#### Abstract

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 , 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 and the fundamental price of the asset . 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. In [4], the model describing the time evolution of the asset price is given by

where 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.

In [4], 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 for (P), which depends both on time delay and the market fraction of the fundamentalists . Using the normal form theory and center manifold theorem [11, 12], we derive sufficient conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Furthermore, taking as a parameter, we establish the existence of periodic solutions for far away from the Hopf bifurcation values using a global Hopf bifurcation result of Wu (see [13] and also [14–18]). A key step in establishing the global extension of the local Hopf branch at the first critical value is to verify that (P) has no nonconstant periodic solutions of period . 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 Section 2, our main results are stated and some numerical simulations are carried out to illustrate the analytic results. In Section 3, results on stability and bifurcations at positive equilibrium are proved, taking and 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

Equation (P) admits a unique solution (Hale and Lunel [10]).Any solution of (P) is nonnegative if and only if , following the fact that

Note that

We conclude that, for any sufficiently small ,

holds for all large . This establishes the ultimate uniform boundedness of solutions for (P).

Equation (P) has two equilibria and . Denote by any of the equilibria. Then the linearization of (P) at is given by

If , then it is unstable since (2.5) becomes

If , then the characteristic equation of (2.5) is

We investigate the dependence of local dynamics of (P) on parameters and .

*Case 1. *Choosing as parameter.

Define
where solves
with the assumption of .

Proposition 2.1. *If , then is globally stable.*

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

Theorem 2.2. *Assume . Then*(i)*the positive equilibrium of (P) is asymptotically stable when , where ;*(ii)*(P) undergoes a Hopf bifurcation at when .*

*Case 2. *Regarding as parameter.

Define
with .

Theorem 2.3. *For (P),*(i)*if , then is asymptotically stable for all ;*(ii)*if , then is asymptotically stable when and becomes unstable when ; moreover, (P) undergoes a Hopf bifurcation at when *

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

Theorem 2.4. *Assume . If , then the bifurcating periodic solutions at 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 is stable (unstable) in the phase space if .*

Corollary 2.5. *When , is stable (unstable) if .*

*Remark 2.6. *See the proof in Section 4 for the explanation of which appears in Theorem 2.4 and Corollary 2.5.

The conclusions above are illustrated in Figure 1, using bifurcation set on the -plane. Here, are Hopf bifurcation curves. When

is asymptotically stable, and when

is unstable. Denote the curve by . Clearly, lies in the boundary of . The stability of when is given by Corollary 2.5.

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 in [4, Figure ]:

we compute by (2.10) that . Therefore, Theorem 2.3 provides insight on the explanation of existence of a limit cycle in [4, Figure ].

Finally, a natural question is that if the bifurcating periodic solutions of (P) exist when is far away from critical values? In the following theorem, we obtain the global continuation of periodic solutions bifurcating from the points (, ) , using a global Hopf bifurcation theorem given by Wu [13].

Theorem 2.7. *If , then for , (P) has at least one periodic solution. Furthermore, if , then (P) has at least one periodic solution for and at least two periodic solutions for . Here, is defined in (2.10).*

Figures 2 and 3 describe the phenomena stated in Theorem 2.7, where and . 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, and under these parameters. Therefore, combining Theorem 2.7 with the two figures, we know that there exists at least one stable periodic solution of (P) with period when . We would like to mention that the graphs in Figure 2 are prepared by using DDE-BIFTOOL developed by Engelborghs et al. [20, 21].

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#### 3. Proof of the Results in Cases 1 and 2

*Case 1. *Choosing as parameter.

When , (P) is a scalar ordinary differential equation and therefore it has no nonconstant periodic solutions. Additionally, the only root of (2.7) is . These complete the proof of Proposition 2.1.

When , is a root of (2.7) if and only if satisfies
It follows that
Denote . Then , and
Therefore, there exists at least one zero of if .

Define as in (2.8), then and is a pair of purely imaginary roots of (2.7) with . Let be the root of (2.7) satisfying . Substituting into (2.7), it follows that
Thus, we have the transversal condition
where is used. Accordingly, a Hopf bifurcation at occurs when .

In summary, if , then there exists at least one value of defined by (2.8) such that all the roots of (2.7) have negative real parts when , in addition, a pair of purely imaginary roots when . This implies Theorem 2.2.

*Case 2. *Regarding as a parameter.

First of all, we know that the root of (2.7) with satisfies that . Therefore, is globally asymptotically stable when .

Let be a root of (2.7). Then
This leads to . makes sense when . Define as in (2.10); then is a pure imaginary root of (2.7) with ….

Similarly, let be the root of (2.7), satisfying
Differentiating both sides of (2.7) gives

Therefore,
This implies that , ….

Note that is not a root of (2.7) when . Therefore, we obtain the following spectral properties of (2.7):

(i)if , then all roots of (2.7) have negative real parts;(ii)if , then there exist a sequence values of defined by (2.10) such that (2.7) has a pair of purely imaginary roots when . Additionally, all roots of (2.7) have negative real parts when , all roots of (2.7), except , have negative real parts when , and (2.7) has at least a pair of roots with positive real parts when .These spectral properties immediately lead to the dynamics near the positive equilibrium described in Theorem 2.3.

#### 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 Theorem 2.4. Normalizing the delay by the time scaling and using the change of variables , we transform (P) into

whose characteristic equation at is

Comparing (4.1) with (2.7), we see for . Therefore, we obtain the following Lemma.

Lemma 4.1. *Assume .*(i)*If , then (4.1) has a pair of purely imaginary roots , where and are defined as before.*(ii)*Let be the root of (4.1), satisfying
then
*(iii)*All roots of (4.1) have negative real parts when , all roots of (4.1), except , have negative real parts when , and (4.1) has at least a pair of roots with positive real parts when .*

Without loss of generality, we denote the critical value at which (P_{0}) undergoes a Hopf bifurcation at . Let , then is the Hopf bifurcation value of (P_{0}). Let and still denote by , so that (P_{0}) can be written as

For , define

By the Riesz representation theorem, there exists a bounded variation function ) such that

In fact, we can choose

Define

Furthermore, define the operator as

then (4.4) is equivalent to the following operator equation:

where for

For , define an operator

and a bilinear form

where , then and are adjoint operators.

From the preceding discussions, we know that are eigenvalues of and therefore they are also eigenvalues of . It is not difficult to verify that the vectors and are the eigenvectors of and corresponding to the eigenvalues and , respectively, where

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

where for ,

Consequently, can be expressed explicitly.

Thus, we can compute the following values:

which determine the properties of bifurcating periodic solutions at the critical value . More specifically, determines the directions of the Hopf bifurcation: if , then the bifurcating periodic solutions exist for ; determines the stability of bifurcating periodic solutions: the bifurcating periodic solutions on the center manifold are stable (unstable) if ; determines the period of the bifurcating periodic solutions: the period increases (decreases) if . Thus Theorem 2.4 follows.

In particular, when . It is well known that the normal form of the restriction of (P_{0}) with on the center manifold is given by

By Liapunov's second method, the zero solution of (4.17) is stable (unstable) if , and so is .

#### 5. Proof of Theorem 2.7

Assume . It is known that (P) undergoes a local Hopf bifurcation at when . Now we show that each bifurcation branch can be continued with respect to the parameter under the conditions in Theorem 2.7. We introduce the following notations:

Let denote the connected component of in , and its projection on component, where and thus is a pair of purely imaginary roots of (2.7) with . By Theorem 2.3, we know that is nonempty.

Lemma 5.1. *All bifurcating periodic solutions of (P) from Hopf bifurcation at are positive if .*

*Proof. *For each , denote by the corresponding nontrivial periodic solution, with its minimum value . From the fact that the periodic solutions bifurcated from the positive equilibrium are continuous with respect to and uniformly in , we have that is continuous with respect to , and the bifurcating periodic solutions are positive when varies in a small neighborhood of . To prove the positivity, it is equivalent to prove when . Otherwise, there exists a such that . Without loss of generality, we assume that and when . By the proof of positivity of solution, we obtain . This implies that is a center of (P) for some . This contradiction completes the proof.

Lemma 5.2. *If , then (P) has no positive periodic solution of period .*

*Proof. *Suppose that is a positive periodic solution to (P) of period . Set
Then is a periodic solution to the following system of ordinary differential equations:
where denotes . Let
For (P), the ultimate uniform boundedness of solutions implies that the periodic solutions of (P) belong to the region . To rule out the -periodic solution to (P), it suffices to prove the nonexistence of nonconstant periodic solutions of (5.3) in the region . To do the latter, we use a general Bendixson's criterion in higher-dimensions developed by Li and Muldowney [19]. More specifically, we shall apply [19, Corollary ]. The Jacobian matrix of (5.3), for , is
For ,
with . The second additive compound matrix of is

Choose a vector norm in as
Then with respect to this norm, the Lozinskii measure of the matrix is (see [22])
with , and . By [19, Corollary ], system (5.3) has no periodic orbits in the region if for all . In fact, when , one can acquire the inequality as below,
This completes the proof.

Lemma 5.3. *Equation (P) has no nonconstant periodic solution of period .*

*Proof. *Equation (P) has no nonconstant periodic solution of period is equivalent to the fact that (P) with has no nonconstant periodic solution. It is straightforward that a first order autonomous ODE has no nonconstant periodic solutions. (P) with is a first-order autonomous ODE, which proves the lemma.

*Proof of Theorem 2.7. *First note that for any , the stationary points of (P) are nonsingular and isolated centers (see Wu [13]) under the assumption ; then the hypothesis () in [13] is satisfied. By (3.9), there exist and a smooth curve , such that
for all , where is defined as (2.7), and
Denote and let
Clearly, if and such that , then , and . Moreover, putting
we have the crossing number
By the local Hopf bifurcation theorem for FDE [10], we conclude that the connected component through in is nonempty. Meanwhile, we have
and thus is unbounded by the global Hopf bifurcation theorem given by Wu [13].

Again, the property of ultimate uniform boundedness implies that the projection of onto the is bounded. Meanwhile, (P) with = 0 has no nonconstant periodic solutions since it is a first-order autonomous ordinary differential equation. Therefore, is bounded below.

By the definition of given in (2.10), we know that, when ,
which implies that
By Lemma 5.3, we have if for . Because is unbounded, must be unbounded. Consequently, include for . The former part of the theorem follows.

Moreover, if , then
where (3.6) is used. This implies that
Thus, we have by Lemma 5.2 that if . Similarly, this shows that in order for to be unbounded, must be unbounded. This implies that includes .

In addition, the first global Hopf branch contains periodic solutions with period between and , which are the slowly oscillating periodic solutions. The th branches, for , since the periods are less than , contain fast-oscillating periodic solutions. Therefore, (P) has at least two periodic solutions for provided . The proof of Theorem 2.7 is complete.

#### Acknowledgments

This research is supported by the NNSF (no. 10771045) of 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.