Global Hopf Bifurcation Analysis for a Time-Delayed Model of Asset Prices

Discrete Dynamics in Nature and Society Volume 2010, Article ID 432821,17pages doi:10.1155/2010/432821

Research Article

Global Hopf Bifurcation Analysis for a Time-Delayed Model of Asset Prices

Ying Qu and Junjie Wei

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Correspondence should be addressed to Junjie Wei,[email protected] Received 7 October 2009; Accepted 13 January 2010

Academic Editor: Xuezhong He

Copyrightq2010 Y. Qu and J. Wei. This 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 time-delayed model of speculative asset markets is investigated to discuss the effect of time delay and market fraction of the fundamentalists on the dynamics of asset prices. It proves that a sequence of Hopf bifurcations occurs at the positive equilibriumv, the fundamental price of the asset, as the parameters vary. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined using normal form method and center manifold theory. Global existence of periodic solutions is established combining a global Hopf bifurcation theorem with a Bendixson’s criterion for higher-dimensional ordinary differential equations.

1. Introduction

Efficient Market Hypothesis EMH is a standard theory of financial market dynamics.

According to the theory, asset prices follow a geometric Brownian motion representing the fundamental value of the asset, and hence asset prices cannot deviate from their fundamental values. The EMH theory, however, cannot explain excess volatility of financial markets such as speculative booms followed by severe crashes. Recently, models have been developed to explain fluctuations in financial markets see1–9 and the references therein. In such models, asset prices follow deterministic paths that can deviate from fundamental values generating what is called a speculative bubble in asset markets. In speculative markets modeling, almost all the previous models have utilized either differential or difference equations methodology.

Dibeh 4 presents a delay-differential equation to describe the dynamics of asset prices. The author divides market participants into chartists and fundamentalists. The fundamentalists follow the EMH theory and base their demand formation on the difference between the actual price of the assetpand the fundamental price of the assetv. The chartists, on the other hand, base their decisions of market participation on the price trend of the asset.

They attempt to exploit the information of past prices to make their decisions of purchasing or selling the asset. In4, the model describing the time evolution of the asset price is given by

dp

dtt 1−mtanh

pt−pt−τ

pt−m

pt−v

pt, P

wherem∈0,1is the market fraction of the fundamentalists. A time delay is introduced for chartists since they base their estimation of the slope of the asset price trend on an adaptive process that takes into account past values of the slope of the price trend.

In4, it is shown by simulation that there may exist limit cycles forP, explaining the persistence of deviations from fundamental values in speculative markets. The aim of the present paper is to provide a detailed theoretical analysis of this phenomenon from the bifurcation point of view. Applying the local Hopf bifurcation theory see 10, we investigate the existence of periodic oscillations forP, which depends both on time delayτ and the market fraction of the fundamentalistsm. Using the normal form theory and center manifold theorem11,12, we derive sufficient conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Furthermore, taking τ as a parameter, we establish the existence of periodic solutions forτ far away from the Hopf bifurcation values using a global Hopf bifurcation result of Wusee13and also14–

18. A key step in establishing the global extension of the local Hopf branch at the first critical valueτ τ0is to verify thatPhas no nonconstant periodic solutions of period 4τ.

This is accomplished by applying a higher-dimensional Bendixson’s criterion for ordinary differential equations given by Li and Muldowney19.

The rest of this paper is organized as follows: in Section2, our main results are stated and some numerical simulations are carried out to illustrate the analytic results. In Section3, results on stability and bifurcations at positive equilibrium are proved, taking m and τ as parameters, respectively. In Section 4, a theorem on the direction and stability of Hopf bifurcation is provided. Finally, a global Hopf bifurcation theorem is proved.

2. Main Results

Given the nonnegative initial condition

pt ϕ≥0, ϕ0>0, t∈−τ,0, 2.1

EquationPadmits a unique solutionHale and Lunel10. Any solutionpt, ϕofPis nonnegative if and only ifϕ0>0, following the fact that

pt ϕ0e^t01−mtanhps−ps−τ−mps−vds. 2.2

Note that

dp dtt≤

1−m mv−mpt

pt. 2.3

We conclude that, for any sufficiently small >0, pt< 1−m

m v 2.4

holds for all larget >0. This establishes the ultimate uniform boundedness of solutions for P.

EquationPhas two equilibria 0 andv. Denote byp^∗any of the equilibria. Then the linearization ofPatp^∗is given by

dp dtt

1−3mp^∗ mv

pt−1−mp^∗pt−τ. 2.5

Ifp^∗0, then it is unstable since2.5becomes dp

dtt mvpt. 2.6

Ifp^∗v, then the characteristic equation of2.5is

Δλ:λ−1−2mv 1−mve^−λτ 0. 2.7 We investigate the dependence of local dynamics ofPon parametersmandτ.

Case 1. Choosingmas parameter.

Define

m_k1− 1

2−cosη_kτ, 2.8

whereη_ksolves

2−cosητ

η−vsinητ0, 2.9

with the assumption ofvτ >1.

Proposition 2.1. Ifm1, thenp^∗vis globally stable.

The case of m 1 represents that asset prices are totally determined by the fundamentalists. Obviously, asset prices cannot deviate from their fundamental values v since the fundamentalists obey the EMH theory.

Theorem 2.2. Assumevτ >1. Then

ithe positive equilibriump^∗ vof Pis asymptotically stable whenm ∈m0,1, where m₀:max{mk};

ii Pundergoes a Hopf bifurcation atvwhenmm_k.

Case 2. Regardingτas parameter.

Define

τj 1 ω₀

arccos1−2m 1−m 2jπ

, j0,1,2, . . . 2.10

withω0 :v m2−3m.

Theorem 2.3. ForP,

iifm∈2/3,1, thenp^∗vis asymptotically stable for allτ ∈R ;

iiifm∈0,2/3, thenp^∗vis asymptotically stable when 0< τ < τ0and becomes unstable when τ > τ0; moreover,Pundergoes a Hopf bifurcation atp^∗ vwhen τ τj j 0,1,2, . . ..

Theorem 2.4gives the direction of Hopf bifurcation and stability of the bifurcating periodic solutions. Similar results hold if we choosemas parameter.

Theorem 2.4. Assumem∈0,2/3. If Rec10<0>0, then the bifurcating periodic solutions atv are asymptotically stable (unstable) on the center manifold and the direction of bifurcation is forward (backward). In particular, the bifurcating periodic solution from the first bifurcation value τ τ₀is stable (unstable) in the phase space if Rec10<0>0.

Corollary 2.5. Whenτ τ0,vis stable (unstable) if Rec10<0>0.

Remark 2.6. See the proof in Section 4 for the explanation of c₁0 which appears in Theorem2.4and Corollary2.5.

The conclusions above are illustrated in Figure1, using bifurcation set on them, τ-plane.

Here,τ₀m, τ1m, τ2m, . . . , τjm, . . .are Hopf bifurcation curves. When m, τ∈D :

m, τ|0< m < 2

3,0≤τ < τ0m

∪

m, τ| 2

3 ≤m <1

, 2.11

p^∗vis asymptotically stable, and when m, τ∈D^c:

m, τ|0< m < 2

3, τ > τ0m

, 2.12

p^∗ vis unstable. Denote the curveτ τ0mbyl. Clearly,llies in the boundary ofD. The stability ofvwhenm, τ∈lis given by Corollary2.5.

The occurrence of Hopf bifurcation explains the persistent deviation of the asset price from the fundamental value and it depends both on the time delay and the market fraction of the fundamentalists. In fact, choosing the same parameters as in4, Figure 2:

m0.62, v5, τ2, 2.13

we compute by 2.10 that τ0 1.5304. Therefore, Theorem 2.3 provides insight on the explanation of existence of a limit cycle in4, Figure 2.

0 20 40 60

τ

1/3 2/3 1

m

Stable Hopf

bifurcation curves

τ₀m τ₁m τ₂m

.. . τjm

.. . Unstable

Figure 1: Hopf bifurcation set onm−τplane.

Finally, a natural question is that if the bifurcating periodic solutions ofPexist when τis far away from critical values? In the following theorem, we obtain the global continuation of periodic solutions bifurcating from the pointsv,τ_j j 0,1,2, . . ., using a global Hopf bifurcation theorem given by Wu13.

Theorem 2.7. If m ∈ 0,2/3, then for τ > τ_jj ≥ 1,P has at least one periodic solution.

Furthermore, ifm∈3 √

2/7,2/3, thenPhas at least one periodic solution forτ > τj j ≥0 and at least two periodic solutions forτ > τ_j j≥1. Here,τ_j j0,1,2, . . .is defined in2.10.

Figures2 and3 describe the phenomena stated in Theorem 2.7, where m 0.65 ∈ 3 √

2/7,2/3andv 5. It is shown that the time delay plays an important role in the dynamics of the nonlinear model. To be concrete, the longer the time delays used by the chartists in estimating the asset price trend, the more likely the market will exhibit persistent deviation of the asset price from the fundamental value by a limit cycle, and the greater the period of a limit cycle becomes. In fact,τ₀ 2.885 andc₁0 −0.856 under these parameters.

Therefore, combining Theorem2.7with the two figures, we know that there exists at least one stable periodic solution ofPwith periodTτ≥2π/ω6.9706 whenτ >2.885. We would like to mention that the graphs in Figure2are prepared by using DDE-BIFTOOL developed by Engelborghs et al.20,21.

3. Proof of the Results in Cases 1 and 2

Case 1. Choosingmas parameter.

When m 1, Pis a scalar ordinary differential equation and therefore it has no nonconstant periodic solutions. Additionally, the only root of 2.7 isλ −v < 0. These complete the proof of Proposition2.1.

Whenm <1,iηη >0is a root of2.7if and only ifηsatisfies sinητ η

1−mv, cosητ 1−2m

1−m. 3.1

It follows that

2−cosητ

η−vsinητ0,

m1− 1

2−cosητ. 3.2

0 τ0 τ1 τ2 τ3 τ4 · · ·τj · · ·

maxpt−minpt

a

τ0 τ1 τ2 τ3 τ4 · · · τj · · ·

Tτ

2π/ω0

0

b

Figure 2: Hopf bifurcation branches through the centersv, τj,2π/ω0.aon theτ, d-plane, whered maxpt−minpt;bon theτ, Tτ-plane, whereTτis the period of periodic solution bifurcated at v.

DenoteGη^def 2−cosητη−vsinητ. ThenG0 0, G∞ ∞, and G

η

ητsinητ 2−1 vτcosητ. 3.3

Therefore, there exists at least one zeroηkofGηifvτ >1.

Definem_kas in2.8, thenm_k ∈0,2/3and±iηkis a pair of purely imaginary roots of2.7withmm_k. Letλmbe the root of2.7satisfyingλmk iη_k. Substitutingλm into2.7, it follows that

dλ dm

ve^−λτ−2v

1−τ1−mve^−λτ. 3.4

Thus, we have the transversal condition Sign

Re dλ

dm

|mmk

Sign

cosη_kτ−2−τv1−m_k

1−2 cosη_kτ

Sign

3τv1−m_k²−2τv1−m_k−1 1−m_k

,

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

1, m_k∈

0,1 3

2−

1 3

τv

,

−1, mk∈ 1

3

2−

1 3

τv

,2 3

,

3.5

where cosηkτ 2−1/1−mkis used. Accordingly, a Hopf bifurcation atp^∗voccurs when mm_k.

4.6 5 5.4

pt

460 470 480 490 500

t a

4.6 5 5.4

pt

400 425 450 475 500

t b

4.6 5 5.4

pt

300 350 400 450 500

t c

4.6 5 5.4

pt

100 200 300 400 500

t d

Figure 3: Stable periodic solution ofP, whereawithτ 2.9> τ0 2.885,bwithτ 10,cwith τ20, anddwithτ50.

In summary, ifvτ >1, then there exists at least one value ofmdefined by2.8such that all the roots of2.7have negative real parts whenm ∈ m0,1, in addition, a pair of purely imaginary roots whenmm_k. This implies Theorem2.2.

Case 2. Regardingτas a parameter.

First of all, we know that the root of 2.7 with τ 0 satisfies thatλ −mv < 0.

Therefore,p^∗vis globally asymptotically stable whenτ 0.

Letiωω >0be a root of2.7. Then

sinωτ ω

1−mv >0, cosωτ 1−2m

1−m . 3.6

This leads toω² mv²2−3m.ω₀ v m2−3mmakes sense whenm∈0,2/3. Define τjas in2.10; theniω0is a pure imaginary root of2.7withτ τj, j 0,1,2, . . ..

Similarly, letλτ ατ iτbe the root of2.7, satisfying

α τj

0, τj

ω0. 3.7

Differentiating both sides of2.7gives dλ

dτ ₋₁

e^λτ

λv1−m −τ

λ. 3.8

Therefore,

Re dλ

dτ ₋₁

|ττj

sinω₀τ_j

ω₀v1−m >0. 3.9

This implies thatατj>0, j 0,1,2,. . ..

Note thatλ0 is not a root of2.7whenm2/3. Therefore, we obtain the following spectral properties of2.7:

iifm∈2/3,1, then all roots of2.7have negative real parts;

iiifm∈ 0,2/3, then there exist a sequence values ofτ defined by2.10such that 2.7has a pair of purely imaginary roots±iω0whenτ τj. Additionally, all roots of2.7have negative real parts whenτ∈0, τ0, all roots of2.7, except±iω0, have negative real parts whenτ τ₀, and2.7has at least a pair of roots with positive real parts whenτ > τ₀.

These spectral properties immediately lead to the dynamics near the positive equilibriumvdescribed in Theorem2.3.

4. Proof of Theorem 2.4

We use the center manifold and normal form theories presented in Hassard et al. 12 to establish the proof of Theorem2.4. Normalizing the delayτby the time scalingt→t/τand using the change of variablespt ptτ, we transformPinto

dp

dtt 1−mτtanh

pt−pt−1

pt−mτ

pt−v

pt, P0

whose characteristic equation atpvis

Δ1z:z−1−2mτv 1−mτve^−z0. 4.1

Comparing 4.1 with 2.7, we see z λτ for τ /0. Therefore, we obtain the following Lemma.

Lemma 4.1. Assumem∈0,2/3.

iIfτ τj j 0,1,2, . . ., then4.1has a pair of purely imaginary roots±iω0τj, whereτj

andω₀are defined as before.

iiLetzτ γτ iζτbe the root of 4.1, satisfying

γ τj

0 andζ τj

ω0τj, 4.2

then

γ τj

τjα τj

>0, j 0,1,2, . . . . 4.3

iiiAll roots of4.1have negative real parts whenτ∈0, τ0, all roots of4.1, except±iω0τ0, have negative real parts whenτ τ₀, and4.1has at least a pair of roots with positive real parts whenτ > τ₀.

Without loss of generality, we denote the critical valueτ^∗ at whichP0undergoes a Hopf bifurcation atv. Letτ τ^∗ μ, thenμ 0 is the Hopf bifurcation value ofP0. Let p₁t pt−vand still denotep₁tbypt, so thatP0can be written as

dp

dtt v

τ^∗ μ

1−2mpt−1−mpt−1 τ^∗ μ

1−2mp²t−1−mptpt−1

− 1 3v

τ^∗ μ

1−m

p³t−3p²tpt−1 3ptp²t−1−p³t−1

O4.

4.4

Forφ∈C−1,0,R, define Lμ

φ v

τ^∗ μ

1−2mφ0−1−mφ−1

. 4.5

By the Riesz representation theorem, there exists a bounded variation functionηθ, μθ ∈

−1,0such that

L_μ φ

₀

−1

dη θ, μ

φθ, forφ∈C−1,0,R. 4.6

In fact, we can choose

η θ, μ

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ v

τ^∗ μ

1−2m, θ0,

0, θ∈−1,0,

v τ^∗ μ

1−m, θ−1.

4.7

Define

A μ

φθ

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ dφθ

dθ , θ∈−1,0,

₀

−1

dη ξ, μ

φξ, θ0,

h μ, φ

τ^∗ μ

1−2mφ²0−1−mφ0φ−1

−1 3v

τ^∗ μ

1−m

φ³0−3φ²0φ−1 3φ0φ²−1−φ³−1

O4.

4.8 Furthermore, define the operatorRμas

R μ

φθ

⎧⎨

⎩

0, θ∈−1,0,

h μ, φ

, θ0; 4.9

then4.4is equivalent to the following operator equation:

˙ x_tA

μ x_t R

μ

x_t, 4.10

wherextxt θforθ∈−1,0.

Forψ∈C¹0,1,R, define an operator

A^∗ψs

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−dψs

ds , s∈0,1,

₀

−1ψ−ξdηξ,0, s0

4.11

and a bilinear form

ψs, φθ

ψ0φ0− ₀

−1

_θ

ξ0ψξ−θdηθφξdξ, 4.12

whereηθ ηθ,0, thenA0andA^∗are adjoint operators.

From the preceding discussions, we know that±iω0τ^∗ are eigenvalues ofA0and therefore they are also eigenvalues ofA^∗. It is not difficult to verify that the vectorsqθ e^iω0τ∗θθ ∈ −1,0 and q^∗s le^iω0τ∗ss ∈ 0,1 are the eigenvectors of A0 and A^∗ corresponding to the eigenvalues iω0τ^∗and−iω0τ^∗, respectively, where

l

1−vτ^∗1−me^−iω0τ∗₋₁

. 4.13

Following the algorithms stated in Hassard et al.12, we obtain the coefficients which will be used in determining the important quantities:

g20 2lτ^∗

1−2m−1−me^−iω0τ∗ , g11 lτ^∗

21−2m−1−m

e^iω0τ∗ e^−iω0τ∗ , g₀₂ 2lτ^∗

1−2m−1−me^iω0τ∗ , g21 2lτ^∗1−2m2W110 W200

−lτ^∗1−m

2W₁₁−1 W₂₀−1 W₂₀0e^iω0τ∗ 2W₁₁0e^−iω0τ∗

−2lτ^∗v1−m

3−3e^−iω0τ∗−e^iω0τ∗ e^{−2iω0τ∗} ,

4.14

where forθ∈−1,0,

W₂₀θ ig₂₀

ω₀τ^∗qθ ig₀₂

3ω₀τ^∗qθ E₁e^2iω0τ∗θ, W11θ − ig11

ω₀τ^∗qθ ig₁₁

ω₀τ^∗qθ E2,

E₁ 2

1−2m−1−me^−iω0τ∗ 2iω0−v1−2m−1−me^{−2iω0τ∗}, E2 21−2m−1−m

e^iω0τ∗ e^−iω0τ∗

mv .

4.15

Consequently,g₂₁can be expressed explicitly.

Thus, we can compute the following values:

c10 i 2ω0τ^∗

g11g20−2g11²− g₀₂² 3

g21

2 , μ₂−Rec10

Reλτ^∗, β22 Rec10,

T2−Imc10 μ2Imλτ^∗

ω₀τ^∗ ,

4.16

which determine the properties of bifurcating periodic solutions at the critical valueτ^∗. More specifically,μ₂ determines the directions of the Hopf bifurcation: if μ₂ > 0μ2 < 0, then the bifurcating periodic solutions exist for τ > τ^∗ τ < τ^∗;β2 determines the stability of bifurcating periodic solutions: the bifurcating periodic solutions on the center manifold are

stable unstableif β2 < 0 β2 > 0;T2 determines the period of the bifurcating periodic solutions: the period increasesdecreasesifT₂>0 T2<0. Thus Theorem2.4follows.

In particular, whenτ τ₀m. It is well known that the normal form of the restriction ofP0withττ0on the center manifold is given by

zt ˙ iω0τ0z c10z²z · · ·. 4.17 By Liapunov’s second method, the zero solution of4.17is stableunstableif Rec10 <

0>0, and so isv.

5. Proof of Theorem 2.7

Assumem ∈ 0,2/3. It is known thatP undergoes a local Hopf bifurcation at p^∗ v whenτ τ_jj 0,1, . . .. Now we show that each bifurcation branch can be continued with respect to the parameterτunder the conditions in Theorem2.7. We introduce the following notations:

XC−τ,0,R, Σ Cl

y, τ, T

:y is a T−periodic solution of P

⊂X×R ×R ,

N

y, τ, T

:y0 orv .

5.1

Let Cv, τj,2π/ω0 denote the connected component of v, τj,2π/ω0 in Σ, and Proj_τv, τj,2π/ω₀its projection onτ component, whereω₀ v m2−3mand thus±iω0

is a pair of purely imaginary roots of 2.7 with τ τ_j. By Theorem 2.3, we know that Cv, τj,2π/ω0is nonempty.

Lemma 5.1. All bifurcating periodic solutions of Pfrom Hopf bifurcation atvare positive ifτ ∈ Proj_τv, τj,2π/ω₀.

Proof. For each τ ∈ Proj_τv, τj,2π/ω0, denote by Xt, τ the corresponding nontrivial periodic solution, with its minimum value inf_t∈R Xt, τ. From the fact that the periodic solutions bifurcated from the positive equilibrium are continuous with respect to τ and limτ→τjXt, τ v uniformly in t ∈ R, we have that inft∈RXt, τ is continuous with respect to τ, and the bifurcating periodic solutions are positive when τ varies in a small neighborhood of τ_j. To prove the positivity, it is equivalent to prove inf_t∈RXt, τ > 0 when τ ∈ Proj_τv, τj,2π/ω0. Otherwise, there exists a τ^∗ ∈ Proj_τv, τj,2π/ω0such that inf_t∈RXt, τ^∗ 0. Without loss of generality, we assume thatτ^∗ > τ_j and inf_t∈RXt, τ > 0 whenτ ∈τj, τ^∗. By the proof of positivity of solution, we obtainXt, τ^∗≡ 0. This implies that0, τ^∗,2π/ωis a center ofPfor someω >0. This contradiction completes the proof.

Lemma 5.2. Ifm∈3 √

2/7,2/3, thenPhas no positive periodic solution of period 4τ.

Proof. Suppose thatytis a positive periodic solution toPof period 4τ. Set ujt y

t− j−1

τ

, j1,2,3,4. 5.2

Thenut u1t, u2t, u3t, u4tis a periodic solution to the following system of ordinary differential equations:

˙

u₁t 1−mtanhu1−u₂u1−mu1−vu1,

˙

u₂t 1−mtanhu2−u₃u2−mu2−vu2,

˙

u3t 1−mtanhu3−u4u3−mu3−vu3,

˙

u₄t 1−mtanhu4−u₁u4−mu4−vu4,

5.3

where·denotes d/dt. Let G

u∈R:u_j∈

0,1−m

m v , j1,2,3,4

. 5.4

ForP, the ultimate uniform boundedness of solutions implies that the periodic solutions of Pbelong to the regionG. To rule out the 4τ-periodic solution toP, it suffices to prove the nonexistence of nonconstant periodic solutions of5.3in the regionG. To do the latter, we use a general Bendixson’s criterion in higher-dimensions developed by Li and Muldowney 19. More specifically, we shall apply19, Corollary 3.5. The Jacobian matrixJ Ju of 5.3, foru∈R⁴, is

Ju

⎛

⎜⎜

⎜⎜

⎜⎝

A₁ B₁ 0 0 0 A₂ B₂ 0 0 0 A₃ B₃ B4 0 0 A4

⎞

⎟⎟

⎟⎟

⎟⎠. 5.5

Fori1,2,3,4,

A_i 1−m

u_itanh¹ui−u_{i 1} tanhui−u_{i 1}

−2mu_i mv, B_i−1−muitanh¹ui−u_{i 1}<0,

5.6

withu₅u₁. The second additive compound matrixJ²uofJuis

J²u

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎝

A₁ A₂ B₂ 0 0 0 0

0 A₁ A₃ B₃ −B1 0 0

0 0 A₁ A₄ 0 B₁ 0

0 0 0 A2 A3 B3 0

−B4 0 0 0 A2 A4 B2

0 −B4 0 0 0 A3 A4

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎠

. 5.7

Choose a vector norm inR⁶as

|x1, x2, x3, x4, x5, x6| max'√

2|x1|,|x2|,√ 2|x3|,√

2|x4|,|x5|,√ 2|x6|(

. 5.8

Then with respect to this norm, the Lozinskii measureμJ²uof the matrixJ²uissee 22

μ

J²u max

A_i A_j−√

2B_i, A_p A_q−

√2 2

B_p B_q

, 5.9

with i, j ∈ {1,4,2,1,3,2,4,3}, and p, q ∈ {1,3,2,4}. By 19, Corollary 3.5, system 5.3 has no periodic orbits in the regionG ifμJ²u < 0 for all u ∈ G. In fact, whenm∈3 √

2/7,2/3, one can acquire the inequality as below,

μ

J²u

<1−m mv )

2 21−3m √

21−m m

*

<0. 5.10

This completes the proof.

Lemma 5.3. EquationPhas no nonconstant periodic solution of periodτ.

Proof. EquationPhas no nonconstant periodic solution of periodτis equivalent to the fact thatP withτ 0 has no nonconstant periodic solution. It is straightforward that a first order autonomous ODE has no nonconstant periodic solutions.Pwithτ0 is a first-order autonomous ODE, which proves the lemma.

Proof of Theorem2.7. First note that for anyτ_j, the stationary pointsv, τj,2π/ω₀ofPare nonsingular and isolated centerssee Wu13under the assumptionm∈0,2/3; then the hypothesisH2in13 is satisfied. By3.9, there exist > 0, δ > 0 and a smooth curve λ:τj−δ, τ_j δ → C, such that

Δλτ Δ_v,τ,Tλτ 0, |λτ−iω₀|< , 5.11

for allτ∈τj−δ, τj δ, whereΔis defined as2.7, and

λ τ_j

iω₀, dReλτ

dτ |

ττj

>0. 5.12

Denotepj 2π/ω0and let

Ω 0, q

: 0< u < ,q−pj<

. 5.13

Clearly, if|τ−τj| ≤δandu, q∈Ωsuch thatΔv,τ,Tu 2πi/q 0, thenτ τj, u0, and qp_j. Moreover, putting

H^±

v, τj,2π ω₀

u, q

Δv,τj±δ,T

u i2π

q

, 5.14

we have the crossing number

γ₁

v, τ_j,2π ω0

deg_B

H⁻

v, τ_j,2π ω0

,Ω

−deg_B

H

v, τ_j,2π ω0

,Ω

−1. 5.15

By the local Hopf bifurcation theorem for FDE 10, we conclude that the connected componentCv, τj,2π/ω₀throughv, τj,2π/ω₀inΣis nonempty. Meanwhile, we have

+ ^{y,τ,T ∈Cv,τj,2π/ω0}

γ1

y, τ, T

<0 5.16

and thusCv, τj,2π/ω0is unbounded by the global Hopf bifurcation theorem given by Wu 13.

Again, the property of ultimate uniform boundedness implies that the projection of Cv, τj,2π/ω0 onto the y-space is bounded. Meanwhile, P with τ 0 has no nonconstant periodic solutions since it is a first-order autonomous ordinary differential equation. Therefore, Proj_τv, τj,2π/ω₀is bounded below.

By the definition ofτjgiven in2.10, we know that, whenm∈0,2/3, 2π < τ_jω₀<2

j 1

π, j ≥1, 5.17

which implies that

τ_j j 1 < 2π

ω₀ < τ_j. 5.18

By Lemma5.3, we have τ/j 1 < T < τ ifx, τ, T ∈ Cv, τj,2π/ω₀forj ≥ 1. Because Cv, τj,2π/ω₀ is unbounded, Proj_τv, τj,2π/ω₀ must be unbounded. Consequently, Proj_τv, τj,2π/ω0includeτj,∞forj≥1. The former part of the theorem follows.

Moreover, ifm∈3 √

2/7,2/3, then π

2 < τ0ω0< π, 5.19

where3.6is used. This implies that

2τ0< 2π

ω₀ <4τ0. 5.20

Thus, we have by Lemma 5.2 that 2τ < T < 4τ if x, τ, T ∈ Cv, τ0,2π/ω0. Similarly, this shows that in order for Cv, τ0,2π/ω₀to be unbounded, Proj_τv, τ0,2π/ω₀must be unbounded. This implies that Proj_τv, τ0,2π/ω₀includesτ0,∞.

In addition, the first global Hopf branch contains periodic solutions with period between 2τand 4τ, which are the slowly oscillating periodic solutions. Thejth branches, for j ≥ 1, since the periods are less thanτ, contain fast-oscillating periodic solutions. Therefore, Phas at least two periodic solutions forτ > τ0providedm∈3 2/7,2/3. The proof of Theorem2.7is complete.

Acknowledgments

This research is supported by the NNSFno. 10771045of China, the Program of Excellent Team in HIT and the Natural Science Foundation of Heilongjiang Province,A200806. The authors are grateful to Professor Xuezhong He and the anonymous referees for their helpful comments and valuable suggestions.

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