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

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: 𝑥̇(𝑡)(𝑡)1=−𝑦+𝑙3𝑎𝑥³(𝑡)+12𝑏𝑥²(𝑡)(𝑡,𝑦̇(𝑡)(𝑡),+𝑐𝑥)=𝑥 (1) where 𝑥(𝑡) is the price at time 𝑡, 𝑦(𝑡) is the amount of supply at time 𝑡, and 𝑎, 𝑏, 𝑐, 𝑙 are the constants.

By introducing the time delay, the above system (1) can be transformed into the following form:𝑥̇(𝑡)(𝑡)1=−𝑦+𝑙3𝑎𝑥³(𝑡)+12𝑏𝑥²(𝑡)(𝑡,𝑦̇(𝑡)(),+𝑐𝑥)=𝑥𝑡−𝜏 (2) where 𝜏 is positive and the other parameters are the same as (1).

In [1], by employing the 𝜏−𝐷 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: 𝑥̇(𝑡)(𝑡)1=−𝑦+𝑙3𝑎𝑥³(𝑡)+12𝑏𝑥²(𝑡)(𝑡,𝑦̇(𝑡)=1+𝑐𝑥)2(𝑡)+1𝑑𝑥2𝑘𝑥(𝑡−𝜏), (3) where 𝑑, 𝑘 refer to [2].

2. The Stability Analysis

In this section, we obtain the domain of the stable equilibrium when time delay 𝜏 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 𝜏=0 and the characteristic equation of (3) has no purely imaginary roots for any 𝜏>0, it is also stable for any 𝜏>0. On the other hand, if the equilibrium of system (3) is stable when 𝜏=0 and there exist some positive values 𝜏 such that the characteristic equation of (3) has a pair of purely imaginary roots, there exists a domain concerning 𝜏 such that the equilibrium of system (3) is stable in the domain.

Obviously, system (3) has the only equilibrium (0,0). And the linearization of system (3) at (0,0) is 𝑥̇(𝑡)(𝑡)(𝑡),𝑦̇(𝑡)=1=−𝑦+𝑙𝑐𝑥2(𝑡)+1𝑑𝑥2𝑘𝑥(𝑡−𝜏), (4) whose characteristic equation is 𝜆²1−𝑙𝑐𝜆+2𝑘𝑒^−𝜆𝜏+12𝑑=0. (5)

When the case 𝜏=0, we have the following.

Lemma 1. (i) When 𝑐<0, the equilibrium (0,0) of system (3) is stable.
(ii) When 𝑐>0, the equilibrium (0,0) of system (3) is unstable.
(iii) When 𝑐=0 and 𝑏<0 (𝑏>0), the equilibrium (0,0) of system (3) is stable (unstable).

The proof is straightforward, and we omit it.

Lemma 2. If 𝑑²<𝑘² is satisfied, the characteristic equation (5) has a pair of purely imaginary roots ±𝑖𝜔₀ when 𝜏=𝜏_𝑗, where 𝜔₀=−𝑙²𝑐²−𝑑+(𝑙²𝑐²−𝑑)²−(𝑑²−𝑘²)2^1/2,𝜏_𝑗=arccos2𝜔₀−𝑑/𝑘+2𝑗𝜋𝜔₀,𝑗=0,1,2,…. (6)

Proof. Let 𝑖𝜔 (𝜔>0) is a root of (5). Then −𝜔²1−𝑙𝑐𝜔𝑖+21𝑑+2𝑘(cos(𝜔𝜏)−𝑖sin(𝜔𝜏))=0. (7)
The separation of the real and imaginary parts yields −𝜔²+12()+1𝑘cos𝜔𝜏21𝑑=0,−𝑙𝑐𝜔−2𝑘sin(𝜔𝜏)=0, (8) which lead to 𝜔⁴−𝑑𝜔²+𝑙²𝑐²𝜔²+14𝑑²−14𝑘²=0. (9)
By solving the second-degree equation (9) concerning 𝜔², we have 𝜔²=−𝑙²𝑐²−𝑑±𝑙²𝑐²−𝑑²−𝑑²−𝑘²2. (10)
We take 𝜔₀=−𝑙²𝑐²−𝑑+(𝑙²𝑐²−𝑑)²−(𝑑²−𝑘²)2^1/2. (11)
And, hence 𝜏₀=arccos2𝜔²₀−𝑑/𝑘𝜔₀. (12)
By setting 𝜏_𝑗=𝜏₀+(2𝑗𝜋/𝜔₀), 𝑗=0,1,2,…. Then 𝜔₀, 𝜏_𝑗 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 𝜏 in which the equilibrium (0,0) is stable.

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

Assume that 𝜆(𝜏)=𝑎(𝜏)+𝑖𝑏(𝜏) is the root of the characteristic equation (5), and it meets 𝑎(𝜏_𝑗)=0, 𝑏(𝜏_𝑗)=𝜔₀. By differentiating both sides of (5) with regard to 𝜏 and solving 𝜆^′(𝜏), we obtain 𝜆^′(𝜏)=(1/2)𝑘𝜆𝑒^−𝜆𝜏2𝜆−𝑙𝑐−(1/2)𝑘𝜏𝑒^−𝜆𝜏. (13)

That is, 𝜆^′𝜏_𝑗=(1/2)𝑘𝜔₀𝑖𝑒^{−𝜔0𝑖𝜏𝑗}2𝜔₀𝑖−𝑙𝑐−(1/2)𝑘𝜏_𝑗𝑒^{−𝜔0𝑖𝜏𝑗}. (14)

We substitute (8) into the above equation and separate the real and imaginary parts, and we have Re𝜆^′𝜏_𝑗=2𝜔⁴₀+(𝑙𝑐−𝑑)𝜔²₀(1/2)𝑑𝜏_𝑗−𝑙𝑐−𝜔²₀𝜏_𝑗²+2𝜔₀−𝑙𝑐𝜔₀𝜏_𝑗². (15)

When 𝑙𝑐>𝑑, Re𝜆^′𝜏_𝑗>0. (∗)

Here, we know that the root of (5) crosses the imaginary axis from the left to the right as 𝜏 continuously varies from a number less than 𝜏_𝑗 to one greater than 𝜏_𝑗. Thus, when 𝜏>𝜏₀, the characteristic equation (5) has at least one root with positive real part. Further, the equilibrium (0,0) is unstable in the interval (𝜏₀,+∞).

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

Theorem 3. If 𝑐<0, 𝑘²>𝑑² and 𝑙𝑐>𝑑 hold, the equilibrium (0,0) of system (3) is asymptotically stable for 𝜏∈[0,𝜏₀) and unstable for 𝜏∈(𝜏₀,+∞). System (3) exhibits the Hopf bifurcation at the equilibrium (0,0) for 𝜏=𝜏_𝑗, 𝑗=0,1,2,….

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 𝜏. 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 𝑡→𝑡/𝜏, system (3) is transformed into the following form: 𝑥̇(𝑡)(𝑡)1=−𝜏𝑦+𝑙𝜏3𝑎𝑥³(𝑡)+12𝑏𝑥²(𝑡)(𝑡,𝑦̇(𝑡)=1+𝑐𝑥)2(𝑡)+1𝜏𝑑𝑥2𝜏𝑘𝑥(𝑡−1). (16)

It is not difficult to show that system (16) also exhibits the Hopf bifurcation at the equilibrium (0,0) for 𝜏=𝜏_𝑗 (𝑗=0,1,2,…). Without loss of generality, we only consider the bifurcation parameter 𝜏₀. For convenience, by setting 𝜏=𝜏₀+𝜇, system (16) is rewriten as 𝑥̇(𝑡)=−𝜏₀+𝜇𝑦(𝑡)+𝑙𝜏₀×1+𝜇3𝑎𝑥³(𝑡)+12𝑏𝑥²(𝑡)(𝑡),𝑦̇(𝑡)=1+𝑐𝑥2𝜏₀(𝑡)+1+𝜇𝑑𝑥2𝜏₀+𝜇𝑘𝑥(𝑡−1). (17)

Clearly, 𝜇=0 is the Hopf bifurcation value of system (17).

Throughout the following section, 𝐶₁=𝐶([−1,0];𝑅²₊) is a phase space and the superscripts “𝑇” and “∗” stand for the transpose and adjoint, respectively.

In 𝐶₁, system (17) can be written as the following equation: 𝑋̇(𝑡)=𝜏₀+𝜇𝐿𝑋_𝑡+𝜏₀+𝜇𝐹𝑋_𝑡, (18) where 𝑋(𝑡) is a vector (𝑥₁(𝑡),𝑥₂(𝑡)), 𝑋_𝑡=𝑋(𝑡+𝜃) for 𝜃∈[−1,0], and 𝐿, 𝐹 are given by 𝐿𝜙=𝑙𝑐𝜙₁(0)−𝜙₂(0)12𝑑𝜙₁(0)+12𝑘𝜙₁(),𝐹𝜙=1−13𝑎𝑙𝜙³₁(0)+12𝑏𝑙𝜙²₁(0)0, (19) for 𝜙=(𝜙₁,𝜙₂)∈𝐶₁.

Now, we consider the abstract functional differential equation [3]: 𝑢̇_𝑡=𝐴𝜇𝑢_𝑡+𝑅𝜇𝑢_𝑡, (20) where 𝑢_𝑡=𝑢(𝑡+𝜃)∈𝐶₁ for 𝜃∈[−1,0].

The operators 𝐴 and 𝑅 are defined as 𝐴𝜇𝜙(𝜃)=𝑑𝜙(𝜃)[),∫𝑑𝜃,𝜃∈−1,0₀₋₁𝑑𝜂𝜙(𝑡)𝑅𝜇𝜙(𝜃)=[),𝑓𝑡,𝜇,𝜃=0,0,𝜃∈−1,0𝜇,𝜃,𝜃=0, (21) where ∫⁰₋₁𝑑(𝜂(𝑡,𝜇)𝜙(𝑡))=(𝜏₀+𝜇)𝐿(𝜙) (here, 𝜂(𝜃,𝜇) is a bounded variation function for 𝜃∈[−1,0]), 𝑓(𝜇,𝜙)=(𝜏₀+𝜇)𝐹(𝜙).

Consider the adjoint bilinear form ⟨⋅,⋅⟩ on 𝐶₁×𝐶^∗₁: 𝜓,𝜙=𝜓(0)𝜙(0)−⁰₋₁^𝜃₀𝜓(𝜉−𝜃)𝑑𝜂(𝜃)𝜙(𝜉)𝑑𝜉, (22) where 𝜂(𝜃)=𝜂(𝜃,0).

According to the adjoint bilinear form ⟨⋅,⋅⟩, we can define an adjoint operator 𝐴^∗(0) corresponding to 𝐴(0) as the following form: 𝐴^∗𝜓(𝑠)=−𝑑𝜓(𝑠)(],∫𝑑𝑠,𝑠∈0,1₀₋₁𝑑𝜂^𝑇(𝑡,0)𝜓(−𝑡),𝑠=0. (23)

To determine the normal form of operator 𝐴, we need to calculate the eigenvectors 𝑞(𝜃) and 𝑞^∗(𝑠) of 𝐴 and 𝐴^∗ corresponding to 𝑖𝜏₀𝜔₀ and −𝑖𝜏₀𝜔₀, respectively.

Proposition 4. Assume that 𝑞(𝜃) and 𝑞^∗(𝑠) are the eigenvector 𝑞(𝜃) and 𝑞^∗(𝑠) of 𝐴 and 𝐴^∗ corresponding to 𝑖𝜏₀𝜔₀ and −𝑖𝜏₀𝜔₀, respectively, satisfying ⟨𝑞^∗,𝑞⟩=1 and ⟨𝑞^∗,𝑞⟩=0.
Then 𝑞(𝜃)=𝛼,𝛽^𝑇𝑒^{𝑖𝜔0𝜏0𝜃}=1,𝑙𝑐−𝜔₀𝑖^𝑇𝑒^{𝑖𝜔0𝜏0𝜃},𝑞^∗(𝑠)=𝐷𝛼^∗,𝛽^∗𝑒^{𝑖𝜔0𝜏0𝑠}=𝜔₀𝑖,1𝑒^{𝑖𝜔0𝜏0𝑠}, (24) where 𝐷=2/(2𝛼𝛼^∗+2𝛽𝛽^∗+𝑘𝜏₀𝛼𝛽^∗𝑒^{−𝑖𝜔0𝜏0}).

Proof. Without loss of generality, we just consider the eigenvector 𝑞(𝜃).
Firstly, when 𝜃∈[−1,0), by the definition of 𝐴 and 𝑞(𝜃), we obtain the form 𝑞(𝜃)=(𝛼,𝛽)^𝑇𝑒^{𝑖𝜔0𝜏0} (here, 𝛼, 𝛽 are unknown parameters).
In what follows, notice that 𝑞(0)=(𝛼,𝛽)^𝑇 and 𝐴𝑞(0)=∫⁰₋₁𝑑(𝜂(𝑡,𝜇)𝜙(𝑡))=𝑖𝜔₀𝜏₀𝑞(0), and we have 𝛼=1, 𝛽=𝑙𝑐−𝜔₀𝑖.
Finally, by ⟨𝑞^∗,𝑞⟩=1, we obtain the parameter 𝐷 (refer to [4, 5]).
The proof is completed.

As in [6], the bifurcating periodic solutions 𝑥(𝑡,𝜇) of system (16) are indexed by a small parameter 𝜀. A solution 𝑥(𝑡,𝜇(𝜀)) has amplitude 𝑂(𝜀), period 𝑇(𝜀), and nonzero Floquet exponent 𝛽(𝜀) with 𝛽(0)=0. Under the present assumptions, 𝜇, 𝑇, and 𝛽 have expansions [7, 8]: 𝜇=𝜇₂𝜀²+𝜇₄𝜀⁴+⋯,𝑇=2𝜋𝜔1+𝑇₂𝜀²+𝑇₄𝜀⁴+⋯,𝛽=𝛽₂𝜀²+𝛽₄𝜀⁴+⋯, (25) where the sign of 𝜇₂ determines the directions of the Hopf bifurcations, the sign of 𝛽₂ determines the stability of the bifurcation periodic solutions, and 𝑇₂ determines the period of the bifurcating periodic solutions.

The purpose of this section is to compute the coefficients 𝜇₂, 𝑇₂, 𝛽₂ in the above expansions.

Next, we construct the coordinates of the center manifold 𝐶₀ at 𝜇=0. Let 𝑧(𝑡)=𝑞^∗,𝑢_𝑡,𝑊(𝑡,𝜃)=𝑢_𝑡(𝜃)−2Re𝑧(𝑡)𝑞(𝜃). (26)

On the center manifold 𝐶₀, we have 𝑊(𝑡,𝜃)=𝑊𝑧(𝑡),𝑧(𝑡),𝜃, (27) where 𝑊(𝑧,𝑧,𝜃)=𝑊₂₀(𝜃)𝑧²2+𝑊₁₁(𝜃)𝑧𝑧+𝑊₀₂𝑧²2+𝑊₃₀𝑧³6⋯. (28)

𝑧 and 𝑧 are local coordinates for the center manifold 𝐶₀ in the direction of 𝑞 and 𝑞^∗, respectively. Since 𝜇=0, we have 𝑧^′(𝑡)=𝑖𝜏₀𝜔₀𝑧(𝑡)+𝑞^∗(𝜃),𝑓𝑊+2Re𝑧(𝑡)𝑞(𝜃)=𝑖𝜏₀𝜔₀𝑧(𝑡)+𝑞^∗(0)𝑓𝑊(𝑧,𝑧,0)+2Re𝑧(𝑡)𝑞(0)≜𝑖𝜏₀𝜔₀𝑧(𝑡)+𝑞^∗(0)𝑓₀(𝑧,𝑧), (29) where 𝑓₀(𝑧,𝑧)=𝑓_𝑧2𝑧²2+𝑓_𝑧²𝑧²2+𝑓_𝑧𝑧𝑧𝑧+⋯. (30)

We rewrite this as 𝑧^′(𝑡)=𝑖𝜏₀𝜔₀𝑧+𝑔(𝑧,𝑧), (31) with 𝑔(𝑧,𝑧)=𝑞^∗(0)𝑓₀(𝑧,𝑧)=𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧2+⋯. (32)

Proposition 5. For (31), one has (𝑖)𝑔₂₀=𝑔₁₁=𝑔₀₂=𝐷𝑏𝑙𝜏₀𝛼^∗𝛼²,(𝑖𝑖)𝑔₂₁=𝐷𝜏₀𝛼^∗2𝑎𝑙+2𝑏𝑙𝑊⁽¹⁾₁₁(0)+𝑏𝑙𝑊⁽¹⁾₂₀(0), (33) where 𝑊⁽¹⁾₁₁(0)=(−𝑖𝑔₁₁/𝜔₀𝜏₀)𝛼+(𝑖𝑔₁₁/𝜔₀𝜏₀)𝛼+𝐸₂⁽¹⁾, 𝑊⁽¹⁾₂₀(0)=(𝑖𝑔₂₀/𝜔₀𝜏₀)𝛼+(𝑖𝑔₀₂/3𝜔₀𝜏₀)𝛼+𝐸₁⁽¹⁾,𝐸₁⁽¹⁾=4𝑙𝑐𝛼²𝜔₀𝜏₀𝑖/(𝑑+𝑘𝑒^{−𝑖𝜔0𝜏0}−4𝜔²₀𝜏²₀−2𝑙𝑐𝜔₀𝜏₀), 𝐸₂⁽¹⁾=2𝛼𝛼.

Proof. (i) Noticing 𝑥_𝑡(𝜃)=(𝑥_1𝑡(𝜃),𝑥_2𝑡(𝜃))=𝑊(𝑡,𝜃)+𝑧𝑞(𝜃)+𝑧𝑞(𝜃) and 𝑞(𝜃)=(𝛼,𝛽)^𝑇𝑒^{𝑖𝜔0𝜏0𝜃}, we have 𝑥_1𝑡(0)=𝑧+𝑧+𝑊⁽¹⁾₂₀(0)𝑧²2+𝑊⁽¹⁾₁₁(0)𝑧𝑧+𝑊⁽¹⁾₀₂(0)𝑧²2+⋯. (34)
By (32), we obtain 𝑔(𝑧,𝑧)=𝑞^∗(0)𝑓₀(𝑧,𝑧)=𝐷𝜏₀𝛼^∗,𝛽^∗13𝑎𝑙𝑥³_1𝑡(0)+12𝑏𝑙𝑥²_1𝑡(0),0^𝑇. (35)
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 [9–11].
This completes the proof.

According to Proposition 5, we can compute the following parameters: 𝐶₁(0)=𝑖2𝜏₀𝜔₀𝑔₂₀𝑔₁₁−2𝑔₁₁²−13𝑔₀₂²+𝑔₂₁2,𝜇₂𝐶=−Re₁(0)Re𝜆^′𝜏₀,𝛽₂=2Re𝐶₁(0,𝑇₂)𝐶=−Im₁(0)+𝜇₂Im𝜆^′𝜏₀𝜔₀. (36)

From the discussion in Section 2 we know that Re{𝜆^′(𝜏₀)}>0. We therefore have the following result.

Theorem 6. If Re{𝐶₁(0)}<0(>0), the direction of the Hopf bifurcation of the system (1) at the equilibrium (0,0) when 𝜏=𝜏₀ 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 𝑑=−7, 𝑙=2, 𝑐=−2, 𝑘=9 in (3). Omitting these complicated expressions, we obtain the numerical results directly by means of the software MatLab: 𝜔₀≐0.5871, 𝜏₀≐0.935, Re{𝜆^′(𝜏₀)}≐2.2695×10⁻⁵. So, we directly compete 𝐶₁(0)≐−0.0431+0.0381𝑖, 𝜇₂≐12.4372, 𝛽₂≐−0.862.

According to Theorem 6, Re{𝐶₁(0)}≐−0.0431<0. That is, the bifurcating periodic solutions of system (3) with the above coefficients are supercritical and orbitally asymptotically stable at 𝜏=𝜏₀.

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

4. Conclusion

Firstly, under the condition of 𝜏=0, we discuss the Reyleigh price model. We know that the stability of price varies with the parameters changing. When 𝑐<0, 𝑘+𝑑>0 and 𝑐=0, 𝑘+𝑑>0, 𝑏<0, 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 𝑑, 𝑘, we more easily control the price such that the price tends to our expected results. For example, when we take 𝑎=0, 𝑏=0, 𝑑=−7, 𝑙=2, 𝑐=−2, 𝑘=9, 𝜏=0.8 in the system (3), the equilibrium (0,0) of system (3) is stable (see Figure 1). By contrast, when we take 𝑎=0, 𝑏=0, 𝑑=−7, 𝑙=2, 𝑐=−2, 𝑘=9, 𝜏=0.97 in the system (3), the (0,0) 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.

Figure 1 

The equilibrium (0,0) of system (3) is stable with 𝜏=0.8<𝜏0=0.935.

Figure 2 

The equilibrium (0,0) of system (3) is unstable with 𝜏=0.97>𝜏0=0.935.

Figure 3 

When 𝜏0=0.935, the periodic solutions occur from the equilibrium (0,0).