# Bifurcation Analysis in a Two-Dimensional Neutral Differential Equation

Authors: Ming Liu; Xiaofeng Xu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The dynamics of a 2-dimensional neural network
model in neutral form are investigated. We prove that a sequence
of Hopf bifurcations occurs at the origin as the delay increases. The
direction of the Hopf bifurcations and the stability of the bifurcating
periodic solutions are determined by using normal form method
and center manifold theory. Global existence of periodic solutions is
established using a global Hopf bifurcation result of Krawcewicz et
al. Finally, some numerical simulations are carried out to support
the analytic results.


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## Body

## 1. Introduction

The fundamental theories and applications of neutral functional differential equations have been largely developed and there are a great deal of important conclusions on the stability, bifurcation theory, and numerical solutions of neutral delay differential equations during the last few decades. It is well known that Wei and Ruan [1] have established a basic theory to analyze the distribution of the zeros of general transcendental equations and applied it to the neutral functional differential equations. One can use the theory to study the existence of Hopf bifurcation to a neutral delay differential equation. In [2], Wang and Wei extend the computation of the properties of Hopf bifurcation, such as the direction of bifurcation and stability of bifurcating periodic solutions, of DDE introduced by Kazarinoff et al. Based on combining the global Hopf bifurcation theory of neutral equations due to Krawcewicz et al. and the higher dimensional Bendixson’s criterion for ordinary differential equations due to Li and Muldowney, the global existence of periodic solutions of neutral differential equations have been studied by Qu et al. [3]. But research findings of bifurcation about high-dimension neutral differential equations can rarely be found.In order to determine the dynamics of artificial neural network, the two-dimensional delay neural network without self-feedback is studied, and some significant results are obtained (see [4–8]). In this paper, we consider the following system:
(1)x˙1(t)=-μx1(t)+af(x1(t-τ))+bx˙2(t-τ),x˙2(t)=-μx2(t)+af(x2(t-τ))+bx˙1(t-τ),
where μ,a,  and b are real numbers with μ>0 and |b|<1. f is assumed adequately smooth, for example, f∈C3, and satisfies the following condition:(H1)
f(0)=0,f′(0)=1, and there exists L>0, such that |f(x)|⩽L for all x∈ℝ,(H2)
f′(x)>0 for all x∈ℝ, xf′′(x)<0 for all x≠0.In the present paper, we provide a detailed analysis of this equation. By applying the local Hopf bifurcation theory (see [9] and also [1, 2, 10, 11]), we investigate the existence of stable periodic oscillations for (1). More specially, we prove that the equilibrium (0,0) loses its stability as τ increases, and a sequence of Hopf bifurcations occurs at the origin. Whereafter, based on the normal form and center manifold theory due to [12, 13], by using the method introduced in Wang and Wei [2], we derive a sufficient condition for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Furthermore, the existence of periodic solutions for τ far away from the Hopf bifurcation values is also established, by using a global Hopf bifurcation result due to Krawcewicz et al. [14] (also see [15, 16]).The rest of this paper is organized as follows: in Section2, taking a and τ as parameters, we give the analysis of stability and bifurcations at equilibria. Section 3 is devoted to establishing the direction and stability of Hopf bifurcation. Finally, a global Hopf bifurcation is established, and some numerical simulations are carried out to illustrate the analytic results in Section 4.


## 2. Stability and Bifurcation Analysis

In this section, we will investigate the stability and bifurcations of the equilibria by takinga and τ as bifurcation parameters.Under the given hypothesis (H1), it is easy to check that(0,0) is an equilibrium point of system (1). At this trivial equilibrium, the linearization of system (1) is
(2)X˙(t)=AX(t)+BX(t-τ)+CX˙(t-τ),
where
(3)A=(-μ00-μ),B=(a00a),C=(0bb0).
And the corresponding characteristic equation is
(4)Δ(λ)=(λ+μ-ae-λτ)2-b2λ2e-2λτ=0,
that is,
(5)(λ+μ-(a-bλ)e-λτ)(λ+μ-(a+bλ)e-λτ)=defΔ1(λ)·Δ2(λ)=0.Case 1.
Choosea as parameter.
For convenience, we give two claims at first.Claim 1.
All the roots of (4) with a=0 have negative real parts for any τ⩾0.Proof.
Whena=0 and τ=0, (4) becomes (λ+μ)2-b2λ2=0, and the roots are λ1,2=-μ/(1±b)<0.
i
v
0
(
v
0
>
0
) is a zero of Δ(λ) with a=0 if and only if v0 solves
(6)(iv0+μ)2+b2v02(cos2v0τ-isin2v0τ)=0.
Separating the real and imaginary parts gives that (b2-1)v02=μ2, which contradicts with |b|<1. Then the conclusion follows Wei and Ruan [1], and the proof is complete.Claim 2.
Equation (4) has at least two positive roots when a>μ.
In fact, the conclusion follows fromΔ1,2(0)=μ-a<0 and limλ→+∞Δ1,2(λ)=+∞.
Whenb≠0,iv(v>0) is a root of (4) if and only if v satisfies
(7)asinvτ=-v(1+bcosvτ),acosvτ=u+bvsinvτ
or
(8)asinvτ=-v(1-bcosvτ),acosvτ=u-bvsinvτ.
Applying the method in [3], we have that there exists a unique value of v, denoted by v1m+ (v1m-), satisfying (7) and a unique value of v, denoted by v2m+ (v2m-), satisfying (8) on the interval  (2(m-1)π/τ,2mπ/τ),m∈ℤ+ when a>0(<0). We also can show that vkm+∈((2m-1)π/τ,2mπ/τ),k=1,2,m∈ℤ+ with a>0 and vkm-∈(2(m-1)π/τ,(2m-1)π/τ),k=1,2,m∈ℤ+ with a<0.
Define(9)akm±=±vkm±2(1-b2)+μ2,k=1,2,m∈ℤ+.akm± make sense since |b|<1, and akm+ and akm- are all dependent on τ. Hence, ±vkm+ (±vkm-) are a pair of purely imaginary roots of (4) with a=akm+(akm-). And we have the transversal condition
(10)SignRe(dλda)|a=akm±={1,a=akm+,-1,a=akm-,k=1,2,m∈ℤ+.
Moreover, it is fulfilled thatak×(m+1)-<akm-<0<akm+<ak×(m+1)+, following the fact that 0<vkm-<vkm+<vk×(m+1)-<vk×(m+1)+. Now we can distinguish two cases. First, b>0. From (7) and (8), we can obtain that a1m-cosv1m-τ>a2m-cosv2m-τ, which implies v1m->v2m- and a1m-<a2m-. Second, b<0. With the similar process as above, it is obtained that v1m-<v2m- and a1m->a2m-.
Summarizing the discussion above and applying Claims1 and 2, we have the following.Lemma 1.
There exist sequence values ofa defined by (9) such that all the roots of (4) have negative real parts when a∈(a1-(τ),μ), and (4) has at least two roots with positive real part when a∈(-∞,a1-(τ))∪(μ,+∞), a pair of purely imaginary roots when a=akm±(τ)(k=1,2,m∈ℤ+), where a1-(τ)=a11-(a21-) when b<0(>0).Lemma 2.
Equation (1) undergoes a pitchfork bifurcation at (0,0) when a=μ.Proof.
Leth(x)=-μx+af(x). Then we have h(0)=0, h(±∞)=∓∞, and h′(x)=-μ+af′(x). From (H1) and (H2), there exists a unique zero x=0 of h(x), and (1) has only one steady-state X=(0,0) when a≤μ. On the other hand, when a>μ, we have h′(0)=a-μ>0 and xh′′(x)<0 when x≠0. Thus, there exists a unique c+∈ℝ+(c-∈ℝ-) satisfying h(x)=0, and (1) has exactly nonzero equilibria (c±,0), (0,c±), and (c±,c±). This completes the proof.In the following, we will investigate the stability of(c±,0), (0,c±), and (c±,c±).Firstly, we consider the stability of(c+,0) when a>μ. (It has similar results with (c-,0) and (0,c±).)The characteristic equation of linearization of (1) at (c+,0) is
(11)Δ(c+,0)(λ)=(λ+μ-ae-λτ)(λ+μ-af′(c+)e-λτ)-b2λ2e-2λτ=0.Equation (11) has at least one positive root for any τ≥0. In fact, we have h′(c±)<0 and 0<af′(c±)<μ from the proof of Lemma 2. Then the conclusion follows from Δ(c+,0)(0)=(μ-a)(μ-af′(c+))<0 and limλ→+∞Δ(c+,0)(λ)=+∞, which implies that (c+,0) is unstable.Secondly, we consider the stability of(c+,c+) when a>μ. (It has similar results with (c+,c-) and (c-,c±)).The characteristic equation of linearization of (1) at (c+,c+) is
(12)Δ(c+,c+)(λ,τ)=(λ+μ-af′(c+)e-λτ)2-b2λ2e-2λτ=0.
In particular,
(13)Δ(c+,c+)(λ,0)=(λ+μ-af′(c+))2-b2λ2=0,
which implies
(14)λ1,2=-μ+af′(c+)1±b<0.Moreover, (12) has no zero root from af′(c+)<μ.In order to prove that(c+,c+) is asymptotically stable for all τ∈ℝ+, what we need is to verify that (12) has no purely imaginary roots. Let iη(η>0) be the root of (12), then η solves
(15)af′(c+)sinητ+bηcosητ=-η,af′(c+)cosητ-bηsinητ=μ
or
(16)af′(c+)sinητ-bηcosητ=-η,af′(c+)cosητ+bηsinητ=μ.
This leads to
(17)η2=(af′(c+))2-μ21-b2<0.
The assertion follows.Applying Lemmas1 and 2, we have the following results.Theorem 3.
(i) The zero solution of (1) is asymptotically stable when a∈(a1-(τ),μ) and unstable when a∈(-∞,a1-(τ))∪(μ,+∞).
(ii) Equation (1) undergoes a pitchfork bifurcation at (0,0) when a=μ. More precisely, some new equilibria bifurcate from zero and (c±,0), (0,c±) are unstable, and (c±,c±) are asymptotically stable for τ∈ℝ+ when a>μ.
(iii) Equation (1) undergoes a Hopf bifurcation at (0,0) when a=akm±(τ)(k=1,2,m∈ℤ+).Case 2.
Regardτ as parameter.
First of all, we know that the root of (4) with τ=0 satisfies that λ1,2=(a-μ)/(1±b)>0 when a>μ, and λ1,2=(a-μ)/(1±b)<0 when a<μ.
Letiω(ω>0) be a root of (4); then it follows that
(18)sinωτ=-ω(a±bμ)μ2+ω2,cosωτ=aμ∓bω2μ2+ω2.
This leads to ω2=(a2-μ2)/(1-b2), ω0=def(a2-μ2)/(1-b2) makes sense when |a|>μ. Define(19)τ1j={1ω0(arccosaμ-bω02μ2+ω02+2jπ),a<-μ,1ω0(-arccosaμ-bω02μ2+ω02+2(j+1)π),a>μ,j=0,1,2,…,τ2j={1ω0(arccosaμ+bω02μ2+ω02+2jπ),a<-μ,1ω0(-arccosaμ+bω02μ2+ω02+2(j+1)π),a>μ,j=0,1,2,….Then iω0 is a purely imaginary root of (4) with τ=τkj,k=1,2 defined by (19).
Letλ(τ)=α(τ)+iϱ(τ) be the root of (4), satisfying
(20)α(τkj)=0,ϱ(τkj)=ω0.
Differentiating both side of (4) gives that
(21)Re(dλdτ)-1|τ=τkj=1-b2a2+b2ω02>0.
This implies that α′(τkj)>0,k=1,2,j=0,1,2,….
Summarizing the discussions above, one can obtain the following.Lemma 4.
(i) If|a|<μ, then all roots of (4) have negative real parts.
(ii) If|a|>μ, then there exist a sequence values of τ defined by (19) such that (4) has a pair of purely imaginary roots ±iω0 when τ=τkj,k=1,2. Additionally, if a<-μ, then all roots of (4) have negative real parts when τ∈[0,τ0), all roots of (4), except ±iω0, have negative real parts when τ=τ0, and (4) has at least a pair of root with positive real parts when τ>τ0, where τ0=τ10(τ20) when b<0(>0); if a>μ, then (4) has at least two positive roots.Spectral properties in Lemma4 immediately lead to the dynamics near the origin described by the following theorem.Theorem 5.
For (1), the following holds.(i)
If|a|<μ, then (0,0) is asymptotically stable for all τ>0.(ii)
Ifa<-μ, then (0,0) is asymptotically stable when τ∈(0,τ0), and unstable when τ>τ0.(iii)
Ifa>μ, then (0,0) is always unstable for all τ>0.(iv)
If|a|>μ, then (1) undergoes a Hopf bifurcation at (0,0) when τ=τkj,k=1,2,j=0,1,2,….


## 3. Properties of Hopf BifurCation

Theorems3(iii) and 5(iv) in the previous section give the sufficient conditions for (1) to undergo Hopf bifurcations with a and τ as bifurcation parameters. In this section, we will investigate the direction of Hopf bifurcations and stability of bifurcating periodic solutions, following the same algorithms as Wang and Wei’s recent work and using a computation process similar to that in [2] (see also [17]). We should mention that we will choose τ as bifurcation parameter, and the similar results follow when choosing other coefficients as bifurcation parameters.Letyi(t)=xi(τt),i=1,2, and Y(t)=(y1(t),y2(t))T, then (1) becomes
(22)ddt[Y(t)-CY(t-1)]=τAY(t)+τBF(Y(t-1)),
where A, B, and C are denoted by (2) and F(Y(t-1))=(f(y1(t-1)),f(y2(t-1)))T. And the corresponding characteristic equation around Y=(0,0)T is
(23)(z+μτ-aτe-z)2-b2z2e-2z=0,
Comparing (23) with (4), if is found that z=λτ for τ≠0. Therefore, combining this fact with Lemma 4, one has the following.Lemma 6.
Assume|a|>μ.(i)
Ifτ=τj,j=0,1,2,…, then (23) has a pair of purely imaginary roots ±iω0τj, where τj=τkj and ω0 are defined by (19).(ii)
Letz(τ)=γ(τ)+iς(τ) be the root of (23), satisfying
(24)γ(τj)=0,ς(τj)=ω0τj,then
(25)γ′(τj)=τjα′(τj)>0,j=0,1,2,….(iii)
Equation (23) has at least a pair of roots with positive real parts when τ=τj for j≥1, and it has two positive roots when a>μ and τ=τ0. All roots of (23) with τ=τ0, except ±iω0τ0, have negative real parts when a<-μ.For convenience, denoteτ=τj+ν. Then we know that (22) undergoes a Hopf bifurcation at origin when ν=0. For ϕ∈C([-1,0],ℝ2), let
(26)D(ϕ)=ϕ(0)-Cϕ(-1),L(ν,ϕ)=A(τj+ν)ϕ(0)+B(τj+ν)ϕ(-1),F(ν,ϕ)=f′′′(0)6(τj+ν)Bϕ3(-1)+⋯.
By the Riesz representation theorem, there exist functions η(θ) and μ(θ) such that
(27)D(ϕ)=ϕ(0)-∫-10‍dμ(θ)ϕ(θ),L(ν,ϕ)=∫-10‍dη(θ)ϕ(θ).
In fact, we can choose
(28)μ(θ)={-C,θ=-1,0,θ∈(-1,0],η(θ)={-B(τj+ν),θ=-1,0,θ∈(-1,0),A(τj+ν),θ=0.Define(29)A(ν)ϕ={dϕ(θ)dθ,θ∈[-1,0),ϕ′(0)-Dϕ′(0)+Lϕ,θ=0,R(ν)ϕ={0,θ∈[-1,0),F(ν,ϕ),θ=0.
Then (22) can be written as
(30)Y˙t=A(ν)Yt+R(ν)Yt.
Clearly, (30) is an abstract ODE on the phase space BC′ of (22), where BC′={ϕ:[-1,0]→ℂ2, ϕ is continuous on [-1,0), and limθ→0ϕ(θ) exists}.The adjoint operatorA*ψ=-dψ/ds with domain
(31)D(A*)={ψ∈C′=defC([0,1],ℝ2):dψds∈C′;Ddψds=-Lψ}.
For (ψ,ϕ)∈C′×C, define a bilinear form:(32)(ψ,ϕ)=ψ-(0)ϕ(0)-∫-10‍d[∫α=0θ‍ψ-(θ-α)dμ(α)]ϕ(θ)-∫-10‍∫θ=0s‍ψ-(θ-s)dη(s)ϕ(θ)dθ.It is not difficult to verify thatq(θ)=q0eiω0τjθ(θ∈[-1,0]) and q*(s)=l-q0*eiω0τjs(s∈[0,1]) are the eigenvectors of A(0) and A* corresponding to the eigenvalues iω0τj and -iω0τj, respectively, where
(33)q0={(1,-1)Tτj=τ1j,(1,1)Tτj=τ2j,q0*={(1,-1)τj=τ1j,(1,1)τj=τ2j,l=1q0*(I-Ce-iω0τj+Bτje-iω0τj)q-0,
and (q*,q)=1.Now we compute the center manifoldC0 at ν=0. Define
(34)z(t)=(q*,Yt),W(t,θ)=Yt(θ)-2Rez(t)q(θ),
then we have
(35)z˙(t)=iω0τjz+q-*(0)F(0,Yt).
Equation (35) can be written in the abbreviated form as
(36)z˙(t)=iω0τjz+g(z,z-),
with
(37)g(z,z-)=g20z22+g11zz-+g02z-22+g21z2z-2+⋯.
Noting that Yt(θ)=W(t,θ)+z(t)q(θ)+z-(t)q-(θ), we have(38)Yt(-1)=(q01e-iω0τjz+q-01eiω0τjz-+W20(1)(-1)z22+W11(1)(-1)zz-+W02(1)(-1)z-22+⋯q02e-iω0τjz+q-02eiω0τjz-+W20(2)(-1)z22+W11(2)(-1)zz-+W02(2)(-1)z-22+⋯),where q0=(q01,q02)T. Therefore, from (26), we have
(39)F(0,Yt)=f′′′(0)6τjYt3(-1)+⋯=defFz2z22+Fzz-zz-+Fz-2z-22+Fz2z-z2z-2+⋯.
Substituting the expression of F(0,Yt) into (35) and comparing its coefficients with that of (36) give that
(40)g20=g02=g11=0,g21=l-q-0*f′′′(0)6τjB(q012q-01q022q-02)e-iω0τj.
It is well known that the coefficient c1(0) of third degree term of Poincaré normal form of (35) is given by (see [12])
(41)c1(0)=i2ω0τj(g20g11-2∥g11∥2-13∥g02∥2)+g212.
Consequently, when τ=τ1j or τ=τ2j,
(42)c1(0)=τkj12∥1+(∓b+aτkj)e-iω0τkj∥2×af′′′(0)[1+(∓b+aτkj)e-iω0τkj]e-iω0τkj.
Hence we obtain
(43)SignRec1(0)=Signaf′′′(0)×[aμ∓bω02μ2+ω02+(∓b+aτkj)(2(aμ∓bω02μ2+ω02)2-1)].
Summarizing the above analysis, we have the following theorem.Theorem 7.
For (22), the direction of Hopf bifurcation at τ=τkj,k=1,2,j=0,1,2,…, is supercritical (subcritical), and the bifurcating periodic solutions are asymptotically stable (unstable) when  Rec1(0)<0 (Rec1(0)>0).


## 4. Global Hopf Bifurcation Analysis

Our objective in this section is to obtain the global continuation of periodic solution bifurcating from the point(0,τkj),k=1,2,j=0,1,2,…, for (1) by using a global Hopf bifurcation theorem given by Krawcewicz et al. [14]. For the reader’s convenience, we copy (22), which is equivalent to (1), as follows
(44)y˙1(t)-by˙2(t-1)=-μτy1(t)+aτf(y1(t-1)),y˙2(t)-by˙1(t-1)=-μτy2(t)+aτf(y2(t-1)).
We have known that (44) undergoes a local Hopf bifurcation at the origin when τ=τkj,k=1,2,j=0,1,2,…. Now we begin to show that each bifurcation branch can be continued with respect to the parameter τ under certain conditions. To bring out the ideas in the results of subsequent part, it is convenient to introduce the following notations:X=C([-1,0],ℝ2),Σ=Cl{(Y,τ,T):Y is a T-periodic solution of (44)}⊂X×ℝ+×ℝ+,N={(Y^,τ,T):μY^=aF(Y^)}.DenoteC(0,τj,2π/ω0τj) the connected component of (0,τj,2π/ω0τj) in Σ, where ω0=(a2-μ2)/(1-b2) and τj(j=0,1,2,…) are defined by Lemma 6.Lemma 8.
Equation (44) has no periodic nonconstant solution when τ=0.Proof.
Letu(t)=y1(t)+y2(t). When τ=0, we from (44) have
(45)ddt[u(t)-bu(t-1)]=0.
Therefore, there exists c1∈ℝ such that
(46)u(t)=c1+bu(t-1).
Which implies that when t∈[n,n+1], we have
(47)u(t)=c1∑i=0n-1‍bi+bnu(t-n).
Hence we obtain limt→+∞u(t)=c1/(1-b).
Letv(t)=y1(t)-y2(t). With the similar process as above, it is obtained that there exists c2∈ℝ such that limt→+∞v(t)=c2/(1+b).
Then the conclusion follows fromlimt→+∞y1(t)=(1/2)[c1/(1-b)+c2/(1+b)], and the proof is complete.Lemma 9.
If|b|<1/2, then all periodic solution of (44) are uniformly bounded.Proof.
LetY(t)=(y1(t),y2(t))T be a periodic solution of (44) and u(t)=y1(t)+y2(t). Then there exist t1 and t2 such that
(48)u(t1)-bu(t1-1)=maxt∈ℝ[u(t)-bu(t-1)],u(t2)-bu(t2-1)=mint∈ℝ[u(t)-bu(t-1)],
together with T0 and T0 such that
(49)u(T0)=maxt∈ℝ+u(t),u(T0)=mint∈ℝ+u(t),
Now we take the case-1/2<b<0 as an example, and it has the same result when 0⩽b<1/2.
By (44), (48), and (H1), we have that
(50)u(ti)=au∑j=12‍f(yj(ti-1))∈[-2|a|Lμ,2|a|Lμ],i=1,2,u(t2)-bu(t2-1)⩽u(t)-bu(t-1)⩽u(t1)-bu(t1-1).
On one hand, we have
(51)u(t)⩽bu(t-1)+[u(t1)-bu(t1-1)]⩽b2u(t-2)+b[u(t2)-bu(t2-1)]+[u(t1)-bu(t1-1)]⩽b2mu(t-2m)+∑i=1m‍b2i-1[u(t2)-bu(t2-1)]+∑i=1m‍b2(i-1)[u(t1)-bu(t1-1)]⩽b1-b2[u(t2)-bu(t2-1)]+11-b2[u(t1)-bu(t1-1)]⩽2|a|L(1+b)μ-b21-b2u(T0)-b1-b2u(T0).
And on the other hand, we can obtain
(52)u(t)⩾-2|a|L(1+b)μ-b1-b2u(T0)-b21-b2u(T0).
It follows that
(53)u(T0)⩾1-b21+b-b2[-2|a|L(1+b)μ-b21-b2u(T0)],u(T0)⩽2|a|L(1+2b)μ.
Similarly, one can prove that
(54)u(T0)⩾-2|a|L(1+2b)μ.
Which implies that
(55)-2|a|L(1+2b)μ⩽u(t)⩽2|a|L(1+2b)μ.
Letv(t)=y1(t)-y2(t). In the same way, it can be verified that
(56)-2|a|L(1+2b)μ⩽v(t)⩽2|a|L(1+2b)μ.
Theny1(t) and y2(t) are uniformly bounded, and the proof is complete.Lemma 10.
Ifa<-μ or a>μ/M, then (44) has no periodic nonconstant solution of period 1, where M=min{f′(-2|a|L/(1+2b)μ),f′(2|a|L/(1+2b)μ)}.Proof.
Let(y1(t),y2(t)) be a periodic solution to (44) of period 1. Then it is a periodic solution to the following system of ordinary differential equations:
(57)u˙1(t)-bu˙2(t)=-μτu1(t)+aτf(u1(t)),u˙2(t)-bu˙1(t)=-μτu2(t)+aτf(u2(t)),
where · denotes d/dt. Denote
(58)G={u∈ℝ2:uj∈[-2|a|L(1+2b)μ,2|a|L(1+2b)μ],j=1,2}.
Lemma 9 shows that the periodic solution of (44) belong to the region G. Clearly, (57) is equivalent to
(59)u˙1=τ1-b2[-μ(u1+bu2)+a(f(u1)+bf(u2))]=defP(u1,u2),u˙2=τ1-b2[-μ(u2+bu1)+a(f(u2)+bf(u1))]=defQ(u1,u2),
Then
(60)∂P∂u1+∂Q∂u2=τ1-b2[-2μ+a(f′(u1)+f′(u2))].
From (H2), it is easy to see that ∂P/∂u1+∂Q/∂u2<0 when a<-μ and ∂P/∂u1+∂Q/∂u2>0 when a>μ/M. Thus, the classical Bendixson’s criterion implies that (57) has no nonconstant periodic solution in the region G, and the proof is complete.Up to now, we have prepared sufficiently to state the following global Hopf bifurcation results.Theorem 11.
If|b|<1/2 and either a<-μ or a>μ/M holds, then (44) has at least one periodic solution for τ>τj(j⩾1).Proof.
First note that the stationary points(0,τj,2π/ω0τj) of (44) are nonsingular and isolate centers (see [14]) under the assumption |a|>μ for each τ>τj(j⩾1). By (25), there exist ϵ>0, δ>0, and a smooth curve λ:(τj-δ,τj+δ)→ℂ, such that
(61)Δ(λ(τ))=Δ(0,τ,T)(λ(τ))=0,|λ(τ)-iω0τj|<ϵ,
for all τ∈[τj-δ,τj+δ], where Δ is defined as (23), and
(62)λ(τj)=iω0τj,dRe(λ(τ))dτ|τ=τj>0.
Let
(63)Ωϵ={(u,p):0<u<ϵ,|p-2πω0τj|<ϵ}.
Clearly, if τ-τj⩽δ and (u,p)∈Ωϵ such that Δ(0,τ,T)(u+i2π/p)=0, then τ=τj,  u=0, and p=2π/ω0τj. Moreover, if we put
(64)H±(0,τj,2πω0τj)(u,p)=Δ(0,τ±δ,T)(u+i2πp),
at m=1, we have, from Reλ′(τj)>0, that the crossing number is
(65)γ1(0,τj,2πω0τj)=degB(H-(0,τj,2πω0τj),Ωϵ)-degB(H+(0,τj,2πω0τj),Ωϵ)=-1.
Using the local Hopf bifurcation theorem for NFDE [14, Theorem 5.6], we conclude that the connected component C(0,τj,2π/ω0τj) through (0,τj,2π/ω0τj) in Σ is nonempty, and thus C(0,τj,2π/ω0τj) is unbounded by the global Hopf bifurcation theorem given by Krawcewicz et al. [14, Theorem 5.14].
Lemma9 implies that the projection of C(0,τj,2π/ω0τj) onto the Y-space is bounded. Meanwhile, the projection of C(0,τj,2π/ω0τj) onto τ-space is bounded below from Lemma 8.
By the definition ofτj, we know that
(66)2π<ω0τj<2(j+1)π,j⩾1,
under the assumptions that |a|>μ and |b|<1, which implies
(67)1j+1<2πω0τj<1.
Applying Lemma10, one has 1/(j+1)<T<1 if (Y,τ,T)∈C(0,τj,2π/ω0τj) for j⩾1, when |b|<1/2 and either a<-μ or a>μ/M holds. Thus in order for C(0,τj,2π/ω0τj) to be unbounded, its projection onto the τ-space must be unbounded. Consequently, the projection of C(0,τj,2π/ω0τj) onto the τ-space include [τj,∞) for j⩾1 when |b|<1/2 and either a<-μ or a>μ/M holds. The proof is complete.Next we carry out some numerical simulations for (1).Letf(x)=arctan(x), and assume that μ=1.2,a=-3, and b=0.2. From Lemma 4, it is obtained that ω0≈2.8062 and τ0≈0.6378. Accordingly, it is known that (0,0) is asymptotically stable for τ∈(0,τ0) and unstable for τ>τ0, and Hopf bifurcation at the origin occurs when τ=τ0 by Theorem 5. From Theorem 7, the direction of the Hopf bifurcation at τ=τ0 is supercritical, and the bifurcating periodic solutions are asymptotically stable. Furthermore, according to Theorem 11, (1) with this set of parameters has at least one periodic solution when τ>τj(j⩾1). The corresponding numerical simulation results are shown in Figures 1, 2, and 3.For system (1), the equilibrium (0,0) is asymptotically stable when τ=0.55<τ0≈0.6378.
(a)
(b)
(c)The system (1) undergoes a Hopf bifurcation at (0,0), and the bifurcating periodic solution is asymptotically stable when τ=2.5∈(τ0,τ1).
(a)
(b)
(c)The system (1) undergoes a Hopf bifurcation at (0,0) with sufficiently large τ=50.
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(b)
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*Source: 367589-2013-04-23.xml*