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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 290497, 6 pages
http://dx.doi.org/10.1155/2013/290497
Research Article

Hopf Bifurcation Analysis for the Modified Rayleigh Price Model with Time Delay

School of Science, Tianjin Polytechnic University, Tianjin 300387, China

Received 9 May 2013; Accepted 28 June 2013

Academic Editor: Sakthivel Rathinasamy

Copyright © 2013 Yanhui Zhai et al. 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.

Abstract

This paper mainly modifies and further develops the Reyleigh price model. By modifying the basic Reyleigh model, we can more accurately illustrate the economic phenomena with price varying. First, we research the dynamics of the modified Reyleigh model with time delay. By employing the normal form theory and center manifold theory, we obtain some testable results on these issues. The conclusion confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. Finally, some numerical simulations are given to illustrate the effectiveness of our results.

1. Introduction

With the rapid development of economic society. As one of the many important economic problems, the price oscillation has been widely accepted by many people, especially some researchers. Since the price is a main factor of affecting supply and demand, many researchers have been devoted to study the price model. Different researchers may apply different price models to solve the practical problems. In this paper, our research is based on the traditional Reyleigh price model [1]. As for this model, many researchers have widely studied it by using different methods and proposed some new ideals. However, there is only a limited number of analytical works on the model with time delay. Although in [1] Lv and Liu have studied the Reyleigh model with time delay, they consider the situation that supply depends on the price of the past only. In order to finely interpret economic phenomena, our research is based on the fact that supply depends not only on the price of the past but also on the present price. So, the main purpose of this study is to provide an insight into these unexplored aspects of the Reyleigh price model with time delay.

The traditional Reyleigh price model is described by the following two-dimensional autonomous system: where is the price at time , is the amount of supply at time , and , , , are the constants.

By introducing the time delay, the above system (1) can be transformed into the following form: where is positive and the other parameters are the same as (1).

In [1], by employing the partitioning approach, Lv and Liu have systematically discussed some complex dynamic behaviors of system (2).

Now, based on the economic meaning and the fact discussed at the beginning of the introduction, we modify system (2) as follows: where , refer to [2].

2. The Stability Analysis

In this section, we obtain the domain of the stable equilibrium when time delay varies from small to large. And further, by applying the Hopf bifurcation theorem, we give the condition of the Hopf bifurcation.

It is known that, on the one hand, if the equilibrium of system (3) is stable when and the characteristic equation of (3) has no purely imaginary roots for any , it is also stable for any . On the other hand, if the equilibrium of system (3) is stable when and there exist some positive values such that the characteristic equation of (3) has a pair of purely imaginary roots, there exists a domain concerning such that the equilibrium of system (3) is stable in the domain.

Obviously, system (3) has the only equilibrium . And the linearization of system (3) at is whose characteristic equation is

When the case , we have the following.

Lemma 1. (i) When , the equilibrium of system (3) is stable.
(ii) When , the equilibrium of system (3) is unstable.
(iii) When and ( ), the equilibrium of system (3) is stable (unstable).

The proof is straightforward, and we omit it.

Lemma 2. If is satisfied, the characteristic equation (5) has a pair of purely imaginary roots when , where

Proof. Let ( ) is a root of (5). Then
The separation of the real and imaginary parts yields which lead to
By solving the second-degree equation (9) concerning , we have
We take
And, hence
By setting , . Then , satisfies the condition of Lemma 2.
The proof is completed.

According to Lemmas 1 and 2 and the assertion at the beginning of this section, we know there must exist some finite interval with regard to in which the equilibrium is stable.

Now we investigate how the real part of the roots of (5) varies as varies in a small neighbourhood of .

Assume that is the root of the characteristic equation (5), and it meets , . By differentiating both sides of (5) with regard to and solving , we obtain

That is,

We substitute (8) into the above equation and separate the real and imaginary parts, and we have

When ,

Here, we know that the root of (5) crosses the imaginary axis from the left to the right as continuously varies from a number less than to one greater than . Thus, when , the characteristic equation (5) has at least one root with positive real part. Further, the equilibrium is unstable in the interval .

By applying Lemmas 1 and 2 and condition , we have the following results.

Theorem 3. If , and hold, the equilibrium of system (3) is asymptotically stable for and unstable for . System (3) exhibits the Hopf bifurcation at the equilibrium for , .

3. Hopf Bifurcation Analysis

In Section 2, we obtain the conditions under which family periodic solutions bifurcate from the steady state at the critical value of . In this section, by applying the normal form and centre manifold theory, we discuss the direction and stability of the bifurcating periodic solutions. Throughout this section, we always assume that system (3) meets the conditions of the Hopf bifurcation.

By time scaling , system (3) is transformed into the following form:

It is not difficult to show that system (16) also exhibits the Hopf bifurcation at the equilibrium for ( ). Without loss of generality, we only consider the bifurcation parameter . For convenience, by setting , system (16) is rewriten as

Clearly, is the Hopf bifurcation value of system (17).

Throughout the following section, is a phase space and the superscripts “ ” and “ ” stand for the transpose and adjoint, respectively.

In , system (17) can be written as the following equation: where is a vector , for , and , are given by for .

Now, we consider the abstract functional differential equation [3]: where for .

The operators and are defined as where (here, is a bounded variation function for ), .

Consider the adjoint bilinear form on : where .

According to the adjoint bilinear form , we can define an adjoint operator corresponding to as the following form:

To determine the normal form of operator , we need to calculate the eigenvectors and of and corresponding to and , respectively.

Proposition 4. Assume that and are the eigenvector and of and corresponding to and , respectively, satisfying and .
Then where .

Proof. Without loss of generality, we just consider the eigenvector .
Firstly, when , by the definition of and , we obtain the form (here, , are unknown parameters).
In what follows, notice that and , and we have , .
Finally, by , we obtain the parameter (refer to [4, 5]).
The proof is completed.

As in [6], the bifurcating periodic solutions of system (16) are indexed by a small parameter . A solution has amplitude , period , and nonzero Floquet exponent with . Under the present assumptions, , , and have expansions [7, 8]: where the sign of determines the directions of the Hopf bifurcations, the sign of determines the stability of the bifurcation periodic solutions, and determines the period of the bifurcating periodic solutions.

The purpose of this section is to compute the coefficients , , in the above expansions.

Next, we construct the coordinates of the center manifold at . Let

On the center manifold , we have where

and are local coordinates for the center manifold in the direction of and , respectively. Since , we have where

We rewrite this as with

Proposition 5. For (31), one has where , , , .

Proof. (i) Noticing and , we have
By (32), we obtain
Substituting (34) into the above equation and comparing the coefficients with (32), we obtain the results.
(ii) The detail procedure of proof refers to [5] and [911].
This completes the proof.

According to Proposition 5, we can compute the following parameters:

From the discussion in Section 2 we know that . We therefore have the following result.

Theorem 6. If , the direction of the Hopf bifurcation of the system (1) at the equilibrium when is supercritical (subcritical) and the bifurcating periodic solutions are orbitally asymptotically stable (unstable).

Finally, we give a concrete example to illustrate the dynamics behaviour of the Raleigh model.

We take the coefficients , , , in (3). Omitting these complicated expressions, we obtain the numerical results directly by means of the software MatLab: , , . So, we directly compete , , .

According to Theorem 6, . That is, the bifurcating periodic solutions of system (3) with the above coefficients are supercritical and orbitally asymptotically stable at .

Thus, the conclusion confirms the effectiveness of our research results.

4. Conclusion

Firstly, under the condition of , we discuss the Reyleigh price model. We know that the stability of price varies with the parameters changing. When , and , , , the price tends to the stability. The other cases are unstable. morever, we discuss the Reyleigh price model with time delay (3). By adjusting the parameters , , we more easily control the price such that the price tends to our expected results. For example, when we take , , , , , , in the system (3), the equilibrium of system (3) is stable (see Figure 1). By contrast, when we take , , , , , , in the system (3), the is unstable and there occurs a periodic solution around (0.0) (see Figures 2 and 3). So, by shortening the time delay between the supply and the demand, we can keep the price stable. On the contrary, the price is unstable and undergoes a periodic oscillation. However our analysis indicates that the dynamics of the Reyleigh price model with time delay can be much more complicated than we may have expected. It is still interesting and inspiring to research the price.

290497.fig.001
Figure 1: The equilibrium of system (3) is stable with .
290497.fig.002
Figure 2: The equilibrium of system (3) is unstable with .
290497.fig.003
Figure 3: When , the periodic solutions occur from the equilibrium .

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