# Equivariant Hopf Bifurcation in a Ring of Identical Cells with Delay

Authors: Dejun Fan; Junjie Wei
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

A kind of delay neural network withn elements is considered. By analyzing the distribution of the eigenvalues, a bifurcation set is given in an appropriate parameter space. Then by using the theory of equivariant
Hopf bifurcations of ordinary differential equations due to Golubitsky et al. (1988) and delay differential equations due to Wu (1998), and combining the normal
form theory of functional differential equations due to Faria and Magalhaes (1995), the equivariant Hopf bifurcation is completely analyzed.


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## Body

## 1. Introduction

The works of Golubitsky et al. [1, Chapter XVIII] have shown that rings of identical cells can lead to many interesting patterns of oscillation, which are predictable based on the theory of equivariant bifurcations. In a series of papers, Wu et al. [2–4] have extended the theory of equivariant Hopf bifurcation to delay differential equations. Recently, there has been interest in applying these results to neural networks, primarily to models related to the Hopfield-Cohen-Grossberg neural networks with time delays [5–9]. In a series of papers [10–23], the authors studied the Hopf bifurcation to a network with one or two delays. Particularly, Wu et al. [24] have investigated the synchronization and stable phase-locking in a delayed network with three identical neurons:
(1.1)ẋi(t)=-xi(t)+αf(xi(t-τ))+β[f(xi-1(t-τ))+f(xi+1(t-τ))],i(mod3)
where f:R→R is a sufficiently smooth sigmoid amplification function, normalized so that f(0)=0 and f′(0)=1,α,β, and τ are regarded as parameters. They have shown that in the region Aas={(α,β):|α-β|<1} of the normalized parameters, system (4.1) is absolutely synchronous in the sense that every solution is convergent to the set of all synchronized phase states independent of the size of the time delay. The ω-limit set of a given orbit can be either a synchronized equilibrium or a synchronized periodic solution, depending on the connection topology of the network, the strength of the self-connection and the neighborhood-interaction, and the size of the delay. On the other hand, they have shown that, in the region α-β<-1, there exists a continuous surface τ=τ(α,β) where Hopf bifurcation of either a stable synchronized periodic solution or two stable phase-locked periodic solutions and six unstable periodic waves take place (more precisely, three mirror-reflecting waves and three standing waves). A question of mathematical and biological interest is whether the dynamics of (4.1) are possible for the following neural network model with n elements:
(1.2)ẋi(t)=-xi(t)+αf(xi(t-τ))+β[f(xi-1(t-τ))+f(xi+1(t-τ))],i(modn).
The purpose of the paper is to provide a detailed analysis of this question in the case that n is an odd number. More precisely, we shall extend the main results of Wu et al. [24] to the system (1.2) with n being an odd number. We are going to regard the delay τ as a parameter to investigate the dynamics of (1.2). According to the properties of the function tanhx, we assume that the transfer function f is adequately smooth, for example, f∈C3, and satisfies the following normalization, monotonicity, concavity, and boundedness conditions:(C1)
f(0)=0,f′(0)=1,andf′(x)>0forallx∈R;(C2)
f′′(0)=0,andxf′′(x)<0forallx≠0;(C3)
-∞<limx→±∞f(x)<∞;(C4)
f′′′(0)<0.At first, by analyzing the distribution of the eigenvalues, we give a bifurcation set in an appropriate parameter space to describe the stability of the equilibrium of the system (1.2) of  how to change as the parameters change. Meanwhile, the equivariant Hopf bifurcations are found. Then, by employing the center manifold and normal form theory, the direction of the bifurcation and the stability of the bifurcating periodic solutions are determined.The rest of this paper is organized as follows. In Section2, the characteristic equation of the linearization of system (1.2) at the zero equilibrium is derived. In Section 3, by analyzing the distribution of the eigenvalues, a bifurcation set is given in the (α,β)-plane. Then by using the theory of equivariant Hopf bifurcation to ordinary and delay differential equations due to Golubitsky et al. [1] and Wu [4], respectively, the local equivariant Hopf bifurcation of the trivial solution is completely analyzed. The appendix contain the detailed calculations of the normal forms on center manifolds of system (1.2) near Hopf bifurcation points.We would like to mention that there are several articles on the bifurcation forn-dimensional neural network models with delays; we refer the reader to [16, 17, 22] and references therein.


## 2. The Characteristic Equations

From conditions(C1)–(C4), we know that the linearization of system (1.2) at an equilibrium (0,0,…,0)T is given by
(2.1)ẋi(t)=-xi(t)+αxi(t-τ)+β[xi-1(t-τ)+xi+1(t-τ)],i(modn).
The characteristic matrix is
(2.2)Mn(0,λ)=(λ+1)Id-αe-λτId-βe-λτδ,
where Id is n×n identity matrix and
(2.3)δ=(0100⋯011010⋯000101⋯000010⋯000001⋯00⋮⋮⋮⋮⋯⋮⋮1000⋯10)n×n.Using a computation process similar to that of Wu et al. [24], Yuan and Campbell [25], we can obtain the following results.Let(2.4)χ=e2πi/n,vj=(1,χj,χ2j,…,χ(n-1)j)T(j=0,1,…,n-1).
From
(2.5)χnj=1,χ(n-1)j=χ-j,χ(n-2)j=χ-2j,…,…,
it follows that
(2.6)δvj=(χj+χ(n-1)j1+χ2jχj+χ3j⋮⋮⋮1+χ(n-2)j)=(χj+χ-j)vj=2cos(2πjn)vj.
Hence,
(2.7)Mn(0,λ)vj=[λ+1-(α+β(χj+χ-j))e-λτ]vj=[λ+1-(α+2βcos2πjn)e-λτ]vj,
and the characteristic equation is
(2.8)detMn(0,λ)=∏j=0n-1[λ+1-(α+2βcos2πjn)e-λτ]=Δ0(λ)×∏j=1n-1Δj(λ)=0,
where
(2.9)Δj(λ)=λ+1-(α+2βcos2πjn)e-λτ,(j=0,1,2,…,n-1).
We observe that (2.7) can be simplified, further elucidating the structure of characteristic equation.(I)
Whenn is an odd number, we obtain
(2.10)detMn(0,λ)=Δ0(λ)×∏j=1(n-1)/2Δj2(λ)=[λ+1-(α+2β)e-λτ]×∏j=1(n-1)/2[λ+1-(α+2βcos2πjn)e-λτ]2.(II)
Whenn is an even number, we obtain
(2.11)detMn(0,λ)=Δ0(λ)Δn/2(λ)×∏j=1(n-2)/2Δj2(λ)=[λ+1-(α+2β)e-λτ][λ+1-(α-2β)e-λτ]×∏j=1(n-2)/2[λ+1-(α+2βcos2πjn)e-λτ]2.In the case whenn is even, that is, n=2m, detMn(0,λ) may be further simplified into two subcases.(i)
Whenm is an odd number, we have
(2.12)detMn(0,λ)=Δ0(λ)Δn/2(λ)×∏j=1(m-1)/2Δj2(λ)Δm-j2(λ)=[λ+1-(α+2β)e-λτ][λ+1-(α-2β)e-λτ]×∏j=1(m-1)/2[λ+1-(α+2βcosπjm)e-λτ]2[λ+1-(α-2βcosπjm)e-λτ]2.(ii)
Whenm is an even number, we have
(2.13)detMn(0,λ)=Δ0(λ)Δn/2(λ)Δn/42(λ)×∏j=1(m-2)/2Δj2(λ)Δm-j2(λ)=[λ+1-(α+2β)e-λτ][λ+1-(α-2β)e-λτ](λ+1-αe-λτ)2×∏j=1(m-2)/2[λ+1-(α+2βcosπjm)e-λτ]2[λ+1-(α-2βcosπjm)e-λτ]2.From (2.8)–(2.13), we obtain the characteristic equation of (1.2) given by the following.(1)
Whenn is an odd number,
(2.14)[λ+1-(α+2β)e-λτ]∏j=1(n-1)/2[λ+1-(α+2βcos2πjn)e-λτ]2=0.(2)
Whenn is an even number and n is not the multiple of 4,
(2.15)[λ+1-(α+2β)e-λτ]∏j=1(n-2)/4[λ+1-(α+2βcos2πjn)e-λτ]2×[λ+1-(α-2β)e-λτ]∏j=1(n-2)/4[λ+1-(α-2βcos2πjn)e-λτ]2=0.(3)
Whenn is the multiple of 4,
(2.16)[λ+1-(α+2β)e-λτ]∏j=1(n-4)/4[λ+1-(α+2βcos2πjn)e-λτ]2×[λ+1-(α-2β)e-λτ]∏j=1(n-4)/4[λ+1-(α-2βcos2πjn)e-λτ]2×(λ+1-αe-λτ)2=0.


## 3. Bifurcation Analysis withn being an Odd Number

In this section, we are going to analyze the distribution of the root of the characteristic equations given in previous section, respectively.In this case, the characteristic equation of (1.2) is
(3.1)[λ+1-(α+2β)e-λτ]∏j=1(n-1)/2[λ+1-(α+2βcos2πjn)e-λτ]2=0.
Clearly, iω(ω>0) is a root of (3.1) if and only if there exists a
(3.2)j∈{0,1,2,…,n-12}
such that ω satisfies
(3.3)iω+1-(α+2βcos2πjn)e-iωτ=0,
that is,
(3.4)cosωτ=1α+2βcos(2πj/n),sinωτ=-ωα+2βcos(2πj/n).
This leads to
(3.5)ωj=(α+2βcos2πjn)2-1.
This shows that the necessary condition for (3.1) possessing purely imaginary roots is that there exists j∈{0,1,2,…,(n-1)/2} such that |α+2βcos(2πj/n)|>1, that is,
(3.6)α+2βcos2πjn>1orα+2βcos2πjn<-1.From (3.1), conclusion then follows immediately.Proposition 3.1.
Equation (3.1) has at most  (n+1)/2 pairs of purely imaginary roots when n is an odd number.
Without loss of generality, one assumes the (3.1) has (n+1)/2 pairs of purely imaginary roots ±ωj(j=0,1,2,…,(n-1)/2) exactly. Define
(3.7)τk(j)=1ωj[arccos1α+2βcos(2πj/n)+2kπ],j=0,1,…,n-12;k=0,1,2,….
Then ±iωj are purely imaginary roots of (3.1) with τ=τk(j)(k=0,1,2,…), respectively. one notes that ωjτk(j)∈(0,π](resp.,∈(π,2π]) when
(3.8)sin(ωjτk(j))=-ωjα+2βcos(2πj/n)>0(resp.<0).We can obtain the following result with no difficulty.Proposition 3.2.
Letλ(τ)=σ(τ)±iω(τ) be the root of (3.1) near τ=τk(j) satisfying σ(τk(j))=0, ω(τk(j))=ωj(j=0,1,…,(n-1)/2;k=0,1,2,…). Then
(3.9)dσ(τ)dτ|τ=τk(j)>0.Proof.
From (3.1) we have
(3.10)λ+1-(α+2βcos2πjn)e-λτ=0.
Differentiating both sides with respect to τ, we have
(3.11)(dλdτ)-1=-eλτλ(α+2βcos(2πj/n))-τλ.
Notice thatλ(τk(j))=iωj, it follows that
(3.12)Redλdτ|τ=τk(j)=dσ(τ)dτ|τ=τk(j)=ωj2(1+τk(j))2+(τk(j)ωj)2>0.
This completes the proof.Whenτ=0, (3.1) becomes
(3.13)[λ+1-(α+2β)]∏j=1(n-1)/2[λ+1-(α+2βcos2πjn)]2=0.
Its roots are given by
(3.14)λ0=α+2β-1,λ1=α+2βcos2πn-1,λ2=α+2βcos4πn-1,⋮⋮⋮λ(n-1)/2=α+2βcos(n-1)πn-1=α-2βcosπn-1.For convenience, denote(3.15)S:={(α,β)∣α<1,1-α2cos((n-1)π/n)<β<1-α2},(3.16)P:={(α,β)∣-1-α2<β<1-α2,1-α2cos((n-1)π/n)<β<-1-α2cos((n-1)π/n)},
and let D:=S/P. One can find out that the region S is a sector, P is a parallelogram, and D is the outside of the parallelogram in the sector S. These are shown in Figure 1.Figure 1
Bifurcation set in(α,β)-plane. The outside of the sector S is unstable region; the parallelogram P is the absolutely stable region; the outside of the parallelogram in the sector is conditionally stable region.Proposition 3.3.
Ifα+2β<1 and α+2βcos((n-1)π/n)<1(α-2βcos(π/n)<1), that is (α,β)∈S, then all roots of (3.13) have negative real parts.From (3.6), Proposition 3, and Corollary 2.4 of Ruan and Wei [19], we have the following result immediately.Proposition 3.4.
If|α+2β|<1 and |α+2βcos((n-1)π/n)|<1(|α-2βcos(π/n)|<1), that is, (α,β)∈P, then all roots of (3.1) have negative real parts for all τ≥0.From Proposition2 and Corollary 2.4 of Ruan and Wei [19], conclusion then follows immediately.Proposition 3.5.
If there existj∈{0,1,2,…,(n-1)/2} such that α+2βcos(2πj/n)>1, that is, (α,β) is located outside of the sector S, then (3.1) has at least one root with positive part for all τ≥0.Proposition 3.6.
Suppose that one of the following conditions is satisfied (i.e.,(α,β)∈D):
(1)
|α+2β|<1andα-2βcos(π/n)<-1;(2)
α+2β<-1and|α-2βcos(π/n)|<1;(3)
α+2β<-1andα-2βcos(π/n)<-1.
Then there exist at most  (n+1)/2 sequences of critical value
(3.17)τk(j),j=0,1,…,n-12;k=0,1,2,…
defined by (3.7) such that(1)
all roots of (3.1) have negative real parts for all τ∈[0,τ0), where
(3.18)τ0=min{τ0(j):0≤j≤n-12};(2)
equation (3.1) has at least one root with positive part when τ>τ0;(3)
atτ=τk(j)(j=0,1,…,(n-1)/2;k=0,1,…), (3.1) has a pair of purely imaginary roots ±iωj.From Propositions3.1–3.6, we have the following results immediately.Theorem 3.7.
(i) If|α+2β|<1 and |α-2βcos(π/n)|<1, then the zero equilibrium of (1.2) is asymptotically stable for all τ≥0.
(ii)
Ifj(j=0,1,2,…,(n-1)/2) exist such that α+2βcos(2πj/n)>1, then the zero equilibrium of (1.2) is unstable for all τ≥0.(iii)
Suppose that one of the following conditions holds:
(1)
|α+2β|<1andα-2βcos(π/n)<-1;(2)
α+2β<-1and|α-2βcos(π/n)|<1;(3)
α+2β<-1andα-2βcos(π/n)<-1.
Then there exist at most (n+1)/2 sequences of critical value τk(j)(j=0,1,…,(n-1)/2;k=0,1,2,…) defined by (3.7) such that the zero equilibrium of (1.2) is asymptotically stable for τ∈[0,τ0), where τ0=min{τ0(j):0≤j≤(n-1)/2}, unstable for all τ>τ0, and the system undergoes a Hopf bifurcation with τ=τk(j)(j=0,1,…,(n-1)/2;k=0,1,…).Proposition3.6 shows that the region D, which is the outside of the parallelogram in the sector shown as in Figure 1, is a conditionally stable region. This means that the stability of the zero equilibrium of (1.2) is dependent on the delay. Namely, the distribution of the roots of (3.1) is dependent on the delay. Particularly, for (α,β)∈D, (3.1) with some critical value τ has simple pure imaginary roots or multiply pure imaginary roots. It is well know that the Hopf bifurcations are different between the simple and multiple pure imaginary roots. So it is necessary to observe the region D. In fact, we have the following conclusions on occurrence of purely imaginary roots of (3.1) for (α,β)∈D.(1) In the region(3.19)E0={(α,β)∣α+2β<-1;α-2βcosπn<1;α+2βcos2πn>-1},
Equation (3.1) with τk(0) has one pair of purely imaginary roots exactly, denoted by ±iω0, which are simple, where ω0=(α+2β)2-1 and
(3.20)τk(0)=1ω0(arccos1α+2β+2kπ)(k=0,1,2,…).(2) In the region(3.21)E1={(α,β)∣α+2βcos2πn<-1;α-2βcosπn<1;α+2βcos4πn>-1},
Equation (3.1) has two pairs of purely imaginary roots exactly, denoted by ±iω0 and ±iω1, when τ=τk(1) and τ=τk(0), respectively. Here ±iω0 are simple, ±iω1 are double, and ω1=(α+2βcos(2π/n))2-1, τk(0) is defined by (3.20) and
(3.22)τk(1)=1ω1(arccos1α+2βcos(2π/n)+2kπ)(k=0,1,2,…).
Furthermore,
(3.23)τ0(0)=1ω0(arccos1α+2β)>τ0(1)=1ω1(arccos1α+2βcos(2π/n)).(3) In the region(3.24)E2={(α,β)∣α+2βcos4πn<-1;α-2βcosπn<1;α+2βcos6πn>-1},
Equation (3.1) has three pairs of purely imaginary roots exactly, denoted by ±iω0, ±iω1, and ±iω2, when τ=τk(0), τ=τk(1), and τ=τk(2), respectively. Here ±iω0 are simple, ±iω1 and ±iω2 are double, ω2=(α+2βcos(4π/n))2-1, τk(0) and τk(1) are defined by (3.20) and (3.22), respectively, and
(3.25)τk(2)=1ω2(arccos1α+2βcos(4π/n)+2kπ)(k=0,1,2,…).
Furthermore,
(3.26)τ0(0)>τ0(1)>τ0(2)=1ω2(arccos1α+2βcos(4π/n)).…………
Naturally, the region E(n-1)/4 (when n-1 is multiple of 4) or E(n-3)/4 (when n-1 is not multiple of 4) can be defined by
(3.27)E(n-1)/4={(α,β)∣α+2βcos(n-1)π2n<-1;α-2βcosπn<1;α+2βcos2πn<-1},E(n-3)/4={(α,β)∣α+2βcos(n-3)π2n<-1;α-2βcosπn<1;α+2βcos2πn<-1}.⋆A: Obviously, in the region E(n-1)/4 (or E(n-3)/4), we have that
(3.28)τ0(0)>τ0(1)>τ0(2)>τ0(3)>⋯>τ0((n-1)/4)(orτ0((n-3)/4)).(4) In the region(3.29)E((n-1)/2)={(α,β)∣α+2βcos(n-1)πn<-1;α+2β<1;α+2βcos(n-3)πn>-1}={(α,β)∣α-2βcosπn<-1;α+2β<1;α-2βcos3πn>-1},
Equation (3.1) has a pair of purely imaginary roots exactly, denoted by ±iω(n-1)/2, which are double, where ω(n-1)/2=(α+2βcos((n-1)π/n))2-1=(α-2βcos(π/n))2-1 and
(3.30)τk((n-1)/2)=1ω(n-1)/2(arccos1α-2βcos(π/n)+2kπ)(k=0,1,2,…).(5) In the region(3.31)E((n-3)/2)={(α,β)∣α-2βcos3πn<-1;α+2β<1;α-2βcos5πn>-1},
Equation (3.1) has two pairs of purely imaginary roots exactly, denoted by ±iω(n-1)/2 and ±iω(n-3)/2, when τ=τk((n-1)/2) and τ=τk((n-3)/2)), respectively, and the two pairs of purely imaginary roots are all double. Here
(3.32)ω(n-3)/2=(α+2βcos(n-3)πn)2-1=(α-2βcos3πn)2-1,τk((n-1)/2) is defined by (3.30), and
(3.33)τk((n-3)/2)=1ω(n-3)/2(arccos1α-2βcos(3π/n)+2kπ)(k=0,1,2,…).
Moreover,
(3.34)τ0((n-1)/2)=1ω(n-1)/2(arccos1α-2βcos(π/n))>τ0((n-3)/2)=1ω(n-3)/2(arccos1α-2βcos(3π/n)).…………
Naturally, the region E(n+3)/4 (when n-1 is multiple of 4) or E(n+1)/4 (when n-1 is not multiple of 4) can be defined by
(3.35)E(n+3)/4={(α,β)∣α+2βcos2πn<-1;α+2β<1;α+2β>-1},E(n+1)/4={(α,β)∣α+2βcos2πn<-1;α+2β<1;α+2β>-1}.⋆B: Obviously, in the region E(n+3)/4 (or E(n+1)/4), we have
(3.36)τ0((n-1)/2)>τ0((n-3)/2)>τ0((n-5)/2)>⋯>τ0((n+3)/4)(orτ0((n+1)/4)).To carry out our work we need some background from the theory of functional differential equation. LetC=C([-τ,0],Rn) denote the Banach space of continuous mappings from [-τ,0] into Rn equipped with the supernorm. Let x(t) be a solution of (1.2) and define xt(θ)=x(t+θ),-τ≤θ≤0. If x(t) is continuous, then xt(θ)∈C. With this structure, we may write the model as the following functional differential equation:
(3.37)ẋ(t)=F(xt),
where F:C→Rn is defined via
(3.38)Fi(ϕ)=-ϕi(0)+αf(ϕi(-τ))+β[f(ϕi-1(-τ))+f(ϕi+1(-τ))],i(modn).
Similarly, the linearization of (3.37) at equilibrium (0,0,…,0)T may be written as
(3.39)u̇(t)=L(τ)ut,
where the linear operator L(τ):C→Rn is defined via
(3.40)L(τ)ϕ=-Idϕ(0)+αIdϕ(-τ)+βδϕ(-τ).
It is well known that a linear functional differential equation such as (3.39) generates a strongly continuous semigroup of linear operators with infinitesimal generator A given by
(3.41)Aϕ=ϕ̇,ϕ∈Dom(A),
where Dom(A)=ϕ∈C,ϕ(0)=L(τ)ϕ.To explore the possible (spatial) symmetry of (1.2), we need to introduce three compact Lie groups. One is the cycle group S1; another is Zn, the cyclic group of order n, which corresponds to rotations of 2π/n, denoting the generator of this group by ρ, then action on Rn is given by (ρx)i=xi+1; the third is the dihedral group Dn of order 2n, which corresponds to the group of symmetries of an n-gon. It can be shown that Dn is generated by ρ and κ, where κ is the flip of order 2 or reflection, and it acts on Rn by (κx)i=xn+2-i.Definition 3.8.
LetF:C→Rn and Γ be a compact group. The system (3.37) is said to be Γ-equivariant if F(γxt)=γF(xt) for all γ∈Γ.Proposition 3.9.
The nonlinear system (3.37) and linear system (3.39) are Dn equivariant.Proof.
We begin with (3.37); that is, we let F be as in (3.38) and ϕ∈C. We need only check the equivariant condition on the generators, ρ, κ, of Dn:
(3.42)Fi(ρϕ)=-(ρϕ)i(0)+αf((ρϕ)i(-τ))+β[f((ρϕ)i-1(-τ))+f((ρϕ)i+1(-τ))]=-ϕi+1(0)+αf(ϕi+1(-τ))+β[f(ϕi(-τ))+f(ϕi+2(-τ))]=Fi+1(ϕ)=ρFi(ϕ),Fi(κϕ)=-(κϕ)i(0)+αf((κϕ)i(-τ))+β[f((κϕ)i-1(-τ))+f((κϕ)i+1(-τ))]=-ϕn+2-i(0)+αf(ϕn+2-i(-τ))+β[f(ϕn+3-i(-τ))+f(ϕn+1-i(-τ))]=Fn+2-i(ϕ)=κFi(ϕ).
Thus (3.37) is Dn equivariant.
From (3.39), we begin by noting that L may be written componentwise as follows:
(3.43)Li(τ)ϕ=-ϕi(0)+aϕi(-τ)+β[ϕi-1(-τ)+ϕi+1(-τ)],i(modn).
The rest of the proof is similar to that for (3.37). This completes the proof.In the case where(α,β)∈E0, we can apply the standard Hopf bifurcation theorem of delay differential equations to obtain a Hopf bifurcation of synchronous periodic solutions. In the case where (α,β)∈Ej (1≤j≤(n-1)/2), the aforementioned standard Hopf bifurcation theorem does not apply since ±iωj are double eigenvalues. On the other hand, the considered system is equivariant with respect to the Dn-action where the Zn subgroup acts by permutation (sending xi to xi+1) and the flip acts by interchanging (sending xi to xn+2-i). This allows us to apply the symmetric Hopf bifurcation theorem for delay differential equations established in [4] (as an extension of the well-known Golubitsky-Stewart Theorem [1] for symmetric ordinary differential equations) to obtain 2(n+1) branches of asynchronous periodic solutions. More precisely, we have the following theorem.Theorem 3.10.
Assume that(α,β)∈D (D=S/P, where the regions S and P are defined by (3.15) and (3.16), resp.). Then
(1)
in case(α,β)∈E0, near τ=τ0(0) there exists a supercritical bifurcation of stable synchronous periodic solutions of period pτ near (2π/ω0), bifurcated from the zero solution of system (1.2);(2)
in case(α,β)∈Ej (1≤j≤(n-1)/2), near τ=τ0(j) there exist 2(n+1) branches of asynchronous periodic solutions of period pτ near (2π/ωj), bifurcated simultaneously from the zero solution of system (1.2), and there are
(a)
two stable phase-locked oscillations:xi(t)=xi+1(t±(pτ/n)), for i(modn) and t∈R,(b)
n unstable mirror-reflecting waves: xi(t)=xn+2k-i(t) for i(modn) and t∈R, where k=1,2,…,n,(c)
n unstable standing waves: xi(t)=xn+2k-i(t-(pτ/2)) for i(modn) and t∈R, where k=1,2,…,n.Proof.
(1) The existence is an immediate application of the standard Hopf bifurcation theorem for functional differential equations. Letτ*=τ0(0), μ=τ-τ0(0). According to the calculations in Part 2 of  the appendix, the normal form of (1.2) on the center manifold can be written in polar coordinates as
(3.44)ρ̇=(a1μ+b1ρ2)ρ+O(μ2ρ)+O(ρ4),
where
(3.45)a1=ω0τ0(0)(1+τ0(0))2+ω02>0,b1=f′′′(0)2τ0(0)(1+τ0(0)+ω0τ0(0))(1+τ0(0))2+ω02<0.
Conclusion (1) then follows immediately.
(2) Letτ*=τ0(j), u*=ωj/τ0(j), μ=τ-τ0(j). We obtain from the calculations in Part 1 of  the appendix the normal form of (1.2) on the center manifolds that is given by
(3.46)(ω̇1ω̇2)=u*τ*(ω2-ω1)+μu*(Im(a-1)ω1+Re(a-1)ω2-Re(a-1)ω1+Im(a-1)ω2)+3bτ*(ρ12+2ρ22)(Re(a-1(1-iu*))ω1-Im(a-1(1-iu*))ω2Im(a-1(1-iu*))ω1+Re(a-1(1-iu*))ω2)+O(μ2|ω|)+O(|ω|4),(ω̇3ω̇4)=u*τ*(ω4-ω3)+μu*(Im(a-1)ω3+Re(a-1)ω4-Re(a-1)ω3+Im(a-1)ω4)+3bτ*(ρ22+2ρ12)(Re(a-1(1-iu*))ω3-Im(a-1(1-iu*))ω4Im(a-1(1-iu*))ω3+Re(a-1(1-iu*))ω4)+O(μ2|ω|)+O(|ω|4),
where
(3.47)ρ1=ω12+ω22,ρ2=ω32+ω42,a=1+τ*-iu*τ*,b=13!f′′′(0).
Introducing the periodic-scaling parameter ω and letting
(3.48)z1(t)=ω1(s)+iω2(s),z2(t)=ω3(s)+iω4(s)
with
(3.49)s=[(1+ω)u*τ*]-1t,
and using a computation process similar to that of Wu et al. [24], we obtain the normal form
(3.50)(1+ω)ż1=-iz1-iμ(τ*)-1a-1z1+3b(u*)-1a-1(1-iu*)(|z1|2+2|z2|2)z1,(1+ω)ż2=-iz2-iμ(τ*)-1a-1z2+3b(u*)-1a-1(1-iu*)(|z2|2+2|z1|2)z2.
Let g:C⊕C⊕R→C⊕C be given so that -g(z1,z2,μ) is the right-hand side of (3.50). Then (3.50) can be written as
(3.51)(1+ω)ż+g(z,μ)=0.
Note that
(3.52)Dzg(0,0)(z1,z2)=i(z1,z2),z=(z1,z2)∈C⊕C.
Also note that g(·,μ):C⊕C→C⊕C is Dn×S1-equivariant with respect to the following Dn×S1-action on C⊕C:
(3.53)ρ(z1,z2)=(ei(2kπ/n)z1,e-i(2kπ/n)z2),Zn=〈ρ〉≤Dn,κ(z1,z2)=(z2,z1),Z2=〈κ〉≤Dn,eiθ(z1,z2)=(eiθz1,eiθz2),eiθ∈S1.
Using a computation process similar to that of Wu et al. [24], we obtain
(3.54)A0=iu*(τ*)-1a-1-iω,AN=-6b(u*)-1a-1(1-iu*),B0=3b(u*)-1a-1(1-iu*).
By the results of [1, page 376], we know that the bifurcation of phase-locked oscillation is supercritical (resp., subcritical) and depends on whether Re(AN+B0)>0 (resp., Re(AN+B0)<0), and these are orbitally asymptotically stable if Re(AN+B0)>0 and ReB0<0.
Note that(3.55)Re(AN+B0)=Re(-3b(u*)-1a-1(1-iu*))=-3b(u*)-1Re[1+τ*+iu*τ*(1+τ*)2+(u*τ*)2(1-iu*)]=-3b(u*)-11+τ*+(u*)2τ*(1+τ*)2+(u*τ*)2>0,Re(B0)=Re(3b(u*)-1a-1(1-iu*))=3b(u*)-11+τ*+(u*)2τ*(1+τ*)2+(u*τ*)2<0.
Consequently, the bifurcation of phase-locked oscillations is supercritical and orbitally asymptotically stable.
Note also that(3.56)Re(2AN+B0)=Re(-9b(u*)-1a-1(1-iu*))=-9b(u*)-11+τ*+(u*)2τ*(1+τ*)2+(u*τ*)2>0,Re(B0)<0.
We infer from the results of [1, page 376] again that the bifurcations of mirror-reflecting waves and standing waves are supercritical and unstable. This completes the proof.Using a proof process similar to that of Yuan and Campbell [25], by using Liapunov's second method we can obtain the followings.Theorem 3.11.
If|α|+2|β|<1, then the trivial solution of (1.2) is global asymptotically stable.


## 4. Computer Simulations

To demonstrate the properties of the Hopf bifurcation in Section3, we carry out some numerical simulations for a particular case of (1.2) as in following form:
(4.1)ẋi(t)=-xi(t)+α[tanh(xi(t-τ))]+β[tanh(xi-1(t-τ))+tanh(xi+1(t-τ))],i(mod3).Clearly, the origin(0,0,0) is a fixed point to (4.1). Choosing α=-2 and β=1/3 we obtain that τ0(0)=2.73273 and τ0(1)=0.955185, thus τ0=0.955185.From Theorem3.7(iii), it follows that the zero equilibrium is asymptotically stable if τ∈[0,τ0), it is unstable if τ>τ0, and the system undergoes a Hopf bifurcation with τ=τk(j)(j=0,1,…,(n-1)/2;k=0,1,…). Simulate the solutions of system (4.1) for τ=0.5 and 1.0. In Figure 2, it is shown that the zero equilibrium is asymptotically stable for τ=0.5<τ0=0.955185. In Figure 3, for the data τ=1.0>τ0=0.99185, it is shown that there exists a periodic orbit which is orbitally asymptotically stable.Figure 2
Simulations for the asymptotically stable equilibrium to system (4.1) with τ=0.5<τ0=0.955185.Figure 3
Simulations for an orbitally asymptotically stable periodic orbit to system (4.1) with τ=1.0>τ0=0.99185.



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*Source: 950251-2009-07-29.xml*