# Stability and Bifurcation Analysis in a Class of Two-Neuron Networks with Resonant Bilinear Terms

Authors: Changjin Xu; Xiaofei He
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

A class of two-neuron networks with resonant bilinear terms is considered. The
stability of the zero equilibrium and existence of Hopf bifurcation is studied. It is shown that the zero
equilibrium is locally asymptotically stable when the time delay is small enough, while change of stability
of the zero equilibrium will cause a bifurcating periodic solution as the time delay passes through a
sequence of critical values. Some explicit formulae for determining the stability and the direction of the
Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal
form theory and center manifold theory. Finally, numerical simulations supporting the theoretical analysis
are carried out.


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## Body

## 1. Introduction

Based on the assumption that the elements in the network can respond to and communicate with each other instantaneously without time delays, Hopfield proposed Hopfield neural networks (HNNs) model in 1980s [1, 2]. During the past several years, the dynamical phenomena of neural networks have been extensively studied because of the widely application in various information processing, optimization problems, and so forth. In particular, the appearance of a cycle bifurcating from an equilibrium of an ordinary or a delayed neural network with a single parameter, which is known as a Hopf bifurcation, has attracted much attention (see [3–13]).In 2008, Yang et al. [14] investigated the Bautin bifurcation of the two-neuron networks with resonant bilinear terms and without delay:(1.1)ẋ1(t)=(α1+a)f(x1)+(α2+b)f(x2)+cx1x2,ẋ2(t)=(α2-b)f(x1)+(α1-a)f(x2)+dx1x2,
where xi(t)(i=1,2) represents the state of the ith neuron at time t, f(xi)(i=1,2) is the connection function between two neurons, and α1,α2,a,b,c,d are real parameters, and obtained a sufficient condition for a Bautin bifurcation to occur for system (1.1) by using the standard normal form theory and with Maple software. It is well known that in the implementation of networks, time delays are inevitably encountered because of the finite switching speed of signal transmission. Motivated by the viewpoint, in the following, we assume that the time delay from the first neuron to the second neuron is τ2 and back to the first neuron is τ1, then we have the following neural networks whose delays are introduced:(1.2)ẋ1(t)=(α1+a)f(x1)+(α2+b)f(x2(t-τ1))+cx1x2,ẋ2(t)=(α2-b)f(x1(t-τ2))+(α1-a)f(x2)+dx1x2,
where xi(t)(i=1,2) represents the state of the i-th neuron at time t, f(xi)(i=1,2) is the connection function between two neurons, α1,α2,a,b,c,d are real parameters, and τ1,τ2 are positive constants. We all know that time delays that occurred in the interaction between neurons will affect the stability of a network by creating instability, oscillation, and chaos phenomena.The purpose of this paper is to discuss the stability and the properties of Hopf bifurcation of model (1.2). To the best of our knowledge, it is the first to deal with the stability and Hopf bifurcation of the system (1.2).This paper is organized as follows. In Section2, the stability of the equilibrium and the existence of Hopf bifurcation at the equilibrium are studied. In Section 3, the direction of Hopf bifurcation and the stability and  periods of bifurcating periodic solutions  on the center manifold are determined. In Section 4, numerical simulations are carried out to illustrate the validity of the main results. Some main conclusions are drawn in Section 5.


## 2. Stability of the Equilibrium and Local Hopf Bifurcations

Throughout this paper, we assume that the functionf satisfies the following conditions:(H1)f∈C3(R),f(0)=0, and uf(u)>0,    for u≠0.Hypothesis (H1) implies thatE*(0,0) is an equilibrium of the system (1.2) and linearized system of (1.2) takes the form(2.1)ẋ1(t)=(α1+a)f′(0)x1+(α2+b)f′(0)x2(t-τ1),ẋ2(t)=(α2-b)f′(0)x1(t-τ2)+(α1-a)f′(0)x2.
The associated characteristic equation of (2.1) is(2.2)λ2-2α1f′(0)λ+(α12-a2)f′2(0)-(α22-b2)f′2(0)e-λτ=0,
where τ=τ1+τ2.In the section, we consider the sum of two delays as the parameter to give some conditions that separate the first quadrant of the(τ1,τ2) plane into two parts, one is the stable region another is the unstable region, and the boundary is the Hopf bifurcation curve.In order to investigate the distribution of roots of the transcendental equation (2.2), the following Lemma that is stated in [15] is useful.Lemma 2.1 (see [15]).
For the transcendental equation(2.3)P(λ,e-λτ1,…,e-λτm)=λn+p1(0)λn-1+⋯+pn-1(0)λ+pn(0)+[p1(1)λn-1+⋯+pn-1(1)λ+pn(1)]e-λτ1+⋯+[p1(m)λn-1+⋯+pn-1(m)λ+pn(m)]e-λτm=0,
as (τ1,τ2,τ3,…,τm) vary, the sum of orders of the zeros of P(λ,e-λτ1,…,e-λτm) in the open right half plane can change and only a zero appears on or crosses the imaginary axis.Now we make the following assumptions:(H2)α1f′(0)<0 and α12-α22-a2+b2>0;(H3)|α12-a2|<|α22-b2|.Lemma 2.2.
If (H1)–(H3) hold, then one has the following.
(i)
When(2.4)τ=τj=def1ω0[arccos2α1ω0(α22-b2)f′2(0)+2jπ],j=0,1,2,….
Equation (2.2) has a simple pair of imaginary roots ±iω0, where
(2.5)ω0=(α12+a2)2-[(α12-a2)2-(α22-b2)2]f′4(0)-(α12+a2).
(ii)
Forτ∈[0,τ0), all roots of (2.2) have strictly negative real parts.
(iii)
Whenτ=τ0, (2.2) has a pair of imaginary roots ±iω0 and all other roots have strictly negative real parts.Proof.
Obviously, by assumption (H2),λ=0 is not the root of (2.2). When τ=0, then (2.2) becomes
(2.6)λ2-2α1f′(0)λ+[α12-α22-a2+b2]f′2(0)=0.
It is easy to see that all roots of (2.6) have negative real parts.
±
i
ω
(
ω
>
0
) is a pair of purely imaginary roots of (2.2) if and only if ω satisfies
(2.7)-ω2-2α1f′(0)ωi+(α12-a2)f′2(0)-(α2-b2)f′2(0)(cos⁡ωτ-isinωτ)=0.
Separating the real and imaginary parts, we get
(2.8)(α22-b2)f′2(0)cos⁡ωτ=(α12-a2)f′2(0)-ω2,(α22-b2)f′2(0)sinωτ=2α1f′(0)ω.
It follows from (2.8) that
(2.9)ω4+2(α12+a2)f′2(0)ω2+[(α12-a2)2-(α22-b2)2]f′4(0)=0.
Thus, we obtain
(2.10)ω=(α12+a2)2-[(α12-a2)2-(α22-b2)2]f′4(0)-(α12+a2).
It is clear that ω is well defined if condition (H3) holds.
Denote(2.11)ω0=(α12+a2)2-[(α12-a2)2-(α22-b2)2]f′4(0)-(α12+a2).
Let
(2.12)τj=1ω0[arccos2α1ω0(α22-b2)f′2(0)+2jπ],j=0,1,2,….
From (2.8), we know that (2.2) with τ=τj(j=0,1,2,…) has a pair of imaginary roots ±iω0, which are simple.
According, the discussion and applying the Lemma2.1 and Cooke and Grossman [16], we obtain the conclusion (ii) and (iii). This completes the proof.Letλj(τ)=αj(τ)+iωj(τ) be a root of (2.2) near τ=τj, and αj(τj)=0,ωj(τj)=ω0,(j=0,1,2…). Due to functional differential equation theory, for every τj,k=0,1,2…, there exists ɛ>0 such that λj(τ) is continuously differentiable in τ for |τ-τj|<ɛ. Substituting λ(τ) into the left-hand side of (2.2) and taking derivative with respect to τ, we have (2.13)[dλdσ]-1=-2λ-2α1f′(0)(α22-b2)f′2(0)e-λτλ-τλ,
which leads to (2.14)Re[dλdσ]τ=τj-1=2α1f′(0)(α22-b2)f′2(0)ω0sinω0τj-2ω02(α22-b2)f′2(0)cos⁡ω0τj[(α22-b2)f′2(0)ω0sinω0τj]2+[ω0(α22-b2)f′2(0)cos⁡ω0τj]2.
By (2.8), we get(2.15)Re[dλdσ]τ=τj-1=2ω02+(3α12+a2)f′2(0)[2ω0α1f′(0)]2+[(α12-a2)f′2(0)-ω02]2>0.
So we have(2.16)signRe[dλdτ]τ=τj=signRe[dλdτ]τ=τj-1>0.From the above analysis, we have the following results.Lemma 2.3.
Letτ=τj, then the following transversality condition:
(2.17)ddτRe[λj(τ)]|τ=τj>0
is satisfied.From Lemma2.3, we can obtain the following lemma.Lemma 2.4.
Assume that (H3) holds. Ifτ>τ0, then (2.2) has at least one root with strictly positive real part.Remark 2.5.
In fact, Applying the lemma in Cooke and Grossman [16] and Lemma 2.3, we can easily see that if τ∈(τj,τj+1),  (2.2) has 2(j+1)(j=0,1,2,…) roots with positive real parts.
From Lemma2.2–2.4, we have the following results on the local stability and Hopf bifurcation for system (1.2).Theorem 2.6.
For system (1.2), let  τ0 be defined by (2.4) and assume that (H1)–(H3) hold.
(i)
Ifτ∈[0,τ0), then the equilibrium point of system (1.2) is asymptotically stable.
(ii)
Ifτ>τ0, then the equilibrium point of system (1.2) is unstable.
(iii)
τ
=
τ
j
(
j
=
0,1
,
2
,
…
) are Hopf bifurcation values for system (1.2).


## 3. Direction and Stability of the Hopf Bifurcation

In the previous section, we obtained some conditions which guarantee that the two-neuron networks with resonant bilinear terms undergo the Hopf bifurcation at some values ofτ=τ1+τ2. In this section, we shall derived the explicit formulae determining the direction, stability, and period of these periodic solutions bifurcating from the equilibrium E*(0,0) at this critical value of τ, by using techniques from normal form and center manifold theory [17], Throughout this section, we always assume that system (2.1) undergoes Hopf bifurcation at the equilibrium E*(0,0) for τ=τ0 and then ±iω0 is corresponding purely imaginary roots of the characteristic equation at the equilibrium E*(0,0).For convenience, letτ=τ0+μ,μ∈R. Then μ=0 is the Hopf bifurcation value of (1.2). Thus, we shall study Hopf bifurcation of small amplitude periodic solutions of (1.2) from the equilibrium point for μ close to 0. Without loss of generality, we assume that τ10>τ20 and let |μ|≤τ10-τ20. Since our analysis is local, where τ0=τ10+τ20 and τ=τ10+(τ20+μ). We can consider the fixed phase space C=C([-τ10,0],R2).For(ϕ1,ϕ2)∈C, define(3.1)Lμϕ=A1ϕ(0)+Bϕ(-τ2)+Cϕ(-τ1),
where (3.2)A1=((α1+a)f′(0)00(α1-a)f′(0)),B=(00(α2-b)f′(0)0),C=(0(α2+b)f′(0)00).
We expand the nonlinear part of the system (1.2) and derive the following expression:(3.3)f(μ,ϕ)=(f1(μ,ϕ)f2(μ,ϕ)),
where(3.4)f1(μ,ϕ)=(α1+a)[f′′(0)2ϕ12(0)+f′′′(0)3!ϕ13(0)]+(α2+b)[f′′(0)2ϕ22(-τ1)+f′′′(0)3!ϕ23(-τ1)]+cϕ1(0)ϕ2(0)+h.o.t.,f2(μ,ϕ)=(α2-b)[f′′(0)2ϕ12(-τ2)+f′′′(0)3!ϕ13(-τ2)]+(α1-a)[f′′(0)2ϕ22(0)+f′′′(0)3!ϕ23(0)]+dϕ1(0)ϕ2(0)+h.o.t..
By the representation theorem, there is a matrix function with bounded variation components η(θ,μ),θ∈[-τ10,0] such that(3.5)Lμϕ=∫-τ100dη(θ,μ)ϕ(θ),forϕ∈C.
In fact, we can choose(3.6)η(θ,μ)={A1,θ=0,Bδ(θ+τ2),θ∈[-τ2,0),-Cδ(θ+τ10),θ∈[-τ10,-τ2),
where δ is the Dirac delta function.Forϕ∈C([-τ10,0],R2), define(3.7)A(μ)ϕ={dϕ(θ)dθ,-τ10≤θ<0,∫-τ100dη(s,μ)ϕ(s),θ=0,R(μ)ϕ={0,-τ10≤θ<0,f(μ,ϕ),θ=0.
Then (1.2) is equivalent to the abstract differential equation(3.8)ẋt=A(μ)xt+R(μ)xt,
where x=(x1,x2)T,xt(θ)=x(t+θ),θ∈[-τ10,0].Forψ∈C([0,τ10],(R2)*), define(3.9)A*ψ(s)={-dψ(s)ds,s∈(0,τ10],∫-τ100dηT(t,0)ψ(-t),s=0.Forϕ∈C([-τ10,0],R2) and ψ∈C([0,τ10],(R2)*), define the bilinear form(3.10)〈ψ,ϕ〉=ψ¯(0)ϕ(0)-∫-τ100∫ξ=0θψ¯(ξ-θ)dη(θ)ϕ(ξ)dξ,
where η(θ)=η(θ,0). We have the following result on the relation between the operators A=A(0) and A*.Lemma 3.1.
A
=
A
(
0
) and  A* are adjoint operators.Proof.
Letϕ∈C1([-τ10,0],R2) and ψ∈C1([0,τ10],(R2)*). It follows from (3.10) and the definitions of A=A(0) and A* that
(3.11)〈ψ(s),A(0)ϕ(θ)〉=ψ¯(0)A(0)ϕ(0)-∫-τ100∫ξ=0θψ¯(ξ-θ)dη(θ)A(0)ϕ(ξ)dξ=ψ¯(0)∫-τ100dη(θ)ϕ(θ)-∫-τ100∫ξ=0θψ¯(ξ-θ)dη(θ)A(0)ϕ(ξ)dξ=ψ¯(0)∫-τ100dη(θ)ϕ(θ)-∫-τ100[ψ¯(ξ-θ)dη(θ)ϕ(ξ)]ξ=0θ+∫-τ100∫ξ=0θdψ¯(ξ-θ)dξdη(θ)ϕ(ξ)dξ=∫-τ100ψ¯(-θ)dη(θ)ϕ(0)-∫-τ100∫ξ=0θ[-dψ¯(ξ-θ)dξ]dη(θ)ϕ(ξ)dξ=A*ψ¯(0)ϕ(0)-∫-τ100∫ξ=0θA*ψ¯(ξ-θ)dη(θ)ϕ(ξ)dξ=〈A*ψ(s),ϕ(θ)〉.
This shows that A=A(0) and A* are adjoint operators and the proof is complete.By the discussions in the Section2, we know that ±iω0 are eigenvalues of A(0) and they are also eigenvalues of A* corresponding to iω0 and -iω0, respectively. We have the following result.Lemma 3.2.
The vector(3.12)q(θ)=(1,γ)Teiω0θ,θ∈[-τ10,0],
where
(3.13)γ=iω0-(α1+a)f′(0)(α2+b)f′(0)e-iω0τ10,
is the eigenvector of A(0) corresponding to the eigenvalue iω0, and
(3.14)q*(s)=D(1,γ*)eiωs,s∈[0,τ10],
where
(3.15)γ*=-iω0+(α1+a)f′(0)(α2-b)f′(0)e-iωτ20,
is the eigenvector of A* corresponding to the eigenvalue -iω0, moreover, 〈q*(s),q(θ)〉=1, where
(3.16)D=1+γ¯γ*¯+γ*τ20(α2-b)f′(0)e-iω0τ20+γ¯τ10(α2+b)f′(0)eiω0τ10.Proof.
Letq(θ) be the eigenvector of A(0) corresponding to the eigenvalue iω0 and q*(s) be the eigenvector of A* corresponding to the eigenvalue -iω0, namely, A(0)q(θ)=iω0q(θ) and A*q*T(s)=-iω0q*T(s). From the definitions of A(0) and A*, we have A(0)q(θ)=dq(θ)/dθ and A*q*T(s)=-dq*T(s)/ds. Thus, q(θ)=q(0)eiω0θ and q*(s)=q*(0)eiω0s. In addition,
(3.17)∫-τ100dη(θ)q(θ)=A1q(0)+Bq(-τ2)+Cq(-τ1)=A(0)q(0)=iω0q(0).
That is,
(3.18)(iω0-(α1+a)f′(0)-(α2+b)f′(0)e-iω0τ10-(α2-b)f′(0)e-iω0τ20iω0-(α1-a)f′(0))q(0)=(00).
Therefore, we can easily obtain
(3.19)γ=iω0-(α1+a)f′(0)(α2+b)f′(0)e-iω0τ10
and so
(3.20)q(0)=(1,iω0-(α1+a)f′(0)(α2+b)f′(0)e-iω0τ10)T
hence
(3.21)q(θ)=(1,iω0-(α1+a)f′(0)(α2+b)f′(0)e-iω0τ10)Teiω0θ.
On the other hand,
(3.22)∫-τ100q*(-t)dη(t)=A1Tq*T(0)+BTq*T(-τ20)+CTq*T(-τ10)=A*q*T(0)=-iω0q*T(0).
Namely,
(3.23)(iω0+(α1+a)f′(0)(α2-b)f′(0)e-iω0τ20(α2+b)f′(0)e-iω0τ10iω0+(α1-a)f′(0))q*(0)=(00).
Therefore, we can easily obtain
(3.24)γ*=-iω0+(α1+a)f′(0)(α2-b)f′(0)e-iω0τ20
and so
(3.25)q*(0)=(1,-iω0+(α1+a)f′(0)(α2-b)f′(0)e-iω0τ20)
hence
(3.26)q*(s)=(1,-iω0+(α1+a)f′(0)(α2-b)f′(0)e-iω0τ20)eiω0s.
In the sequel, we will verify that 〈q*(s),q(θ)〉=1. In fact, from (3.10), we have
(3.27)〈q*(s),q(θ)〉=D¯(1,γ*¯)(1,γ)T-∫-τ100∫ξ=0θD¯(1γ*¯)e-iω0(ξ-θ)dη(θ)(1,γ)Teiω0ξdξ=D¯[1+γγ*¯-∫-τ100(1,γ*¯)θeiω0θdη(θ)(1,γ)T]=D¯{1+γγ*¯-(1,γ*¯)[B(-τ20)e-iω0τ20+C(-τ10)e-iω0τ10](1,γ)T}=D¯[1+γγ*¯+γ¯*τ20(α2-b)f′(0)eiω0τ20+γτ10(α2+b)f′(0)e-iω0τ10]=1.Next, we use the same notations as those in Hassard et al. [17] and we first compute the coordinates to describe the center manifold C0 at μ=0. Let xt be the solution of (1.2) when μ=0.Define(3.28)z(t)=〈q*,xt〉,W(t,θ)=xt(θ)-2Re{z(t)q(θ)}
on the center manifold C0, and we have(3.29)W(t,θ)=W(z(t),z¯(t),θ),
where(3.30)W(z(t),z¯(t),θ)=W(z,z¯)=W20z22+W11zz¯+W02z¯22+⋯
and z and z¯ are local coordinates for center manifold C0 in the direction of q* and q¯*. Noting that W is also real if xt is real, we consider only real solutions. For solutions xt∈C0 of (1.2),(3.31)ż(t)=〈q*(s),ẋt〉=〈q*(s),A(0)xt+R(0)xt〉=〈q*(s),A(0)xt〉+〈q*(s),R(0)xt〉=〈A*q*(s),xt〉+q*¯(0)R(0)xt-∫-τ100∫ξ=0θq*¯(ξ-θ)dη(θ)A(0)R(0)xt(ξ)dξ=〈iω0q*(s),xt〉+q*¯(0)f(0,xt(θ))=defiω0z(t)+q*¯(0)f0(z(t),z¯(t)).
That is,(3.32)ż(t)=iω0z+g(z,z¯),
where(3.33)g(z,z¯)=g20z22+g11zz¯+g02z¯22+g21z2z¯2+⋯.
Hence, we have(3.34)g(z,z¯)=q¯*(0)f0(z,z¯)=f(0,xt)=D¯(1,γ*)(f1(0,xt),f2(0,xt))T,
where (3.35)f1(0,xt)=(α1+a)[f′′(0)2x1t2(0)+f′′′(0)3!x1t3(0)]+(α2+b)[f′′(0)2x2t2(-τ10)+f′′′(0)3!x2t3(-τ10)]+cx1t(0)x2t(0)+h.o.t.,f2(0,xt)=(α2-b)[f′′(0)2x1t2(-τ20)+f′′′(0)3!x1t3(-τ20)]+(α1-a)[f′′(0)2x2t2(0)+f′′′(0)3!x2t3(0)]+dx1t(0)x2t(0)+h.o.t..
Noticing xt(θ)=(x1t(θ),x2t(θ))T=W(t,θ)+zq(θ)+z¯q¯(θ) and q(θ)=(1,γ)Teiω0θ, we have (3.36)x1t(0)=z+z¯+W20(1)(0)z22+W11(1)(0)zz¯+W02(1)(0)z¯22+⋯,x2t(0)=γz+γ¯z¯+W20(2)(0)z22+W11(2)(0)zz¯+W02(2)(0)z¯22+⋯,x1t(-τ20)=e-iω0τ20z+eiω0τ20z¯+W20(1)(-τ20)z22+W11(1)(-τ20)zz¯+W02(1)(-τ20)z¯22+⋯,x2t(-τ10)=γe-iω0τ10z+γ¯eiω0τ10z¯+W20(2)(-τ10)z22+W11(2)(-τ10)zz¯+W02(2)(-τ10)z¯22+⋯.
From (3.33) and (3.34), we have (3.37)g(z,z¯)=q¯*(0)f0(z,z¯)=D¯[f1(0,xt)+γ*¯f2(0,xt)]={D¯[(α1+a)f′′(0)2+(α2+b)f′′(0)2γ2+cγ]+D¯γ*¯[(α2-b)f′′(0)2e-2iω0τ20+(α1-a)f′′(0)2γ2+dγ]}z2+{D¯[(α1+a)f′′(0)+(α2+b)f′′(0)2γγ¯+2cRe{γ}]+D¯γ*¯[(α2-b)f′′(0)+(α1-a)f′′(0)γγ¯+2dRe{γ}]}zz¯+{D¯[(α1+a)f′′(0)2+(α2+b)f′′(0)2γ¯2+cγ¯]+D¯γ*¯[(α2-b)f′′(0)2e2iω0τ20+(α1-a)f′′(0)2γ¯2+dγ¯]}z¯2+{D¯[(α1+a)f′′(0)2(2W11(1)(0)+W20(1)(0))+(α1+a)f′′′(0)2+(α2+b)f′′(0)γe-iω0τ10W11(2)(-τ10)+(α2+b)f′′′(0)2γ2γ¯e-iω0τ10+12c(W20(2)(0)+γ¯W20(1)(0))f′′(0)2(2W11(1)(0)+W20(1)(0))]+D¯γ*¯[(α2-b)f′′(0)2(W20(1)(-τ20))eiω0τ20+2e-iω0τ20W11(1)(-τ20)+(α2-b)f′′′(0)2e-iω0τ20+(α1-a)f′′(0)2(2γW11(2)(0))+W20(2)(0)γ¯+(α1-a)f′′(0)2γ2γ¯+12d(W20(2)(0)+γ¯W20(1)(0))][(α1+a)f′′(0)2(2W11(1)(0)+W20(1)(0))+(α1+a)f′′′(0)2}z2z¯+h.o.t.
and we obtain (3.38)g20=D¯[(α1+a)f′′(0)+(α2+b)f′′(0)γ2+cγ]+D¯γ*¯[(α2-b)f′′(0)e-2iω0τ20+(α1-a)f′′(0)γ2+dγ],g11=D¯[(α1+a)f′′(0)+(α2+b)f′′(0)aa¯+2cRe{γ}]+D¯γ*¯[(α2-b)f′′(0)+(α1-a)f′′(0)γγ¯+2dRe{γ}],g02=D¯[(α1+a)f′′(0)+(α2+b)f′′(0)γ¯2+2cγ¯]+D¯γ*¯[(α2-b)f′′(0)e2iω0τ20+(α1-a)f′′(0)γ¯2+2dγ¯],g21=D¯[(α1+a)f′′(0)(2W11(1)(0)+W20(1)(0))+(α1+a)f′′′(0)+2(α2+b)f′′γe-iω0τ10W11(2)(-τ10)+(α2+b)f′′′(0)γ2γ¯e-iω0τ10]+c(W20(2)(0)+γ¯W20(1)(0))+D¯γ*¯[(α2-b)f′′(0)(W20(1)(-τ20))eiω0τ20+2e-iω0τ20W11(1)(-τ20)+(α2-b)f′′′(0)e-iω0τ20+(α1-a)f′′(0)(2γW11(2)(0))+W20(2)(0)γ¯+(α1-a)f′′(0)γ2γ¯+d(W20(2)(0)+γ¯W20(1)(0))].
For unknown (3.39)W20(1)(0),W20(1)(-τ20),W11(1)(0),W11(2)(0),W11(2)(-τ10),W11(2)(-τ20)
in g21, we still need to compute them.Form (3.8), (3.32), we have (3.40)W′={AW-2Re{q¯*(0)fq(θ)},-τ10≤θ<0,AW-2Re{q¯*(0)fq(θ)}+f,θ=0=defAW+H(z,z¯,θ),
where(3.41)H(z,z¯,θ)=H20(θ)z22+H11(θ)zz¯+H02(θ)z¯22+⋯.
Comparing the coefficients, we obtain(3.42)(A-2iω0)W20=-H20(θ),(3.43)AW11(θ)=-H11(θ),⋯.
And we know that, for θ∈[-τ10,0),(3.44)H(z,z¯,θ)=-q¯*(0)f0q(θ)-q*(0)f¯0q¯(θ)=-g(z,z¯)q(θ)-g¯(z,z¯)q¯(θ).
Comparing the coefficients of (3.41) with (3.44) gives that(3.45)H20(θ)=-g20q(θ)-g¯02q¯(θ),(3.46)H11(θ)=-g11q(θ)-g¯11q¯(θ).
From (3.42), (3.45), and the definition of A, we get(3.47)Ẇ20(θ)=2iω0W20(θ)+g20q(θ)+g¯02q¯(θ).
Noting that q(θ)=q(0)eiω0θ, we have(3.48)W20(θ)=ig20ω0q(0)eiω0θ+ig¯023ω0q¯(0)e-iω0θ+E1e2iω0θ,
where E1=(E1(1),E1(2))T is a constant vector.Similarly, from (3.43), (3.46), and the definition of A, we have(3.49)Ẇ11(θ)=g11q(θ)+g¯11q¯(θ),(3.50)W11(θ)=-ig11ω0q(0)eiω0θ+ig¯11ω0q¯(0)e-iω0θ+E2,
where E2=(E2(1),E2(2))T is a constant vector.In what follows, we will seek appropriateE1, E2 in (3.48), (3.50), respectively. It follows from the definition of A and (3.45), (3.46) that(3.51)∫-τ100dη(θ)W20(θ)=2iω0W20(0)-H20(0)(3.52)∫-10dη(θ)W11(θ)=-H11(0),
where η(θ)=η(0,θ).From (3.42), we have(3.53)H20(0)=-g20q(0)-g¯02q¯(0)+(H1,H2)T,
where (3.54)H1=(α1+a)f′′(0)2+(α2+b)f′′(0)2γ2,H2=(α2-b)f′′(0)2e-2iω0τ20+(α1-a)f′′(0)2γ2.
From (3.43), we have(3.55)H11(0)=-g11q(0)-g¯11(0)q¯(0)+(P1,P2)T,
where (3.56)P1=(α1+a)f′′(0)+(α2+b)f′′(0)γγ¯,P2=(α2-b)f′′(0)+(α1-a)f′′(0)γγ¯.
Noting that(3.57)(iω0I-∫-τ100eiω0θdη(θ))q(0)=0,(-iω0I-∫-τ100e-iω0θdη(θ))q¯(0)=0
and substituting (3.48) and (3.53) into (3.51), we have(3.58)(2iω0I-∫-τ100e2iω0θdη(θ))E1=(H1,H2)T.
That is,(3.59)(2iω0I-A1-Be-2iω0τ20-Ce-2iω0τ10)E1=(H1,H2)T,
then(3.60)(2iω0-(α1+a)f′(0)-(α2+b)f′(0)e-iω0τ10-(α2-b)f′(0)e-iω0τ202iω0-(α1-a)f′(0))(E1(1)E1(2))=(H1H2).
Hence, (3.61)E1(1)=Δ11Δ1,E1(2)=Δ12Δ1,
where (3.62)Δ1=det⁡(2iω0-(α1+a)f′(0)-(α2+b)f′(0)e-iω0τ10-(α2-b)f′(0)e-iω0τ202iω0-(α1-a)f′(0)),Δ11=det⁡(H1-(α2+b)f′(0)e-iω0τ10H22iω0-(α1-a)f′(0)),Δ12=det⁡(2iω0-(α1+a)f′(0)H1-(α2-b)f′(0)e-iω0τ20H2).
Similarly, substituting (3.49) and (3.55) into (3.52), we have(3.63)(∫-τ100dη(θ))E2=(P1,P2)T.
Then,(3.64)(A1+B+C)E2=(-P1,-P2)T.
That is,(3.65)((α1+a)f′(0)(α2+b)f′(0)(α2-b)f′(0)(α1-a)f′(0))(E2(1)E2(2))=(-P1-P2).
Hence, (3.66)E2(1)=Δ21Δ2,E2(2)=Δ22Δ2,
where (3.67)Δ2=det⁡((α1+a)f′(0)(α2+b)f′(0)(α2-b)f′(0)(α1-a)f′(0)),Δ21=det⁡(-P1(α2+b)f′(0)-P2(α1-a)f′(0)),Δ22=det⁡((α1+a)f′(0)-P1(α2-b)f′(0)-P2).
From (3.48), (3.50), we can calculate g21 and derive the following values: (3.68)c1(0)=i2ω0τ0(g20g11-2|g11|2-|g02|23)+g212,μ2=-Re{c1(0)}Re{λ′(τ0)},β2=2Re(c1(0)),T2=-Im⁡{c1(0)}+μ2Im⁡{λ′(τ0)}ω0τ0.
These formulae give a description of the Hopf bifurcation periodic solutions of (1.2) at τ=τ0 on the center manifold. From the discussion above, we have the following result.Theorem 3.3.
The periodic solution is supercritical (subcritical) ifμ2>0(μ2<0); The bifurcating periodic solutions are orbitally asymptotically stable with asymptotical phase (unstable) if β2<0(β2>0); the  periods of bifurcating periodic solutions  increase (decrease) if T2>0(T2<0).


## 4. Numerical Examples

In this section, we present some numerical results to verify the analytical predictions obtained in the previous section. As an example, we consider the following special case of the system (1.2) with the parameters α1=-0.03,α2=-0.1,a=1.5,b=2,c=0,d=-4,andf(x)=tanh(x). Then, the system (1.2) becomes(4.1)ẋ1(t)=1.47tanh(x1(t))+1.9tanh(x2(t-τ1)),ẋ2(t)=-2.1tanh(x1(t-τ2))-1.53tanh(x2(t))-4x1(t)x2(t).
By some complicated computation by means of Matlab 7.0, we get ω0≈1.3211,τ0≈0.01,λ′(τ0)≈ 0.0140-1.4926i. We can easily obtain g20=-0.2501+2.3128i,g11=1.2377+0.3484i,g02=0.4533-0.5693i,g21=-2.3022+4.3015i. Thus, we can calculate the following values:  c1(0)=-1.0617-1.7138i,μ2=75.8357,β2=-2.1234,T2=86.9780. We obtain that the conditions indicated in Theorem 2.6 are satisfied. Furthermore, it follows that μ2>0 and β2<0. Thus, the equilibrium E*(0,0) is stable when τ<τ0 as illustrated by the computer simulations (see Figures 1, 2, and 3). When τ passes through the critical value τ0, the equilibrium E*(0,0) loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcations from the equilibrium E*(0,0). Since μ2>0 and β2<0, the direction of the Hopf bifurcation is τ>τ0 and these bifurcating periodic solutions from E*(0,0) at τ0 are stable, which are depicted in Figures 4, 5, and 6.Figure 1
Trajectories graphs of the system (4.1) with τ1=0.003,τ2=0.004, and τ1+τ2=0.007<τ0≈0.01. The equilibrium E*(0,0) is asymptotically stable. The initial value is (0.05, 0.01).Figure 2
Trajectories graphs of the system (4.1) with τ1=0.003,τ2=0.004, and τ1+τ2=0.007<τ0≈0.01. The equilibrium E*(0,0) is asymptotically stable. The initial value is (0.05, 0.01).Figure 3
Trajectories graphs of the system (4.1) with τ1=0.003,τ2=0.004, and τ1+τ2=0.007<τ0≈0.01. The equilibrium E*(0,0) is asymptotically stable. The initial value is (0.05, 0.01).Figure 4
Trajectories graphs of thesystem (4.1) with τ1=0.01,τ2=0.04, and τ1+τ2=0.05>τ0≈0.01. Hopf bifurcation occurs from the equilibrium E*(0,0). The initial value is (0.05, 0.01).Figure 5
Trajectories graphs of the system (4.1) with τ1=0.01,τ2=0.04, and τ1+τ2=0.05>τ0≈0.01. Hopf bifurcation occurs from the equilibrium E*(0,0). The initial value is (0.05, 0.01).Figure 6
Trajectories graphs of the system (4.1) with τ1=0.01,τ2=0.04, and τ1+τ2=0.05>τ0≈0.01. Hopf bifurcation occurs from the equilibrium E*(0,0). The initial value is (0.05, 0.01).


## 5. Conclusions

In this paper, we have analyzed a two-neuron networks with resonant bilinear terms. Firstly, we obtained the sufficient conditions to ensure local stability of the equilibriumE*(0,0) and the existence of local Hopf bifurcation. Moreover, we note also that, if the two-neuron networks with resonant bilinear terms begin with a stable equilibrium, but then become unstable due to delay, then it will likely be destabilized by means of a Hopf bifurcation which leads to periodic solutions with small amplitudes. Finally, the direction of Hopf bifurcation and the stability of the bifurcating periodic orbits are discussed by applying normal form theory and center manifold theorem.



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*Source: 697630-2011-06-21.xml*