ωF11ω=Fc1ω,sinωτ1=F22ωF21ω=Fs1ω,where A1,A2,A3,A4,B1,B2,B3,B4,F11,F12,F21, and F22 are given Appendix A. Obviously, from fist to second equation of system (14), it can be implied that(15)Fc12ω+Fs12ω=1.From equation (13), one can obtain(16)τ1l=1warccosF12wF11w+2lπ,l=0,1,2,….Remark 2.
This is an inhomogeneous system of linear equations (13), and the independent variables are cos2ωτ1,sin2ωτ1,cosωτ1,sinωτ1, respectively. According Cramer’s rule of linear equation, if the coefficient determinant of the system of linear equations is not equal to 0, we can easily solve solutions of linear equations (13). That is, we can obtain cosωτ1 and sinωτ1 or cosω2τ1 and sin2ωτ1.
Define the bifurcation point of fractional neural network with multiple delays (6) as(17)τ10∗=minτ1l,l=0,1,2,….
Ifτ1 vanishes, then equation (9) becomes(18)H1s+H2se−sτ2=0,where(19)H1s=s4ϕ+4s3ϕ+6s2ϕ+4sϕ+1,H2s=−m3m6s2ϕ−2m3m6sϕ−m3m6−m2m3m5sϕ−m2m3m5−m1m2m3m4.
Ifτ2=0, then the equation (18) becomes(20)0=s4ϕ+4s3ϕ+6s2ϕ+4sϕ+1−m3m6s2ϕ−2m3m6sϕ−m3m6−m2m3m5sϕ−m2m3m5−m1m2m3m4.
Suppose that all rootss of the equation (18) obey Lemma 1, then we get that both roots λi in equation (18) have negative real parts.
The imaginary and real parts ofHjsj=1,2 can be denoted by HjIandHjR, respectively. Multiplying e2sτ2 on both sides of equation (18), we can obtain(21)H1sesτ2+H2s=0.
Also, lets=iv=vcosπ/2+isinπ/2v>0 be a purely imaginary root of equation (11) if and only if(22)H1Rcosvτ2−H1Isinvτ2=−H2R,H1Icosvτ2+H1Rsinvτ2=−H2I.
This leads to form(23)cosvτ2=−H2RH2R+H1IH2IH1R2+H1I2=fc1v,sinvτ1=−−H2RH1I+H1RH2IH1R2+H1R2=fs1v.
It is not difficult to see that(24)fc12w+fs12w=1.
Additionally, we will give the following assumptions which hold true.
C2 The equation (24) has at least a positive real root.
From equation (24), the values ofv can be obtained according to Mathematics software Mathematica 10.0, and then the Hopf bifurcation point τ20 of fractional order recurrent neural network (6) with τ1=0 can be derived.To demonstrate our main results, we further present the following hypothesis: C3Υ1Ω1+Υ2Ω2/Ω12+Ω22≠0, where(25)Υ1=w0A2sinw0τ10−B2cosw0τ10+2A3sin2w0τ10−B2cos2w0τ10+3A4cos3w0τ10+B4sin3w0τ10,Υ2=w0A2cosw0τ10+B2sinw0τ10+2A3cos2w0τ10+B2sin2w0τ10+3A4cos3w0τ10+B4sin3w0τ10,Ω1=A1′+A2′−τ1A2cosw0τ10+B2′−τ1B2sinw0τ10+A3′−2τ1A3cos2w0τ10+B2′−τ1B3sin2w0τ10+A4′−3τ1A4cos3w0τ10+B4′−3τ1B4sin3w0τ10,Ω2=B1′+B2′−τ1B2cosw0τ10−A2′−τ1A2sinw0τ10+B3′−2τ1B3cos2w0τ10−A2′−τ1A3sin2w0τ10+B4′−3τ1B4cos3w0τ10−A4′−3τ1A4sin3w0τ10.Lemma 2.
Letsτ1=ντ10+iwτ1 be a root of equation (9) near τ1=τ1j satisfying ντ1j=0, wτ1j=w0, then the following transversality condition is satisfied.(26)Redsdτ1|w=w0,τ1=τ10≠0.Proof.
With implicit function theorem, we can differentiate equation (9) with respect to τ1, and thus we get(27)0=K1′sdsdτ1+K2′se−sτ1dsdτ1+K2se−sτ1−τ1dsdτ1−s+K3′se−2sτ1dsdτ1+K3se−2sτ1−2τ1dsdτ1−2s+K4′se−3sτ1dsdτ1+K4se−3sτ−1−3τdsdτ1−3s.dsdτ1=ΥsΩs,where(28)Υs=sK2se−sτ1+2K3se2−sτ1+3K4se−3sτ1,Ωs=K1′s+K2′s−τ1K2se−2sτ1+K3′s−2τ1K3se−2sτ1+K4′s−3τ1K4se−3sτ1.
We further suppose thatΥ1 and Υ2 are the real and imaginary parts of Υs, respectively, and Ω1 and Ω2 are the real and imaginary parts of Ωs, respectively, then(29)Redsdτ|τ=τ0∗,w=w0∗=Υ1Ω1+Υ2Ω2Ω12+Ω22.
FromC3, we conclude that the transversality condition holds true. This completes the proof of Lemma 2.
From the above investigation, we can obtain the following results.Theorem 1.
assumptionsC1–C3 hold true, then the following results can be given:(i)
The zero equilibrium point of fractional order four-neuron recurrent neural network with multiple delays (6) is asymptotically stable when τ1∈0,τ10∗.(ii)
Ifτ1∈0,τ10∗, then fractional order four neurons recurrent neural network with multiple delays (6) causes Hopf bifurcation at the origin when τ1=τ10∗. That is, a branch of periodic solutions can bifurcate from the zero equilibrium point at τ1=τ10∗.


### 4.2. Bifurcation Depending onτ2 in Equation (6)

As in the previous subsection, next we change another delayτ2 to the bifurcation parameter to account for the bifurcation of the model (6). It is hard to point out that equation (8) changes as follows:(30)q1s+q2se−sτ2=0,where(31)q1s=1+4s2ϕ+6s2ϕ+4s3ϕ+s4ϕ,q2s=−m3m61+2sϕ+s2ϕe−sτ1−m2m3m51+sϕe−2sτ1−m1m2m3m4e−3sτ1.Multiplyinge2sτ2 on both sides of equation (30), we can obtain(32)q1sesτ2+q2s=0.Supposeq1s=a1+ib1 and q2s=a2+ib2, and from equation (32), we have(33)a1+ib1esτ2+a2+ib2=0,where a1,a2,b1,b2 are given in Appendix B.Takes=iw˜=w˜cosπ/2+isinπ/2ω˜>0 as a root of equation (33) if and only if(34)a1cosw˜τ2−b1sinw˜τ2=−a2,b1cosw˜τ2+a1sinw˜τ2=−b2,that is,(35)cosw˜τ2=−a1a2+b1b2a12+b12=ρw˜,sinw˜τ2=−−a2b1+a1b2a12+b12=ϱw˜.It is simple to derive the following equation.(36)ρ2w˜+ϱ2w˜=1.From (35), one can obtain(37)τ2l=1w˜arccosϱw˜+2lπ,l=0,1,2,….The bifurcation point is defined byωkk=1,2,3C3Υ1Ω1+Υ2Ω2Ω12+Ω22≠0(38)τ20∗=minτ˜2l,l=0,1,2,….C5α1β1+α2β2/α12+α22≠0 here τ2lis defined by equation (38)Ifτ2=0, then the equation (32) becomes(39)M1s+M2se−sτ1+M3se−2sτ1+M4se−3sτ1=0,where(40)M1s=1+4sϕ+6s2ϕ+4s3ϕ+s4ϕ,M2s=−m3m61+2sϕ+s2ϕM3s=−m2m3m51+sϕ,M4s=−m1m2m3m4.Assume that all rootss of equation (39) observe Lemma 1, then we get that both roots of equation (39) have negative real parts.The imaginary and real parts ofMisi=1,2,3,4 can be expressed as MilandMiR, respectively. Multiplying both sides of the equation (39) by e2sτ1 and esτ1 yields(41)M1se2sτ1+M2sesτ1+M3s+M4se−sτ1=0,M1sesτ1+M2s+M3se−sτ1+M4se−2sτ1=0.Lets=iv˜=v˜cosπ/2+isinπ/2v˜>0 be a solution of equation (41). Substituting s into equation (41) and separating the imaginary and real units yields the following equations:(42)M1Rcos2v˜τ1−M1Isin2v˜τ1+M2R+M4Rcosv˜τ1+M4I−P2Isinv˜τ1=−M3R,M1Icos2v˜τ1+M1Rsin2v˜τ1+M2I+M4Icosv˜τ1+M2R−P4Rsinv˜τ1=−M3I,P1Rcos2v˜τ1−M1Isin2v˜τ1+M2R+P4Rcosv˜τ1+M4I−P2Isinv˜τ1=−M3R,P1Icos2v˜τ1+M1Rsin2v˜τ1+M2I+P4Rcosv˜τ1+M2R−P4Rsinv˜τ1=−M3I,which lead to(43)cosv˜τ1=E12v˜E11v˜=Cv˜2,sinv˜τ1=E22v˜E21v˜=Sv˜2.Obviously, from first and second equation of system (43), we get(44)Cv˜2+Sv˜2=1.To theoretically gain the sufficient conditions for the Hopf bifurcation, we assume that the following assumptions hold true:C4 Equation (36) has at least a positive real root.By means of equation (36), the values of ω˜ can be obtained according to mathematical software Mathematica 10.0, and then the bifurcation point τ10 of recullrent fractional four-neuron neural networks (6) with τ2=0 can be derived.As a summary of our main results, we provide the following assumption: C5α1β1+α2β2/α12+α22≠0, where(45)α1=a1′+a2′−τ20a2cosw˜0τ20+b2′−τ2b2sinw˜0τ20,α2=b1′+b2′−τ20b2cosw˜0τ20−a2′−τ2a2sinw˜0τ20,β1=w˜0a2sinw˜0τ20−b2cosw˜0τ20,β2=w˜0a2cosw˜0τ20+b2sinw˜0τ20.Lemma 3.
Letsτ2=ητ2+iw˜τ2 be a root of equation (9) near τ2=τ2j satisfying ητ2j=0, wτ2j=w˜0, then we get the following transversality condition(46)Redsdτ2|w˜=w˜0,τ2=τ20≠0.Proof.
Similar to Lemma2, by utilizing the implicit function theorem and differentiating (9) with respect to τ2, we get(47)0=q1′sdsdτ2+q2′se−sτ2dsdτ2+q2se−sτ2−τ2dsdτ2−s,dsdτ2=βsαs,where(48)βs=sq2se−sτ2,αs=q1′s+q2′se−sτ2−τ2q2se−sτ2.
We further suppose thatα1 and α2 are the real and imaginary units of αs, respectively, and β1 and β2 are the real and imaginary parts of βs, respectively, then we get(49)Redsdτ2|w˜=w˜0,τ2=τ20=α1β1+α2β2α12+α22.
As a direct consequence ofC5, we can conclude that the transversality condition is satisfied. Then the proof of Lemma 3 is complete.
Based on the above analysis, the following conclusions can be drawn.Theorem 2.
By assuming that assumptionsC1, C4, and C5 are valid, the following conditions can be inferred:(i)
The zero equilibrium point of fractional order four-neuron recurrent neural network with multiple delays (6) is asymptotically stable when τ2∈0,τ20∗)(ii)
The fractional order four-neuron recurrent neural network with multiple delays (6) experiences a Hopf bifurcation at its origin when τ2=τ20∗; that is, a family of periodic solutions can bifurcate from the zero equilibrium point near τ2=τ20∗


## 4.1. Bifurcation Depending onτ1 in Equation (6)

In this subsection, we first study the effects ofτ1 on bifurcations of system (6) by establishing τ2.Applying Taylor series formula, the following form of equation (6) at the origin is(7)Dϕx1t=−x1t+m1x2t−τ1,Dϕx2t=−x2t+m2x3t−τ1,Dϕx3t=−x3t+m3x4t−τ1,Dϕx4t=−x4t+m4x1t−τ2+m5x2t−τ2+m6x3t−τ2.By applying Laplace transformation, its characteristic equation is given as(8)detsϕ+1−m1esτ1000sϕ+1−m2e−sτ1000sϕ+1−m3e−sτ1−m4e−sτ2−m5e−sτ2−m6e−sτ2sϕ+1=0,where mk=f′0k=1,2,3,mk=ωjf′0j=1,2,3,k=4,5,6.From (8), we have(9)K1s+K2se−sτ1+K3se−2sτ1+K4se−3sτ1=0,where(10)K1s=s4ϕ+4s3ϕ+6s2ϕ+4sϕ+1,K2s=−m3m6s2ϕ+2sϕ+1e−sτ2,K3s=−m2m3m5sϕ+1e−sτ2,K4s=−m1m2m3m4e−sτ2.Multiplyingesτ1 and e2sτ1 on both sides of equation (9), respectively, we can obtain(11)K1se2sτ1+K2sesτ1+K3s+K4se−sτ1=0,K1sesτ1+K2s+K3se−sτ1+K4se−2sτ1=0.LetK1s=A1+iB1, K2s=A2+iB2, K3s=A3+iB3, K4s=A4+iB4, and from equation (9), we have(12)A1+iB1e2sτ1+A2+iB2esτ1+A3+iB3+A4+iB4e−sτ1=0,A1+iB1esτ1+A2+iB2+A3+iB3e−sτ1+A4+iB4e−2sτ1=0.Takes=iw=wcosπ/2+isinπ/2ω>0 be a purely imaginary root of equation (11). Apply inserting s into equation (11) and separating the imaginary and real parts yields the following equations:(13)A1cos2ωτ1−B1sin2ωτ1+A2+A4cosωτ1+B4−B2sinωτ1=−A3,B1cos2ωτ1+A1sin2ωτ1+B2+B4cosωτ1+A2−A4sinωτ1=−B3,A1cos2ωτ1−B1sin2ωτ1+A2+A4cosωτ1+B4−B2sinωτ1=−A3,B1cos2ωτ1+A1sin2ωτ1+B2+B4cosωτ1+A2−A4sinωτ1=−B3.Evidently,(14)cosωτ1=F12ωF11ω=Fc1ω,sinωτ1=F22ωF21ω=Fs1ω,where A1,A2,A3,A4,B1,B2,B3,B4,F11,F12,F21, and F22 are given Appendix A. Obviously, from fist to second equation of system (14), it can be implied that(15)Fc12ω+Fs12ω=1.From equation (13), one can obtain(16)τ1l=1warccosF12wF11w+2lπ,l=0,1,2,….Remark 2.
This is an inhomogeneous system of linear equations (13), and the independent variables are cos2ωτ1,sin2ωτ1,cosωτ1,sinωτ1, respectively. According Cramer’s rule of linear equation, if the coefficient determinant of the system of linear equations is not equal to 0, we can easily solve solutions of linear equations (13). That is, we can obtain cosωτ1 and sinωτ1 or cosω2τ1 and sin2ωτ1.
Define the bifurcation point of fractional neural network with multiple delays (6) as(17)τ10∗=minτ1l,l=0,1,2,….
Ifτ1 vanishes, then equation (9) becomes(18)H1s+H2se−sτ2=0,where(19)H1s=s4ϕ+4s3ϕ+6s2ϕ+4sϕ+1,H2s=−m3m6s2ϕ−2m3m6sϕ−m3m6−m2m3m5sϕ−m2m3m5−m1m2m3m4.
Ifτ2=0, then the equation (18) becomes(20)0=s4ϕ+4s3ϕ+6s2ϕ+4sϕ+1−m3m6s2ϕ−2m3m6sϕ−m3m6−m2m3m5sϕ−m2m3m5−m1m2m3m4.
Suppose that all rootss of the equation (18) obey Lemma 1, then we get that both roots λi in equation (18) have negative real parts.
The imaginary and real parts ofHjsj=1,2 can be denoted by HjIandHjR, respectively. Multiplying e2sτ2 on both sides of equation (18), we can obtain(21)H1sesτ2+H2s=0.
Also, lets=iv=vcosπ/2+isinπ/2v>0 be a purely imaginary root of equation (11) if and only if(22)H1Rcosvτ2−H1Isinvτ2=−H2R,H1Icosvτ2+H1Rsinvτ2=−H2I.
This leads to form(23)cosvτ2=−H2RH2R+H1IH2IH1R2+H1I2=fc1v,sinvτ1=−−H2RH1I+H1RH2IH1R2+H1R2=fs1v.
It is not difficult to see that(24)fc12w+fs12w=1.
Additionally, we will give the following assumptions which hold true.
C2 The equation (24) has at least a positive real root.
From equation (24), the values ofv can be obtained according to Mathematics software Mathematica 10.0, and then the Hopf bifurcation point τ20 of fractional order recurrent neural network (6) with τ1=0 can be derived.To demonstrate our main results, we further present the following hypothesis: C3Υ1Ω1+Υ2Ω2/Ω12+Ω22≠0, where(25)Υ1=w0A2sinw0τ10−B2cosw0τ10+2A3sin2w0τ10−B2cos2w0τ10+3A4cos3w0τ10+B4sin3w0τ10,Υ2=w0A2cosw0τ10+B2sinw0τ10+2A3cos2w0τ10+B2sin2w0τ10+3A4cos3w0τ10+B4sin3w0τ10,Ω1=A1′+A2′−τ1A2cosw0τ10+B2′−τ1B2sinw0τ10+A3′−2τ1A3cos2w0τ10+B2′−τ1B3sin2w0τ10+A4′−3τ1A4cos3w0τ10+B4′−3τ1B4sin3w0τ10,Ω2=B1′+B2′−τ1B2cosw0τ10−A2′−τ1A2sinw0τ10+B3′−2τ1B3cos2w0τ10−A2′−τ1A3sin2w0τ10+B4′−3τ1B4cos3w0τ10−A4′−3τ1A4sin3w0τ10.Lemma 2.
Letsτ1=ντ10+iwτ1 be a root of equation (9) near τ1=τ1j satisfying ντ1j=0, wτ1j=w0, then the following transversality condition is satisfied.(26)Redsdτ1|w=w0,τ1=τ10≠0.Proof.
With implicit function theorem, we can differentiate equation (9) with respect to τ1, and thus we get(27)0=K1′sdsdτ1+K2′se−sτ1dsdτ1+K2se−sτ1−τ1dsdτ1−s+K3′se−2sτ1dsdτ1+K3se−2sτ1−2τ1dsdτ1−2s+K4′se−3sτ1dsdτ1+K4se−3sτ−1−3τdsdτ1−3s.dsdτ1=ΥsΩs,where(28)Υs=sK2se−sτ1+2K3se2−sτ1+3K4se−3sτ1,Ωs=K1′s+K2′s−τ1K2se−2sτ1+K3′s−2τ1K3se−2sτ1+K4′s−3τ1K4se−3sτ1.
We further suppose thatΥ1 and Υ2 are the real and imaginary parts of Υs, respectively, and Ω1 and Ω2 are the real and imaginary parts of Ωs, respectively, then(29)Redsdτ|τ=τ0∗,w=w0∗=Υ1Ω1+Υ2Ω2Ω12+Ω22.
FromC3, we conclude that the transversality condition holds true. This completes the proof of Lemma 2.
From the above investigation, we can obtain the following results.Theorem 1.
assumptionsC1–C3 hold true, then the following results can be given:(i)
The zero equilibrium point of fractional order four-neuron recurrent neural network with multiple delays (6) is asymptotically stable when τ1∈0,τ10∗.(ii)
Ifτ1∈0,τ10∗, then fractional order four neurons recurrent neural network with multiple delays (6) causes Hopf bifurcation at the origin when τ1=τ10∗. That is, a branch of periodic solutions can bifurcate from the zero equilibrium point at τ1=τ10∗.


## 4.2. Bifurcation Depending onτ2 in Equation (6)

As in the previous subsection, next we change another delayτ2 to the bifurcation parameter to account for the bifurcation of the model (6). It is hard to point out that equation (8) changes as follows:(30)q1s+q2se−sτ2=0,where(31)q1s=1+4s2ϕ+6s2ϕ+4s3ϕ+s4ϕ,q2s=−m3m61+2sϕ+s2ϕe−sτ1−m2m3m51+sϕe−2sτ1−m1m2m3m4e−3sτ1.Multiplyinge2sτ2 on both sides of equation (30), we can obtain(32)q1sesτ2+q2s=0.Supposeq1s=a1+ib1 and q2s=a2+ib2, and from equation (32), we have(33)a1+ib1esτ2+a2+ib2=0,where a1,a2,b1,b2 are given in Appendix B.Takes=iw˜=w˜cosπ/2+isinπ/2ω˜>0 as a root of equation (33) if and only if(34)a1cosw˜τ2−b1sinw˜τ2=−a2,b1cosw˜τ2+a1sinw˜τ2=−b2,that is,(35)cosw˜τ2=−a1a2+b1b2a12+b12=ρw˜,sinw˜τ2=−−a2b1+a1b2a12+b12=ϱw˜.It is simple to derive the following equation.(36)ρ2w˜+ϱ2w˜=1.From (35), one can obtain(37)τ2l=1w˜arccosϱw˜+2lπ,l=0,1,2,….The bifurcation point is defined byωkk=1,2,3C3Υ1Ω1+Υ2Ω2Ω12+Ω22≠0(38)τ20∗=minτ˜2l,l=0,1,2,….C5α1β1+α2β2/α12+α22≠0 here τ2lis defined by equation (38)Ifτ2=0, then the equation (32) becomes(39)M1s+M2se−sτ1+M3se−2sτ1+M4se−3sτ1=0,where(40)M1s=1+4sϕ+6s2ϕ+4s3ϕ+s4ϕ,M2s=−m3m61+2sϕ+s2ϕM3s=−m2m3m51+sϕ,M4s=−m1m2m3m4.Assume that all rootss of equation (39) observe Lemma 1, then we get that both roots of equation (39) have negative real parts.The imaginary and real parts ofMisi=1,2,3,4 can be expressed as MilandMiR, respectively. Multiplying both sides of the equation (39) by e2sτ1 and esτ1 yields(41)M1se2sτ1+M2sesτ1+M3s+M4se−sτ1=0,M1sesτ1+M2s+M3se−sτ1+M4se−2sτ1=0.Lets=iv˜=v˜cosπ/2+isinπ/2v˜>0 be a solution of equation (41). Substituting s into equation (41) and separating the imaginary and real units yields the following equations:(42)M1Rcos2v˜τ1−M1Isin2v˜τ1+M2R+M4Rcosv˜τ1+M4I−P2Isinv˜τ1=−M3R,M1Icos2v˜τ1+M1Rsin2v˜τ1+M2I+M4Icosv˜τ1+M2R−P4Rsinv˜τ1=−M3I,P1Rcos2v˜τ1−M1Isin2v˜τ1+M2R+P4Rcosv˜τ1+M4I−P2Isinv˜τ1=−M3R,P1Icos2v˜τ1+M1Rsin2v˜τ1+M2I+P4Rcosv˜τ1+M2R−P4Rsinv˜τ1=−M3I,which lead to(43)cosv˜τ1=E12v˜E11v˜=Cv˜2,sinv˜τ1=E22v˜E21v˜=Sv˜2.Obviously, from first and second equation of system (43), we get(44)Cv˜2+Sv˜2=1.To theoretically gain the sufficient conditions for the Hopf bifurcation, we assume that the following assumptions hold true:C4 Equation (36) has at least a positive real root.By means of equation (36), the values of ω˜ can be obtained according to mathematical software Mathematica 10.0, and then the bifurcation point τ10 of recullrent fractional four-neuron neural networks (6) with τ2=0 can be derived.As a summary of our main results, we provide the following assumption: C5α1β1+α2β2/α12+α22≠0, where(45)α1=a1′+a2′−τ20a2cosw˜0τ20+b2′−τ2b2sinw˜0τ20,α2=b1′+b2′−τ20b2cosw˜0τ20−a2′−τ2a2sinw˜0τ20,β1=w˜0a2sinw˜0τ20−b2cosw˜0τ20,β2=w˜0a2cosw˜0τ20+b2sinw˜0τ20.Lemma 3.
Letsτ2=ητ2+iw˜τ2 be a root of equation (9) near τ2=τ2j satisfying ητ2j=0, wτ2j=w˜0, then we get the following transversality condition(46)Redsdτ2|w˜=w˜0,τ2=τ20≠0.Proof.
Similar to Lemma2, by utilizing the implicit function theorem and differentiating (9) with respect to τ2, we get(47)0=q1′sdsdτ2+q2′se−sτ2dsdτ2+q2se−sτ2−τ2dsdτ2−s,dsdτ2=βsαs,where(48)βs=sq2se−sτ2,αs=q1′s+q2′se−sτ2−τ2q2se−sτ2.
We further suppose thatα1 and α2 are the real and imaginary units of αs, respectively, and β1 and β2 are the real and imaginary parts of βs, respectively, then we get(49)Redsdτ2|w˜=w˜0,τ2=τ20=α1β1+α2β2α12+α22.
As a direct consequence ofC5, we can conclude that the transversality condition is satisfied. Then the proof of Lemma 3 is complete.
Based on the above analysis, the following conclusions can be drawn.Theorem 2.
By assuming that assumptionsC1, C4, and C5 are valid, the following conditions can be inferred:(i)
The zero equilibrium point of fractional order four-neuron recurrent neural network with multiple delays (6) is asymptotically stable when τ2∈0,τ20∗)(ii)
The fractional order four-neuron recurrent neural network with multiple delays (6) experiences a Hopf bifurcation at its origin when τ2=τ20∗; that is, a family of periodic solutions can bifurcate from the zero equilibrium point near τ2=τ20∗


## 5. Numerical Examples

To demonstrate the validity and feasibility of the conclusions reached in this paper, we provide two examples. The simulations were based on a prediction and correction scheme [53] of Adama–Bashforth–Moulton and step-size h=0.01.


### 5.1. Example 1

Consider the four-neuron fractional recurrent neural networks with multiple delays as(50)Dϕx1t=−x1t+fx2t−τ1,Dϕx2t=−x2t+fx3t−τ1,Dϕx3t=−x3t+fx4t−τ1,Dϕx4t=−x4t+ω1fx1t−τ2+ω2fx2t−τ2+ω3fx3t−τ2.Choose parametersϕ=0.9,ω1=2,ω2=ω3=−2, action function f⋅=tanh⋅; therefore, f0=tanh0=0,f′0=1.Let the initial values be selected asx10,x20,x30,x40=0.15,−0.14,0.1,0.2 for the system (50). First, taking fixed τ2 such that τ2=0.6 by complex computing, we get ω10=5.23599, and then τ10=0.312709. Obviously, it is easy to verify that the conditions in Theorem 1 are satisfied. The numerical simulations in Figures 1 and 2 that the zero equilibrium point of system (50) is locally asymptotically stable when τ1=0.25<τ10=0.312709. Moreover, Figures 3 and 4 simulates that the zero equilibrium point of system (50) is unstable, and Hopf bifurcation occurs when τ1=0.35>τ10=0.312709. The bifurcation diagrams are plotted in Figure 5, which illustrates the theoretical results.Figure 1
Time responses of system (50) with ϕ=0.9, τ1=0.25<τ10=0.312709.Figure 2
Phase diagrams of system (50) with ϕ=0.9, τ1=0.25<τ10=0.312709.Figure 3
Time responses of system (50) with ϕ=0.9, τ1=0.36>τ10=0.312709.Figure 4
Phase diagrams of system (50) with ϕ=0.9, τ1=0.36>τ10=0.312709.Figure 5
Bifurcation diagram of system (50) with ϕ=0.9, τ1=0.36>τ10=0.312709.


### 5.2. Example 2

The same as example 1, letϕ=0.95, and now we consider the following four-neurons fractional current network with double different delays:(51)D0.95x1t=−x1t+fx2t−τ1,D0.95x2t=−x2t+fx3t−τ1,D0.95x3t=−x3t+fx4t−τ1,D0.95x4t=−x4t+ω1fx1t−τ2+ω2fx2t−τ2+ω3fx3t−τ2.Takingω1=1,ω2=ω3=−1.5,ϕ=0.95, action function f⋅=tanh⋅, then f0=tanh0=0,f′0=1, and we first also set τ1=0.8, in the next step, we apply a complex calculation, and it obtains a ω˜20=1.02089 and τ20=0.329454. Thus, Theorem 2 yields that the zero solution 0,0,0,0 of the system (51) is locally asymptotically stable when τ2=0.22<τ20, which is simulated in Figures 6 and 7 which describes the impact of fractional order on τ20. In addition, the zero equilibrium point of the system (51) is unstable, and Hopf bifurcation occurs when τ2=0.38>τ20, as shown in Figures 8 and 9. Moreover, the bifurcation diagrams are plotted in Figure 10, which illustrates the theoretical results.Figure 6
Time responses of system (51) with ϕ=0.95, τ1=0.22<τ20=0.329454.Figure 7
Phase diagrams of system (51) with ϕ=0.95, τ1=0.22<τ20=0.329454.Figure 8
Time responses of system (51) with ϕ=0.95, τ1=0.38>τ20=0.329454.Figure 9
Phase diagrams of system (51) with ϕ=0.95, τ1=0.38>τ20=0.329454.Figure 10
Bifurcation diagrams of system (51) with ϕ=0.95, τ1=0.38>τ20=0.329454.Remark 3.
In fact, in order to better reflect the influence of different time delays at the bifurcation point of the systems (50) and (51), the corresponding bifurcation point τ10 and τ20 and τ10∗ and τ20∗ can be determined by changing the order of ϕ. This means that systems (50) and (51) involving different two delays are prone to earlier Hopf bifurcation for some fixed fractional order ϕ.


## 5.1. Example 1

Consider the four-neuron fractional recurrent neural networks with multiple delays as(50)Dϕx1t=−x1t+fx2t−τ1,Dϕx2t=−x2t+fx3t−τ1,Dϕx3t=−x3t+fx4t−τ1,Dϕx4t=−x4t+ω1fx1t−τ2+ω2fx2t−τ2+ω3fx3t−τ2.Choose parametersϕ=0.9,ω1=2,ω2=ω3=−2, action function f⋅=tanh⋅; therefore, f0=tanh0=0,f′0=1.Let the initial values be selected asx10,x20,x30,x40=0.15,−0.14,0.1,0.2 for the system (50). First, taking fixed τ2 such that τ2=0.6 by complex computing, we get ω10=5.23599, and then τ10=0.312709. Obviously, it is easy to verify that the conditions in Theorem 1 are satisfied. The numerical simulations in Figures 1 and 2 that the zero equilibrium point of system (50) is locally asymptotically stable when τ1=0.25<τ10=0.312709. Moreover, Figures 3 and 4 simulates that the zero equilibrium point of system (50) is unstable, and Hopf bifurcation occurs when τ1=0.35>τ10=0.312709. The bifurcation diagrams are plotted in Figure 5, which illustrates the theoretical results.Figure 1
Time responses of system (50) with ϕ=0.9, τ1=0.25<τ10=0.312709.Figure 2
Phase diagrams of system (50) with ϕ=0.9, τ1=0.25<τ10=0.312709.Figure 3
Time responses of system (50) with ϕ=0.9, τ1=0.36>τ10=0.312709.Figure 4
Phase diagrams of system (50) with ϕ=0.9, τ1=0.36>τ10=0.312709.Figure 5
Bifurcation diagram of system (50) with ϕ=0.9, τ1=0.36>τ10=0.312709.


## 5.2. Example 2

The same as example 1, letϕ=0.95, and now we consider the following four-neurons fractional current network with double different delays:(51)D0.95x1t=−x1t+fx2t−τ1,D0.95x2t=−x2t+fx3t−τ1,D0.95x3t=−x3t+fx4t−τ1,D0.95x4t=−x4t+ω1fx1t−τ2+ω2fx2t−τ2+ω3fx3t−τ2.Takingω1=1,ω2=ω3=−1.5,ϕ=0.95, action function f⋅=tanh⋅, then f0=tanh0=0,f′0=1, and we first also set τ1=0.8, in the next step, we apply a complex calculation, and it obtains a ω˜20=1.02089 and τ20=0.329454. Thus, Theorem 2 yields that the zero solution 0,0,0,0 of the system (51) is locally asymptotically stable when τ2=0.22<τ20, which is simulated in Figures 6 and 7 which describes the impact of fractional order on τ20. In addition, the zero equilibrium point of the system (51) is unstable, and Hopf bifurcation occurs when τ2=0.38>τ20, as shown in Figures 8 and 9. Moreover, the bifurcation diagrams are plotted in Figure 10, which illustrates the theoretical results.Figure 6
Time responses of system (51) with ϕ=0.95, τ1=0.22<τ20=0.329454.Figure 7
Phase diagrams of system (51) with ϕ=0.95, τ1=0.22<τ20=0.329454.Figure 8
Time responses of system (51) with ϕ=0.95, τ1=0.38>τ20=0.329454.Figure 9
Phase diagrams of system (51) with ϕ=0.95, τ1=0.38>τ20=0.329454.Figure 10
Bifurcation diagrams of system (51) with ϕ=0.95, τ1=0.38>τ20=0.329454.Remark 3.
In fact, in order to better reflect the influence of different time delays at the bifurcation point of the systems (50) and (51), the corresponding bifurcation point τ10 and τ20 and τ10∗ and τ20∗ can be determined by changing the order of ϕ. This means that systems (50) and (51) involving different two delays are prone to earlier Hopf bifurcation for some fixed fractional order ϕ.


## 6. Conclusion

This paper examines the Hopf bifurcation problem of fractional recurrent neural networks with four neurons and two delays. Using time delay as the bifurcation parameter, several criteria are destabilized in order to ensure the Hopf bifurcation for the fractional four-neuron of recurrent neural networks. Based on our analysis, different communication time delays and order effects have quantitatively changed the dynamic behavior of the system (6). These results can contribute to our understanding of delayed fractional recurrent neural networks as a continuation of the previous work. The results of the simulations are illustrated by two numerical examples.



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