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.. The dire
Trang 1Volume 2010, Article ID 432821, 17 pages
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,weijj@hit.edu.cn
Received 7 October 2009; Accepted 13 January 2010
Academic Editor: Xuezhong He
Copyrightq 2010 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 equilibrium v, 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 see 1 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 asset p and the fundamental price of the asset v The chartists,
on the other hand, base their decisions of market participation on the price trend of the asset
Trang 2They 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
dt t 1 − mtanhp t − pt − τp t − mp t − vp t, P
where m ∈ 0, 1 is 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 for P, 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 delay τ and the market fraction of the fundamentalists m 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 for τ far away from the
Hopf bifurcation values using a global Hopf bifurcation result of Wusee 13 and also 14–
18 A key step in establishing the global extension of the local Hopf branch at the first
critical value τ τ0is to verify thatP has 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 Muldowney 19
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
EquationP admits a unique solution Hale and Lunel 10 Any solution pt, ϕ of P is
nonnegative if and only if ϕ0 > 0, following the fact that
p t ϕ0et01−mtanhps−ps−τ−mps−vds 2.2 Note that
dp
Trang 3We conclude that, for any sufficiently small > 0,
p t < 1− m
holds for all large t > 0 This establishes the ultimate uniform boundedness of solutions for
P
EquationP has two equilibria 0 and v Denote by p∗any of the equilibria Then the linearization ofP at p∗is given by
dp
dt t 1 − 3mp∗ mvp t − 1 − mp∗p t − τ. 2.5
If p∗ 0, then it is unstable since 2.5 becomes
dp
If p∗ v, then the characteristic equation of 2.5 is
We investigate the dependence of local dynamics ofP on parameters m and τ.
Case 1 Choosing m as parameter.
Define
where η ksolves
with the assumption of vτ > 1.
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 Assume vτ > 1 Then
i the positive equilibrium p∗ v of P is asymptotically stable when m ∈ m0, 1 , where
m0: max{mk };
ii P undergoes a Hopf bifurcation at v when m m k
Trang 4Case 2 Regarding τ as parameter.
Define
τ j 1
ω0
arccos1− 2m
1− m 2jπ
with ω0 : v m 2 − 3m.
i if m ∈ 2/3, 1, then p∗ v is asymptotically stable for all τ ∈ R ;
ii if m ∈ 0, 2/3, then p∗ v is asymptotically stable when 0 < τ < τ0and becomes unstable when τ > τ0; moreover,P undergoes a Hopf bifurcation at p∗ v when τ τ j j
0, 1, 2, .
Theorem 2.4gives the direction of Hopf bifurcation and stability of the bifurcating
periodic solutions Similar results hold if we choose m as parameter.
at v 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 Re c10 < 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, , τ j m, are Hopf bifurcation curves When
m, τ ∈ D : m, τ | 0 < m < 2
3, 0 ≤ τ < τ0m
∪ m, τ | 2
3 ≤ m < 1
p∗ v is asymptotically stable, and when
m, τ ∈ D c: m, τ | 0 < m < 2
3, τ > τ0m
p∗ v is unstable Denote the curve τ τ0m by l Clearly, l lies in the boundary of D The stability of v when m, τ ∈ l is 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:
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
Trang 50 20 40 60
τ
m
Stable Hopf
bifurcation curves
τ0m
τ1m
τ2m
.
τ jm
Unstable
Figure 1: Hopf bifurcation set on m − τ plane.
Finally, a natural question is that if the bifurcating periodic solutions ofP exist when
τ is 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
Furthermore, if m∈ 3 √2/7, 2/3, then P has at least one periodic solution for τ > τ j j ≥ 0
and at least two periodic solutions for τ > τ j j ≥ 1 Here, τ j j 0, 1, 2, is defined in 2.10.
Figures2 and3 describe the phenomena stated in Theorem 2.7, where m 0.65 ∈
3 √2/7, 2/3 and v 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 and c10 −0.856 under these parameters.
Therefore, combining Theorem2.7with the two figures, we know that there exists at least one stable periodic solution ofP with period Tτ ≥ 2π/ω 6.9706 when τ > 2.885 We would
like to mention that the graphs in Figure2are prepared by using DDE-BIFTOOL developed
by Engelborghs et al.20,21
Case 1 Choosing m as parameter.
When m 1, P is a scalar ordinary differential equation and therefore it has no nonconstant periodic solutions Additionally, the only root of 2.7 is λ −v < 0 These
complete the proof of Proposition2.1
When m < 1, iη η > 0 is a root of 2.7 if and only if η satisfies
sin ητ 1 − mv η , cos ητ 1− 2m
It follows that
2− cos ητη − v sin ητ 0,
2− cos ητ .
3.2
Trang 60 τ0 τ1 τ2 τ3 τ4 · · · τ j · · ·
a
τ0 τ1 τ2 τ3 τ4 · · · τ j · · ·
2π/ω0
0
b
max pt − min pt; b on the τ, Tτ-plane, where Tτ is the period of periodic solution bifurcated at
v.
Denote Gηdef 2 − cos ητη − v sin ητ Then G0 0, G∞ ∞, and
G
η
Therefore, there exists at least one zero η k of Gη if vτ > 1.
Define m kas in2.8, then m k ∈ 0, 2/3 and ±iη kis a pair of purely imaginary roots
of2.7 with m m k Let λm be the root of 2.7 satisfying λm k iη k Substituting λm
into2.7, it follows that
dλ dm
ve −λτ − 2v
Thus, we have the transversal condition
Sign
dm
|
m m k
Sign cos η k τ − 2 − τv1 − m k1− 2 cos η k τ
Sign
3τv1 − m k2− 2τv1 − m k − 1
1− m k
,
⎧
⎪
⎪
⎪
⎪
⎪
⎪
0,1
3
2−
1 3
τv
,
1 3
2−
1 3
τv
,2
3
,
3.5
where cos η k τ 2 − 1/1 − m k is used Accordingly, a Hopf bifurcation at p∗ v occurs when
m m k
Trang 75
5.4
t
a
4.6
5
5.4
t
b
4.6
5
5.4
t
c
4.6
5
5.4
t
d
τ 20, and d with τ 50.
In summary, if vτ > 1, then there exists at least one value of m defined by2.8 such that all the roots of2.7 have negative real parts when m ∈ m0, 1, in addition, a pair of
purely imaginary roots when m m k This implies Theorem2.2
Case 2 Regarding τ as a parameter.
First of all, we know that the root of 2.7 with τ 0 satisfies that λ −mv < 0 Therefore, p∗ v is globally asymptotically stable when τ 0.
Let iωω > 0 be a root of 2.7 Then
sin ωτ 1 − mv ω > 0, cos ωτ 1− 2m
This leads to ω2 mv22 − 3m ω0 v m 2 − 3m makes sense when m ∈ 0, 2/3 Define
τ jas in2.10; then iω0is a pure imaginary root of2.7 with τ τ j , j 0, 1, 2,
α
τ j
τ j
Trang 8
Differentiating both sides of 2.7 gives
dλ dτ
−1
λv 1 − m −
τ
Therefore,
Re
dλ dτ
−1
|
τ τ j
sin ω0τ j
This implies that α τ j > 0, j 0, 1, 2,
Note that λ 0 is not a root of 2.7 when m 2/3 Therefore, we obtain the following
spectral properties of2.7:
i if m ∈ 2/3, 1, then all roots of 2.7 have negative real parts;
ii if m ∈ 0, 2/3, then there exist a sequence values of τ defined by 2.10 such that
2.7 has a pair of purely imaginary roots ±iω0when τ τ j Additionally, all roots
of2.7 have negative real parts when τ ∈ 0, τ0, all roots of 2.7, except ±iω0, have
negative real parts when τ τ0, and2.7 has at least a pair of roots with positive
real parts when τ > τ0
These spectral properties immediately lead to the dynamics near the positive
equilibrium v described in Theorem2.3
We use the center manifold and normal form theories presented in Hassard et al 12 to establish the proof of Theorem2.4 Normalizing the delay τ by the time scaling t
using the change of variables pt ptτ, we transform P into
dp
dt t 1 − mτ tanhp t − pt − 1p t − mτp t − vp t, P0
whose characteristic equation at p v is
Comparing 4.1 with 2.7, we see z λτ for τ / 0 Therefore, we obtain the following
Lemma
Lemma 4.1 Assume m ∈ 0, 2/3.
i If τ τ j j 0, 1, 2, , then 4.1 has a pair of purely imaginary roots ±iω0τ j , where τ j
and ω0are defined as before.
Trang 9ii Let zτ γτ iζτ be the root of 4.1, satisfying
γ
τ j
0 and ζτ j
then
γ
τ j
τ j α
τ j
> 0, j 0, 1, 2, 4.3
iii All roots of 4.1 have negative real parts when τ ∈ 0, τ0, all roots of 4.1, except ±iω0τ0, have negative real parts when τ τ0, and4.1 has at least a pair of roots with positive real
parts when τ > τ0.
Without loss of generality, we denote the critical value τ∗ at whichP0 undergoes a
Hopf bifurcation at v Let τ τ∗ μ, then μ 0 is the Hopf bifurcation value of P0 Let
p1t pt − v and still denote p1t by pt, so that P0 can be written as
dp
dt t vτ∗ μ1 − 2mpt − 1 − mpt − 1
τ∗ μ1 − 2mp2t − 1 − mptpt − 1
− 1
3v
τ∗ μ1 − mp3t − 3p2tpt − 1 3ptp2t − 1 − p3t − 1 O4.
4.4
For φ ∈ C−1, 0, R, define
L μ
φ
vτ∗ μ1 − 2mφ0 − 1 − mφ−1. 4.5
By the Riesz representation theorem, there exists a bounded variation function ηθ, μθ ∈
−1, 0 such that
L μ
φ
0
−1
dη
θ, μ
In fact, we can choose
η
θ, μ
⎧
⎪
⎨
⎪
⎩
v
τ∗ μ1 − 2m, θ 0,
v
τ∗ μ1 − m, θ −1.
4.7
Trang 10A
μ
φ θ
⎧
⎪
⎨
⎪
⎩
dφθ
0
−1
dη
ξ, μ
φ ξ, θ 0,
h
μ, φ
τ∗ μ1 − 2mφ20 − 1 − mφ0φ−1
−1
3v
τ∗ μ1 − mφ30 − 3φ20φ−1 3φ0φ2−1 − φ3−1 O4.
4.8
Furthermore, define the operator Rμ as
R
μ
φ θ
⎧
⎨
⎩
h
μ, φ
then4.4 is equivalent to the following operator equation:
˙x t Aμ
x t Rμ
where x t xt θ for θ ∈ −1, 0.
For ψ ∈ C10, 1, R, define an operator
A∗ψ s
⎧
⎪
⎨
⎪
⎩
−dψs
0
−1ψ −ξdηξ, 0, s 0
4.11
and a bilinear form
ψ s, φθ ψ0φ0 −
0
−1
θ
where ηθ ηθ, 0, then A0 and A∗are adjoint operators
From the preceding discussions, we know that±iω0τ∗ are eigenvalues of A0 and therefore they are also eigenvalues of A∗ It is not difficult to verify that the vectors qθ
e iω0τ∗θ θ ∈ −1, 0 and q∗s le iω0τ∗s s ∈ 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
... such that2.7 has a pair of purely imaginary roots ±iω0when τ τ j Additionally, all roots
of 2.7 have negative real parts when... 0, τ0, all roots of 2.7, except ±iω0, have
negative real parts when τ τ0, and2.7 has at least a pair of roots with... class="text_page_counter">Trang 7
5
5.4
t
a
4.6