# Bifurcation Analysis in a Kind of Fourth-Order Delay Differential Equation

Authors: Xiaoqian Cui; Junjie Wei
Journal: Discrete Dynamics in Nature and Society
 (2009)

---

## Abstract

A kind of fourth-order delay differential equation is considered. Firstly, the linear stability is investigated by analyzing the associated characteristic equation. It is found that there are stability switches for time delay and Hopf bifurcations when time delay cross through some critical values. Then the direction and stability of the Hopf bifurcation are determined, using the
normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the analytic results.


---

## Body

## 1. Introduction

Sadek [1] has considered the following fourth-order delay
differential equation:(1.1)x(4)(t)+α1x⃛(t)+α2x¨(t)+ϕ(x˙(t−τ))+f(x(t))=0.By constructing Lyapunov
functionals, it was given a group of conditions to ensure that the zero
solution of (1.1) is globally asymptotically stable when the delay τ is suitable
small, but if the sufficient conditions are not satisfied, what are the behaviors
of the solutions? This is a interesting question in mathematics. The purpose of
the present paper is to study the dynamics of (1.1) from bifurcation. We
will give a detailed analysis on the above mentioned question. By regarding the delay τ as a
bifurcation parameter, we analyze the distribution of the roots of the
characteristic equation of (1.1) and obtain the existence of stability
switches and Hopf bifurcation when the delay varies. Then by using the center
manifold theory and normal form method, we derive an explicit algorithm for
determining the direction of the Hopf bifurcation and the stability of the
bifurcating periodic solutions.We would like to mention that there are several
articles on the stability of fourth-order delay differential equations, we
refer the readers to [1–8] and the references cited therein.The rest of this paper is organized as follows. In
Section2, we firstly focus mainly on the local stability of the zero solution.
This analysis is performed through the study of a characteristic equation,
which takes the form of a fourth-degree exponential polynomial. Using the
approach of Ruan and Wei [9], we show that the stability of the zero solution can
be destroyed through a Hopf bifurcation. In Section 3, we investigate the
stability and direction of bifurcating periodic solutions by using the normal
form theory and center manifold theorem presented in Hassard et al. [10]. In Section 4, we illustrate
our results by numerical simulations. Section 5 with conclusion completes the
paper.


## 2. Stability and Hopf Bifurcation

In this section, we will study the stability of the
zero solution and the existence of Hopf bifurcation by analyzing the
distribution of the eigenvalues. For convenience, we give the following
assumptions:(H1)τ>0,α1>0,α2>0,ϕ(0)=0,f(0)=0,with ϕ and f are both
continuous functions and those three-order differential quotients at origin are
existent. We rewrite (1.1) as the following form:(2.1)x˙=y,y˙=u,u˙=v,v˙=−α2u−α1v−f(x)−ϕ(y(t−τ)).It is easy to see that (0,0,0,0) is the only
trivial solution to the system (2.1) and the linearization around (0,0,0,0) is given
by(2.2)x˙=y,y˙=u,u˙=v,v˙=−f′(0)x−α2u−α1v−ϕ′(0)y(t−τ).Its characteristic equation
is(2.3)λ4+α1λ3+α2λ2+ϕ′(0)λe−λτ+f′(0)=0.Lemma 2.1.
Suppose (H1) and(H2)α1α2−ϕ′(0)>0,f′(0)>0,ϕ′(0)(α1α2−ϕ′(0))−α12f′(0)>0are satisfied. Then the trivial
solution (0,0,0,0) is
asymptotically stable when τ=0.Proof.
Whenτ=0, (2.3) becomes(2.4)λ4+α1λ3+α2λ2+ϕ′(0)λ+f′(0)=0.By Routh-Hurwitz criterion, all
roots of (2.4) have negative real parts if and only if(2.5)α1>0,α1α2−ϕ′(0)>0,f′(0)>0,ϕ′(0)(α1α2−ϕ′(0))−α12f′(0)>0.The conclusion follows from (H1) and (H2).
Letiω(ω>0) be a root of
(2.3), then we have(2.6)ω4−iα1ω3−α2ω2+ϕ′(0)(iω)e−iωτ+f′(0)=0.Separating the real and
imaginary parts gives(2.7)−ω4+α2ω2−f′(0)=ϕ′(0)ωsinωτ,α1ω3=ϕ′(0)ωcosωτ.Adding up the squares of both
equations yields(2.8)ω8+(α12−2α2)ω6+(α22+2f′(0))ω4−(ϕ′2(0)+2α2f′(0))ω2+f′2(0)=0.Let V=ω2, and denote(2.9)P=α12−2α2,Q=α22+2f′(0),K=−ϕ′2(0)−2α2f′(0).Then (2.8)
becomes(2.10)V4+PV3+QV2+KV+f′2(0)=0.Set(2.11)h(V)=V4+PV3+QV2+KV+f′2(0).Then we have(2.12)h′(V)=4V3+3PV2+2QV+K.Consider(2.13)4V3+3PV2+2QV+K=0.Let U=V+(3/4)P. Then (2.13) becomes(2.14)U3+P1U+Q1=0,where(2.15)P1=Q2−316P2,Q1=132P3−18PQ+K.Define(2.16)M=(Q12)2+(P13)3,σ=−1+3i2,U1=−Q12+M3+−Q12−M3,U2=−Q12+M3σ+−Q12−M3σ2,U2=−Q12+M3σ2+−Q12−M3σ,Vi=Ui−34P,i=1,2,3.Then by Lemma 2.2 in Li and Wei
[11], we have the
following results on the distribution of the roots of (2.10).Lemma 2.2.
(i) IfM≥0, then (2.10) has positive roots if and
only if V1>0 and h(V1)<0.
(ii) IfM<0, then (2.10) has positive roots if and
only if there exists at least one V*∈{V1,V2,V3}, such that V*>0 and h(V*)≤0.Without loss of generality, we assume that equationh(V)=0 has four
positive roots denoted by V1,V2,V3, and V4, respectively. Then (2.8) also has four
positive roots, say ωi=Vi,i=1,2,3,4.From (2.7), and conditions (H1) and (H2), we have
that(2.17)cosωτ=α1ω2ϕ′(0)>0.Hence, we define(2.18)τkj=1ωk[arccosα1ωk2ϕ′(0)+2jπ],k=1,2,3,4,j=0,1,…,when(2.19)−ωk4+α2ωk2−f′(0)ϕ′(0)ωk>0,τkj=1ωk[−arccosα1ωk2ϕ′(0)+2(j+1)π],k=1,2,3,4,j=0,1,…,when(2.20)−ωk4+α2ωk2−f′(0)ϕ′(0)ωk<0.Let(2.21)λ(τ)=α(τ)+iβ(τ)be the root of (2.3)
satisfying α(τkj)=0,β(τkj)=ωk.Lemma 2.3.
Supposeh′(Vi)≠0(i=1,2,3,4). If τ=τkj, then ±iωk is a pair of
simple purely imaginary roots of (2.3); and Re(dλ(τkj)/dτ)>0 when k=2,4; and Re(dλ(τkj)/dτ)<0 when k=1,3.Proof.
Substitutingλ(τ) into (2.3)
and differentiating with respet to τ gives(2.22)dλdτ=ϕ′(0)λ2e−λτ4λ3+3α1λ2+2α2λ+ϕ′(0)e−λτ−τϕ′(0)λe−λτ=−λ2(λ4+α1λ3+α2λ2+f′(0))τλ5+(3+τα1)λ4+(3α1+τα2)λ3+α2λ2+f′(0)τλ−f′(0).Then(2.23)Redλ(τj)dτ=(ωk6−α2ωk4+f′(0)ωk2)((3+τα1)ωk4−α2ωk2−f′(0))−α1ωk5(τωk5−(3α1+τα2)ωk3+f′(0)τωk)((3+τα1)ωk4−α2ωk2−f′(0))2+(τωk5−(3α1+τα2)ωk3+f′(0)τωk)2=ωk2Δh′(Vk),where(2.24)Δ=((3+τα1)ωk4−α2ωk2−f′(0))2+(τωk5−(3α1+τα2)ωk3+f′(0)τωk)2;and for h′(Vk)≠0(k=1,2,3,4), h(0)=f′2(0)>0 and limV→±∞h(V)=∞, we can know
that Re(dλ(τkj)/dτ)>0 when k=2,4; and Re(dλ(τkj)/dτ)<0 when k=1,3. This completes the proof.Fromh(0)=f′2(0)>0 and limV→±∞h(V)=∞, it is easy to
know that: if h(V) satisfies h′(Vi)≠0(i=1,2,3,4), if the equation h(V)=0 has positive
roots, then the number of the roots must be even; and from Lemma 2.3, we have
that the sign of α′(τkj) changes as τkj varies, and
then the stability switches may happen.From Lemmas2.1–2.3 and the theory in [9], we have the following.Lemma 2.4.
Suppose that (H1), (H2) and h′(Vi)≠0(i=1,2,3,4) are
satisfied.
(i)
If conditions (i) and (ii) in Lemma2.2 are not satisfied, then all the roots of (2.3) have negative real parts for all τ≥0.(ii)
If one of conditions (i) and (ii) in Lemma2.2 is satisfied, let τ*=min(τ20,τ40), then all roots
of (2.3) have negative real parts when τ∈[0,τ*); and there may
exist an integer m≥0 such that 0<τ1<τ2<⋯<τm−1<τm<τm+1<⋯, and all the roots of (2.3) have negative
real parts when τ∈[0,τ1)∪(τ2,τ3)∪⋯∪(τm−1,τm), and (2.3) has at least a pair of roots with positive real parts when τ∈(τ1,τ2)∪(τ3,τ4)∪⋯∪(τm,∞), where τm∈{τkj}.From Lemma2.4 and applying the Hopf bifurcation
theorem for functional differential equations [12, Chapter 11, Theorem 1.1],
we have the following results.Theorem 2.5.
Suppose (H1), (H2), and h′(Vi)≠0(i=1,2,3,4) are
satisfied.
(i)
If conditions (i) and (ii) in Lemma2.2 are not satisfied, then the trivial solution (0,0,0,0) of system (2.1)
is asymptotically stable when τ>0.(ii)
If one of conditions (i) and (ii) in Lemma2.2 is satisfied, let τ*=min{τ20,τ40}, then the trivial solution (0,0,0,0) of system (2.1)
is asymptotically stable when τ∈[0,τ*); and there may
exist an integer m≥0 such that 0<τ1<τ2<⋯<τm−1<τm<τm+1<⋯, and the trivial solution (0,0,0,0) of system (2.1)
is asymptotically stable when τ∈[0,τ1)∪(τ2,τ3)∪⋯∪(τm−1,τm), and is unstable when τ∈(τ1,τ2)∪(τ3,τ4)∪⋯∪(τm,∞), where τm∈{τkj}.(iii)
The system (2.1)
undergoes a Hopf bifurcation at the origin when τ=τkj, with k=1,2,3,4;j=0,1,2,….


## 3. Direction and Stability of the Hopf Bifurcation

In this
section, we will study the direction, stability, and the period of the
bifurcating periodic solution. The method we used is based on the normal form
method and the center manifold theory presented by Hassard et al. [10].We first rescale the time byt→t/τ to normalize
the delay so that system (2.1) can be written as the form(3.1)x˙=τy,y˙=τu,u˙=τv,v˙=−α2τu−α1τv−τf(x)−τϕ(y(t−1)).The linearization around (0,0,0,0) is given
by(3.2)x˙=τy,y˙=τu,u˙=τv,v˙=−τf′(0)x−α2τu−α1τv−τϕ′(0)y(t−1);and the nonlinear term
is(3.3)F=(000−f′′(0)2x2−f′′′(0)x36−τϕ′′(0)y2(t−1)2−τϕ′′′(0)y3(t−1)6−⋯).The characteristic equation
associated with (3.2) is(3.4)γ4+α1τγ3+α2τ2γ2+ϕ′(0)τ3γe−γ+τ4f′(0)=0.Comparing (3.4) with
(2.3), one can find out that γ=τλ, and hence, (3.4) has a pair of imaginary roots ±iτkjωk, when τ=τkj for k=1,2,3,4, j=0,1,2,…, and the
transversal condition holds.Letτ=τ0+μ, μ∈R where τ0∈{τkj}, ω0∈{ωk}, k=1,2,3,4, j=0,1,2,…. Then μ=0 is the Hopf
bifurcation value for (3.1). Let iτ0ω0 be the root of (3.4).Forφ∈C([−1,0],R4),
let(3.5)Lμφ=(τ0+μ)(010000100001−f′(0)0−α2−α1)φ(0)−(τ0+μ)ϕ′(0)(0000000000000100)φ(−1),F(μ,φ)=(000−(τ0+μ)f′′(0)φ12(0)2−(τ0+μ)f′′′(0)φ13(0)6−(τ0+μ)ϕ′′(0)φ22(−1)2−(τ0+μ)ϕ′′′(0)φ23(−1)6−⋯).By the Riesz representation
theorem, there exists a matrix whose components are bounded variation functions η(θ,μ) in θ∈[−1,0] such
that(3.6)Lμφ=∫−10dη(θ,μ)φ(θ),φ∈C([−1,0],R4).In fact, we
choose(3.7)η(θ,μ)=(τ0+μ)(010000100001−f′(0)0−α2−α1)δ(θ)+(τ0+μ)ϕ′(0)(0000000000000100)δ(θ+1),where (3.8)δ(θ)={1,θ=0,0,θ≠0.Forφ∈C1([−1,0],C4), define(3.9)A(μ)φ={dφ(θ)dθ,θ∈[−1,0),∫−10dη(t,μ)φ(t),θ=0,R(μ)φ={0,θ∈[−1,0),F(μ,φ),θ=0.Hence, we can rewrite (3.1) in the following form:(3.10)w˙t=A(μ)wt+R(μ)wt,where w=(x,y,u,v)T, wt=w(t+θ) for θ∈[−1,0].Forψ∈C1([0,1],C4),
define(3.11)A*ψ(s)={−dψ(s)ds,s∈(0,1],∫−10ψ(−t)dη(t,0),s=0.For φ∈C([−1,0],C4) and ψ∈C([0,1],C4), define the
bilinear form(3.12)〈ψ,φ〉=ψ¯(0)φ(0)−∫−10∫ξ=0θψ¯T(ξ−θ)dη(θ)φ(ξ)dξ,where η(θ)=η(θ,0). Then A* and A(0) are adjoint
operators, and ±iτ0ω0 are eigenvalues
of A(0). Thus, they
are also eigenvalues of A*.By direct computation, we obtain that(3.13)q(θ)=(1iω0−ω02−iω03)eiτ0ω0θis the eigenvector of A(0) corresponding
to iτ0ω0,
and(3.14)q*(θ)=D(−iω03−ω02α1+iω0α2+ϕ′(0)e−iτ0ω0−ω02+iω0α1+α2iω0+α11)Teiτ0ω0sis the eigenvector of A* corresponding
to −iτ0ω0.
Moreover,(3.15)〈q*,q〉=1,〈q*,q¯〉=0,where(3.16)D=1−ω02α1+ϕ′(0)e−iτ0ω0−iτ0ω0ϕ′(0)eiτ0ω0.Using the same notation as in Hassard et al. [10], we first compute the
coordinates to describe the center manifold 𝒞0 at μ=0. Let wt be the solution
of (3.1) when μ=0.Define(3.17)zt=〈q*,wt〉,W(t,θ)=wt(θ)−2Re{z(t)q(θ)}.On the center manifold 𝒞0, we
have(3.18)W(t,θ)=W(z(t),z¯(t),θ),where(3.19)W(z,z¯,θ)=W20(θ)z22+W11(θ)zz¯+W02(θ)z¯22+W30(θ)z36+⋯,z and z¯ are local
coordinates for center manifold 𝒞0 in the
direction of q* and q¯*. Note that W is real if wt is real. We
consider only real solutions.For solutionwt in 𝒞0 of (3.1), since μ=0,(3.20)z˙(t)=iτ0ω0z+〈q*(θ),F(W(z,z¯,θ)+2Re{z(t)q(θ)})〉=iτ0ω0z+q¯*(0)F(W(z,z¯,0)+2Re{z(t)q(0)})=defiτ0ω0z+q¯*(0)F0(z,z¯).We rewrite this
as(3.21)z˙(t)=iτ0ω0z(t)+g(z,z¯),where(3.22)F0(z,z¯)=Fz2z22+Fz¯2z¯22+Fzz¯zz¯+Fz2z¯z2z¯2+⋯,(3.23)g(z,z¯)=q¯*(0)F(W(z,z¯,0)+2Re{z(t)q(0)})=g20z22+g11zz¯+g02z¯22+g21z2z¯2+⋯.Compare the coefficients of
(3.20) and (3.21), noticing (3.23), we have(3.24)g20=q¯*(0)Fz2,g11=q¯*(0)Fzz¯,g02=q¯*(0)Fz¯2,g21=q¯*(0)Fz2z¯.By (3.10) and (3.21), it follows
that(3.25)W˙=w˙t−z˙q−z¯˙q¯={AW−2Re{q¯*(0)F0q(θ)}−1≤θ<0AW−2Re{q¯*(0)F0q(0)}+F0θ=0=defAW+H(z,z¯,θ),where(3.26)H(z,z¯,θ)=H20z22+H11(θ)zz¯+H02(θ)z¯22+⋯.Expanding the above series and
comparing the coefficients, we obtain(3.27)(A−2iτ0ω0I)W20(θ)=−H20(θ),AW11(θ)=−H11(θ),(A−2iτ0ω0)W02(θ)=−H02(θ).Notice that(3.28)wt(θ)=W(z,z¯,θ)+zq(θ)+zq¯(θ),that is,(3.29)x(t)=z+z¯+W(1)(z,z¯,0)=z+z¯+W20(1)(0)z22+W11(1)(0)zz¯+W02(1)(0)z¯22+⋯,y(t−1)=iω0e−iτ0ω0z−iω0eiτ0ω0z¯+W(2)(z,z¯,−1)=iω0e−iτ0ω0z−iω0eiτ0ω0z¯+W20(2)(−1)z22+W11(2)(−1)zz¯+W02(2)(−1)z¯22+⋯.Thus(3.30)x2(x)=2z22+2zz¯+2z¯22+(4W111(0)+2W201(0))z2z¯2+⋯,x3(x)=6z2z¯2+⋯,y2(t−1)=−2ω02e−2iτ0ω0z22+2ω02zz¯−2ω02e2iτ0ω0z¯22+(4iω0W11(2)(−1)e−iτ0ω0−iω02W20(2)(−1)eiτ0ω0)z2z¯2+⋯,y3(t−1)=6iω03e−iτ0ω0z2z¯2+⋯;and we have(3.31)F(0,ωt)=(000−τ0f′′(0)x2(t)2−τ0f′′′(0)x3(t)6−τ0ϕ′′(0)y2(t−1)2−τ0ϕ′′(0)y3(t−1)6−⋯),q*(0)=D(−iω03−ω02α1+iω0α2+ϕ′(0)e−iτ0ω0−ω02+iω0α1+α2iω0+α11)T,g(z,z¯)=q¯*(0)F0(z,z¯).Then we have(3.32)g20=−τ0D¯f′′(0)+τ0D¯ϕ′′(0)ω2e−2iτ0ω0,g11=−τ0D¯f′′(0)−τ0D¯ϕ′′(0)ω2,g02=−τ0D¯f′′(0)+τ0D¯ϕ′′(0)ω2e2iτ0ω0,g21=−τ0D¯f′′(0)[2W11(1)(0)+W20(1)(0)]−τ0D¯f′′′(0)−τ0D¯ϕ′′(0)[2iω0W11(2)(−1)e−iτ0ω0−iω0W20(2)(−1)eiτ0ω0]−τ0D¯ϕ′′′(0).So we only need to find out W11(1)(0), W20(1)(0), W11(2)(−1), and W20(2)(−1) to obtain g21.Whenθ∈[−1,0), we have(3.33)H(z,z¯,θ)=−2Re{q¯*(0)F0q(θ)}=−q¯*(0)F0q(θ)−q*(0)F0q¯(θ)=−gq(θ)−gq¯(θ).Comparing the coefficients with
(3.26), we get(3.34)H20(θ)=−g20q(θ)−g¯02q¯(θ),H11(θ)=−g11q(θ)−g¯11q¯(θ).From (3.27), (3.32),
(3.33), and (3.34), we derive(3.35)W˙20(θ)=2iτ0ω0W20(θ)+g20q(θ)+g¯02q¯(θ),W˙11(θ)=g11q(θ)+g¯11q¯(θ).Then we can get(3.36)W20(θ)=ig20τ0ω0q(0)eiτ0ω0θ+ig¯023τ0ω0q¯(0)e−iτ0ω0θ+E1e2iτ0ω0θ,W11(θ)=g11iτ0ω0q(0)eiτ0ω0θ−g¯11iτ0ω0q¯(0)e−iτ0ω0θ+E2.Notice that(3.37)(2iτ0ω0I−∫−10e2iτ0ω0θdη(θ))E1=Fz2,(∫−10dη(θ))E2=−Fzz¯.We obtain(3.38)E1=(E*2iω0E*−4ω02E*−8iω03E*),E2=(−f′′(0)+ϕ′′(0)f′(0)000),where
E*=−f′′(0)+ϕ′′(0)ω02e−2iτ0ω0f′(0)+2iω0e−2iτ0ω0−4ω02α2−8iω03(2iω0+α1).Hence(3.39)W20(1)(0)=ig20τ0ω0q1(0)+ig¯023τ0ω0q¯1(0)+E*e−2iτ0ω0=i(−D¯f′′(0)+D¯ϕ′′(0)ω02e−2iτ0ω0)ω0+i(−Df′′(0)+Dϕ′′(0)ω02e−2iτ0ω0)3ω0+E*e−2iτ0ω0,W20(2)(−1)=ig20τ0ω0q2(0)e−iτ0ω0+ig¯023τ0ω0q¯2(0)eiτ0ω0+(2iω0)E*e−2iτ0ω0=−(−D¯f′′(0)+D¯ϕ′′(0)ω02e−2iτ0ω0)e−iτ0ω0+−Df′′(0)+Dϕ′′(0)ω02e−2iτ0ω03eiτ0ω0+(2iω0)E*e−2iτ0ω0,W11(1)(0)=g11iτ0ω0q1(0)−g¯11iτ0ω0q¯1(0)−f′′(0)+ω02ϕ′(0)f′(0)=−D¯f′′(0)−D¯ϕ′′(0)ω02iω0+Df′′(0)+Dϕ′′(0)ω02iω0−f′′(0)+ω02ϕ′(0)f′(0),W11(2)(−1)=g11iτ0ω0q2(0)e−iτ0ω0−g¯11iτ0ω0q¯2(0)eiτ0ω0=(−D¯f′′(0)−D¯ϕ′′(0)ω02)e−iτ0ω0+(Df′′(0)+Dϕ′′(0)ω02)eiτ0ω0.Consequently, from
(3.32),(3.40)g21=−τ0D¯f′′(0)[2W11(1)(0)+W20(1)(0)]−τ0D¯f′′′(0)−τ0D¯ϕ′′(0)[2iω0W11(2)(−1)e−iτ0ω0−iω0W20(2)(−1)eiτ0ω0]−τ0D¯ϕ′′′(0)=−2τ0D¯f′′(0)[−D¯f′′(0)−D¯ϕ′′(0)ω02iω0+Df′′(0)+Dϕ′′(0)ω02iω0−f′′(0)+ω02ϕ′′(0)f′(0)]−τ0D¯f′′(0)×[i(−D¯f′′(0)+D¯ϕ′′(0)ω02e−2iτ0ω0)ω0+i(−Df′′(0)+Dϕ′′(0)ω02e−2iτ0ω0)3ω0+E*e−2iτ0ω0]−τ0D¯f′′′(0)−2τ0D¯ϕ′′(0)(iω0)[(−D¯f′′(0)−D¯ϕ′′(0)ω02)e−iτ0ω0+(Df′′(0)+Dϕ′′(0)ω02)eiτ0ω0]e−iτ0ω0+τ0D¯ϕ′′(0)(iω0)[−(−D¯f′′(0)+D¯ϕ′′(0)ω02e−2iτ0ω0)e−iτ0ω0]eiτ0ω0+τ0D¯ϕ′′(0)(iω0)[−Df′′(0)+Dϕ′′(0)ω02e−2iτ0ω03eiτ0ω0+(2iω0)E*e−2iτ0ω0]eiτ0ω0−τ0D¯ϕ′′′(0).Substituting g20, g11, g02, and g21 into(3.41)C1(0)=i2τ0ω0(g20g11−2|g11|2−13|g02|2)+g212,we can obtain ReC1(0). Then we
obtain the sign of(3.42)β2=2ReC1(0),μ2=−ReC1(0)α′(τ0).By the general theory due to Hassard et al. [10], we
know that the quantity of β2 determines the
stability of the bifurcating periodic solutions on the center manifold, and μ2 determines the
direction of the bifurcation; and we have the following.Theorem 3.1.
(i) Ifμ2>0(<0), then the Hopf
bifurcation at the origin of system (1.1) is supercritical (subcritical).
(ii) Ifβ2<0(>0), then the bifurcating periodic solutions of system
(1.1) are asymptotically stable (unstable).


## 4. An Example and Numerical Simulations

In this section, we give an example and present some
numerical simulations to illustrate the analytic results.Example 4.1.
Consider the following equation:(4.1)x(4)(t)+x⃛(t)+14x¨(t)+12sinx˙(t−τ)+x3(t)+18x(t)=0.Clearly,(4.2)α1=1,α2=14,ϕ(x˙(t−τ))=12sinx˙(t−τ),f(x)=x3+18x,ϕ(0)=ϕ′′(0)=0,ϕ′(0)=12,ϕ′′′(0)=−12,f(0)=f′′(0)=0,f′(0)=18,f′′′(0)=6.By direct computation, we know (H1) and (H2) are satisfied.
That is, the data satisfy the conditions of Lemma 2.1. The characteristic
equation is(4.3)λ4+λ3+14λ2+12λe−λτ+18=0;and we can
obtain(4.4)h(V)=V4−27V3+232V2−648V+324. As shown in Figure 1, the equation h(V)=0 has four roots
as(4.5)V1=0.633,V2=4.166,V3=10.464,V4=11.737;and (4.6)h′(V1)<0,h′(V2)>0,h′(V3)<0,h′(V4)>0.Hence(4.7)ω1=0.796,ω2=2.041,ω3=3.235,ω4=3.426,τ10=5.988,τ20=0.596,τ30=0.158,τ40=0.061.For τ40<τ30<τ20<τ10, we obtain
that the zero solution of system (4.1) is asymptotically stable when τ∈[0,0.061)∪(0.158,0.596).Figure 1
The curve of functionh(V)=V4−27V3+232V2−648V+324.According to the formula given in Section3, we can
obtain that(4.8)D4=−1.898+0.133i,D3=−0.323+0.056i,D2=−0.061+0.027i,g21(4)=−0.697+0.049i,g21(3)=−0.306+0.053i,g21(2)=−0.217+0.098i,g02=g11=g02=E*=0.Then we have(4.9)C1(0)=g212.Hence, when τ∈{τ10,τ20,τ30}, we
have(4.10)β2=2ReC1(0)<0,μ2=−ReC1(0)α′(τ0)>0.Conclusion of (4.1)
The zero solution of
system (4.1) is asymptotically stable when τ∈[0,0.061)∪(0.158,0.596). The Hopf
bifurcation at the origin when τ0=τk0 is
supercritical, and the bifurcating periodic solutions are asymptotically stable.
The following is the results of numerical simulations
to system (4.1).
(i)
We chooseτ=0.4∈(0.158,0.596), then the zero
solution of system (4.1) is asymptotically stable, as shown in Figure 2.(ii)
We chooseτ=0.64 being near to τ20=0.596, a periodic solution bifurcates from the origin and
is asymptotically stable, as shown in Figure 3.The zero solution of system (4.1) is
asymptotically stable when τ=0.4.
(a)
y-x phase plots(b)
y waveform(c)
x waveform(d)
v-u phase plots(e)
v waveform(f)
u waveformFor (4.1) with τ=0.64>τ20 and sufficiently near τ20=0.596, the bifurcating periodic solution from zero
solution occurs and is asymptotically stable.
(a)
y-x phase plots(b)
y waveform(c)
x waveform(d)
v-u phase plots(e)
v waveform(f)
u waveform


## 5. Conclusion

In this paper,
we consider a certain fourth-order delay differential equation. The linear
stability is investigated by analyzing the associated characteristic equation.
It is found that there may exist the stability switches when delay varies, and
the Hopf bifurcation occurs when the delay passes through a sequence of
critical values. Then the direction and the stability of the Hopf bifurcation
are determined using the normal form method and the center manifold theorem.
Finally, an example is given and numerical simulations are carried out to
illustrate the results. By using Lyapunov's second method, Sadek [1] investigated the stability
of system (1.1). The main result is as the following.Theorem 5.1.
Suppose that the following hold.
(i)
There are
constantsα1>0,α2>0,α3>0,α4>0, and Δ>0 such that(5.1)(α1α2−ϕ′(y))α3−α12α4≥Δfor all y.(ii)
f(0)=0,xf(x)>0(x≠0),F(x)=∫0xf(ξ)dξ→∞     as |x|→∞, and(5.2)0≤α4−f′(x)≤εd0α10for all x, where ε is a positive
constant such that(5.3)ε=min(1α1,Δ4α1α3d0,α34α4d0(2Δα4α1α32−δ1))with d0=α1α2+α2α3α4−1.(iii)
ϕ(0)=0 and ϕ′(y)≥α3>0 for all y, and 0≤ϕ′(y)−ϕ(y)/y≤δ1<2Δα4/α1α32 for all y≠0. Then the zero
solution of (1.1) is asymptotically stable, provided that(5.4)τ<min{εd2α2,Δ2α1α3(α2α3+2μ),α1εd1α2α3},with μ=(α2α3/2)(d1+d2+1)>0.Comparing Theorem5.1 with Theorem 2.5 obtained in
Section 2, one can find out that if the sufficient conditions to ensure the
globally asymptotical stability of system (1.1) given in [10] are not satisfied, we can
also get the stability of system (1.1), but here the stability means local
stability, and the system undergoes a Hopf bifurcation at the origin.
Otherwise, here we just need to give the condition on the origin of f(x) and ϕ(x), the condition
is relatively weak.



---

*Source: 235038-2009-04-26.xml*