Bifurcation analysis in a scalar delay differential equation

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IOP PUBLISHING NONLINEARITY

Nonlinearity 20 (2007) 2483–2498 doi:10.1088/0951-7715/20/11/002

Bifurcation analysis in a scalar delay differentialequation

Junjie Wei

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republicof China

E-mail: weijj@hit.edu.cn

Received 13 November 2006, in final form 2 September 2007Published 28 September 2007Online at stacks.iop.org/Non/20/2483

Recommended by D Trescher

AbstractThe dynamics of a scalar delay differential equation, which includes Mackey–Glass equations, are investigated. We prove that a sequence of Hopf bifurcationsoccurs at the equilibrium as the delay increases. An explicit algorithm fordetermining the direction of the Hopf bifurcations and the stability of thebifurcating periodic solutions is derived, using the theory of normal formand centre manifold. The global existence of multiple periodic solutions isestablished using a global Hopf bifurcation result given by Wu (1998 Trans.Am. Math. Soc. 350 4799) and a Bendixson criterion for higher dimensionalordinary differential equations given by Li and Muldowney (1993 J. Diff. Eqns106 27).

Mathematics Subject Classification: 34K-xx, 15-xx

1. Introduction

The equations

p(t) = βθn

θn + pn(t − τ)− γp(t), t � 0 (1)

and

p(t) = βθnp(t − τ)

θn + pn(t − τ)− γp(t), t � 0 (2)

were proposed by Mackey and Glass [6, 17] to describe a physiological control system. Herep(t) denotes the density of mature cells in blood circulation, τ is the time delay between the

0951-7715/07/112483+16$30.00 © 2007 IOP Publishing Ltd and London Mathematical Society Printed in the UK 2483

2484 J Wei

production of immature cells in the bone marrow and their maturation for release in circulatingbloodstreams and β, θ and γ are all positive constants. Similarly, the equation

p(t) = −δp(t) + ρe−γp(t−τ), t � 0 (3)

was proposed by Wazewska-Czyzewska and Lasota [29] to describe the survival of red bloodcells in animals, where p(t) denotes the number of red blood cells at time t, δ is the rate ofthe red blood cells, ρ and γ describe the production of cells per unit time and τ is the timerequired to produce a red blood cell. Equations (1), (2) and (3) have been extensively studiedin the literature. A majority of results on them deal with the global attractivity of the positiveequilibrium and oscillatory behaviours of solutions (see Gopalsamy et al [7, 8], Gyori andLadas [9], Karakostas et al [12], Kubiaczyk and Saker [14], Kulenovic et al [13], Zaghroutet al [38]). Meanwhile, there are several papers on the complex dynamics to equation (2).For example, Mackey and Glass [17] and Namajunas et al [19] studied the chaotic behaviourwhile, Losson et al [16] investigated the multistability. Recently, Song et al [26] and Wei andFan [30] have studied the bifurcation and global existence of periodic solutions to equations (1)and (3), respectively, using the centre manifold and normal form theory introduced by Hassardet al [11] and a global Hopf bifurcation theory given by Wu [37]. Clearly, (1), (2) and (3) canbe regarded as a special form of the following delay differential equation:

x(t) = −γ x(t) + βf (x(t − τ)). (4)

For equation (4) with τ = 1, under the assumption that f satisfies ξf (ξ) < 0 for ξ �= 0 and isbounded from above or below, and f ′(ξ) < 0 for all ξ , Walther [27] has derived a Poincare–Bendixson theorem. Recently, Skubachevskii and Walther [22,23] have studied hyperbolicityand, in particular, the stability of periodic solutions of equation (4) with τ = 1. For this areduction to a system of ordinary differential equations in the case of an arbitrary period wasused. For the studies on the existence and stability of periodic solutions of equation (4) werefer to [23] and references therein.

The purpose of the paper is to study the general delay equation (4). In fact, we providea detailed analysis of this equation. Regarding the delay τ as a parameter and applying thelocal and global Hopf bifurcation theory (see, e.g. Hale [10] and Wu [37]), we investigate theexistence of stable periodic oscillations for equation (4). More specifically, we prove that, asthe delay τ increases, the equilibrium k loses its stability and a sequence of Hopf bifurcationsoccurs at k. Furthermore, by using the normal form and centre manifold theory, we derivean explicit algorithm and sufficient conditions for the stability of the bifurcating periodicsolutions. The existence of periodic solutions for τ far away from the Hopf bifurcationvalues is also established, using a global Hopf bifurcation result of Wu [37]. A key stepin establishing the global extension of the local Hopf branch at τ = τ0 is to verify thatequation (4) has no nonconstant periodic solutions of period 4τ . This is accomplished byapplying a higher dimensional Bendixson criterion for ordinary differential equations givenby Li and Muldowney [15]. As examples, we apply the results for equation (4) to the specialmodels (2) and (3) giving the conditions to describe their dynamics.

We would like to mention that there are several papers on the bifurcation from a branchof periodic orbits of a family functional differential equations, see Dormaer [3–5] and Walther[28], and the global existence of periodic solutions in delayed differential equations based onthe global Hopf bifurcation theory given by Wu [37], for example, see the works given by Ruanand Wei [21], Song and Wei [24], Song, Wei and Han [25], Song, Wei and Yuan [26], Wei andLi [31, 32], Wei and Ruan [33], Wei and Yuan [34], Wei and Zou [35], Wen and Wang [36]and Wu [37].

The rest of the paper is organized as follows: in section 2, we investigate the stability of thepositive equilibrium and the occurrence of Hopf bifurcations. In section 3, the direction and

Bifurcation analysis in a scalar delay differential equation 2485

stability of the Hopf bifurcation are established, and certain numerical simulations are carriedout. A global Hopf bifurcation is established in section 4. Finally, the results are applied to(2) and (3) in section 5.

2. Stability and Hopf bifurcation analysis

In this section, we study the stability of the positive equilibrium and the existence of localHopf bifurcations. We make the following assumption on f .

(H1)f ∈ C3, there is a x0 and a neighbourhood of x0, denoted by N,

such that f (x0) = 0, and f (x) �= 0 for x ∈ N and x �= x0.

Under the assumption (H1), x0 is a fixed point to equation (4), and the linearization ofequation (4) around x0 is

x(t) = −γ x(t) + βf ′(x0)x(t − τ). (5)

The characteristic equation associated with (5) is

λ = −γ + βf ′(x0)e−λτ . (6)

For ω > 0, iω is a root of (6) if and only if

iω = −γ + βf ′(x0)(cos ωτ − i sin ωτ).

Separating the real and imaginary parts, we get

γ = βf ′(x0) cos ωτ,

ω = −βf ′(x0) sin ωτ,(7)

which leads to

γ 2 + ω2 = β2f ′2(x0),

namely,

ω =√

β2f ′2(x0) − γ 2. (8)

This is possible if and only if |βf ′(x0)| > γ .For |βf ′(x0)| > γ , let

τk =

1√β2f ′2(x0) − γ 2

[arccos

(γ

βf ′(x0)

)+ 2kπ

], for f ′(x0) < 0,

1√β2f ′2(x0) − γ 2

[(2π − arccos

(γ

βf ′(x0)

))+ 2kπ

], for f ′(x0) > 0,

(9)

where k = 0, 1, . . . .

Set

ω0 =√

β2f ′2(x0) − γ 2. (10)

Let λk(τ ) = αk(τ )+iωk(τ) be a root of (6) near τ = τk satisfying αk(τk) = 0 and ωk(τk) = ω0.We have the following result.

2486 J Wei

Lemma 2.1. α′k(τk) > 0.

Proof. Differentiating both sides of (6) with respect to τ , it follows thatdλ

dτ= βf ′(x0)e

−λτ

[−λ − τ

dλ

dτ

].

Therefore, noting that βf ′(x0)e−λτ = λ + γ, we havedλ

dτ= βf ′(x0)λe−λτ

1 + βf ′(x0)τe−λτ= − λ(λ + γ )

1 + τ(λ + γ )

and hencedλ

dτ

∣∣τ=τk

= ω20 − iω0γ

(1 + τγ ) + iτω0.

This implies that

α′(τk) = ω20

�,

where � = (1 + τγ )2 + τ 2ω20, completing the proof.

Lemma 2.2.(i) If γ > |βf ′(x0)|, then all roots of the characteristic equation (6) have negative real parts.

(ii) If |βf ′(x0)| > γ , then there exists a sequence of values of τ

0 < τ0 < τ1 < · · · < τk < · · · ,such that (1) equation (6) has a pair of simple imaginary roots ±iω0 when τ = τk(k =0, 1, 2, . . .), (2) if βf ′(x0) < −γ and τ ∈ [0, τ0), all the roots of equation (6) havenegative real parts, if τ = τ0, all roots of (6) except ±iω0 have negative real parts, and ifτ ∈ (τk, τk+1] for k = 0, 1, 2, . . . , equation (6) has 2(k + 1) roots with positive real parts;(3) if βf ′(x0) > γ , then equation (6) has at least one root with positive real part for allτ � 0.

Proof. At first, we have that the only root of equation (6) with τ = 0 is negative whenγ − βf ′(x0) > 0 and positive when γ − βf ′(x0) < 0.

From the analysis leading to relation (8) we know that, if |βf ′(x0)| < γ , equation (6) hasno purely imaginary root iω with ω > 0. Since λ = 0 is not a root of (6), we know that, forany τ � 0, equation (6) has no roots on the imaginary axis. Applying a result of Ruan andWei [20, corollary 2.4], we arrive at the conclusion (i).

If |βf ′(x0)| > γ , ω0 makes sense in (9). Then τk in (9) is defined well. From (7) and (8),we have that equation (6) has purely imaginary roots ±iω0 if and only if τ = τk and ω0 is givenin (10). The statement on the number of eigenvalues with positive real parts in (2)(respectively,in (3)) follows from the fact γ + βf ′(x0) < 0 (respectively βf ′(x0) > γ ) and lemma 2.1 andRouche’s theorem (Dieudonne [2, theorem 9.17.4]).

The spectral properties in lemma 2.2 immediately lead to the stability properties of thefixed point x0 of equation (4).

Theorem 2.3. For equation (4), the following hold.

(i) If |βf ′(x0)| < γ , then x = x0 is asymptotically stable for any τ � 0.

(ii) If βf ′(x0) < −γ , then x = x0 is asymptotically stable for τ ∈ [0, τ0) and unstable forτ > τ0.

(iii) If βf ′(x0) > γ , then x = x0 unstable for τ � 0.

(iv) If |βf ′(x0)| > γ , then equation (4) undergoes a Hopf bifurcation at x0 when τ = τk , fork = 0, 1, 2, . . ..

Bifurcation analysis in a scalar delay differential equation 2487

3. Stability and direction of the Hopf bifurcation

In the previous section, we obtained conditions for the Hopf bifurcation to occur whenτ = τk, k = 0, 1, 2, . . .. We know that, under the hypothesis βf ′(x0) < −γ , all thecharacteristic values, except ±iω0, have negative real parts when τ = τ0. By the centremanifold theorem, the stability of the bifurcating periodic solutions from τ = τ0 is the sameas that of the projection on the centre manifold. In this section we study the direction of theHopf bifurcation and the stability of the bifurcating periodic solutions when τ = τ0, usingtechniques from normal form and centre manifold theory (see, e.g. Hassard et al [11]).

Let y(t) = x(τ t) − x0. Then equation (4) becomes

y(t) = −γ τ(y(t) + x0) + τβf (y(t − 1) + x0). (11)

Correspondingly, the characteristic equation (6) becomes

z = −τγ + τβf ′(x0)e−z, (12)

with z = λτ for τ �= 0. From the conclusion (ii) of lemma 2.2 we know that, if nax(n−1)∗

(1+xn∗ )2 > γ

and τ = τ0, all roots of equation (11) except ±iτ0ω0 have negative real parts. Furthermore,by lemma 2.1, the root of equation (11)

z(τ ) = τα(τ) + iτω(τ)

with α(τ0) = 0 and ω(τ0) = ω0 satisfies

d(τα(τ))

dτ

∣∣τ=τ0

= τ0α′(τ0) > 0.

Set τ = τ0 + µ, µ ∈ R. Then µ = 0 is a Hopf bifurcation value for equation (11). Rewriteequation (11) as

y(t) = −(τ0 + µ)[γy(t) − βf ′(x0)x(t − 1)] + β(τ0 + µ)f ′′(x0)

2y2(t − 1)

+ β(τ0 + µ)f ′′′(x0)

3!y3(t − 1) + O(u4). (13)

For ϕ ∈ C([−1, 0], R), let

Lµϕ = −(τ0 + µ)[γ ϕ(0) − βf ′(x0)ϕ(−1)]

and

F(µ, ϕ) = β(τ0 + µ)f ′′(x0)

2ϕ2(−1) +

β(τ0 + µ)f ′′′(x0)

3!ϕ3(−1) + O(ϕ4(−1)). (14)

By the Riesz representation theorem, there exists a function η(θ, µ) of the bounded variationfor θ ∈ [−1, 0], such that

Lµϕ =∫ 0

−1dη(θ, µ)ϕ(θ), for ϕ ∈ C. (15)

In fact, we can choose

η(θ, µ) = −(τ0 + µ)γ δ(θ) − (τ0 + µ)βf ′(x0)δ(θ + 1). (16)

For ϕ ∈ C1([−1, 0], R), we set

A(µ)ϕ =

dϕ(θ)

dθ, θ ∈ [−1, 0),

∫ 0

−1dη(ξ, µ)ϕ(ξ), θ = 0

(17)

2488 J Wei

and

N(µ)ϕ ={

0, θ ∈ [−1, 0),

F (µ, ϕ), θ = 0.(18)

Then (13) can be rewritten as

yt = A(µ)yt + N(µ)yt , (19)

where yt (θ) = y(t + θ) for θ ∈ [−1, 0]. For ψ ∈ C1([0, 1], R), define

A∗ψ =

−dψ(s)

ds, s ∈ (0, 1],∫ 0

−1dη(t, 0)ψ(−t), s = 0.

For ϕ ∈ C[−1, 0] and ψ ∈ C[0, 1], using the bilinear form

〈ψ, ϕ〉 = ψ(0)ϕ(0) −∫ 0

−1

∫ θ

ξ=0ψ(ξ − θ) dη(θ)ϕ(ξ) dξ, (20)

we know that A∗ and A = A(0) are adjoint operators. By the discussion at the beginningof this section, we know that ±iτ0ω0 are eigenvalues of A(0). Thus they are eigenvalues ofA∗. Clearly, q(θ) = eiτ0ω0θ is an eigenvector of A corresponding to the eigenvalue iτ0ω0, andq∗(s) = Deiτ0ω0s is an eigenvector of A∗ corresponding to the eigenvalue −iτ0ω0. Furthermore,

〈q∗, q〉 = 1, 〈q∗, q〉 = 0,

where

D = 1

1 + τ0γ − iτ0ω0. (21)

Following the algorithms given by Hassard et al [11] and using a computation process similar tothat given by Wei and Li [32], we can obtain the coefficients which will be used in determiningthe important quantities:

g20 = Dβτ0f′′(x0)e

−2iτ0ω0 ,

g11 = Dβτ0f′′(x0),

g02 = Dβτ0f′′(x0)e

2iτ0ω0 ,

g21 = Dβτ0[f ′′(x0)(e−iω0τ0W11(−1) + eiω0τ0W20(−1)) + f ′′′(x0)e

−iω0τ0 ], (22)

where

W20(−1) = − g20

iτ0ω0e−iτ0ω0 − g02

3iτ0ω0eiτ0ω0 + E1e−2iτ0ω0 ,

W11(−1) = g11

iτ0ω0e−iτ0ω0 − g11

iτ0ω0eiτ0ω0 + E2,

E1 = βf ′′(x0)e−2iτ0ω0

2iω0 + γ − βf ′(x0)e−2iτ0ω0

and

E2 = βf ′′(x0)

γ − βf ′(x0).

Bifurcation analysis in a scalar delay differential equation 2489

Because each gij in (22) is expressed by the parameters and delay in (11), we can compute thefollowing quantities:

c1(0) = i

2ω0τ0

(g11g20 − 2|g11|2 − |g02|2

3

)+

g21

2,

µ2 = − Re(c1(0))

Re(λ′(τ0)),

β2 = 2Re(c1(0)),

T2 = − Im(c1(0)) + µ2Im(λ′(τ0))

ω0.

(23)

It is well known that µ2 determines the direction of the Hopf bifurcation: if µ2 > 0 (µ2 < 0),then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutionsexist for τ > τ0 (τ < τ0); β2 determines the stability of bifurcating periodic solutions: thebifurcating periodic solutions are orbitally asymptotically stable (unstable) if β2 < 0 (β2 > 0),and T2 determines the period of the bifurcating periodic solutions: the period increases(decreases) if T2 > 0 (T2 < 0).

From the discussion in section 2, we know that Re(λ′(τ0)) > 0; therefore we have thefollowing result.

Theorem 3.1. The direction of the Hopf bifurcation of the system (4) at the equilibrium x0

when τ = τ0 is supercritical (subcritical) and the bifurcating periodic solutions are orbitallyasymptotically stable (unstable) if Re(c1(0)) < 0 (> 0).

Corollary 3.2. If f ′′(x0) = 0, then the direction of the Hopf bifurcation is supercritical(subcritical) and the bifurcating periodic solutions are orbitally asymptotically stable(unstable) when f ′′′(x0)

f ′(x0)< 0(> 0).

In fact, from (22) it follows that g11 = g20 = g02 = 0, and g21 = Dβτ0f′′′(x0)e−iω0τ0 .

Substituting D = 11+τ0γ +iτ0ω0

into g21 and noting e−iω0τ0 = iω0+γ

βf ′(x0), it follows that

g21 = f ′′′(x0)

f ′(x0)

τ0

�{γ (1 + τ0γ ) + τ0ω20 + i(ω0(1 + τ0γ ) − τ0γω0)},

where � = (1+τ0γ )2 +τ 20 ω2

0. And hence from (23) and theorem 3.1, the conclusion is reached.

Theorem 3.1 provides an explicit algorithm for determining the direction and stabilityof the Hopf bifurcation at the first bifurcation value τ0. We shall perform some numericalsimulations in section 4 to illustrate the application of theorem 3.1.

4. Global existence of periodic solutions

In this section, we shall study the global continuation of positive periodic solutions bifurcatingfrom the point (x0, τk), k = 0, 1, 2, . . . for equation (4) by using a global Hopf bifurcationtheorem given by Wu [37]. For the reader’s convenience, we copy equation (4) in the following.

x(t) = −γ x(t) + βf (x(t − τ)). (24)

At first, we make a further assumption on f (x).

(H2) Either f (x) is positive (respectively, negative) definite and bounded for x ∈ R+

(respectively, R−), or f (x) is bounded for x ∈ R.

2490 J Wei

For convenience, we introduce some notation:

X = C([−τ, 0], R),� = Cl{(x, τ, p) : x is a p-periodic solution of (24)} ⊂ X × R+ × R+,N = {(x, τ, p) : γ x = βf (x)}.Let C(x0, τk, 2π/ω0) denote the connected component of (x0, τk, 2π/ω0) in �, where τk

and ω0 are defined in (9) and (10), respectively.

Lemma 4.1. If (H2) is satisfied, then all periodic solutions to (24) are uniformly bounded.

Proof. Let x(t) be a nonconstant periodic solution to (24) and x(t1) = M, x(t2) = m be itsmaximum and minimum, respectively. Then x ′(t1) = x ′(t2) = 0, and by (24),

M = β

γf (x(t1 − τ)) and m = β

γf (x(t2 − τ)).

Hence, 0 � m < M <β

γB when f (x) is positive definite, where B is an upper bound

of f (x). Similarly, when f (x) is negative definite and B is its lower bound, it follows thatβ

γB < m < M � 0. If f (x) is bounded, there exists a positive constant C such that |f (x)| < C

and hence − β

γC < m < M <

β

γC. Summarizing the above discussion, the conclusion follows.

Remark 1. From the proof one can see that the periodic solution of (24) is positive (respectively,negative) when f (x) is positive (respectively, negative) definite.

Lemma 4.2. If (H2) and

|f ′(x)| <

√2γ

β, for x ∈ [B, U ], (25)

are satisfied, then equation (24) has no periodic solutions of period 4τ , where B and U arethe lower and upper bounds of the periodic solutions of equation (24), respectively.

Proof. From lemma 4.1, B and U are well defined. Let x(t) be a periodic solution to (24) ofperiod 4τ . Set

uj (t) = x(t − (j − 1)τ ), j = 1, 2, 3, 4.

Then u(t) = (u1(t), u2(t), u3(t), u4(t)) is a periodic solution to the following system ofordinary differential equations:

u1(t) = −γ u1(t) + βf (u2(t)),

u2(t) = −γ u2(t) + βf (u3(t)),

u3(t) = −γ u3(t) + βf (u4(t)),

u4(t) = −γ u4(t) + βf (u1(t)).

(26)

From lemma 4.1, the orbit of (26) belongs to the region

G = {u ∈ R4 : B < uk < U, k = 1, 2, 3, 4}. (27)

To rule out the 4τ -periodic solution to (24), it suffices to prove the nonexistence of nonconstantperiodic solutions of (26) in the region G. To do the latter, we use a general Bendixson’scriterion in higher dimensions developed by Li and Muldowney [15]. More specifically, we

Bifurcation analysis in a scalar delay differential equation 2491

shall apply corollary 3.5 in that given by Li and Muldowney [15]. The Jacobian matrixJ = J (u) of (26), for u ∈ R4, is

J (u) =

−γ βf ′(u2) 0 00 −γ βf ′(u3) 00 0 −γ βf ′(u4)

βf ′(u1) 0 0 −γ

.

The second additive compound matrix J [2](u) of J (u) is, see (Muldowney [18]),

J [2](u) =

−2γ βf ′(u3) 0 0 0 00 −2γ βf ′(u4) βf ′(u2) 0 00 0 −2γ 0 βf ′(u2) 00 0 0 −2γ βf ′(u4) 0

−βf ′(u1) 0 0 0 −2γ βf ′(u3)

0 −βf ′(u1) 0 0 0 −2γ

.

Choose a vector norm in R6 as

|(x1, x2, x3, x4, x5, x6)| = max{√

2|x1|, |x2|,√

2|x3|,√

2|x4|, |x5|,√

2|x6|}.Then, with respect to this norm, the Lozinski l measure µ(J [2](u)) of the matrix J [2](u) is, see(Coppel [1]),

µ(J [2](u)) = max{√

2(−√

2γ + β|f ′(u3)|),√

2(−√

2γ + β|f ′(u4)|/2 + β|f ′(u2)|/2),√2(−

√2γ + β|f ′(u2)|),

√2(−

√2γ + β|f ′(u4)|),√

2(−√

2γ + β|f ′(u1)|/2 + β|f ′(u3)|/2),√

2(−√

2γ + β|f ′(u1)|)}. (28)

By corollary 3.5 in (Li and Muldowney [15]), system (26) has no periodic orbits in G ifµ(J [2](u)) < 0 for all u ∈ G. From (28), one can see that µ(J [2](u)) < 0 if and only if

β|f ′(uj )| <√

2γ, j = 1, 2, 3, 4, (29)

for u ∈ G.From (25), for u ∈ G, (29) is satisfied, completing the proof.

Lemma 4.3. Equation (24) has no periodic solutions of period τ or 2τ .

Proof. First note that any nonconstant τ -periodic solution u(t) of (24) is also a nonconstantperiodic solutions of the ordinary differential equation

u(t) = −γ u(t) + βf (x(t)). (30)

Clearly, equation (30) has no nonconstant periodic solutions.As in the proof of lemma 4.2, if u(t) is a 2τ -periodic solution of (24), then u1(t) = u(t)

and u2(t) = u(t − τ) are periodic solutions of the system of ordinary differential equations

u1(t) = −γ u1(t) + βf (u2(t)),

u2(t) = −γ u2(t) + βf (u1(t)).(31)

Let (P (u1, u2), Q(u1, u2)) denote the vector field of (31), then∂P

∂u1+

∂Q

∂u2= −2γ < 0

for all (u1, u2) ∈ R2. Thus the classical Bendixson’s negative criterion implies that (31) hasno nonconstant periodic solutions. This completes the proof.

2492 J Wei

Theorem 4.4. Suppose that β|f ′(x0)| > γ , (H2), and either f ′(x0) < 0 and (25) orf ′(x0) > 0 is satisfied. Then, for each τ > τk, k = 0, 1, 2, . . ., equation (24) has atleast k + 1 positively periodic solutions, where τk is defined in (9).

Proof. First note that

F(xt , τ, p) = −γ x(t) + βf (x(t − τ))

satisfies the hypotheses (A1), (A2) and (A3) in (Wu [37]), with

(x0, α0, p0) =(

x0, τk,2π

ω0

)and

�(x0,τ,p)(λ) = λ + γ − βf ′(x0)e−λτ .

It can also be verified that (x0, τk,2πω0

) are isolated centres, then by lemmas 2.1 and 2.2, thereexist ε > 0, δ > 0 and a smooth curve λ : (τk − δ, τk + δ) −→ C such that

�(λ(τ)) = 0, |λ(τ) − iω0| < ε

for all τ ∈ [τk − δ, τk + δ] and

λ(τk) = iω0,dRe(λ(τ ))

dτ|τ=τk

> 0.

Denote pk = 2π/ω0, and let

�ε = {(0, p) : 0 < u < ε, |p − pk| < ε}.Clearly, if |τ −τk| � δ and (u, p) ∈ �ε such that �(x0,τ,p)(u+2π i/p) = 0, then τ = τk, u = 0and p = pk . This verifies the assumption (A4) in (Wu [31]) for m = 1. Moreover, putting

H±(x0, τk, 2π/ω0)(u, p) = �(x0,τk±δ,p)(u + i2π/p),

we have the crossing number

γ1(x0, τk, 2π/ω0) = degB(H−(x0, τk, 2π/ω0), �ε) − degB(H +(x0, τk, 2π/ω0), �ε) = −1.

By theorem 3.2 given by Wu [37], we conclude that the connected component C(x0, τk, 2π/ω0)

through (x0, τk, 2π/ω0) in � is nonempty. Meanwhile, we have∑(x,τ,p)∈C(x0,τk,2π/ω0)

γ1(x, τ, p) < 0,

and hence, by theorem 3.3 given by Wu [37], C(x0, τk, 2π/ω0) is unbounded.Lemma 4.1 implies that the projection of C(x0.τk, 2π/ω0) onto the x-space is bounded.

Similar to lemma 4.3, one can get that equation (23) with τ = 0 has no nonconstant periodicsolutions. Therefore, the projection of C(x0.τk, 2π/ω0) onto the τ -space is bounded below.

From the definition of τk in (8) and (6) we know that when f ′(x0) < 0π

2< τ0ω0 < π and 2π < τkω0 < (2k + 1)π, k � 1

and when f ′(x0) > 0

3π

2< τ0ω0 < 2π and 2π < τkω0 < (2k + 1)π, k � 1.

Therefore,

2τ0 <2π

ω0< 4τ0 and

τk

k + 1<

2π

ω0< τk, k � 1 (32)

Bifurcation analysis in a scalar delay differential equation 2493

when f ′(x0) < 0 and

τ0 <2π

ω0<

4τ0

3< 2τ0 and

τk

k + 1<

2π

ω0< τk, k � 1 (33)

when f ′(x0) > 0.Applying lemmas 4.2 and 4.3 we know that (1) when f ′(x0) < 0, 2τ < p < 4τ if

(x, τ, p) ∈ C(x0, τ0, 2π/ω0) and τk+1 < p < τ if (x, τ, p) ∈ C(x0, τk, 2π/ω0) for k � 1,

(2) when f ′(x0) > 0, τ < p < 2τ if (x, τ, p) ∈ C(x0, τ0, 2π/ω0) and τk+1 < p < τ if

(x, τ, p) ∈ C(x0, τk, 2π/ω0) for k � 1. This shows that in order for C(x0, τk, 2π/ω0) to beunbounded, its projection onto the τ -space must be unbounded. Consequently, the projectionof C(x0, τk, 2π/ω0) onto the τ -space includes [τk, ∞). This shows that, for each τ > τk ,equation (23) has k + 1 nonconstant periodic solutions. The positive property of the periodicsolutions follows from lemma 4.1. This completes the proof.

Remarks.

1. From the proof of theorem 4.4, we know that the first global Hopf branch contains periodicsolutions of period between 2τ and 4τ when f ′(x0) < 0 and between τ and 2τ whenf ′(x0) > 0. They are the slowly oscillating periodic solutions. The kth branches, fork � 1, since the periods are less than τ , contain fast-oscillating periodic solutions.

2. For k � 1,τk

k + 1<

2π

ω0< τk

automatically holds. The bounds on the period p for (x, τ, p) ∈ C(x0, τk, 2π/ω0) holdwithout resulting in lemma 4.2. Thus, the global extension of the τk-branch for k � 1 canbe proved without restriction (25).

5. Applications and numerical simulations

In this section we apply the results obtained in the previous sections to the Mackey–Glassequations (2) and (3). Equation (1) has been studied by Wei and Fan [30] in the same way.For the reader’s convenience, we copy equation (2) in the following:

p(t) = βθnp(t − τ)

θn + pn(t − τ)− γp(t), t � 0. (34)

The change in variables p(t) = θx(t) transforms equation (34) to the delay differentialequation:

x(t) = −γ x(t) +βx(t − τ)

1 + xn(t − τ), t � 0. (35)

Throughout this section we assume that

(H3) β > γ > 0 and n, τ ∈ R+.

Then equation (35) has a unique positive equilibrium x0, which is given by

x0 =(

β − γ

γ

)1/n

. (36)

We should point out that equation (35) exhibits a complex dynamical behaviour since thenonlinearity in equation (35) is a so-called single-humped function of x(t − τ) which can leadto chaotic dynamics. We refer to (Mackey and Glass [17]) and (Namajunas et al [19]).

2494 J Wei

Notice that

f (x) = x

1 + xn

and hence

f ′(x) = 1 − (n − 1)xn

(1 + xn)2(37)

and

f ′′(x) = nxn−1

(1 + xn)3[−(n + 1) + (n − 1)xn]. (38)

From (36) and (37) it follows that

f ′(x0) = γ

β2[nγ − (n − 1)β]. (39)

Applying theorem 2.3, we have the following conclusions.

Theorem 5.1. For equation (35), under the assumption (H3), the following hold.

(i) If γ >(n−2)β

n, then the positive equilibrium x0 is asymptotically stable for all τ � 0.

(ii) If γ <(n−2)β

n, then there exists a sequence of values of τ ,

0 < τ0 < τ1 < · · · < τk < · · ·such that the equilibrium x0 is asymptotically stable when τ ∈ [0, τ0) and unstable when τ > τ0

as well as equation (35) undergoes a Hopf bifurcation at x0 when τ = τk, k = 0, 1, 2, . . .,where

τk = β

γ√

[nγ − (n − 1)β]2 − β2

[arccos

(β

nγ − (n − 1)β

)+ 2kπ

], k = 0, 1, . . . . (40)

In order to get the global existence of periodic solutions to equation (35), we assume thatn > 1 and n = p/q with p an even number. Then we know that there exists a positive constantC such that |f (x)| = |x|

1+xn < C for all x ∈ R. This implies that f (x) = x1+xn satisfies (H2).

From (38) it follows that |f ′(x)| takes its maximum at x = 0 or x = ( n+1n−1 )

1n . Notice

f ′(0) = 1 and |f ′(( n+1n−1 )

1n )| = (n−1)2

4n. Applying theorem 4.4, the following conclusion holds.

Theorem 5.2. Suppose that (H3), n = p

qwith q an even number, γ <

(n−2)β

nand

max

{1,

(n − 1)2

4n

}<

√2γ

β(41)

are satisfied. Then equation (35) has at least k + 1 nonconstant periodic solutions forτ > τk, (k = 0, 1, 2, . . .), where τk is defined as in (40).

Remark. The global extension of the τk-branch for k � 1 can be obtained withoutrestriction (41).

Now let us consider equation (3). We copy it as follows

p(t) = −γp(t) + βe−δp(t−τ), (42)

where γ > 0, β > 0 and δ > 0 are all constants. Set

g(x) = −γ x + βe−δx .

From g(0) = β > 0, g′(x) = −γ − βδe−δx < 0 and limx→∞ g(x) = −∞, it follows thatequation (40) has a unique equilibrium, denoted by p0, which is positive. Set

f (x) = e−δx,

Bifurcation analysis in a scalar delay differential equation 2495

then

f ′(x) = −δe−δx . (43)

Applying theorem 2.3, the following conclusions hold.

Theorem 5.3. For equation (42), the following hold.

(i) If βδe−δp0 < γ , then p = p0 is asymptotically stable for any τ � 0.

(ii) If βδe−δp0 > γ , then there exists a sequence of values of τ ,

0 < τ0 < τ1 < · · · < τk < · · · ,such that the equilibrium p = p0 is asymptotically stable for τ ∈ [0, τ0), unstable forτ > τ0 and equation (42) undergoes a Hopf bifurcation at p = p0 when τ = τk, k =0, 1, 2, . . ., where

τk = 1√β2δ2e−2δp0 − γ 2

[arccos

(γ

βδe−δp0

)+ 2kπ

], k = 0, 1, · · · . (44)

Clearly, f (x) = e−δx > 0 satisfies (H2). Hence, by lemma 4.1, the periodic solutionsequation (42) are positive and uniformly bounded. In fact, if p(t) is a periodic solution ofequation (42), it follows that

β

γe− δβ

γ < p(t) <β

γ.

Hence,

|f ′(x)| = δe−δx < δe−δβ

γe− δβ

γ

, for x ∈(

β

γe− δβ

γ ,β

γ

). (45)

Applying theorem 4.4, we have the following.

Theorem 5.4. Suppose that δe−δp0 >γ

βand

δe−δβ

γe− δβ

γ

<

√2γ

β(46)

are satisfied. Then, for each τ > τk, k = 0, 1, 2, . . ., equation (42) has at least k +1 positivelyperiodic solutions, where τk is defined in (44).

In fact, (45) and (46) imply that (25) is satisfied. Hence the conditions of theorem 4.4 areall satisfied, and the conclusion follows.

Remarks.

1. Restriction (46) is not needed for the global extension of the τk−branch for k � 1.

2. Song et al [26] have investigated the global existence of periodic solutions to equation (42).But they did not obtain the global extension of the τ0-branch.

Next we shall carry out numerical simulations to equations (35) and (42), respectively.Consider equation (35); assume n = 4, β = 0.3, γ = 0.1; obviously, this system

has a unique positive equilibrium x0 = 1.1892 and f ′(x0) = −0.5556 < 0. For0.1 = γ < β(n − 2)/n = 0.15, applying theorem 5.1 (ii), we get τk

.= 16.6072 + 47.1239k.Hence the positive equilibrium is asymptotically stable when τ ∈ (0, τ0) and unstable whenτ ∈ (τ0, +∞). From theorem 5.3, we know that equation (35) has at least one periodic solutionwhen τ > τ0.

2496 J Wei

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t

x

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t

x

0 500 1000 1500 2000 2500 30000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t

x

0 1000 2000 3000 4000 5000 6000 7000 80000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t

x

(a) (b)

(c) (d)

Figure 1. Matlab simulations of (35) with n = 4, β = 0.3, γ = 0.1, where (a) withτ = 15 < τ0 = 16.6072, (b) with τ = 18 ∈ (τ0, τ1), (c) with τ = 65 ∈ (τ1, τ2), (d) withτ = 150 > τ3 and the initial value x = 1.5.

Now, we compute the properties of the Hopf bifurcation at the first critical valueτ0 = 16.6072. By direct computation, we get f ′′(x0) = 0.2492, f ′′′(x0) = 7.3330, E1 =0.1596 + 0.0690i and E2 = 0.2803. Substituting these values into equation (22), weobtain g20

.= 0.1430 + 0.3288i, g11.= 0.2756 − 0.2294i, g02

.= −0.2974 − 0.2004i, g21.=

−10.4162 − 2.4222i. Thus, c1(0).= −5.2211 − 1.2529i. This implies µ2

.= 212.2389 >

0, β2.= −10.4422 < 0, T2

.= 174.4445, and hence the Hopf bifurcation is supercritical and thebifurcating solution is orbitally asymptotically stable, and the period of the solution increases.All the above results are shown in figure 1.

Consider equation (42). Let δ = 3, β = 7 and γ = 5, then we have p0 = 0.4096,βδe−δp0

.= 6.1447 > γ = 5. The conditions of 5.3 (ii) are satisfied. Moreover, we get aτk

.= 0.7059 + 1.7592k, k = 0, 1, 2, . . .. Hence the equilibrium p = p0 is asymptoticallystable for τ ∈ [0, τ0) and unstable for τ > τ0, as well as equation (42) has at least one periodicsolution when τ > τ0.

Now, we compute the properties of the Hopf bifurcation at the first critical valueτ = 0.7059. Under the parameters that we give above, we get f ′(x0) = −0.8778, f ′′(x0) =2.6334, f ′′′(x0) = −7.9003 and E1 = 1.2352 + 0.2053i, E2 = 1.6541. Substituting theseparameters into equation (22), we get g20

.= 1.8661 + 1.6789i, g11.= 2.1933 − 1.2209i, g02

.=−0.4437 − 2.4707i, g21

.= 0.3341 + 2.8513i. Thus, c1(0).= −0.1114 − 0.2718i. This

implies µ2 = 0.3324 > 0, β2.= −0.2228 < 0, T2

.= 0.3525; so the Hopf bifurcation issupercritical and the bifurcating solution is orbitally asymptotically stable; the period of thesolution increases. These are shown in figure 2.

Bifurcation analysis in a scalar delay differential equation 2497

0 50 100 150 200 250 3000

0.5

1

1.5

t

x

0 20 40 60 80 100 1200

0.5

1

1.5

t

x

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

t

x

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

t

x

(a) (b)

(c) (d)

Figure 2. Matlab simulations of (42) with δ = 3, β = 7, γ = 5, where (a) with τ = 0.7 < τ0 =0.7059, (b) with τ = 0.75 ∈ (τ0, τ1), (c) with τ = 3.0 ∈ (τ1, τ2), (d) with τ = 5.0 > τ3 and theinitial value x = 1.5.

Acknowledgments

The author wishes to thank the referee for his or her valuable comments and suggestions that ledto truly significant improvement of the manuscript. This research is supported by the NationalNatural Science Foundation of China and Specialized Research Fund for the Doctoral Programof Higher Education.

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