DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi Publishing Corporation23503810.1155/2009/235038235038Research ArticleBifurcation Analysis in a Kind of Fourth-Order Delay Differential EquationCuiXiaoqian1WeiJunjie1, 2ZhangBinggen1Department of MathematicsHarbin Institute of TechnologyHarbinHeilongjiang 150001Chinahit.edu.cn2Department of MathematicsHarbin Institute of Technology (Weihai)WeihaiShandong 264209Chinahitwh.edu.cn200904032009200922122008180220092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Sadek  has considered the following fourth-order delay differential equation:x(4)(t)+α1x(t)+α2x¨(t)+ϕ(x˙(tτ))+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  and the references cited therein.

The rest of this paper is organized as follows. In Section 2, 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 , 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. . 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: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:x˙=y,y˙=u,u˙=v,v˙=α2uα1vf(x)ϕ(y(tτ)).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 byx˙=y,y˙=u,u˙=v,v˙=f(0)xα2uα1vϕ(0)y(tτ).Its characteristic equation isλ4+α1λ3+α2λ2+ϕ(0)λeλτ+f(0)=0.

Lemma 2.1.

Suppose (H1) andα1α2ϕ(0)>0,f(0)>0,ϕ(0)(α1α2ϕ(0))α12f(0)>0are satisfied. Then the trivial solution (0,0,0,0) is asymptotically stable when τ=0.

Proof.

When τ=0, (2.3) becomesλ4+α1λ3+α2λ2+ϕ(0)λ+f(0)=0.By Routh-Hurwitz criterion, all roots of (2.4) have negative real parts if and only ifα1>0,α1α2ϕ(0)>0,f(0)>0,ϕ(0)(α1α2ϕ(0))α12f(0)>0.The conclusion follows from (H1) and (H2).

Let iω(ω>0) be a root of (2.3), then we haveω4iα1ω3α2ω2+ϕ(0)(iω)eiωτ+f(0)=0.Separating the real and imaginary parts givesω4+α2ω2f(0)=ϕ(0)ωsinωτ,α1ω3=ϕ(0)ωcosωτ.Adding up the squares of both equations yieldsω8+(α122α2)ω6+(α22+2f(0))ω4(ϕ2(0)+2α2f(0))ω2+f2(0)=0.Let V=ω2, and denoteP=α122α2,Q=α22+2f(0),K=ϕ2(0)2α2f(0).Then (2.8) becomesV4+PV3+QV2+KV+f2(0)=0.Seth(V)=V4+PV3+QV2+KV+f2(0).Then we haveh(V)=4V3+3PV2+2QV+K.Consider4V3+3PV2+2QV+K=0.Let U=V+(3/4)P. Then (2.13) becomesU3+P1U+Q1=0,whereP1=Q2316P2,Q1=132P318PQ+K.DefineM=(Q12)2+(P13)3,σ=1+3i2,U1=Q12+M3+Q12M3,U2=Q12+M3σ+Q12M3σ2,U2=Q12+M3σ2+Q12M3σ,Vi=Ui34P,i=1,2,3.Then by Lemma 2.2 in Li and Wei , we have the following results on the distribution of the roots of (2.10).

Lemma 2.2.

(i) If M0, then (2.10) has positive roots if and only if V1>0 and h(V1)<0.

(ii) If M<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 equation h(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 thatcosωτ=α1ω2ϕ(0)>0.Hence, we defineτkj=1ωk[arccosα1ωk2ϕ(0)+2jπ],k=1,2,3,4,j=0,1,,whenωk4+α2ωk2f(0)ϕ(0)ωk>0,τkj=1ωk[arccosα1ωk2ϕ(0)+2(j+1)π],k=1,2,3,4,j=0,1,,whenωk4+α2ωk2f(0)ϕ(0)ωk<0.Letλ(τ)=α(τ)+iβ(τ)be the root of (2.3) satisfying α(τkj)=0,β(τkj)=ωk.

Lemma 2.3.

Suppose h(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.

Substituting λ(τ) into (2.3) and differentiating with respet to τ givesdλdτ=ϕ(0)λ2eλτ4λ3+3α1λ2+2α2λ+ϕ(0)eλττϕ(0)λeλτ=λ2(λ4+α1λ3+α2λ2+f(0))τλ5+(3+τα1)λ4+(3α1+τα2)λ3+α2λ2+f(0)τλf(0).ThenRedλ(τj)dτ=(ωk6α2ωk4+f(0)ωk2)((3+τα1)ωk4α2ωk2f(0))α1ωk5(τωk5(3α1+τα2)ωk3+f(0)τωk)((3+τα1)ωk4α2ωk2f(0))2+(τωk5(3α1+τα2)ωk3+f(0)τωk)2=ωk2Δh(Vk),whereΔ=((3+τα1)ωk4α2ωk2f(0))2+(τωk5(3α1+τα2)ωk3+f(0)τωk)2;and for h(Vk)0(k=1,2,3,4), h(0)=f2(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.

From h(0)=f2(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 Lemmas 2.12.3 and the theory in , we have the following.

Lemma 2.4.

Suppose that (H1), (H2) and h(Vi)0(i=1,2,3,4) are satisfied.

If conditions (i) and (ii) in Lemma 2.2 are not satisfied, then all the roots of (2.3) have negative real parts for all τ0.

If one of conditions (i) and (ii) in Lemma 2.2 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 m0 such that 0<τ1<τ2<<τm1<τm<τm+1<, and all the roots of (2.3) have negative real parts when τ[0,τ1)(τ2,τ3)(τm1,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 Lemma 2.4 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.

If conditions (i) and (ii) in Lemma 2.2 are not satisfied, then the trivial solution (0,0,0,0) of system (2.1) is asymptotically stable when τ>0.

If one of conditions (i) and (ii) in Lemma 2.2 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 m0 such that 0<τ1<τ2<<τm1<τm<τm+1<, and the trivial solution (0,0,0,0) of system (2.1) is asymptotically stable when τ[0,τ1)(τ2,τ3)(τm1,τm), and is unstable when τ(τ1,τ2)(τ3,τ4)(τm,), where τm{τkj}.

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. .

We first rescale the time by tt/τ to normalize the delay so that system (2.1) can be written as the formx˙=τy,y˙=τu,u˙=τv,v˙=α2τuα1τvτf(x)τϕ(y(t1)).The linearization around (0,0,0,0) is given byx˙=τy,y˙=τu,u˙=τv,v˙=τf(0)xα2τuα1τvτϕ(0)y(t1);and the nonlinear term isF=(000f(0)2x2f(0)x36τϕ(0)y2(t1)2τϕ(0)y3(t1)6).The characteristic equation associated with (3.2) isγ4+α1τγ3+α2τ2γ2+ϕ(0)τ3γeγ+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.

Let τ=τ0+μ, μR 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).

For φC([1,0],R4), letLμφ=(τ0+μ)(010000100001f(0)0α2α1)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 thatLμφ=10dη(θ,μ)φ(θ),φC([1,0],R4).In fact, we chooseη(θ,μ)=(τ0+μ)(010000100001f(0)0α2α1)δ(θ)+(τ0+μ)ϕ(0)(0000000000000100)δ(θ+1),where δ(θ)={1,θ=0,0,θ0.

For φC1([1,0],C4), defineA(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:w˙t=A(μ)wt+R(μ)wt,where w=(x,y,u,v)T, wt=w(t+θ) for θ[1,0].

For ψC1([0,1],C4), defineA*ψ(s)={dψ(s)ds,s(0,1],10ψ(t)dη(t,0),s=0.For φC([1,0],C4) and ψC([0,1],C4), define the bilinear formψ,φ=ψ¯(0)φ(0)10ξ=0θψ¯T(ξθ)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 thatq(θ)=(1iω0ω02iω03)eiτ0ω0θis the eigenvector of A(0) corresponding to iτ0ω0, andq*(θ)=D(iω03ω02α1+iω0α2+ϕ(0)eiτ0ω0ω02+iω0α1+α2iω0+α11)Teiτ0ω0sis the eigenvector of A* corresponding to iτ0ω0. Moreover,q*,q=1,q*,q¯=0,whereD=1ω02α1+ϕ(0)eiτ0ω0iτ0ω0ϕ(0)eiτ0ω0.

Using the same notation as in Hassard et al. , 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.

Definezt=q*,wt,W(t,θ)=wt(θ)2Re{z(t)q(θ)}.On the center manifold 𝒞0, we haveW(t,θ)=W(z(t),z¯(t),θ),whereW(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 solution wt in 𝒞0 of (3.1), since μ=0,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 asz˙(t)=iτ0ω0z(t)+g(z,z¯),whereF0(z,z¯)=Fz2z22+Fz¯2z¯22+Fzz¯zz¯+Fz2z¯z2z¯2+,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 haveg20=q¯*(0)Fz2,g11=q¯*(0)Fzz¯,g02=q¯*(0)Fz¯2,g21=q¯*(0)Fz2z¯.By (3.10) and (3.21), it follows thatW˙=w˙tz˙qz¯˙q¯={AW2Re{q¯*(0)F0q(θ)}1θ<0AW2Re{q¯*(0)F0q(0)}+F0θ=0=defAW+H(z,z¯,θ),whereH(z,z¯,θ)=H20z22+H11(θ)zz¯+H02(θ)z¯22+.Expanding the above series and comparing the coefficients, we obtain(A2iτ0ω0I)W20(θ)=H20(θ),AW11(θ)=H11(θ),(A2iτ0ω0)W02(θ)=H02(θ).Notice thatwt(θ)=W(z,z¯,θ)+zq(θ)+zq¯(θ),that is,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(t1)=iω0eiτ0ω0ziω0eiτ0ω0z¯+W(2)(z,z¯,1)=iω0eiτ0ω0ziω0eiτ0ω0z¯+W20(2)(1)z22+W11(2)(1)zz¯+W02(2)(1)z¯22+.Thusx2(x)=2z22+2zz¯+2z¯22+(4W111(0)+2W201(0))z2z¯2+,x3(x)=6z2z¯2+,y2(t1)=2ω02e2iτ0ω0z22+2ω02zz¯2ω02e2iτ0ω0z¯22+(4iω0W11(2)(1)eiτ0ω0iω02W20(2)(1)eiτ0ω0)z2z¯2+,y3(t1)=6iω03eiτ0ω0z2z¯2+;and we haveF(0,ωt)=(000τ0f(0)x2(t)2τ0f(0)x3(t)6τ0ϕ(0)y2(t1)2τ0ϕ(0)y3(t1)6),q*(0)=D(iω03ω02α1+iω0α2+ϕ(0)eiτ0ω0ω02+iω0α1+α2iω0+α11)T,g(z,z¯)=q¯*(0)F0(z,z¯).Then we haveg20=τ0D¯f(0)+τ0D¯ϕ(0)ω2e2iτ0ω0,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)eiτ0ω0iω0W20(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.

When θ[1,0), we haveH(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 getH20(θ)=g20q(θ)g¯02q¯(θ),H11(θ)=g11q(θ)g¯11q¯(θ).From (3.27), (3.32), (3.33), and (3.34), we deriveW˙20(θ)=2iτ0ω0W20(θ)+g20q(θ)+g¯02q¯(θ),W˙11(θ)=g11q(θ)+g¯11q¯(θ).Then we can getW20(θ)=ig20τ0ω0q(0)eiτ0ω0θ+ig¯023τ0ω0q¯(0)eiτ0ω0θ+E1e2iτ0ω0θ,W11(θ)=g11iτ0ω0q(0)eiτ0ω0θg¯11iτ0ω0q¯(0)eiτ0ω0θ+E2.Notice that(2iτ0ω0I10e2iτ0ω0θdη(θ))E1=Fz2,(10dη(θ))E2=Fzz¯.We obtainE1=(E*2iω0E*4ω02E*8iω03E*),E2=(f(0)+ϕ(0)f(0)000),where E*=f(0)+ϕ(0)ω02e2iτ0ω0f(0)+2iω0e2iτ0ω04ω02α28iω03(2iω0+α1).HenceW20(1)(0)=ig20τ0ω0q1(0)+ig¯023τ0ω0q¯1(0)+E*e2iτ0ω0=i(D¯f(0)+D¯ϕ(0)ω02e2iτ0ω0)ω0+i(Df(0)+Dϕ(0)ω02e2iτ0ω0)3ω0+E*e2iτ0ω0,W20(2)(1)=ig20τ0ω0q2(0)eiτ0ω0+ig¯023τ0ω0q¯2(0)eiτ0ω0+(2iω0)E*e2iτ0ω0=(D¯f(0)+D¯ϕ(0)ω02e2iτ0ω0)eiτ0ω0+Df(0)+Dϕ(0)ω02e2iτ0ω03eiτ0ω0+(2iω0)E*e2iτ0ω0,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)ω02iω0f(0)+ω02ϕ(0)f(0),W11(2)(1)=g11iτ0ω0q2(0)eiτ0ω0g¯11iτ0ω0q¯2(0)eiτ0ω0=(D¯f(0)D¯ϕ(0)ω02)eiτ0ω0+(Df(0)+Dϕ(0)ω02)eiτ0ω0.Consequently, from (3.32),g21=τ0D¯f(0)[2W11(1)(0)+W20(1)(0)]τ0D¯f(0)τ0D¯ϕ(0)[2iω0W11(2)(1)eiτ0ω0iω0W20(2)(1)eiτ0ω0]τ0D¯ϕ(0)=2τ0D¯f(0)[D¯f(0)D¯ϕ(0)ω02iω0+Df(0)+Dϕ(0)ω02iω0f(0)+ω02ϕ(0)f(0)]τ0D¯f(0)×[i(D¯f(0)+D¯ϕ(0)ω02e2iτ0ω0)ω0+i(Df(0)+Dϕ(0)ω02e2iτ0ω0)3ω0+E*e2iτ0ω0]τ0D¯f(0)2τ0D¯ϕ(0)(iω0)[(D¯f(0)D¯ϕ(0)ω02)eiτ0ω0+(Df(0)+Dϕ(0)ω02)eiτ0ω0]eiτ0ω0+τ0D¯ϕ(0)(iω0)[(D¯f(0)+D¯ϕ(0)ω02e2iτ0ω0)eiτ0ω0]eiτ0ω0+τ0D¯ϕ(0)(iω0)[Df(0)+Dϕ(0)ω02e2iτ0ω03eiτ0ω0+(2iω0)E*e2iτ0ω0]eiτ0ω0τ0D¯ϕ(0).Substituting g20, g11, g02, and g21 intoC1(0)=i2τ0ω0(g20g112|g11|213|g02|2)+g212,we can obtain ReC1(0). Then we obtain the sign ofβ2=2ReC1(0),μ2=ReC1(0)α(τ0).

By the general theory due to Hassard et al. , 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) If μ2>0(<0), then the Hopf bifurcation at the origin of system (1.1) is supercritical (subcritical).

(ii) If β2<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:x(4)(t)+x(t)+14x¨(t)+12sinx˙(tτ)+x3(t)+18x(t)=0.Clearly,α1=1,α2=14,ϕ(x˙(tτ))=12sinx˙(tτ),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 isλ4+λ3+14λ2+12λeλτ+18=0;and we can obtainh(V)=V427V3+232V2648V+324. As shown in Figure 1, the equation h(V)=0 has four roots asV1=0.633,V2=4.166,V3=10.464,V4=11.737;and h(V1)<0,h(V2)>0,h(V3)<0,h(V4)>0.Henceω1=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).

The curve of function h(V)=V427V3+232V2648V+324.

According to the formula given in Section 3, we can obtain thatD4=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 haveC1(0)=g212.Hence, when τ{τ10,τ20,τ30}, we haveβ2=2ReC1(0)<0,μ2=ReC1(0)α(τ0)>0.

Conclusion of (<xref ref-type="disp-formula" rid="eq4.1">4.1</xref>)

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).

We choose τ=0.4(0.158,0.596), then the zero solution of system (4.1) is asymptotically stable, as shown in Figure 2.

We choose τ=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.

y-x phase plots

y waveform

x waveform

v-u phase plots

v waveform

u waveform

For (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.

y-x phase plots

y waveform

x waveform

v-u phase plots

v waveform

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  investigated the stability of system (1.1). The main result is as the following.

Theorem 5.1.

Suppose that the following hold.

There are constants α1>0,α2>0,α3>0,α4>0, and Δ>0 such that(α1α2ϕ(y))α3α12α4Δfor all y.

f(0)=0,xf(x)>0(x0),F(x)=0xf(ξ)dξ as |x|, and0α4f(x)εd0α10for all x, where ε is a positive constant such thatε=min(1α1,Δ4α1α3d0,α34α4d0(2Δα4α1α32δ1))with d0=α1α2+α2α3α41.

ϕ(0)=0 and ϕ(y)α3>0 for all y, and 0ϕ(y)ϕ(y)/yδ1<2Δα4/α1α32 for all y0.

Then the zero solution of (1.1) is asymptotically stable, provided thatτ<min{εd2α2,Δ2α1α3(α2α3+2μ),α1εd1α2α3},with μ=(α2α3/2)(d1+d2+1)>0.

Comparing Theorem 5.1 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  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.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (No10771045), and Program of Excellent Team in Harbin Institute of Technology.

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