# Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays

Authors: Zizhen Zhang; Huizhong Yang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

A four-dimensional recurrent neural network with two delays is considered. The main result is given in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the zero equilibrium and existence of the Hopf bifurcation with respect to both delays are obtained by analyzing the distribution of the roots of the associated characteristic equation. In particular, explicit formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form theory and center manifold theory. Some numerical examples are also presented to verify the theoretical analysis.


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## Body

## 1. Introduction

In recent years, neural networks have attracted many scholars’ attention all over the world and have been applied in different areas such as signal processing [1], pattern recognition [2–4], optimization [5], and automatic control [6–8]. In particular, the appearance of a cycle bifurcating from an equilibrium of an ordinary or a delayed neural network with a single parameter has been widely investigated [9–17]. In [18], Ruiz et al. studied the following recurrent neural network for the first time:
(1)x˙1(t)=-x1(t)+f(x2(t)),⋮x˙n-1=-xn-1(t)+u(t),x˙n=-xn(t)+w1f(x1(t))+⋯+wn-1f(xn-1(t)),y(t)=f(xn(t)),
where x(t)∈Rn is the state, wi∈R, i=1,…,n-1 are the network parameters or weights, u(t) is the input, y(t) is the output, and f(·) is the transfer function of the neurons. The three-node network of system (1) in the feedback configuration, with u(t)=y(t), has been studied in [12, 18, 19]; that is
(2)x˙1(t)=-x1(t)+f(x2(t)),x˙2(t)=-x2(t)+f(x3(t)),x˙3(t)=-x3(t)+w1f(x1(t))+w2f(x2(t)).It is well known that time delays can play a complicated role on neural networks. They can be the source of instabilities and bifurcation in neural networks. Based on this fact, Hajihosseini et al. [11] considered system (2) with distributed delays and f(·)=tanh(·). It is shown that a Hopf bifurcation takes place in the delayed system as the mean delay passes a critical value where a family of periodic solutions bifurcate from the equilibrium. The existence and stability of such solutions are determined by the Hopf bifurcation theorem in the frequency domain and the generalized Nyquist stability criterion.As far as we know, there are some papers on the bifurcations of neural network with two or multiple delays [20–22]. Motivated by the work in [11, 20–22] and considering that when the number of neurons is large, the simplified model can reflect the really large neural networks more closely, we consider the following four-dimensional recurrent neural network with two discrete delays that occur in the interaction between the neurons:
(3)x˙1(t)=-x1(t)+f(x2(t-τ1)),x˙2(t)=-x2(t)+f(x3(t-τ1)),x˙3(t)=-x3(t)+f(x4(t-τ1)),x˙4(t)=-x4(t)+w1f(x1(t-τ2))+w2f(x2(t-τ2))+w3f(x3(t-τ2)),
where τ1≥0, τ2≥0 are time delays that occur in the interaction between the neurons.This paper is organized as follows. In Section2, the stability of the zero equilibrium of system (3) and the existence of local Hopf bifurcation with respect to possible combinations of the two delays are investigated. In Section 3, the properties of the Hopf bifurcation such as the direction and the stability are determined by using the normal form theory and center manifold theory. Some numerical simulations are also included in Section 4 to illustrate the validity of the main results.


## 2. Stability of the Zero Equilibrium and Local Hopf Bifurcation

Throughout this paper we make the following assumption on the transfer functionf(·):(H)
f∈C4(R),  f(0)=0, and  f′(0)≠0.Clearly,E0=(0,0,0,0)T is the zero equilibrium of system (3). Linearization of system (3) at the zero equilibrium is
(4)x˙1(t)=-x1(t)+f′(0)x2(t-τ1),x˙2(t)=-x2(t)+f′(0)x3(t-τ1),x˙3(t)=-x3(t)+f′(0)x4(t-τ1),x˙4(t)=-x4(t)+w1f′(0)x1(t-τ2)+w2f′(0)x2(t-τ2)+w3f′(0)x3(t-τ2).
The characteristic equation of the linearized system (4) is
(5)(λ+1)4+A(λ+1)2e-λ(τ1+τ2)+B(λ+1)e-λ(2τ1+τ2)+Ce-λ(3τ1+τ2)=0,
where
(6)A=-w3f′2(0),B=-w2f′3(0),C=-w1f′4(0).In order to study the local stability of the zero equilibrium of system (3), we investigate the distribution of the roots of (5) in the following.Case 1 (τ1=τ2=0).
Equation (5) reduces to
(7)λ4+d1λ3+d2λ2+d3λ+d4=0,
where
(8)d1=4,d2=A+6,d3=2A+B+4,d4=B+C+1.
Obviously,D1=d1>0. Therefore, by the Routh-Hurwitz criterion, the zero equilibrium E0(0,0,0,0)T of system (3) is locally asymptotically stable if the following condition (H1) holds:
(9)D2=(d11d3d2)>0,D3=(d110d3d2d10d4d3)>0,D4=(d1100d3d2d110d4d3d2000d4)>0.Case 2 (τ1>0, τ2=0).
On substitutingτ2=0, (5) becomes
(10)(λ+1)4+A(λ+1)2e-λτ1+B(λ+1)e-2λτ1+Ce-3λτ1=0.
Multiplying eλτ1 on both sides of (10), it is easy to obtain
(11)A(λ+1)2+(λ+1)4eλτ1+B(λ+1)e-λτ1+Ce-2λτ1=0.
Let λ=iω1(ω1>0) be a root of (11). Then, we can get
(12)A11cosτ1ω1+A12sinτ1ω1+A13=A14,A21cosτ1ω1+A22sinτ1ω1+A23=A24,
where
(13)A11=ω14-6ω12+B-1,A12=4ω13+Bω1-4ω1,A13=A(1-ω12),A14=-Ccos2τ1ω1,A21=Bω1+4ω1-4ω13,A22=ω14-6ω12-B+1,A23=2Aω1,A24=Csin2τ1ω1.
Squaring both sides of the two equations in (12) and adding them up we obtain
(14)(A11cosτ1ω1+A12sinτ1ω1+A13)2+(A21cosτ1ω1+A22sinτ1ω1+A23)2=C2.
According to sinτ1ω1=±1-cos2τ1ω1, we consider the two cases:
(I) ifsinτ1ω1=1-cos2τ1ω1, then (14) takes the following form:
(15)(A11cosτ1ω1+A121-cos2τ1ω1+A13)2+(A21cosτ1ω1+A221-cos2τ1ω1+A23)2=C2,
which is equivalent to
(16)p1cos4τ1ω1+p2cos3τ1ω1+p3cos2τ1ω1+p4cosτ1ω1+p5=0,
where
(17)p1=(A112+A212-A122-A222)2+4(A11A12+A21A22)2,p2=4(A112+A212-A122-A222)(A11A13+A21A23)+8(A11A12+A21A22)(A12A13+A22A23),p3=4(A11A13+A21A23)2+4(A12A13+A22A23)2-4(A11A12+A21A22)2+2(A112+A212-A122-A222)×(A122+A132+A222+A232-C2),p4=4(A11A13+A21A23)(A122+A132+A222+A232-C2)-8(A11A12+A21A22)(A12A13+A22A23),p5=(A122+A132+A222+A232-C2)2-(A12A13+A22A23)2.
Let r=cosτ1ω1, and denote that
(18)f(r)=r4+p2p1r3+p3p1r2+p4p1r+p5p1.
Thus,
(19)f′(r)=4r3+3p2p1r2+2p3p1r+p4p1.
Let
(20)4r3+3p2p1r2+2p3p1r+p4p1=0.
Let y=r+(p2/4p1). Then, (20) becomes
(21)y3+γ1y+γ0=0,
where
(22)γ1=p32p1-3p2216p12,γ0=p2332p13-p2p38p12+p44p1.
Define
(23)β1=(γ22)2+(γ13)3,β2=-1+i32.
Then, we can get
(24)y1=-γ22+β13+-γ22-β13,y2=-γ22+β13β2+-γ22-β13β-2,y3=-γ22+β13β-2+-γ22-β13β2.
Then, we can get the expression of cosτ1ω1 and we denote f1(ω1)=cosτ1ω1. Substituting f1(ω1)=cosτ1ω1 into (12), we can get the expression of sinτ1ω1 and we denote f2(ω1)=sinτ1ω1. Thus, a function with respect to ω1 can be established by
(25)f12(ω1)+f22(ω1)=1.
We assume  that (H21), (25),  has finite positive roots, which are denoted by ω11,…,ω1k. For every fixed ω1i(1≤i≤k), the corresponding critical value of time delay is
(26)τ1i(j)=1ω1iarccosf1(ω1i)+2jπω1i,2jπω1ii=1,2,…,k;j=0,1,2,….
Then, ±ω1i are a pair of purely imaginary roots of (11) with τ1=τ1i(j). Let
(27)τ10=min{τ1i(j)},ω10=ω1i0,ω1i0i=1,2,…,k,j=0,1,2….
(II) Ifsinτ1ω1=-1-cos2τ1ω1, then (14) can be transformed into the following form:
(28)(A11cosτ1ω1-A121-cos2τ1ω1+A13)2+(A21cosτ1ω1-A221-cos2τ1ω1+A23)2=C2.Thus, similar as the process in case (I), we can get the expression ofcosτ1ω1 and sinτ1ω1. Let
(29)f1*(ω1)=cosτ1ω1,f2*=sinτ1ω1.
Therefore,
(30)f1*2(ω1)+f2*2(ω1)=1.Then, we can get the critical value of time delay corresponding to every fixed positive rootω1i′ of (30):
(31)τ1i(j)′=1ω1i′arccosf1(ω1i′)+2jπω1i′,i=1,2,…,k;j=0,1,2….
Let
(32)τ10=min{τ1i(j)′},ω10=ω1i0′,ω1i′i=1,2,…,k,j=0,1,2,….
Next, we verify the transversality. Taking the derivative of λ with respect to τ1 in (11), we obtain
(33)[dλdτ1]-1=2A(λ+1)+Be-λτ1+4(λ+1)3eλτ1λ[(λ+1)4eλτ1-B(λ+1)e-λτ1-2Ce-2λτ1]-τ1λ.
Thus,
(34)Re[dλdτ1]τ1=τ10-1=PRQR+PIQIQR2+QI2,
where
(35)PR=(4-12ω102+B)cosτ10ω10-(3ω10-ω103)sinτ10ω10+2A,PI=(4-12ω102-B)sinτ10ω10+(3ω10-ω103)cosτ10ω10+2Aω10,QR=(4ω104+Bω102-4ω102)cosτ10ω10-(Bω10+ω10-6ω103-ω105)sinτ10ω10-2Cω10sin2τ10ω10,QI=(4ω104-Bω102-4ω102)sinτ10ω10+(ω10-Bω10-6ω103-ω105)cosτ10ω10-2Cω10cos2τ10ω10.Obviously, if the condition(H22): PRQR+PIQI≠0 holds, then Re[dλ/dτ1]τ1=τ10-1≠0. Namely, if the condition (H22) holds, then the transversality condition is satisfied. By the discussion above and the Hopf bifurcation theorem in [23], it is easy to obtain the following results.Theorem 1.
If the condition(H21)  means that  (25) has finite positive roots and (H22)  means that  PRQR+PIQI≠0 holds, then the zero equilibrium E0 of system (3) is asymptotically stable for τ1∈[0,τ10), system (3) undergoes a Hopf bifurcation at E0 when τ1=τ10, and a branch of periodic solutions bifurcates from the zero equilibrium near τ1=τ10.Case 3 (τ2>0,  τ1=0).
Whenτ1=0, (5) becomes the following form:
(36)(λ+1)4+[Aλ2+(2A+B)λ+A+B+C]e-λτ2=0.
Let λ=iω2(ω2>0) be a root of (36). Substituting it into (36) and separating the real and imaginary parts, we obtain
(37)(2A+B)ω2sinτ2ω2+(A+B+C-Aω22)cosω2τ2=6ω22-ω24-1,(2A+B)ω2cosτ2ω2-(A+B+C-Aω22)sinω2τ2=4ω23-4ω2.
It follows that
(38)ω28+c3ω26+c2ω22+c1ω2+c0=0,
where
(39)c0=1-(A+B+C)2,c2=6-A2,c3=4.c1=2A(A+B+C)-(2A+B)2+4.
Let ω22=z, then (38) can be transformed into
(40)z4+c3z3+c2z2+c1z+c0=0.
Next, we make the following assumption.(H31)
means that  (40) has at least one positive root.Without loss of generality, we assume that (40) has four positive roots, which are denoted by z1, z2, z3, and z4. Thus, (38) has four positive roots ω2k=zk,  k=1,2,3,4. The corresponding critical value of time delay is
(41)τ2k(j)=1ω2karccos(×(6ω2k2-ω2k4-1)2)-1(Aω2k6+(A+3B-C)ω2k4+(2B+6C-A)ω2k2-(A+B+C))×((6ω2k2-ω2k4-1)2(2A+B)2ω2k2+(A+B+C-Aω2k2)2×(6ω2k2-ω2k4-1)2)-1)+2jπω2k,bbbbbbbbbbbbbbbbbbbbk=1,2,3,4;j=0,1,2….Then,±iω2k are a pair of purely imaginary roots of (36) with τ2=τ2k(j). Let
(42)τ20=min{τ2k(0)},k=1,2,3,4,ω20=ω2k0.
Taking the derivative of λ with respect to τ2 in (36), we can get
(43)[dλdτ2]-1=-4λ3+12λ2+12λ+4λ(λ4+4λ3+6λ2+4λ+1)+2Aλ+2A+Bλ[Aλ2+(2A+B)λ+A+B+C]-τ2λ.
Then, we can get
(44)Re[dλdτ2]τ=τ20-1=4ω206+12ω204+12ω202+4ω208+4ω206+6ω204+4ω202+1-(2A2ω202+(2A+B)2-2A(A+B+C))×(A2ω204+[(2A+B)2-2A(A+B+C)]×ω202+(A+B+C)2)-1.
From (38), we have
(45)ω208+4ω206+6ω204+4ω202+1=A2ω204+[(2A+B)2-2A(A+B+C)]ω202+(A+B+C)2.
Thus,(46)Re[dλdτ2]τ=τ20-1=g′(z0)ω208+4ω206+6ω204+4ω202+1,
where
(47)g(z)=z4+c3z3+c2z2+c1z+c0,z0=ω202.
Therefore, if the condition (H32): g′(z0)≠0, then Re[dλ/dτ2]τ=τ20-1≠0. From the analysis above and by the Hopf bifurcation theorem in [23], we have the following results.Theorem 2.
If the condition(H31)   means that   (40) has at least one positive root and (H32)  means that  g′(z0)≠0 holds, then the zero equilibrium E0 of system (3) is asymptotically stable for τ2∈[0,τ20), system (3) undergoes a Hopf bifurcation at E0 when τ2=τ20, and a branch of periodic solutions bifurcates from the zero equilibrium near τ2=τ20.Case 4 (τ1=τ2=τ>0).
Forτ1=τ2=τ>0, (5) can be transformed into the following form:
(48)(λ+1)4+A(λ+1)2e-2λτ+B(λ+1)e-3λτ+Ce-4λτ=0.
Multiplying e2λτ on both sides of (48), we obtain
(49)A(λ+1)2+(λ+1)4e2λτ+Ce-2λτ+B(λ+1)e-λτ=0.
Let λ=iω be a root of (49); then we have
(50)A-11cos2τ1ω1-A-12sin2τ1ω1+A-13=A-14,A-21sin2τ1ω1+A-22cos2τ1ω1+A-23=A-24,
where
(51)A-11=ω4-6ω2+C+1,A-12=4ω-ω3,A-13=A-Aω2,A-14=Bcosτω-Bωsinτω,A-21=ω4-6ω2-C+1,A-22=4ω-ω3,A-23=2Aω,A-24=Bcosτω+Bωsinτω.
Then, we get
(52)(A-11cos2τ1ω1-A-12sin2τ1ω1+A-13)2+(A-21sin2τ1ω1+A-22cos2τ1ω1+A-23)2=B2(1+ω2).Similar as in Case2, we can obtain the expression of cos2τω and sin2τω, which is denoted as g1(ω) and g2(ω), respectively. Further we can get a function with respect to ω(53)g12(ω)+g22(ω)=1.Next, we make the following assumption.(H41): Equation (53) has finite positive real roots, which are denoted by ω1,…,ωk, respectively. For every fixed positive root of (53), the corresponding critical value of time delay is
(54)τi(j)=12ωiarccosg1(ωi)+2jπ2ωi,2jπ2ωii=1,…,k;j=0,1,2,….
Then, ±iωi are a pair of purely imaginary roots of (49) with τ=τi(j). Let
(55)τ0=min{τi(j)},ω0=ωi0,i.i=1,2,…,k,j=0,1,2,….
Differentiating both sides of (49) with respect to t, we can obtain
(56)[dλdτ]-1=2A(λ+1)+4(λ+1)3e2λτ+Be-λτBλ(λ+1)e-λτ+2Cλe-2λτ-2λ(λ+1)4e2λτ-τλ.
Thus,
(57)Re[dλdτ2]τ=τ0-1=PR′QR′+PI′QI′QR′2+QI′2,
where
(58)PR′=(1-3ω02)cos2τ0ω0-(3ω0-ω03)sin2τ0ω0+Bcosτ0ω0+2A,PI′=(1-3ω02)sin2τ0ω0+(3ω0-ω03)cos2τ0ω0-Bcosτ0ω0+2Aω0,QR′=Bω0sinτ0ω0-Bω02cosτ0ω0+(ω05+(1+2C)ω0-6ω06)sin2τ0ω0-8(ω04-ω02)cos2τ0ω0,QI′=Bω0cosτ0ω0+Bω02sinτ0ω0-(ω05+(1+2C)ω0-6ω06)cos2τ0ω0-8(ω04-ω02)sin2τ0ω0.Obviously, if the condition(H42): PR′QR′+PI′QI′≠0 holds, then Re[dλ/dτ2]τ=τ0-1≠0. Namely, if the condition (H42) holds, the transversality condition is satisfied. Thus, by the Hopf bifurcation theorem in [23] we have the following results.Theorem 3.
If the condition(H41)   means that   (53) has finite positive real roots and (H42)  means that  PR′QR′+PI′QI′≠0 holds, then the zero equilibrium E0 of system (3) is asymptotically stable for τ∈[0,τ0), system (3) undergoes a Hopf bifurcation at E0 when τ=τ0, and a branch of periodic solutions bifurcates from the zero equilibrium near τ=τ0.Case 5 (τ1>0 and τ2>0).
We consider (5) with τ1 in its stable interval and τ2 is considered as a parameter. Without loss of generality, we consider (5) under Case 2.Letλ=iω2*(ω2*>0) be a root of (5). Then, we can get
(59)ω8+4ω6+6ω4+4ω2+1+2B(Aω3+Cω)sinτ1ω-2B(A+C)cosτ1ω+2AC(ω2-1)cos2τ1ω+2ACωsin2τ1ω=0.
Suppose that (H51)   means that   (59) has finite positive real roots, which are denoted as ω21*,ω22*,…,ω2k*. For every positive real root ω2i*(i=1,2,…,k), their exists a sequence {τ2i*(j)∣j=0,1,2,…}, such that (59) has a pair of purely imaginary roots ±iω2i* when τ2=τ2i*(j).Letτ2*=min{τ2i*(j)∣j=0,1,2,…}, and when τ2=τ2* (59) has a pair of purely imaginary roots ±iω2*. In the following, we make the following assumption.(H52)
:Re[dλ/dτ2]τ2=τ2*-1≠0.Through the analysis above and by the Hopf bifurcation theorem in [23], we have the following results.Theorem 4.
If the condition(H51) means that  (59) has finite positive real roots and (H52)  means that  Re[dλ/dτ2]τ2=τ2*-1≠0 holds, and τ1∈(0,τ10), then the zero equilibrium E0 of system (3) is asymptotically stable for τ2∈[0,τ2*), system (3) undergoes a Hopf bifurcation at E0 when τ2=τ2*, and a branch of periodic solutions bifurcates from the zero equilibrium near τ2=τ2*.


## 3. Stability of Bifurcated Periodic Solutions

In this section, the formulae for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions of system (3) with respect to τ2 for τ1∈(0,τ10) are derived by using the normal form method and center manifold theorem introduced by Hassard et al. [23]. Throughout this section, it is considered that system (3) undergoes Hopf bifurcation at τ2=τ2* and τ1∈(0,τ10). Without loss of generality, we assume that τ1*<τ2*, where τ1*∈(0,τ10).For convenience, lett=sτ2,u-i(t)=ui(τ2t),  (i=1,2,3,4). Drop the bars for simplification of notations. Then system (3) becomes
(60)u˙(t)=Lμut+F(μ,ut),
where u(t)=(u1(t),u2(t),u3(t),u4(t))T∈C=C([-1,0],R4) and Lμ:C→R4, F:R×C→R4 are given, respectively, by
(61)Lμϕ=(τ2*+μ)(A′ϕ(0)+B′ϕ(-τ1*τ2*)+C′ϕ(-1)),F(μ,ϕ)=(τ2*+μ)(F1,F2,F3,F4)T,
with
(62)ϕ(θ)=(ϕ1(θ),ϕ2(θ),ϕ3(θ),ϕ4(θ))T∈C([-1,0],R4),A′=(-10000-10000-10000-1),B′=(0f′(0)0000f′(0)0000f′(0)0000),C′=(000000000000w1f′(0)w2f′(0)w3f′(0)0),F1=f′′2!ϕ22(-τ1*τ2*)+f′′′3!ϕ23(-τ1*τ2*)+⋯,F2=f′′2!ϕ32(-τ1*τ2*)+f′′′3!ϕ33(-τ1*τ2*)+⋯,F3=f′′2!ϕ42(-τ1*τ2*)+f′′′3!ϕ43(-τ1*τ2*)+⋯,F4=w1f′′(0)2!ϕ12(-1)+w1f′′′(0)3!ϕ13(-1)+w2f′′(0)2!ϕ22(-1)+w2f′′′(0)3!ϕ23(-1)+w3f′′(0)2!ϕ12(-1)+w3f′′′(0)3!ϕ33(-1)+⋯.Therefore, according to the Riesz representation theorem, there exists a4×4 matrix function η(θ,μ):[-1,0]→R4 whose elements are of bounded variation such that
(63)Lμϕ=∫-10‍dη(θ,μ)ϕ(θ),ϕ∈C([-1,0],R4).
In fact, we choose
(64)η(θ,μ)={(τ2*+μ)(A′+B′+C′),θ=0,(τ2*+μ)(B′+C′),θ∈[-τ1*τ2*,0),(τ2*+μ)C′,θ∈(-1,-τ1*τ2*),0,θ=-1.
For ϕ∈C([-1,0],R4), we define
(65)A(μ)ϕ={dϕ(θ)dθ,-1≤θ<0,∫-10‍dη(θ,μ)ϕ(θ),θ=0,R(μ)ϕ={0,-1≤θ<0,F(μ,ϕ),θ=0.
Then system (60) can be transformed into the following operator equation:
(66)u˙(t)=A(μ)ut+R(μ)ut,
where ut=u(t+θ) for θ∈[-1,0].Forφ∈C′([0,1],(R4)*), where (R4)* is the 4-dimensional space of row vector, we define the adjoint operator A* of A:
(67)A*(φ)={-dφ(s)ds,0<s≤1,∫-10‍dηT(s,0)φ(-s),s=0,
and a bilinear inner product
(68)〈φ(s),ϕ(θ)〉=φ-(0)ϕ(0)-∫θ=-10‍∫ξ=0θ‍φ-(ξ-θ)dη(θ)ϕ(ξ)dξ,
where η(θ)=η(θ,0).ThenA(0) and A*(0) are adjoint operators. From the discussion above, we know that ±iω2*τ2* are eigenvalues of A(0) and they are also eigenvalues of A*(0). Let q(θ)=(1,q2,q3,q4)Teiω2*τ2*θ be the eigenvector of A(0) corresponding to the eigenvalue +iω2*τ2*, and let q*(s)=D(1,q2*,q3*,q4*)eiω2*τ2*s be the eigenvector of A*(0) corresponding to the eigenvalue -iω2*τ2*. Then, we have
(69)A(0)q(θ)=iω2*τ2*q(θ),A*(0)q*(0)=-iω2*τ2*q*(θ).
By a simple computation, we can obtain
(70)q2=iω2*+1f′(0)e-iω2*τ1*,q3=q2f′(0)e-iω2*τ1*-iω2*,q4=f′(0)(w1+w2q2+w3q3)(iω2*+1)eiω2*τ2*,q4*=1-iω2*w1f′(0)eiω2*τ2*,q2*=eiω2*τ1*+w2q4*eiω2*τ2*1-iω2*f′(0),q3*=q2*eiω2*τ1*+w3q4*eiω2*τ2*1-iω2*f′(0)
and 〈q*,q〉=1, 〈q*,q-〉=0.From (68), we can get
(71)D-=[e-iω2*τ1*1+q2q-2*+q3q-3*+q4q-4*c+τ1*f′(0)(q2+q-2*q3+q-3*q4)e-iω2*τ1*c+τ2*f′(0)q-4*(w1+w2q2+w3q3)e-iω2*τ2*]-1.Following the algorithms given in [23] and using similar computation process in [24], we can get the coefficients which can be used to determine direction of the Hopf bifurcation and stability of the bifurcating periodic solutions:
(72)g20=f′′(0)D-[e-2iω2*τ1*(q22+q-2*q32+q-3*q42)+q-4*e-2iω2*τ2*(w1+w2q22+w3q32)],g11=f′′(0)D-[q2q-2+q-2*q3q-3+q-3*q4q-4+q-4*(w1+w2q2q-2+w3q3q-3)],g02=f′′(0)D-[e2iω2*τ1*(q-22+q-2*q-32+q-3*q-42)+q-4*e2iω2*τ2*(w1+w2q-22+w3q-32)],g21=D-[f′′(0)(2W11(2)(-τ1*τ2*)q2e-iω2*τ1*+W20(2)(-τ1*τ2*)q-2eiω2*τ1*)+f′′′(0)q22q-2e-iω2*τ1*+q-2*(f′′(0)(2W11(3)(-τ1*τ2*)q3e-iω2*τ1*+W20(3)(-τ1*τ2*)q-3eiω2*τ1*)+f′′′(0)q32q-3e-iω2*τ1*(-τ1*τ2*))+q-3*(f′′(0)(2W11(4)(-τ1*τ2*)q2e-iω2*τ1*+W20(4)(-τ1*τ2*)q-2eiω2*τ1*)+(-τ1*τ2*)f′′′(0)q42q-4e-iω2*τ1*)+q-4*(w1f′′(0)(2W11(1)(-1)e-iω2*τ2*+W20(1)(-1)eiω2*τ2*)+w1f′′′(0)e-iω2*τ2*+w2f′′(0)(2W11(2)(-1)q2e-iω2*τ2*+W20(2)(-1)q-2eiω2*τ2*)+w2f′′′(0)q22q-2e-iω2*τ2*+w3f′′(0)(2W11(3)(-1)q3e-iω2*τ2*+W20(3)(-1)q-3eiω2*τ2*)(-τ1*τ2*)+w3f′′′(0)q32q-3e-iω2*τ2*)],
with
(73)W20(θ)=ig20q(0)ω2*τ2*eiω2*τ2*θ+ig-02q-(0)3ω2*τ2*e-iω2*τ2*θ+E1e2iω2*τ2*θ,W11(θ)=-ig11q(0)ω2*τ2*eiω2*τ2*θ+ig-11q-(0)ω2*τ2*e-iω2*τ2*θ+E2,
where E1 and E2 can be computed by the following equations, respectively:
(74)(2iω2*+1α120002iω2*+1α230002iω2*+1α34α41α42α432iω2*+1)E1=(E1(1)E1(2)E1(3)E1(4)),(1-f′(0)0001-f′(0)0001-f′(0)-w1f′(0)-w2f′(0)-w3f′(0)1)E2=-(E2(1)E2(2)E2(3)E2(4)),
with
(75)α12=α23=α34=-f′(0)e-2iω2*τ1*,α41=-w1f′(0)e-2iω2*τ2*,α42=-w2f′(0)e-2iω2*τ2*,α43=-w3f′(0)e-2iω2*τ2*.E1(1)=f′′(0)q22e-2iω2*τ1*,E1(2)=f′′(0)q32e-2iω2*τ1*,E1(3)=f′′(0)q42e-2iω2*τ1*,E1(4)=f′′(0)(w1+w2q22+w3q32)e-2iω2*τ2*,E2(1)=f′′(0)q2q-2,E2(2)=f′′(0)q3q-3,E2(3)=f′′(0)q4q-4,E2(4)=f′′(0)(w1+w2q2q-2+w3q3q-3).Therefore, we can calculate the following values:(76)C1(0)=i2ω2*τ2*(g11g20-2|g11|2-|g02|23)+g212,μ2=-Re{C1(0)}Re{λ′(τ2*)},β2=2Re{C1(0)},T2=-Im{C1(0)}+μ2Im{λ′(τ2*)}ω2*τ2*.Based on the discussion above, we can obtain the following results.Theorem 5.
For system (3),(i)
μ2 determines the direction of the Hopf bifurcation. If μ2>0(μ2<0); then the Hopf bifurcation is supercritical (subcritical);(ii)
β2 determines the stability of the bifurcating periodic solutions. If β2<0(β2>0); then the bifurcating periodic solutions are stable (unstable);(iii)
T2 determines the period of the bifurcating periodic solutions. If T2>0(T2<0); then the period of the bifurcating periodic solutions increases (decreases).


## 4. Numerical Simulation

In this section, we present some numerical simulations to support the theoretical analysis in Sections2 and 3. As an example, we consider the following special case of system (3) with the parameters w1=1, w2=-1, w3=-1, and f(x)=tanh(x). Then f(0)=0, f′(0)=1, and system (3) becomes
(77)x˙1(t)=-x1(t)+tanh(x2(t-τ1)),x˙2(t)=-x2(t)+tanh(x3(t-τ1)),x˙3(t)=-x3(t)+tanh(x4(t-τ1)),x˙4(t)=-x4(t)+tanh(x1(t-τ2))-tanh(x2(t-τ2))-tanh(x3(t-τ2)).Obviously,E0(0,0,0,0) is the equilibrium of system (77). By a simple computation, we get D2=21>0, D3=131>0, and D4=131>0. That is, the condition (H1) holds.Forτ1>0,τ2=0. We can obtain ω10=1.7216,τ10=1.4022 by some complicated computations. From Theorem 1, we know that E0(0,0,0,0) is asymptotically stable when τ1<τ10 as illustrated by Figures 1 and 2. When τ1 passes through, the critical value τ10, E0(0,0,0,0) becomes unstable and a Hopf bifurcation occurs and a branch of periodic solutions bifurcate from E0(0,0,0,0), which can be seen from Figures 3 and 4. Similarly, we have ω20=0.5194,τ20=3.1610 for τ1=0,τ2>0. The corresponding waveforms and the phase plots are shown in Figures 5, 6, 7, and 8.The trajectory ofx1, x2, x3, and x4 when τ1=1.3<1.4022=τ10.
(a)
(b)
(c)
(d)Figure 2
The phase plot ofx1, x2, and x3 when τ1=1.3<1.4022=τ10.The trajectory ofx1, x2, x3, and x4 when τ1=1.5>1.4022=τ10.
(a)
(b)
(c)
(d)Figure 4
The phase plot ofx1, x2, and x3 when τ1=1.5>1.4022=τ10.The trajectory ofx1, x2, x3, and x4 when τ2=2.75<3.1610=τ20.
(a)
(b)
(c)
(d)Figure 6
The phase plot ofx1, x2, and x3 when τ2=2.75<3.1610=τ20.The trajectory ofx1, x2, x3, and x4 when τ2=3.5>3.1610=τ20.
(a)
(b)
(c)
(d)Figure 8
The phase plot ofx1, x2, and x3 when τ2=3.5>3.1610=τ20.Forτ1=τ2=τ>0, we obtain ω0=2.0967,τ0=0.7915. From Theorem 3, when τ increases from zero to the critical value τ0, E0(0,0,0,0) is asymptotically stable, then it will lose its stability and a Hopf bifurcation occurs once τ>τ0. These properties can be shown in Figures 9, 10, 11, and 12.The trajectory ofx1, x2, x3, and x4 when τ=0.7<0.7915=τ0.
(a)
(b)
(c)
(d)Figure 10
The phase plot ofx1, x2, and x3 when τ=0.7<0.7915=τ0.The trajectory ofx1, x2, x3, and x4 when τ=0.85>0.7915=τ0.
(a)
(b)
(c)
(d)Figure 12
The phase plot ofx1, x2, and x3 when τ=0.85>0.7915=τ0.Lastly, forτ2>0 and τ1*=0.35∈(0,τ10), we get ω2*=1.3743, τ2*=1.7488. By Theorem 4, E0(0,0,0,0) is asymptotically stable when τ2∈[0,τ2*), and E0(0,0,0,0) is unstable when τ2>τ2* and a Hopf bifurcation occurs, which can be illustrated by Figures 13, 14, 15, and 16.The trajectory ofx1, x2, x3, and x4 when τ2=1.65<1.7488=τ2* and τ1*=0.35.
(a)
(b)
(c)
(d)Figure 14
The phase plot ofx1, x2, and x3 when τ2=1.65<1.7488=τ2* and τ1*=0.35.The trajectory ofx1, x2, x3, and x4 when τ2=1.85>1.7488=τ2* and τ1*=0.35.
(a)
(b)
(c)
(d)Figure 16
The phase plot ofx1, x2, and x3 when τ2=1.85>1.7488=τ2* and τ1*=0.35.


## 5. Conclusion

In this paper, we have investigated a four-dimensional recurrent neural network with two discrete delays. Compared with the literature [11], we consider the neural network model which can reflect the really large neural networks more closely. By regarding the possible combinations of the two delays as the bifurcation parameter, sufficient conditions for the local stability of the zero equilibrium and the existence of Hopf bifurcation are obtained. If the conditions are satisfied, then there exists a critical value of the time delay below which the system is stable and above which the system is unstable. The results have shown that the two delays can play a complicated role on the model. And from the numerical simulations, we find that τ1 is marked in the model because the critical value of τ1 is much smaller than that of τ2 when we only consider them, respectively. Furthermore, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are discussed by the normal form theory and center manifold theory. Finally, some numerical simulations are also presented to support the theoretical analysis.



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*Source: 436254-2013-11-21.xml*