# Bogdanov-Takens and Triple Zero Bifurcations of a Delayed Modified Leslie-Gower Predator Prey System

Authors: Xia Liu; Jinling Wang
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

A delayed modified Leslie-Gower predator prey system with nonlinear harvesting is considered. The existence conditions that an equilibrium is Bogdanov-Takens (BT) or triple zero singularity of the system are given. By using the center manifold reduction, the normal form theory, and the formulae developed by Xu and Huang, 2008 and Qiao et al., 2010, the normal forms and the versal unfoldings for this singularity are presented. The Hopf bifurcation of the system at another interior equilibrium is analyzed by taking delay (small or large) as bifurcation parameter.


---

## Body

## 1. Introduction

For a more detailed study on the properties of the predator prey systems, the multiple bifurcations for some systems (ODE) with more interior equilibria are investigated by many authors, see [1–4] for example. To deal with this type of systems, the more difficult problem is how to obtain the normal form of the system at its degenerate equilibrium, that is, BT bifurcation.When introducing the time delay into this type of systems, using the methods developed by [5], the authors in [6, 7] have researched the BT bifurcation of some predator prey systems (DDE). Their results show that time delay may have an effect or not on the BT bifurcation.Recently, papers [8–10] have considered the triple zero bifurcation of some delay differential equations, depending on the parameters in the original system; some interesting bifurcation results are obtained. But we find that there are few results about the triple zero bifurcation for predator prey systems.Summarizing the above references, we will consider the following predator prey system with Michaelis-Menten type (nonlinear) prey harvesting:(1)x˙=rx(1-xK)-a1xyn+x-qExk1E+k2x,y˙=s1y(1-a2y(t-τ)n+x(t-τ)),
where x and y denote the prey and predator populations, respectively. τ represents the negative feedback of the predator’s density. For complete reason, we give the biological meaning of the parameters, one can be seeing them in [11]. r and K are intrinsic growth rate and environmental carrying capacity for the prey, respectively. a1 is the maximum value of the per capita reduction rate of prey, n measures the extent to which the environment provides protection to prey and predator respectively, s1 measures the growth rate of the predator species, and s1a2 is the maximum value of the per capita reduction rate of predator. p(x)=qEx/(k1E+k2x) represents Michaelis-Menten type harvesting, q is the catchability coefficient, E is the effort applied to harvest the prey species, and k1 and k2 are suitable constants.The authors in [12, 13] have studied system (1) without prey harvesting p(x), respectively, in [12], the global stability and persistence of the system are investigated. In [13], by using the Hopf bifurcation theorem and taking the delay as a parameter of bifurcation for small and large cases, the existence of the bifurcated limit cycle around a boundary equilibrium or an interior equilibrium is mainly considered.For system (1) with τ=0, the authors in [11] have given detailed analysis about the existence of the multiple bifurcations (including BT bifurcation) depending on the parameters of the system.For computation simplicity, we first rescale system (1).Lett-=rt, x-(t-)=x(t)/K, y-(t-)=a1y(t)/K, and  τ-=rτ; then dropping the bars we obtain
(2)x˙=x(1-x-αym+x-hc+x),y˙=ρy(1-βy(t-τ)m+x(t-τ)),
where α=1/r, β=a2/a1, m=n/K, ρ=s1/r, h=qE/rk2K, and c=k1E/k2K.System (2) with initial conditions is
(3)ϕ~(ϑ)⩾0,ψ~(ϑ)⩾0,ϑ∈[-τ,0),ϕ~(0)>0,ψ~(0)>0,
where ϕ~(ϑ) and ψ~(ϑ) are all continuous bounded functions in the interval [-τ,0).From [11], we know that (2) has interior equilibrium E-=(x-,y-) if
(4)h=h*=14(1-αβ+c)2=(c+x-)2,αβ+c<1,
where x-=(1/2)(1-c-α/β), y-=(m+x-)/β and interior equilibrium E*=(x*,y*) if h<h*, where x*=(1/2)(1-c-α/β+Δ), y*=(m+x*)/β, and Δ=(α/β+c-1)2-4c(α/β+h/c-1).In this paper, for system (2), we will mainly consider the BT and triple zero bifurcations at E- and the Hopf bifurcation at E*. It is easy to see that this system is with six parameters which will let our work become more challenging. When dealing with the BT and triple bifurcations of the delay systems, the core problem is to change delay systems as ordinary differential systems (ODEs).The concrete organization of the paper is as follows: in Section2, we will give the conditions under which the equilibrium E- is a BT singularity, and a universal unfolding will be exhibited; in Section 3, when E- is a triple zero, the universal unfolding will be presented, and in Section 4, some Hopf bifurcation results at E* will be obtained.


## 2. Bogdanov-Takens Bifurcation

System (2) also can be written as
(5)x˙=τx(1-x-αym+x-hc+x),y˙=τρy(1-βy(t-1)m+x(t-1)).
Linearizing system (5) at E- yields the following linear system
(6)x˙1=ταx-β(m+x-)x1(t)-ταx-m+x-x2(t),x˙2=τρβx1(t-1)-τρx2(t-1),
the corresponding characteristic equation is
(7)F(λ)=λ2+(τρe-λ-ταx-β(m+x-))λ.
Evidently, λ=0 is a double zero eigenvalue if
(8)τ≠1ρ,ρ=αx-β(m+x-);λ=0 is a triple zero eigenvalue if
(9)τ=1ρ,ρ=αx-β(m+x-).
It is easy to prove that in the above two cases the rest eigenvalues all have negative real parts.Under the conditions (4) and (8), let x1=x-x-, and let x2=y-y-; then the Taylor expansion system (5) at E- is
(10)x˙1=τρx1(t)-τβρx2(t)+∑i+j≥2‍1i!j!fij(1)x1i(t)x2j(t),x˙2=τρβx1(t-1)-τρx2(t-1)+∑i+j+k≥2‍1i!j!k!fijk(2)x1i(t-1)x2j(t-1)x2k(t),
where i,j,k≥0,  fij(1)=(∂i+jf(1)/∂ix∂jy)|(x-,y-), fijk(2)=(∂i+j+kf(2)/∂ix∂jy∂ky0)|(x-,y-), f(1)=τx(1-x-αy/(m+x)-h/(c+x)), f(2)=τy0(1-βy/(m+x)).In the following, we first give the normal form of the system (10) at the singularity (0,0). Reference [6], we first rewrite system (10) as X˙(t)=L(Xt), here X(t)=(x1(t),x2(t)),  L(ϕ)=L(ϕ1(-1)ϕ2(0)), and ϕ=(ϕ1,ϕ2). By the normal form theory developed by Faria and Magalhaes [5], one can obtain the center manifold of this system at the origin which is two-dimensional and system can be reduced to an ODE in the plane.DefineA0 to be the infinitesimal generator of system. Consider λ={0} and let P denote the invariant space of A0 associated with the eigenvalue λ=0, using the formal adjoint theory in [5], the phase space C1 can be decomposed by λ as C1=P⊕Q. Let Φ and Ψ be the bases for P and P*, respectively, and be let them normalized such that 〈Ψ,Φ〉=I, Φ˙=ΦJ,and Ψ˙=-JΨ, where Φ and Ψ are 2×2 matrices, where J=(0100).Next, we will find theΦ(θ) and Ψ(s) based on the techniques developed by [14].Lemma 1 (see Xu and Huang [14]).
The bases ofP and their dual space P* have the following representations:
(11)P=spanΦ,Φ(θ)=(φ1(θ),φ2(θ)),-1≤θ≤0,P*=spanΨ,Ψ(s)=col(ψ1(s),ψ2(s)),0≤s≤1,
where φ1(θ)=φ10∈Rn∖{0}, φ2(θ)=φ20+φ10θ,φ20∈Rn, and ψ2(s)=ψ20∈Rn*∖{0}, ψ1(s)=ψ10-sψ20, ψ10∈Rn*, which satisfy(1)
(A+B)φ10=0,(2)
(A+B)φ20=(B+I)φ10,(3)
ψ20(A+B)=0,(4)
ψ10(A+B)=ψ20(B+I),(5)
ψ20φ20-(1/2)ψ20Bφ10+ψ20Bφ20=1,(6)
ψ10φ20-(1/2)ψ10Bφ10+ψ10Bφ20+(1/6)ψ20Bφ10-(1/2)ψ20Bφ20=0.By system (6), we know that
(12)A=(ταx-β(m+x-)-ταx-m+x-00)=(τρ-τβρ00),B=(00τρβ-τρ).By Lemma1, we have
(13)Φ(θ)=(ββ[2-(τρ-2)2]2τρ(τρ-1)+βθ12-τρ2(τρ-1)+θ)≜(βm1+βθ1m2+θ),-1≤θ≤0,Ψ(s)=(-τρβ(1-τρ)s1-τρτρ-1sτρβ(1-τρ)τρτρ-1),0≤s≤1,
where m1=β[2-(τρ-2)2]/2τρ(τρ-1), m2=(2-τρ)/2(τρ-1),
(14)Φ(θ)z=(βm1+βθ1m2+θ)(z1z2)=(βz1+(m1+βθ)z2z1+(m2+θ)z2),
therefore,
(15)ϕ1(0)=βz1+m1z2,ϕ2(0)=z1+m2z2,ϕ1(-1)=βz1+(m1-β)z2,ϕ2(-1)=z1+(m2-1)z2.Taking a similar theory in [6], system (6) in the center manifold is
(16)z˙=Jz+Ψ(0)F(Φz),
where
(17)F(ϕ)=(τ∑i+j≥2‍1i!j!fij(1)[ϕ1(0)]i[ϕ2(0)]jτρ∑i+j+k≥2‍1i!j!k!fijk(2)ϕ1i(-1)ϕ2j(-1)ϕ2k(0)).Using (10) and (15), system (16) also can be written as
(18)z˙1=z2+β(τρ-1)τρ(m+x-)z22+h.o.t,z˙2=g1z12+g2z1z2+g3z22+h.o.t,
where g1=-τ2βρx-/(1-τρ)(c+x-), g2=ταm/(1-τρ)(m+x-)2-2m1ρτ2x-/(1-τρ)(c+x-), g3=(τρ/β(1-τρ))[αm1m/ρ(m+x-)2-τx-m12/(c+x-)]+β/(m+x-). Through a series of transformation, system (18) becomes
(19)z˙1=z2+h.o.t,z˙2=d1z12+d2z22+h.o.t,
where d1=g1 and d2=g2. If d1d2≠0, we have the following theorem.Theorem 2.
Let (4), (8), and τ≠βm(c+x-)/2m1x-2(m+x-) hold. Then, the equilibrium E- of system (5) is a BT singularity.Next, we are interested in giving a versal unfolding for system (5) at BT singularity. Choosing h and ρ as bifurcation parameters and incorporating h+λ1 and ρ+λ2 to system (5), where λ1 and λ2 vary in a small neighborhood of (0,0), we obtain
(20)x˙=τx(1-x-αym+x-h+λ1c+x),y˙=τ(ρ+λ2)y(1-βy(t-1)m+x(t-1)).Lety1=x-x- and y2=y-y-. Then, system (20) becomes
(21)y˙1=-τx-λ1c+x-+τ[αx-β(m+x-)+x-λ1h-λ1c+x-]y1(t)-ταx-m+x-y2(t)+[ταmβ(m+x-)2+τ(cλ1-hx-)h(c+x-)]y12(t)-ταm(m+x-)2y1(t)y2(t)+h.o.t,y˙2=τ(ρ*+λ2)×[1βy1(t-1)-y2(t-1)+1m+x-y1(t-1)y2(t)+1m+x-y1(t-1)y2(t-1)-1β(m+x-)y12(t-1)-βm+x-y2(t)y2(t-1)]+h.o.t.
Then, system (21) can be decomposed as
(22)z˙=B1+B2z+Ψ0G(Φz+u),u˙=AQ′u+(I-π)X0×[B0+B2(Φ0z+u(0))+G(Φz+u)],
where
(23)X0(θ)={I,θ=00,-1≤θ<0,B0=(-τx-λ1c+x-0),B1=Ψ(0)B0=(01τρβ(1-τρ)τρτρ-1)(-τx-λ1c+x-0)=(0-τ2ρx-λ1β(c+x-)(1-τρ)),G(ϕ)=(τ∑i+j≥2‍1i!j!gij(1)[ϕ1(0)]i[ϕ2(0)]jτ(ρ+λ2)∑i+j+k≥2‍1i!j!k!gijk(2)ϕ1i(-1)ϕ2j(-1)ϕ2k(0)),gij(1)=∂i+jg(1)∂ix∂jy|(x-,y-,λ1),gijk(2)=∂i+j+kf(2)∂ix∂jy∂ky0|(x-,y-),g(1)=x(1-x-αym+x-h*+λ1c+x),g(2)=y0(1-βym+x),B2=(01+λ2ρτ2ρcλ1h(τρ-1)τ2ρτρ-1(cm1λ1hβ+λ2τρ)).Then, the normal form of system (21) at E- is z˙=B1+B2z+Ψ(0)G(Φz); that is,
(24)z˙1=z2+λ2ρz2+β(ρ+λ2)(τρ-1)τρ2(m+x-)z22+h.o.t,z˙2=-τ2x-ρλ1β(c+x-)(1-τρ)+τ2cρλ1h(τρ-1)z1+τ2ρτρ-1(cm1λ1hβ+λ2τρ)z2+P1z12+P2z1z2+P3z22+h.o.t,
where
(25)P1=τ2βρ(cλ1-hx-)h(1-τρ)(c+x-),P2=ταm(1-τρ)(m+x-)2+2m1ρτ2(cλ1-hx-)h(1-τρ)(c+x-),P3=τρβ(1-τρ)[αm1mρ(m+x-)2+τ(cλ1-hx-)m12h(c+x-)]+β(ρ+λ2)ρ(m+x-).Following the normal form formula in Kuznetsov [15], system (24) can be reduced to
(26)z˙1=z2+h.o.t,z˙2=γ1+γ2z1+d1z12+d2z1z2+h.o.t,
where
(27)γ1=-τ2x-ρλ1β(c+x-)(1-τρ),γ2=[ρ3τ3(m1mβ2-τm12αx-3(c+x-))αx-β2(1-τρ)2(c+x-)2τ2x-c+x-(2β2x-2-τcm1αx-2(c+x-)ρmβ2(c+x-)-2m1ταx-3)+ρ3τ3(m1mβ2-τm12αx-3(c+x-))αx-β2(1-τρ)2(c+x-)]λ1-2ατ2x-3ρ(1-τρ)[mβ(c+x-)-2m1τx-2(m+x-)]λ2.Then, system (5) exists in the following bifurcation curves in a small neighborhood of the origin in the (λ1,λ2) plane.Theorem 3.
Let (4), (8), and τ≠βm(c+x-)/2m1x-2(m+x-) hold. System (5) admits the following bifurcations:(i)
a saddle-node bifurcation curveSN={(λ1,λ2);γ1=(1/4d1)γ22};(ii)
a Hopf bifurcation curveH={(λ1,λ2);λ2=0,γ2<0};(iii)
a homoclinic bifurcation curveHL={(λ1,λ2);γ1=-(6/25d1)γ22,γ2<0}.


## 3. Triple-Zero Bifurcation

From Section2, we know that under the conditions (4) and (9) the equilibrium E- of system (5) is a triple zero singularity. In the following reference, from the work of [8, 9] we will give the triple zero bifurcation at E-.Letx1=x-x- and x2=y-y-; then, system (5) becomes
(28)x˙1=τ[∑i+j≥2‍1i!j!(1-2x--αmβ(m+x-)-hc(c+x-)2)x1(t)-ρβx2(t)+∑i+j≥2‍1i!j!fij(1)x1i(t)x2j(t)],x˙2=τρ[∑i+j+k≥2‍1i!j!k!1βx1(t-1)-x2(t-1)+∑i+j+k≥2‍1i!j!k!fijk(2)x1i(t-1)x2j(t-1)x2k(t)].To determine a versal unfolding for the original system (28) at E-, we choose τ, ρ, and h as bifurcation parameters, and let them become τ+μ1,ρ+μ2, and h+μ3, respectively, where μ1,μ2, and μ3 vary in a small neighborhood of (0,0,0); then (28) can be written as
(29)x˙1=(τ+μ1)×[αm(m+x-)2(ρ-cμ3h)x1(t)-ρβx2(t)+(αmβ(m+x-)2+(cμ3-hx-)h(c+x-))x12(t)-αm(m+x-)2x1(t)x2(t)]+h.o.t,x˙2=(τ+μ1)(ρ+μ2)×[1βx1(t-1)-x2(t-1)+1m+x-x1(t-1)x2(t)+1m+x-x1(t-1)x2(t-1)-1β(m+x-)x12(t-1)-βm+x-x2(t)x2(t-1)]+h.o.t.Next, we need to find the expressions ofΦ(θ) and Ψ(s) based on the techniques developed by [9].Lemma 4 (see Qiao et al. [9]).
The bases ofP and their dual space P* have the following representations:
(30)P=spanΦ,Φ(θ)=(φ1(θ),φ2(θ),φ3(θ)),-1≤θ≤0,P*=spanΨ,Ψ(s)=col(ψ1(s),ψ2(s),ψ3(s)),0≤s≤1,
where φ1(θ)=φ10∈Rn∖{0}, φ2(θ)=φ20+φ10θ, φ3(θ)=φ30φ20θ+(1/2)φ10θ2, φ20,φ30∈Rn, and ψ3(s)=ψ30∈Rn*∖{0}, ψ2(s)=ψ20-sψ30, ψ1(s)=ψ10-sψ20+(1/2)s2ψ30, ψ10,ψ20∈Rn*, which satisfy (1)
(A+B)φ10=0,(2)
(A+B)φ20=(B+I)φ10,(3)
(A+B)φ30=(B+I)φ20-(1/2)Bφ10,(4)
ψ30(A+B)=0,(5)
ψ20(A+B)=ψ30(B+I),(6)
ψ10(A+B)=ψ20(B+I)-(1/2)ψ30B,(7)
ψ30(B+I)φ30-(1/2)ψ30Bφ20+(1/6)ψ30Bφ10=1,(8)
ψ20(B+I)φ30-(1/2)ψ20Bφ20+(1/6)ψ20Bφ10-(1/2)ψ30Bφ30+(1/6)ψ30Bφ20-(1/24)ψ30Bφ10=0,(9)
ψ10(B+I)φ30-(1/2)ψ10Bφ20+(1/6)ψ10Bφ10-(1/2)ψ20Bφ30+(1/6)ψ20Bφ20-(1/24)ψ20Bφ10 + (1/6)ψ30Bφ30-(1/24)ψ30Bφ20 + (1/120)ψ30Bφ10=0.For system (29), one can see that
(31)A+B=τρ(1-β1β-1)=(1-β1β-1).
By Lemma 4, we can obtain
(32)Φ(θ)=(ββθβ(1+12θ2)1-1+θ1-θ+12θ2),-1≤θ≤0,Ψ(s)=(-1718β-2s3β+s2β3518+2s3-s223β-2sβ-23+2s2β-2),0≤s≤1,
such that 〈Ψ,Φ〉=I, Φ˙=ΦJ, and Ψ˙=-JΨ, where J is given by J=(010001000).Using the similar methods as used in [9], system (29) can be rewritten as
(33)x˙(t)=A(μ)x(t)+B(μ)x(t-1)+F(x(t),x(t-1),μ),
where μ=(μ1,μ2,μ3)∈R3 is a parameter vector, x∈R2. Let
(34)F^(x(t),x(t-1),μ)=(A(λ)-A(0))x(t)+(B(μ)-B(0))x(t-1)+F(x(t),x(t-1),μ),
then, we can expand F^(x(t), x(t-1), μ)=A1μ1x(t) + A2μ2x(t) + A3μ3x(t) + B1μ1x(t-1) + B2μ2x(t-1) + B3μ3x(t-1) + E1x1(t)x(t-1) + E2x2(t)x(t-1) + F1x1(t)x(t) + F2x2(t)x(t) + G1x1(t-1)x(t-1) + G2x2(t-1)x(t-1) + h.o.t, where
(35)A1=(ρ-βρ00),A2=(0000),A3=(-cτh000),B1=(00ρβ-ρ),B2=(00τβ-τ),B3=(0000),E1=(0000),E2=(001m+x--βm+x-),F1=(ταmβ(m+x-)2-τx-c+x--ταm2(m+x-)200),F2=(-ταm2(m+x-)2000),G1=(00-1β(m+x-)12(m+x-)),G2=(0012(m+x-)0),φ10=(β1),φ20=(0-1),φ30=(β1),ψ10=(-1718β3518),ψ20=(23β-23),ψ30=(2β-2).Following the formula of Theorem 3.1 in [9], the normal form with versal unfolding of system (29) on the center manifold takes the following form:
(36)z˙1=z2,z˙2=z3,z˙3=l1z1+l2z2+l3z3+η1z12+η2z22+η3z1z2+η4z1z3+h.o.t,
where
(3)l1=-2cτhμ3,l2=-4τμ2-2cτ3hμ3,l3=2ρμ1+4τ3μ2-19cτ18hμ3,η1=-2τx-βc+x-,η2=17τx-β9(c+x-)+2ταm3(m+x-)2,η3=-4τx-β3(c+x-)+2ταm(m+x-)2,η4=2ταm3(m+x-)2-19τx-β9(c+x-).
Referring [8], we know that if
(38)(∂(l1,l2,l3)∂(μ1,μ2,μ3))|μ=0≠0,η1η2η3η4≠0,
that is,
(39)ρ≠2x-2(m+x-)3m(c+x-),ρ≠19x-2(m+x-)6m(c+x-),
for the unfolding normal form (36) then there exist the following results.Theorem 5.
Let (4), (9), and (39) hold. For the parameters μ1,μ2, and μ3 are sufficiently small,(i)
system (36) undergoes a transcritical bifurcation at the origin on the curve
(40)T={(μ1,μ2,μ3):μ3=0},(ii)
system (36) undergoes a Hopf bifurcation at the origin on the curve
(41)H1={(μ1,μ2,μ3):2αx-β(m+x-)μ1=19cτ18(c+x-)2μ3-4τ3μ2+3cμ36(c+x-)2μ2+cμ3,μ2>-c6(c+x-)2μ3},(iii)
system (36) undergoes a Hopf-bifurcation at the nontrivial equilibrium point on the curve
(42)H2={(μ1,μ2,μ3):l3=(η4η1-η1η3l1-η1l2)l1,η1η3l1-η1l2>0},(iv)
system (36) undergoes a Bogdanov-Takens bifurcation at the origin on the curve
(43)B={(μ1,μ2,μ3):μ3=0,μ2=-c6(c+x-)2μ3},(v)
system (36) undergoes a zero-Hopf bifurcation at the origin on the curve
(44)H3={(μ1,μ2,μ3):μ3=0,μ1=2τβ(m+x-)3αx-μ2,μ2>0}.


## 4. Hopf Bifurcation

The Jacobian matrix of system (2) at E* takes the following form:
(45)J(E*)=(x*(-1+αy*(m+x*)2+h(c+x*)2)-αx*m+x*ρβe-λτ-ρe-λτ),
then, the characteristic equation is
(46)F(λ)=λ2+λx*(1-αβ(m+x*)-h(c+x*)2)+(ρλ+ρx*(1-h(c+x*)2))e-λτ=λ2+λx*(Δc+x*-αβ(m+x*))+(ρλ+ρx*Δc+x*)e-λτ.
When τ=0, by Routh-Hurwitz criterion, all roots of (46) have negative real part if
(47)ρ>x*(αβ(m+x*)-Δc+x*).
Using the formula in Theorem 2.4 of [13] or Lemma 2.2 of [16], we have the following theorem.Theorem 6.
Let (47) hold. Then, there exist
(48)w0=22ρ2-A2+(ρ2-A2)2+4B2,τk=1w0arccos(B-Aρ)w02B2+ρ2w02+2kπw0,k=0,1,…
such that the positive equilibrium E* is asymptotically stable for 0⩽τ<τ0, moreover, system (2) undergoes Hopf bifurcation at E* for τ=τk, where A=ρ+x*(Δ/(c+x*)-α/β(m+x*)) and B=ρx*Δ/(c+x*).In the following, using the Hopf bifurcation theorem for a retarded differential system introduced by [17], the Hopf bifurcations at E* for small delay and large delay are presented.Using the same methods as the ones used in [13], for small delay, let e-λτ≃1-λτ, together with (46), the following bifurcation results can be obtained.Theorem 7.
Let0<τ<τ~=1/ρ, ρ<αx*/β(m+x*), τs=A/B and ws=B/(1-ρτs). Then, there exists ϵs>0 such that for each 0⩽ϵ<ϵs, system (2) near E* has a family or periodic solutions γs(ϵ) with period Ts=Ts(ϵ) for τ=τ(ϵ) such that γs(0)=E*, Ts(0)=2π/ws, and τ(0)=τs.For large delay, by [13] we have the following results.Theorem 8.
Let (47) hold. Then there exists ϵ0>0 such that for each 0⩽ϵ<ϵ0 system (2) near E* has a family or periodic solutions γl(ϵ) with period Tl=Tl(ϵ) for τ=τ(ϵ) such that γl(0)=E*, Tl(0)=2π/w0 and τ(0)=τ0.Remark. Because the proofs of Theorems 7 and 8 are the same as the proofs of Theorems 3.4 and 3.5 in [13], we omit them here.



---

*Source: 605471-2013-10-21.xml*