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

Authors: Yanhui Zhai; Haiyun Bai; Ying Xiong; Xiaona Ma
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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:(1)x˙(t)=-y(t)+l(13ax3(t)+12bx2(t)+cx(t)),y˙(t)=x(t),
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:(2)x˙(t)=-y(t)+l(13ax3(t)+12bx2(t)+cx(t)),y˙(t)=x(t-τ),
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:
(3)x˙(t)=-y(t)+l(13ax3(t)+12bx2(t)+cx(t)),y˙(t)=12dx(t)+12kx(t-τ),
where d, k refer to [2].


## 2. The Stability Analysis

In this section, we obtain the domain of the stable equilibrium when time delayτ 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
(4)x˙(t)=-y(t)+lcx(t),y˙(t)=12dx(t)+12kx(t-τ),
whose characteristic equation is
(5)λ2-lcλ+12ke-λτ+12d=0.When the caseτ=0, we have the following.Lemma 1.
(i) Whenc<0, the equilibrium (0,0) of system (3) is stable.
(ii) Whenc>0, the equilibrium (0,0) of system (3) is unstable.
(iii) Whenc=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.
Ifd2<k2 is satisfied, the characteristic equation (5) has a pair of purely imaginary roots ±iω0 when τ=τj, where
(6)ω0=(-(l2c2-d)+(l2c2-d)2-(d2-k2)2)1/2,τj=arccos((2ω0-d)/k)+2jπω0,j=0,1,2,….Proof.
Letiω (ω>0) is a root of (5). Then
(7)-ω2-lcωi+12d+12k(cos(ωτ)-isin(ωτ))=0.
The separation of the real and imaginary parts yields(8)-ω2+12kcos(ωτ)+12d=0,-lcω-12ksin(ωτ)=0,
which lead to
(9)ω4-dω2+l2c2ω2+14d2-14k2=0.
By solving the second-degree equation (9) concerning ω2, we have
(10)ω2=-(l2c2-d)±(l2c2-d)2-(d2-k2)2.
We take(11)ω0=(-(l2c2-d)+(l2c2-d)2-(d2-k2)2)1/2.
And, hence(12)τ0=arccos((2ω02-d)/k)ω0.
By settingτj=τ0+(2jπ/ω0), j=0,1,2,…. Then ω0, τj satisfies the condition of Lemma 2.
The proof is completed.According to Lemmas1 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 τj.Assume thatλ(τ)=a(τ)+ib(τ) is the root of the characteristic equation (5), and it meets a(τj)=0, b(τj)=ω0. By differentiating both sides of (5) with regard to τ and solving λ′(τ), we obtain
(13)λ′(τ)=(1/2)kλe-λτ2λ-lc-(1/2)kτe-λτ.That is,(14)λ′(τj)=(1/2)kω0ie-ω0iτj2ω0i-lc-(1/2)kτje-ω0iτj.We substitute (8) into the above equation and separate the real and imaginary parts, and we have
(15)Re{λ′(τj)}=2ω04+(lc-d)ω02((1/2)dτj-lc-ω02τj)2+(2ω0-lcω0τj)2.Whenlc>d,
(*)Re{λ′(τj)}>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 τj to one greater than τj. 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 Lemmas1 and 2 and condition (*), we have the following results.Theorem 3.
Ifc<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=τj, j=0,1,2,….


## 3. Hopf Bifurcation Analysis

In Section2, 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 scalingt→t/τ, system (3) is transformed into the following form:
(16)x˙(t)=-τy(t)+lτ(13ax3(t)+12bx2(t)+cx(t)),y˙(t)=12τdx(t)+12τkx(t-1).It is not difficult to show that system (16) also exhibits the Hopf bifurcation at the equilibrium (0,0) for τ=τj (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
(17)x˙(t)=-(τ0+μ)y(t)+l(τ0+μ)×(13ax3(t)+12bx2(t)+cx(t)),y˙(t)=12(τ0+μ)dx(t)+12(τ0+μ)kx(t-1).Clearly,μ=0 is the Hopf bifurcation value of system (17).Throughout the following section,C1=C([-1,0];R+2) is a phase space and the superscripts “T” and “*” stand for the transpose and adjoint, respectively.InC1, system (17) can be written as the following equation:
(18)X˙(t)=(τ0+μ)L(Xt)+(τ0+μ)F(Xt),
where X(t) is a vector (x1(t),x2(t)), Xt=X(t+θ) for θ∈[-1,0], and L, F are given by
(19)L(ϕ)=(lcϕ1(0)-ϕ2(0)12dϕ1(0)+12kϕ1(-1)),F(ϕ)=(13alϕ13(0)+12blϕ12(0)0),
for ϕ=(ϕ1,ϕ2)∈C1.Now, we consider the abstract functional differential equation [3]:
(20)u˙t=A(μ)ut+R(μ)ut,
where ut=u(t+θ)∈C1 for θ∈[-1,0].The operatorsA and R are defined as
(21)A(μ)ϕ(θ)={dϕ(θ)dθ,θ∈[-1,0),∫-10‍d(η(t,μ)ϕ(t)),θ=0,R(μ)ϕ(θ)={0,θ∈[-1,0),f(μ,θ),θ=0,
where ∫-10‍d(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×C1*:
(22)〈ψ,ϕ〉=ψ-(0)ϕ(0)-∫-10‍∫0θ‍ψ-(ξ-θ)dη(θ)ϕ(ξ)dξ,
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:
(23)A*ψ(s)={-dψ(s)ds,s∈(0,1],∫-10‍d(ηT(t,0)ψ(-t)),s=0.To determine the normal form of operatorA, we need to calculate the eigenvectors q(θ) and q*(s) of A and A* corresponding to iτ0ω0 and -iτ0ω0, respectively.Proposition 4.
Assume thatq(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(24)q(θ)=(α,β)Teiω0τ0θ=(1,lc-ω0i)Teiω0τ0θ,q*(s)=D(α*,β*)eiω0τ0s=(ω0i,1)eiω0τ0s,
where D¯=2/(2αα*+2ββ*+kτ0αβ*¯e-iω0τ0).Proof.
Without loss of generality, we just consider the eigenvectorq(θ).
Firstly, whenθ∈[-1,0), by the definition of A and q(θ), we obtain the form q(θ)=(α,β)Teiω0τ0 (here, α, β are unknown parameters).
In what follows, notice thatq(0)=(α,β)T and Aq(0)=∫-10‍d(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]:
(25)μ=μ2ε2+μ4ε4+⋯,T=2πω(1+T2ε2+T4ε4+⋯),β=β2ε2+β4ε4+⋯,
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 coefficientsμ2, T2, β2 in the above expansions.Next, we construct the coordinates of the center manifoldC0 at μ=0. Let
(26)z(t)=〈q*,ut〉,W(t,θ)=ut(θ)-2Re{z(t)q(θ)}.On the center manifoldC0, we have
(27)W(t,θ)=W(z(t),z(t)¯,θ),
where
(28)W(z,z-,θ)=W20(θ)z22+W11(θ)zz-+W02z-22+W30z36⋯.z and z- are local coordinates for the center manifold C0 in the direction of q and q*, respectively. Since μ=0, we have
(29)z′(t)=iτ0ω0z(t)+〈q*(θ),f(W+2Re{z(t)q(θ)})〉=iτ0ω0z(t)+q*(0)¯f(W(z,z-,0)+2Re{z(t)q(0)})≜iτ0ω0z(t)+q*(0)¯f0(z,z-),
where
(30)f0(z,z-)=fz2z22+fz-2z-22+fzz-zz-+⋯.We rewrite this as(31)z′(t)=iτ0ω0z+g(z,z-),
with
(32)g(z,z-)=q*-(0)f0(z,z-)=g20z22+g11zz-+g02z-22+g21z2z-2+⋯.Proposition 5.
For (31), one has
(33)(i)g20=g11=g02=D¯blτ0α*¯α2,(ii)g21=D¯τ0α*(2al+2blW11(1)(0)+blW20(1)(0)),
where W11(1)(0)=(-ig11/ω0τ0)α+(ig-11/ω0τ0)α-+E2(1), W20(1)(0)=(ig20/ω0τ0)α+(ig02¯/3ω0τ0)α¯+E1(1),E1(1)=4lcα2ω0τ0i/(d+ke-iω0τ0-4ω02τ02-2lcω0τ0), E2(1)=2αα-.Proof.
(i) Noticingxt(θ)=(x1t(θ),x2t(θ))=W(t,θ)+zq(θ)+z-q(θ)¯ and q(θ)=(α,β)Teiω0τ0θ, we have
(34)x1t(0)=z+z-+W20(1)(0)z22+W11(1)(0)zz-+W02(1)(0)z¯22+⋯.
By (32), we obtain
(35)g(z,z-)=q¯*(0)f0(z,z-)=D¯τ0(α*¯,β*¯)(13alx1t3(0)+12blx1t2(0),0)T.
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 Proposition5, we can compute the following parameters:
(36)C1(0)=i2τ0ω0(g20g11-2|g11|2-13|g02|2)+g212,μ2=-Re{C1(0)}Re{λ′(τ0)},β2=2Re{C1(0)},T2=-Im{C1(0)}+μ2(Im{λ′(τ0)})ω0.From the discussion in Section2 we know that Re{λ′(τ0)}>0. We therefore have the following result.Theorem 6.
IfRe{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 coefficientsd=-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 Theorem6, 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 ofτ=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
Whenτ0=0.935, the periodic solutions occur from the equilibrium (0,0).



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*Source: 290497-2013-08-13.xml*