Dynamics Analysis of a Class of Delayed Economic Model

Abstract

This investigation aims at developing a methodology to establish stability and bifurcation dynamics generated by a class of delayed economic model, whose state variable is described by the scalar delay differential equation of the form d²𝑝(𝑡)/d𝑡²=−𝜇𝛿(𝑝(𝑡))(d𝑝(𝑡)/d𝑡)−𝜇𝑏𝑝(𝑡−𝜏₁)−𝜇(𝑎₀𝑝(𝑡−𝜏₂)/(𝑎₁+𝑝(𝑡−𝜏₂)))+𝜇(𝑑₀−𝑔₀). At appropriate parameter values, linear stability and Hopf bifurcation including its direction and stability of the economic model are obtained. The main tools to obtain our results are the normal form method and the center manifold theory introduced by Hassard. Simulations show that the theoretically predicted values are in excellent agreement with the numerically observed behavior. Our results extend and complement some earlier publications.

1. Introduction

Trade cycles, business cycles, and fluctuations in the price and supply of various commodities have attracted the attention of economists for well over one hundred years and possible more than thousands of years [1]. In the case of the most models discussed earlier in the literature, it is assumed that each economic agent has instantaneous information about its own as well as its rivals' behavior. Many authors often attributed these fluctuations to instantaneous information factors. This assumption is mathematically convenient but does not fully describe real economic situations in which there are always time delays between the times when information is obtained and the times when the decisions are implemented. In recent years it has been recognized in continuous-time economic dynamics that a delay differential equation is useful to describe the periodic and aperiodic behavior of economic variables [2–7]. Time delays usually cause the models to generate not only periodic cycles but also chaotic behavior for certain values of the shape parameter of the production function. With the infinite dimensionality created by a fixed-time delay, even a single first-order equation is transformed into an equation with a sufficient number of degrees of freedom to permit the occurrence of complex dynamics involving chaotic phenomena. This finding indicates that fixed-time delay models of a dynamic economy may explain various complex dynamic behaviors of the economic variables. For decades, a lot of effort has been devoted to deepen the understanding of economic complexity including chaotic behavior, stability, and basins of attractions. In [2], Matsumoto and Szidarovszky and in [3], Akio and Ferenc investigate a continuous-time neoclassical growth model with time delay and study the dynamics of the delayed model. In [4], Bélair and Mackey develop a model of price adjustment with production delays. In [5], Zhang et al. consider a differential-algebraic biological economic system with time delay and harvesting where the dynamics is logistic with carrying capacity proportional to prey population. Howroyd and Russel detect the stability conditions of delay output adjustment processes in a general N-firm oligopoly with fixed time delays [6]. Matsumoto and Szidarovszky introduce a fixed delay in production and a mound-shaped production function into the neoclassical one-sector growth model and show the birth of complex dynamics [2]. Considering about a delay in the production process, Li establish the single commodity price-inventory control model [7].

It is well known that persistent oscillations are one of the most ubiquitous forms by which economic phenomena may be observed [8]. Limit cycles are the simplest nonlinear phenomena, for example, they are the simplest example of how the interaction between economic forces may compel a system to abandon its steady state and start to steadily oscillate. Just as reported by Manfredi and Fanti [9], the detection of stable oscillations, for example, stable limit cycles, in continuous-time systems, is intimately related with the notion of Hopf bifurcation. Hopf bifurcation is important in economics. There are at least three reasons. First, it is always the outcome of a fully endogenous interaction between (nonlinear) economic forces. Second, it is a local bifurcation, thus much in spirit with the common belief of our science by which economic systems are generally close to their equilibrium state. Third, because it implieslocal oscillations, which are the normal route through which disequilibrium manifests itself when the equilibrating forces operating in the economy are relaxed (e.g., the adjustment process of a Walrasian market). For instance, when oscillations persist in a market normally in equilibrium (in the absence of stochastic and seasonal perturbations), it is very likely that these oscillations are the outcome of a Hopf bifurcation. So it is necessary for us to research the Hopf bifurcation in the economic system.

In the case of the most models discussed in the mentioned literature, only one delay appears in the models. Considering that the consumer memory plays an important role in the process of economic activities, just as pointed out by Li [7], bringing another delay might be better candidates for some purposes and would be of great interest. In this paper, we will generalize an economic model with the help of bringing two delays. In specifying how consumer behavior affects commodity demand, we will assume that the behavior is influenced not only by the instantaneous price, but also by the information regarding past prices. As the fact that the quantity supplied may not increase infinitely during the price increasing, similar to [7], we also assume the supply function to be fractional linear function. This investigation aims at developing a methodology to establish stability and bifurcation dynamics generated by the new delayed economic model.

The organization of the rest of this paper is as follows. In Section 2, we establish an economic model with two delays combing with the model considered in [7]. In Section 3, we take the delays as the parameters and use the distribution theory of the transcendental equation root [10] and the theory of Hopf bifurcation about functional differential equation [11] to discuss the stability of the equilibrium point for economic system and the existence of Hopf bifurcation. In Section 4, we apply the normal form theory and center manifold theory to investigate the bifurcation direction and the stability of periodic solution. In Section 5, an example with numerical simulations is arranged to illustrate the obtained results.

2. Characterization of a Generalized Delayed Economic Model

Considering a single commodity market, the quantity of supplied and demanded can be regarded as the function of time, namely, 𝐺(𝑡) and 𝐷(𝑡). The inventory and the level of inventory are recorded, respectively, as 𝑆(𝑡) and 𝑆.

Let 𝑝(𝑡) denote the price at time 𝑡, so that the rate of price increase is in proportion to the difference between 𝑆(𝑡) and 𝑆, namely, d𝑝(𝑡)d𝑡=−𝜇𝑆−𝑆(𝑡),𝜇>0, (1) where 𝜇 is a positive real number depending on the speed of price adjustment, recording 𝑆(𝑡) as 𝑆(𝑡)=𝑆₀+^𝑡₀(𝐺(𝜀)−𝐷(𝜀))𝑑𝜀. (2)

In the traditional cobweb model, demand function is a function of price. If we consider price as the only factor that influences the quantity demanded, there will be certain limitations to reflect the regularity of price change. We should consider other factors influencing the demand such as the rate of the price increase. In [7], Li assume the demand function as 𝐷(𝑡)=𝑑₀(𝑡)𝑝(𝑡)(𝑡)−𝑏𝑝−𝛿d𝑝,d𝑡 (3) where 𝑑₀>0, 𝑏>0, 𝑏 represents the sensitive degree of consumers to the increase of commodity price. 𝛿(𝑝(𝑡)) is the level of price relying on the rate of increase. In terms of consumer behavior, we consider a class of consumers who base buying decisions on the past prices and recent price. Therefore, in this paper, we introduce a delay in demand function 𝐷(𝑡) and set up the demand function as shown later 𝐷(𝑡)=𝑑₀−𝑏𝑝𝑡−𝜏₁𝑝(𝑡)(𝑡)−𝛿d𝑝.d𝑡 (4)

In generally, supply function is monotone increasing about price, but consider that as price goes up, the supply could not unlimitedly increase, one can assume supply function as a fractional linear function as the following: 𝐺(𝑡)=𝑔₀+𝑎₀𝑝(𝑡)𝑎₁(𝑡),+𝑝 (5) where 𝑎₀>0, 𝑎₁>0, 𝑔₀≥0.

Noticing the delay in the production process, supply function should be a function of past price, therefore, we can introduce another delay in supply function and record it as follows: 𝐺(𝑡)=𝑔₀+𝑎₀𝑝𝑡−𝜏₂𝑎₁+𝑝𝑡−𝜏₂, (6) where 𝑎₀, 𝑎₁, 𝑔₀, 𝜏₁, 𝜏₂ are constants and 𝑎₀>0, 𝑎₁>0, 𝑔₀≥0, 𝜏₂≥𝜏₁≥0.

Substituting (2) to (1), calculating the derivation of both sides about time in (1), one can get 𝑑²𝑝(𝑡)d𝑡²=−𝜇(𝐺(𝑡)−𝐷(𝑡)). (7) Substituting (4) and (6) to (7), we can establish a single commodity price inventory control model with two delays take the following form: 𝑑²𝑝(𝑡)d𝑡²𝑝(𝑡)(𝑡)=−𝜇𝛿d𝑝d𝑡−𝜇𝑏𝑝𝑡−𝜏₁𝑎−𝜇⁰𝑝𝑡−𝜏₂𝑎₁+𝑝𝑡−𝜏₂+𝜇𝑑₀−𝑔₀. (8) Let 𝐶¹=𝐶¹([−𝜏₂,0],𝑅⁺) denote the Banach space of continuous and differentiable mapping from [−𝜏₂,0] into 𝑅⁺ equipped with the Supremum Norm ‖𝜙‖=sup_{𝑠∈[−𝜏2,0]}{|𝜙(𝑠)|,|𝜙^′(𝑠)|} for 𝜙∈𝐶¹. The initial condition of (8) is 𝑝(𝜃)=𝜓(𝜃)>0,𝜃∈−𝜏₂,0,𝜓∈𝐶¹. (9) We will consider the following basic assumptions to further investigate the stability and bifurcation dynamics of model (8).(H₁) The inequality hold: 𝑑₀>𝑔₀.(H₂) The inequality hold: 𝛿(𝑝^∗)>0, where 𝑝^∗ is a positive equilibrium point.(H₃) The inequality hold: 𝑏<𝑎₀𝑎₁/(𝑎₁+𝑝^∗)², where 𝑝^∗ is a positive equilibrium point.

3. Stability Analysis and the Existence of Hopf Bifurcation

At first, we will show that system (8) has only one positive equilibrium point under some assumption. We state the following theorem.

Theorem 1. If the inequality (H₁) holds: 𝑑₀>𝑔₀, then system (8) has only one positive equilibrium point.

Proof. Without loss of generality, we may assume that d𝑝(𝑡)(𝑡),d𝑡=𝑞 (10) then model (8) can be rewritten as the following: d𝑝(𝑡)(𝑡),(𝑡)d𝑡=𝑞d𝑞𝑝(𝑡)𝑞(𝑡)d𝑡=−𝜇𝛿−𝜇𝑏𝑝𝑡−𝜏₁𝑎−𝜇⁰𝑝𝑡−𝜏₂𝑎₁+𝑝𝑡−𝜏₂+𝜇𝑑₀−𝑔₀. (11)
Assume 𝑝^∗ to be the equilibrium point of system (8), one can show that (𝑝^∗,0) is the equilibrium point of system (11). Therefore, one can obtain 𝑏𝑝^∗2+𝑎₁𝑏+𝑎₀−𝑑₀−𝑔₀𝑝^∗−𝑎₁𝑑₀−𝑔₀=0. (12) Obviously, if 𝑑₀>𝑔₀, we have Δ=𝑎₁𝑏+𝑎₀−𝑑₀−𝑔₀²+4𝑎₁𝑏𝑑₀−𝑔₀−𝑎>0,₁𝑑₀−𝑔₀𝑏<0. (13) Then (8) has only one positive equilibrium point. This completes the proof.

In real life, 𝑝^∗ is the equilibrium price. With the help of coordinate translation 𝑥(𝑡)=𝑝(𝑡)−𝑝^∗,𝑦(𝑡)=𝑞(𝑡), (14) system (11) can be further rewritten as the following form: d𝑥(𝑡)(𝑡),(𝑡)d𝑡=𝑦d𝑦𝑥(𝑡)d𝑡=−𝜇𝛿+𝑝^∗𝑦(𝑡)−𝜇𝑏𝑥𝑡−𝜏₁+𝑝^∗𝑎−𝜇⁰𝑥𝑡−𝜏₂+𝑝^∗𝑎₁+𝑥𝑡−𝜏₂+𝑝^∗+𝜇𝑑₀−𝑔₀. (15) Then the linearized system at (0,0) is d𝑥(𝑡)(𝑡),(𝑡)d𝑡=𝑦d𝑦𝑝d𝑡=−𝜇𝛿^∗𝑦(𝑡)−𝜇𝑏𝑥𝑡−𝜏₁−𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗2𝑥𝑡−𝜏₂. (16) The characteristic equation of the linearized system (16) at (0,0) takes the following form: 𝜆²+𝜇𝛿𝑝^∗𝜆+𝜇𝑏𝑒^−𝜆𝜏1+𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗2𝑒^−𝜆𝜏2=0. (17)

It is well known that the equilibrium (0,0) is asymptotically stable if all roots of the characteristic equation (17) have negative real parts. Now we reach the position to study the distribution of the roots of (17). We will consider three cases as follows: (Case 1): 𝜏₁=0, 𝜏₂=0; (Case 2): 𝜏₁=0, 𝜏₂>0; and (Case 3): 𝜏₂>𝜏₁>0.

Case 1. 𝜏₁=0, 𝜏₂=0.

Proposition 2. If 𝜏₁=0, 𝜏₂=0 and the inequality (H₂) hold: 𝛿(𝑝^∗)>0, then the equilibrium point (0,0) of system (15) is asymptotically stable.

Proof. As the inequality (H₂) holds, then the characteristic equation (17) turns to be 𝜆²+𝜇𝛿𝑝^∗𝜆+𝜇𝑏+𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗2=0. (18) It is obvious that 𝜇𝑎₀𝑎₁/(𝑎₁+𝑝^∗)²>0; from Hurwitz criterion, all roots of this equation have negative real parts; therefore, the equilibrium point (0,0) of system (15) is asymptotically stable.

Case 2 (𝜏₁=0,𝜏₂>0). If 𝜏₁=0, 𝜏₂>0, then the characteristic equation (17) takes the following form: 𝜆²+𝜇𝛿𝑝^∗𝜆+𝜇𝑏+𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗2𝑒^−𝜆𝜏2=0. (19) Let 𝐸(𝜆)=𝜆²+𝜇𝛿(𝑝^∗)𝜆+𝜇𝑏+(𝜇𝑎₀𝑎₁/(𝑎₁+𝑝^∗)²)𝑒^−𝜆𝜏2; then we obtain the following results.

Lemma 3. If the inequality (H₃) holds: 𝑏<𝑎₀𝑎₁/(𝑎₁+𝑝^∗)², and 𝜏₂=𝜏_2𝑛, then 𝐸(𝜆)=0 has the only pair of purely imaginary roots 𝜆=±𝑖𝜔, where𝜏₂=𝜏_2𝑛=1𝜔arccos−𝜔²+𝜇𝑏𝑎₁+𝑝^∗2𝜇𝑎₀𝑎₁𝜔=+2𝑛𝜋𝑛=0,1,2…,2𝜇𝑏−𝜇²𝛿²𝑝^∗+2𝜇𝑏−𝜇²𝛿²𝑝^∗2−4𝜇²𝑏²−𝜇𝑎₀𝑎₁²𝑎₁+𝑝^∗4×(2)^−11/2. (20)

Proof. If 𝜆=𝑖𝜔(𝜏₂)(𝜔(𝜏₂)>0) is a root of (19), then −𝜔²+𝑖𝜇𝛿𝑝^∗𝜔+𝜇𝑏+𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗×cos𝜔𝜏₂−𝑖sin𝜔𝜏₂+𝐶=0. (21) Separating the real and imaginary parts, we have −𝜔²+𝜇𝑏+𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗2cos𝜔𝜏₂𝑝=0,𝜇𝛿^∗𝜔−𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗2sin𝜔𝜏₂⇓=0.cos𝜔𝜏₂=𝜔²−𝜇𝑏𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗2,sin𝜔𝜏₂=𝛿𝑝^∗𝜔𝑎₀𝑎₁𝑎₁+𝑝^∗2. (22) Then we can obtain 𝜔⁴+𝜇²𝛿²𝑝^∗−2𝜇𝑏𝜔²+𝜇²𝑏²−𝜇𝑎₀𝑎₁²𝑎₁+𝑝^∗4=0. (23) The root of (23) can be expressed as follows: 𝜔=2𝜇𝑏−𝜇²𝛿²𝑝^∗+2𝜇𝑏−𝜇²𝛿²𝑝^∗2−4𝜇²𝑏²−𝜇𝑎₀𝑎₁²𝑎₁+𝑝^∗4×(2)^−11/2. (24) Then we obtain 𝜏₂=𝜏_2𝑛=1𝜔arccos−𝜔²+𝜇𝑏𝑎₁+𝑝^∗2𝜇𝑎₀𝑎₁+2𝑛𝜋,𝑛=0,1,2…. (25) Let 𝐸(𝜆)=0; if 𝜆=±𝑖𝜔 is not the only pair of purely imaginary roots, then we have 𝜕𝐸𝜕𝜆^{𝜏=𝜏2𝑛}_{𝜆=𝑖𝜔}=0. (26) From (17), one can get 2𝑖𝜔+𝜇𝛿𝑝^∗−𝜏_2𝑛𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗2𝑒^{−𝑖𝜔𝜏2𝑛}=0,−𝜔²+𝑖𝜇𝛿𝑝^∗𝜔+𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗2𝑒^{−𝑖𝜔𝜏2𝑛}+𝜇𝑏=0. (27) That is to say, (𝜔²−𝜇𝑏)𝜏_2𝑛=𝜇𝛿(𝑝^∗), which contradicts (25). This completes the proof.

Denote 𝜆(𝜏₂)=𝛼(𝜏₂)+𝑖𝜔(𝜏₂); then the root of (19) satisfies: 𝛼(𝜏_2𝑛)=0,  𝜔(𝜏_2𝑛)=𝜔.

Lemma 4. If one chooses 𝜏₂=𝜏_2𝑛=1𝜔arccos−𝜔²+𝜇𝑏𝑎₁+𝑝^∗2𝜇𝑎₀𝑎₁+2𝑛𝜋,𝑛=0,1,2… (28) and 𝐸(𝜆)=0 has the only pair of purely imaginary roots 𝜆=±𝑖𝜔, then one has dRe𝜆𝜏₂d𝜏_{2𝜏2=𝜏2𝑛}>0. (29)

Proof. Taking the derivative of 𝐸(𝜆)=0 with respect to 𝜏₂, we get 2𝜆d𝜆d𝜏₂+𝜇𝛿𝑝^∗d𝜆d𝜏₂+𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗2𝑒^−𝜆𝜏2−𝜆−𝜏₂d𝜆d𝜏₂=0,Red𝜆d𝜏₂⁻¹_{𝜏2=𝜏2𝑛}=2𝜇𝑏−𝜇²𝛿²𝑝^∗2−4𝜇²𝑏²−𝜇𝑎₀𝑎₁²(𝑎₁+𝑝^∗)^41/2>0. (30) Therefore 𝜏sgnRed𝜆₂d𝜏₂⁻¹_{𝜏2=𝜏2𝑛}𝜆𝜏=sgndRe₂d𝜏_{2𝜏2=𝜏2𝑛}>0. (31) This completes the proof.

Based on the lemmas presented previous and the classical Hopf-Bifurcation-Theorem (see, [11, pages 245–249]), we have the following result.

Theorem 5. Assume that the inequalities (H₁)–(H₃) hold, then one has the following result.(1)If 𝜏₂∈[0,𝜏₂₀), all roots of (19) have negative real parts. Namely, the equilibrium (0,0) of system (15) is locally asymptotically stable.(2)If 𝜏₂=𝜏₂₀, (19) have a pair of purely imaginary roots ±𝑖𝜔, all the other roots have negative real parts. That is to say, system (15) undergoes a Hopf bifurcation at 𝜏₂=𝜏₂₀.(3)If 𝜏₂>𝜏₂₀, (19) has roots of positive real parts. Namely the equilibrium (0,0) of system (15) is unstable.

Case 3 (𝜏₂>𝜏₁>0). We now discuss the stability of equilibrium (0,0) when 𝜏₁>0, 𝜏₂=𝜏∈[0,𝜏₂₀).
Let 𝜆=𝑖𝜔(𝜏₁)(𝜔(𝜏₁)>0) be the root of (17) and substitute it into (17), we have −𝜔²+𝑖𝜇𝛿𝑝^∗𝜔+𝜇𝑏cos𝜔𝜏₁−𝑖sin𝜔𝜏₁+𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗2(cos𝜔𝜏+𝑖sin𝜔𝜏)=0. (32) Separating the real and imaginary parts, we have 𝜇𝑏cos𝜔𝜏₁=𝜔²−𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗2cos𝜔𝜏,𝜇𝑏sin𝜔𝜏₁=𝜇𝛿𝑝^∗𝜔+𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗2sin𝜔𝜏. (33) Then we have 𝜔⁴+𝜇²𝛿²𝑝^∗𝜔²+𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗22−𝜇²𝑏²−2𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗2𝜔²cos𝜔𝜏−2𝛿𝑝^∗𝜇²𝑎₀𝑎₁𝜔𝑎₁+𝑝^∗2sin𝜔𝜏=0. (34)
If (34) has no root or negative root, all the roots of (17) have negative real part. If it has positive roots, we know that the number of positive roots is finite. Denote them to be 𝜔_𝑖(𝑖=1,2,3…,𝑁). From (33), we can get 𝜏^(𝑗)_1𝑖=1𝜔_𝑖𝜔arccos^2𝑖−𝜇𝑎₀𝑎₁/𝑎₁+𝑝^∗2cos𝜔_𝑖𝜏.𝜇𝑏+2𝑗𝜋𝑗=0,1,2,… (35) Let 𝜏⁰₁=𝜏⁽⁰⁾_1𝑖0=min{𝜏⁰_1𝑖}, let 𝑖∈{0,1,2,…,𝑁}, and let 𝜔₀=𝜔_𝑖0; if 𝜏₂=𝜏∈[0,𝜏₂₀), (16) has a pair of purely imaginary roots ±𝑖𝜔₀ at 𝜏₁=𝜏⁰₁, we can also prove that the purely imaginary roots ±𝑖𝜔₀ are simple. Taking the derivative with respect to 𝜏₁, we can get d𝜆𝜏2𝜆¹d𝜏₁+𝜇𝛿𝑝^∗d𝜆𝜏₁d𝜏₁+𝜇𝑏𝑒^−𝜆𝜏1−𝜆−𝜏₁d𝜆𝜏₁d𝜏₁−𝜇𝑎₀𝑎₁𝜏𝑎₁+𝑝^∗2𝑒^−𝜆𝜏d𝜆𝜏₁d𝜏₁=0. (36) Then, we have Red𝜆d𝜏₁^{−1𝜏=𝜏0}_{1𝜆=𝑖𝜔0}=2cos𝜔₀𝜏⁰₁+𝛿𝑝𝜇𝑏^∗sin𝜔₀𝜏⁰₁𝑏𝜔₀−𝑎₀𝑎₁𝜏/𝑎₁+𝑝^∗2sin𝜔₀𝜏⁰₁−𝜏𝑏𝜔₀. (37) Therefore, we have the following theorem.

Theorem 6. Assume that the inequalities (H₁)–(H₃) hold, then one has the following result.(1)If 𝜏₁∈[0,𝜏⁰₁), then all the roots of (17) have negative real parts. One can get that the equilibrium (0,0) of system (15) is locally asymptotically stable.(2)If 𝜏₁=𝜏⁰₁, 𝜏₂=𝜏∈[0,𝜏₂₀) and 2cos𝜔₀𝜏⁰₁/𝜇𝑏−𝛿(𝑝^∗)sin𝜔₀𝜏⁰₁/𝑏𝜔₀+((𝑎₀𝑎₁𝜏/(𝑎₁+𝑝^∗)²)sin𝜔₀(𝜏⁰₁−𝜏))/𝑏𝜔₀≠0, then system (15) undergoes Hopf bifurcations at (0,0).

4. Direction and Stability of the Bifurcation

In this section, formula for determining the direction of Hopf bifurcation and the stability of bifurcation periodic solution of system (11) at 𝜏₁=𝜏⁰₁, 𝜏₂=𝜏∈[0,𝜏₂₀) will be presented by employing the normal form method and center manifold theorem introduced by Hassard et al. in [12]. More precisely, we will compute the reduced system on the center manifold with the pair of conjugate complex, purely imaginary solution of the characteristic equation (17). By this reduction, we can determine the Hopf bifurcation direction, that is, to answer the question of whether the bifurcation branch of periodic solution exists locally for supercritical bifurcation or subcritical bifurcation.

Let 𝜏₁=𝜏⁰₁+𝑘, and let 𝑘∈𝑅; then 𝑘=0 is a critical value of Hopf bifurcation of system (11). With the translation 𝑡→𝑡/𝜏₁, let 𝜇₁(𝑡)=𝑥(𝜏₁𝑡), 𝜇₂(𝑡)=𝑦(𝜏₁𝑡), we rewrite system (15) as follows: d𝜇₁(𝑡)=𝜏d𝑡⁰¹+𝑘𝑦(𝑡),d𝜇₂(𝑡)=𝜏d𝑡⁰¹×𝜇+𝑘−𝜇𝛿₁(𝑡)+𝑝^∗𝜇₂(𝑡)−𝜇𝑏𝜇₁(𝑡−1)+𝑝^∗𝑎−𝜇⁰𝜇₁𝑡−𝜏^∗+𝑝^∗𝑎₁+𝜇₁(𝑡−𝜏^∗)+𝑝^∗+𝜇𝑑₀−𝑔₀, (38) where 𝜏^∗=𝜏₂/𝜏₁. Furthermore, we can obtain the linear system of (38) as mentioned later d𝜇₁(𝑡)=𝜏d𝑡⁰¹+𝑘𝜇₂(𝑡)d𝜇₂(𝑡)=𝜏d𝑡⁰¹+𝑘−𝜇𝑏𝜇₁(𝑡−1)−𝜇𝛿𝑝^∗𝜇₂(𝑡)−𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗2𝜇₁𝑡−𝜏^∗, (39) while the nonlinear term is 𝑓=𝜏⁰₁×+𝑘_{0−𝜇𝛿′𝑝∗𝜇1(𝑡)𝜇2(𝑡)−𝜇𝛿′′𝑝∗𝜇2!21(𝑡)𝜇2(𝑡)+𝜇𝑎0𝑎1𝑎1+𝑝∗3𝜇21𝑡−𝜏∗−𝜇𝑎0𝑎1𝑎1+𝑝∗4𝜇31𝑡−𝜏∗+⋯}. (40)

Let 𝜇(𝑡)=(𝜇₁(𝑡),𝜇₂(𝑡))^𝑇, let 𝜇_𝑡(𝜃)=𝜇(𝑡+𝜃), let 𝜃∈[−𝜏^∗,0], let Φ=(Φ₁,Φ₁)^𝑇∈𝐶[−𝜏^∗,0], and let 𝐿_𝑘Φ=𝐴₁(𝑘)Φ(0)+𝐴₂(𝑘)Φ(−1)+𝐴₃(𝑘)Φ(−𝜏^∗), where 𝐿_𝑘 is a linear operator, 𝐴₁(𝑘)=𝜏⁰₁𝑃+𝑘010−𝜇𝛿^∗,𝐴₂(𝑘)=𝜏⁰₁,𝐴+𝑘00−𝜇𝑏0₃(𝑘)=𝜏⁰₁−+𝑘00𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗20. (41)

As 𝐿_𝑘 is a one-parameter family of bounded linear operator in 𝐶[−𝜏^∗,0], by the Rise Representation Theorem, there exists a matrix whose components are bounded variation functions 𝜂(𝜃,𝑘), 𝜃∈[−𝜏^∗,0] such that 𝐿_𝑘Φ=∫⁰_−𝜏∗Φ(𝜃)d𝜂(𝜃,𝑘), Φ∈𝐶[−𝜏^∗,0]. Actually, we can take 𝜂(𝜃,𝑘)=𝐴₁(𝑘)𝛿(𝜃)+𝐴₂(𝑘)𝛿(𝜃+1)+𝐴₃(𝑘)𝛿(𝜃+𝜏^∗), where 𝛿 is Dirac delta function.

Next, we define 𝐴(𝑘)(𝜃)Φ=dΦd𝜃,𝜃∈−𝜏^∗∫,0_0−𝜏∗d𝜂(𝜉,𝑘)Φ(𝜉)=𝐿_𝑘𝑅(𝑘)Φ,𝜃=0,Φ=0,𝜃∈−𝜏^∗𝑓(),0𝑘,Φ,𝜃=0, (42)(43) where Φ∈𝐶([−𝜏^∗,0],𝑅²). We can rewrite (38) as 𝜇^′_𝑡=𝐴(𝑘)𝜇_𝑡+𝑅(𝑘)𝜇_𝑡, (44) where 𝜇_𝑡=𝜇_𝑡(𝜃).

Denote 𝐴(0)=𝐴, 𝑅(0)=𝑅, 𝐴_𝑖(0)=𝐴_𝑖(𝑖=1,2,3), 𝜂(𝜃,0)=𝜂(𝜃).

For Ψ∈𝐶¹([0,𝜏^∗],𝑅²), the adjoint operator 𝐴^∗ of 𝐴 is defined as 𝐴^∗Ψ(𝑠)=−dΨ(𝑠)d𝑠,𝑠∈0,𝜏^∗,∫⁰_−𝜏∗d𝜂^𝑇(𝑡)Ψ(−𝑡),𝑠=0, (45) where 𝜂^𝑇 is the transpose of 𝜂.

We define the bilinear form ⟨Φ(𝑠),Ψ(𝜃)⟩=Ψ(0)^𝑇Φ(0)−⁰_−𝜏∗^𝜃₀Ψ(𝜉−𝜃)^𝑇d𝜂(𝜃)Φ(𝜉)d𝜉, (46) where Φ∈𝐶([−𝜏^∗,0],𝐶²) and Ψ∈𝐶([−𝜏^∗,0],(𝐶²)^∗).

We easily obtain that ±𝑖𝜏₀𝜔₀ are eigenvalues of (17) by the translation 𝑡→𝑡/𝜏₁. Then we have the following lemma.

Lemma 7. 𝑞(𝜃)=(1,𝛼)^𝑇𝑒^{𝑖𝜏01𝜔0𝜃} is the eigenvector corresponding to 𝑖𝜏₀𝜔₀ and 𝑞^∗(𝑠)=𝐷(1,𝛽)^𝑇𝑒^{𝑖𝜏01𝜔0𝑠} is the eigenvector of 𝐴^∗ corresponding to −𝑖𝜏₀𝜔₀. ⟨𝑞^∗(𝑠),𝑞(𝜃)⟩=1, ⟨𝑞^∗(𝑠),𝑞^∗(𝜃)⟩=0, where 𝛼=𝜇𝑏𝑒^{−𝑖𝜔0𝜏01}+𝜇𝑎₀𝑎₁/𝑎₁+𝑝^∗2𝑒^{−𝑖𝜔0𝜏01𝜏∗}𝜇𝛿𝑝^∗+𝑖𝜔₀,1𝛽=−𝜇𝛿𝑝^∗−𝑖𝜔₀,1𝐷=𝛽+𝛼−𝜇𝑏𝜏⁰₁𝑒^{−𝑖𝜔0𝜏01}−𝜏⁰₁𝜏^∗𝜇𝑎₀𝑎₁/𝑎₁+𝑝^∗2𝑒^{−𝑖𝜔0𝜏01𝜏∗}. (47)

Proof. Letting 𝑞(𝜃) be the eigenvector of corresponding to 𝑖𝜏₀𝜔₀, we have (𝜃)=(𝜃)𝐴𝑞d𝑞d𝜃=𝑖𝜏⁰₁𝜔₀𝑞(𝜃),𝜃∈−𝜏^∗,0. (48) Calculating (48), we obtain 𝑞(𝜃)=(1,𝛼)^𝑇𝑒^{𝑖𝜏01𝜔0𝜃},𝜃∈−𝜏^∗,0, (49) where 𝛼 is constant.
From (43), we can obtain 𝐴𝑞(0)=𝑖𝜏⁰₁𝜔₀𝑞(0)=⁰_−𝜏∗d𝜂(𝜉)(1,𝛼)^𝑇𝑒^{𝑖𝜏01𝜔0𝜉},𝜃=0. (50) Therefore, one can show that 𝛼=−𝜇𝑏𝑒^{−𝑖𝜔0𝜏01}+𝜇𝑎₀𝑎₁/𝑎₁+𝑝^∗2𝑒^{−𝑖𝜔0𝜏01𝜏∗}𝜇𝛿𝑝^∗+𝑖𝜔₀. (51)
Letting 𝑞^∗(𝑠)=𝐷(1,𝛽)^𝑇𝑒^{𝑖𝜏0𝜔0𝑠} be the eigenvector of 𝐴^∗ corresponding to −𝑖𝜏₀𝜔₀, based on (48), (50), we have 𝐴^∗𝑞^∗(0)=−𝑖𝜏⁰₁𝜔₀𝑞^∗(0)=⁰_−𝜏∗d𝜂^𝑇(𝑡)𝑞(−𝑡),𝜃=0. (52) It is easy to obtain that 𝛽=−1/(𝜇𝛿(𝑝^∗)−𝑖𝜔₀) and 𝑞^∗(0)=𝐷(1,−1/(𝜇𝛿(𝑝^∗)−𝑖𝜔₀)).
Now we compute ⟨𝑞^∗(𝑠),𝑞(𝜃)⟩ as the following: 𝑞^∗(𝑠)(𝜃=,𝑞)𝑞^∗(0)𝑞(0)−⁰_−𝜏∗^𝜃₀𝑞^∗()(𝜃)𝑞(𝜉)=𝜉−𝜃d𝜂d𝜉𝐷1,𝛽1𝛼−⁰_−𝜏∗𝐷1,𝛽𝑒^{𝑖𝜏01𝜔0(𝜃−𝜉)}(𝜃)×()d𝜂1,𝛼^𝑇𝑒^{𝑖𝜏01𝜔0𝜉}=d𝜉𝐷1,𝛽1𝛼−⁰_−𝜏∗𝐷1,𝛽d𝜂(𝜃)(1,𝛼)^𝑇𝜃𝑒^{𝑖𝜏01𝜔0𝜃}=𝐷1,𝛽1𝛼−[𝐷1,𝛽−𝜏^∗𝑒^{−𝑖𝜏01𝜔0𝜏∗}𝜏⁰₁×−00𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗201𝛼+𝐷1,𝛽(−1)𝑒^{−𝑖𝜏01𝜔0}𝜏⁰₁×1𝛼00−𝜇𝑏0=0. (53) Then we have 𝐷=1𝛽+𝛼−𝜇𝑏𝜏⁰₁𝑒^{−𝑖𝜔0𝜏01}−𝜏⁰₁𝜏^∗𝜇𝑎₀𝑎₁/𝑎₁+𝑝^∗2𝑒^{−𝑖𝜔0𝜏01𝜏∗}. (54) On the other hand, −𝑖𝜏⁰₁𝜔₀𝑞^∗(𝑠),𝑞(𝜃=𝑞_∗)(𝑠),𝐴𝑞(𝜃)=𝐴^∗𝑞^∗(𝑠),𝑞(𝜃=)−𝑖𝜏⁰₁𝜔₀𝑞^∗(𝑠),𝑞(𝜃)=𝑖𝜏⁰₁𝜔₀𝑞^∗(𝑠),𝑞(𝜃), (55) therefore, we have ⟨𝑞^∗(𝑠),𝑞(𝜃)⟩=0. This completes the proof.

In the remainder of this section, we use the same notation as in [12]. We first compute the center manifold 𝐶₀ at 𝑘=0.

Let 𝜇_𝑡 be the solution of (44) when 𝑘=0, and define 𝑊(𝑧,𝑧,𝜃)=𝜇_𝑡(𝜃)𝑧(𝑡)𝑞(𝜃,𝑧(𝑡)=𝑞−2Re)_∗,𝜇_𝑡. (56)(57) We have 𝑧^′(𝑡)=𝑖𝜏⁰₁𝜔₀𝑧+𝑞^∗(0)^𝑇𝑓₀(𝑧,𝑧), (58) where 𝑓₀(𝑧,𝑧)=𝑓0,𝑊(𝑧,𝑧,𝜃)+2Re𝑧𝑞(𝜃). (59) On the center manifold, we have 𝑊(𝑡,𝜃)=𝑊(𝑧,𝑧,𝜃)=𝑊₂₀(𝜃)𝑧²2+𝑊₁₁(𝜃)𝑧𝑧+𝑊₀₂(𝜃)𝑧²2⋯. (60) In fact, 𝑧, 𝑧 are local coordinates for center manifold 𝐶₀ in the direction of 𝑞^∗, 𝑞^∗. Noting that 𝑊 is real if 𝜇_𝑡 is real, we consider only real solutions in this paper. We rewrite (58) as follows: 𝑧^′(𝑡)=𝑖𝜏⁰₁𝜔₀𝑧+𝑔(𝑧,𝑧), (61) where 𝑔(𝑧,𝑧)=𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧2+⋯. (62) From (56), we have 𝜇_𝑡(𝜃)=𝑊(𝑧,)𝑧(𝑡)𝑞(𝜃(𝑧,𝜃+2Re)=𝑊𝑧,𝑧,𝜃)+𝑧(𝑡)𝑞(𝜃)+𝑧(𝑡)𝑞(𝜃). (63) Then we have 𝜇_1𝑡(𝜃)𝜇_2𝑡(𝜃)=1𝛼𝑒^{𝑖𝜏01𝜔0𝜃}𝑧(𝑡)+1𝛼𝑒^{−𝑖𝜏01𝜔0𝜃}𝑧(𝑡)+𝑊⁽¹⁾₂₀(𝜃)𝑊⁽²⁾₂₀(𝜃)𝑧²2+𝑊⁽¹⁾₁₁(𝜃)𝑊⁽²⁾₁₁(𝜃)𝑧𝑧+𝑊⁽¹⁾₀₂(𝜃)𝑊⁽²⁾₀₂(𝜃)𝑧²2+⋯. (64) It is easy to obtain 𝜇₁(𝑡)=𝑧(𝑡)+𝑧(𝑡)+𝑊⁽¹⁾₂₀(0)𝑧²2+𝑊⁽¹⁾₁₁(0)𝑧𝑧+𝑊⁽¹⁾₀₂(0)𝑧²2𝜇+⋯,₁(𝑡−1)=𝑧(𝑡)𝑒^{−𝑖𝜏01𝜔0}+𝑧(𝑡)𝑒^{𝑖𝜏01𝜔0}+𝑊⁽¹⁾₂₀()𝑧−1²2+𝑊⁽¹⁾₁₁(−1)𝑧𝑧+𝑊⁽¹⁾₀₂(−1)𝑧²2𝜇+⋯,₁𝑡−𝜏^∗=𝑧(𝑡)𝑒^{−𝑖𝜏01𝜔0𝜏∗}+𝑧(𝑡)𝑒^{𝑖𝜏01𝜔0𝜏∗}+𝑊⁽¹⁾₂₀−𝜏^∗𝑧²2+𝑊⁽¹⁾₁₁−𝜏^∗𝑧𝑧+𝑊⁽¹⁾₀₂−𝜏^∗𝑧²2𝜇+⋯,₂(𝑡)=𝛼𝑧+𝛼𝑧+𝑊⁽²⁾₂₀(0)𝑧²2+𝑊⁽²⁾₁₁(0)𝑧𝑧+𝑊⁽²⁾₀₂(0)𝑧²2+⋯. (65) We can also obtain 𝑞^∗(0)^𝑇𝑓₀(𝑧,𝑧)=𝐷1,𝛽𝜏⁰₁×0−𝜇𝛿^′𝑝^∗𝜇₁(𝑡)𝜇₂(𝑡)−𝜇𝛿^′′𝑝^∗𝜇2!²¹(𝑡)𝜇₂(𝑡)+𝜇𝑎₀𝑎₁3!𝑎₁+𝑝^∗3𝜇²₁𝑡−𝜏^∗. (66) Expanding (66) and comparing the coefficients, we obtain 𝑔₂₀=2𝐷𝜏₀−𝜇𝛿^′𝑝^∗𝛼+𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗3𝑒^{−2𝑖𝜏01𝜔0𝜏∗},𝑔₁₁=𝐷𝜏⁰₁−𝜇𝛿^′𝑝^∗(𝛼+𝛼)+2𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗3,𝑔₀₂=2𝐷𝜏⁰₁−𝜇𝛿^′𝑝^∗𝛼+𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗3𝑒^{2𝑖𝜏01𝜔0𝜏∗},𝑔₂₁=2𝐷𝜏⁰₁−𝜇𝛿^′𝑝^∗12𝑊⁽¹⁾₂₀(0)𝛼+𝑊⁽¹⁾₁₁(0)𝛼+𝑊⁽²⁾₁₁(0)+𝑊⁽²⁾₂₀(0)2−𝜇𝛿^′′𝑝^∗(2!2𝛼+𝛼)+𝜇𝑎₀𝑎₁𝑎₁+𝑃^∗3×𝑊⁽¹⁾₂₀−𝜏^∗𝑒^{𝑖𝜏01𝜏∗𝜔0}+2𝑊⁽¹⁾₁₁−𝜏^∗𝑒^{−𝑖𝜏01𝜏∗𝜔0}−3𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗4𝑒^{−𝜏01𝜏∗𝜔0}. (67)

We still need to compute 𝑊₂₀(𝜃) and 𝑊₁₁(𝜃). Noticing that 𝑊^′=𝜇^′_𝑡−𝑧^′𝑞(𝜃)−𝑧𝑞(𝜃)=𝐴𝜇_𝑡+𝑅𝜇_𝑡−𝑖𝜏⁰₁𝜔₀𝑧+𝑞^∗(0)𝑓₀(𝑧,𝑧)𝑞(𝜃)−−𝑖𝜏⁰₁𝜔₀𝑧+𝑞^∗(0)𝑓₀(𝑧,𝑧)𝑞(𝜃)=𝐴𝜇_𝑡+𝑅𝜇_𝑡−𝑖𝜏⁰₁𝜔₀𝑧𝑞(𝜃)−𝑞^∗(0)𝑓₀(𝑧,𝑧)𝑞(𝜃)+𝑖𝜏⁰₁𝜔₀𝑧𝑞(𝜃)−𝑞^∗(0)𝑓₀(𝑧,𝑧)𝑞(𝜃)=𝐴𝑊−2Re𝑞^∗(0)𝑓₀(𝑧,𝑧)𝑞(𝜃),𝜃∈−𝜏^∗{,0𝐴𝑊−2Re𝑞^∗(0)𝑓₀(𝑧,𝑧)𝑞(𝜃)+𝑓₀(𝑧,𝑧),𝜃=0, (68) we rewrite 𝑊^′ as the following: 𝑊^′=𝐴𝑊+𝐻(𝑧,𝑧,𝜃), (69) where 𝐻(𝑧,𝑧,𝜃)=𝐻₂₀(𝜃)(𝑧²/2)+𝐻₁₁(𝜃)𝑧𝑧+𝐻₀₂(𝜃)(𝑧²/2)+𝐻₃₀(𝜃)(𝑧³/6)+⋯

As 𝑊^′=𝑊_𝑧𝑧^′+𝑊_𝑧𝑧^′, we can get 𝑊^′=𝑊₂₀(𝜃)𝑧𝑖𝜏⁰₁𝜔₀𝑧+𝑔(𝑧,𝑧)+𝑊₁₁(𝜃)𝑧×−𝑖𝜏⁰₁𝜔₀𝑧+𝑔(𝑧,𝑧)+𝑊₁₁(𝜃)𝑧𝑖𝜏⁰₁𝜔₀𝑧+𝑔(𝑧,𝑧)+𝑊₀₂(𝜃)𝑧−𝑖𝜏⁰₁𝜔₀𝑧+𝑔(𝑧,𝑧)+⋯=𝑖𝜏⁰₁𝑊₂₀(𝜃)𝜔₀𝑧²+𝑊₂₀(𝜃)𝑧×𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧2⋯−𝑖𝜏⁰₁𝑊₁₁(𝜃)𝜔₀𝑧𝑧+𝑊₁₁(𝜃)𝑧×𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧2⋯+𝑖𝜏⁰₁𝑊₁₁(𝜃)𝜔₀𝑧𝑧+𝑊₁₁(𝜃)𝑧×𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧2⋯−𝑖𝜏⁰₁𝑊₀₂(𝜃)𝑧²𝜔₀+𝑊₀₂(𝜃)𝑧×𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧2⋯. (70) Comparing the coefficients with (69), we can get 𝐴−2𝑖𝜏⁰₁𝜔₀𝑊₂₀(𝜃)=−𝐻₂₀(𝜃),𝐴𝑊₁₁(𝜃)=−𝐻₁₁(𝜃). (71) From (69), 𝐻(𝑧,𝑧,𝜃)=−𝑔(𝑧,𝑧)𝑞(𝜃)−𝑔(𝑧,𝑧)𝑞(𝜃), where 𝜃∈[−𝜏^∗,0). Then we have 𝐻₂₀(𝜃)=−𝑔₂₀𝑞(𝜃)−𝑔₀₂𝑞(𝜃),𝐻₁₁(𝜃)=−𝑔₁₁𝑞(𝜃)−𝑔₁₁𝑞(𝜃). (72) From (43) and (71), we have 𝑊^′₂₀(𝜃)=2𝑖𝜏₀𝜔₀𝑊₂₀(𝜃)+𝑔₂₀𝑞(𝜃)+𝑔₀₂𝑞(𝜃),𝑊^′₁₁(𝜃)=𝑔₁₁𝑞(𝜃)+𝑔₁₁𝑞(𝜃), (73) where 𝜃∈[−𝜏^∗,0). Computing (73), we have 𝑊₂₀(𝜃)=𝑖𝑔₂₀𝑞(0)𝜏₀𝜔₀𝑒^{𝑖𝜏0𝜔0𝜃}+𝑖𝑔₀₂𝑞(0)3𝜏₀𝜔₀𝑒^{−𝑖𝜏0𝜔0𝜃}+𝐸₁𝑒^{2𝑖𝜏0𝜔0𝜃},𝑊₁₁(𝜃)=−𝑖𝑔₁₁𝑞(0)𝜏₀𝜔₀𝑒^{𝑖𝜏0𝜔0𝜃}+𝑖𝑔₁₁𝑞(0)𝜏₀𝜔₀𝑒^{−𝑖𝜏0𝜔0𝜃}+𝐸₂, (74)(75) where 𝐸₁=^𝐸1(1)_𝐸1(2), 𝐸₂=^𝐸2(1)_𝐸2(2).

𝐸₁, 𝐸₂ can be determined by setting 𝜃=0 in 𝐻(𝑧,𝑧,𝜃). As 𝑊₂₀(𝜃) and 𝑊₁₁(𝜃) are continuous on [−𝜏^∗,0], then we have 𝑊₂₀(0)=𝑖𝑔₂₀𝑞(0)𝜏₀𝜔₀+𝑖𝑔₀₂𝑞(0)3𝜏₀𝜔₀+𝐸₁,𝑊₁₁(0)=−𝑖𝑔₁₁𝑞(0)𝜏₀𝜔₀+𝑖𝑔₁₁𝑞(0)𝜏₀𝜔₀+𝐸₂. (76)(77) From (43), we obtain 𝐴𝑊₂₀(0)=⁰_−𝜏∗d𝜂(𝜉)𝑊₂₀(𝜃),𝐴𝑊₁₁(0)=⁰_−𝜏∗d𝜂(𝜉)𝑊₁₁(𝜃). (78) Substitute (74)–(77) to (78), we can obtain 2𝑖𝜔₀𝜏⁰₁𝐼−𝜏⁰₁𝐴₁+𝐴₂𝑒^{−2𝑖𝜏01𝜔0}+𝐴₃𝑒^{−2𝑖𝜏01𝜔0𝜏∗}𝐸₁=0−𝜇𝛿^′𝑝^∗𝛼+𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗3𝑒^{−2𝑖𝜏01𝜔0𝜏∗},𝜏⁰₁𝐴₁+𝐴₂+𝐴₃𝐸₂=0−𝜇𝛿^′𝑝^∗(𝛼+𝛼)+2𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗3. (79) Therefore, we obtain 𝐸₁=2𝑖𝜔₀𝜏⁰₁𝐼−𝜏⁰₁𝐴₁+𝐴₂𝑒^{−2𝑖𝜏01𝜔0}+𝐴₃𝑒^{−2𝑖𝜏01𝜔0𝜏∗−1}×0−𝜇𝛿^′𝑝^∗𝛼+𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗3𝑒^{−2𝑖𝜏01𝜔0𝜏∗},𝐸₂=𝜏⁰₁𝐴₁+𝐴₂+𝐴₃⁻¹×0−𝜇𝛿^′𝑝^∗(𝛼+𝛼)+2𝜇𝑎₀𝑎₁𝑎₁+𝑝^∗3. (80) Based on the previous analysis,we can see that each 𝑔_𝑖𝑗 in (67) is determined by the parameters in system (11). Thus we can compute the following values: 𝐶₁(0)=𝑖2𝜏⁰₁𝜔₀𝑔₁₁𝑔₂₀−2𝑔₁₁²−𝑔₀₂²3+𝑔₂₁2,𝜇₂𝐶=−Re₁(0)Re𝜆^′𝜏⁰₁,𝛽₂=2Re𝐶₁(0),𝑇₂𝐶=−Im₁(0)+𝜇₂Im𝜆^′𝜏₀𝜏⁰₁𝜔₀, (81) which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value 𝑘=0; that is, we thus have the following theorem.

Theorem 8. Using (81), one can compute the values of 𝐶₁(0), 𝜇₂, 𝛽₂, 𝑇₂. Therefore, we can answer the question of whether the bifurcation branch of periodic solution exists locally for supercritical bifurcation or subcritical bifurcation as the following.(i)𝜇₂ determines the directions of the Hopf bifurcation: if 𝜇₂>0 (𝜇₂<0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exists for 𝑘=0.(ii)𝛽₂ determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if 𝛽₂<0 (𝛽₂>0).(iii)𝑇₂ determines the period of the bifurcating periodic solutions: the period increases (decreases), if 𝑇₂>0 (𝑇₂<0).

5. Numerical Simulation Example

Example 9. Consider a single commodity market model as follows: d𝑥(𝑡)(𝑡),(𝑡)d𝑡=𝑦d𝑦((𝑡))𝑦(𝑡)d𝑡=−𝜇2𝑥+1−𝜇𝑏𝑥𝑡−𝜏₁𝑎−𝜇⁰𝑥𝑡−𝜏₂𝑎₁+𝑥𝑡−𝜏₂+𝜇𝑑₀−𝑔₀, (82) where 𝑥(𝑡) denotes the price at time 𝑡, and we choose 𝑑₀=125, 𝑏=2, 𝜇=0.08, 𝑔₀=85, 𝑎₀=180, 𝑎₁=12. Using Theorem 1, a quick computation revealed that 𝑝^∗=𝑥^∗=2.8292.
If one choose 𝜏₁=0, 𝜏₂>0, from Lemma 3, we can calculate that 𝜔=0.8875, 𝜏₂₀=0.7237, and 𝜏₂₀ is the the critical value for Hopf bifurcation. It follows from Theorem 5 that if 𝜏₂∈(0,0.7237), (𝑝^∗,0) is asymptotically stable. System (82) undergoes a Hopf bifurcation at 𝜏₂=0.7237. These conclusions are verified by the numerical simulation in Figures 1, 2, 3, and 4.
When 𝜏₂>𝜏₁>0, we can choose 𝜏₂=𝜏=0.68; from (34) and (35), we can calculate that 𝜔₀=0.9203, 𝜏⁰₁=0.2046; from (37), we have 𝛼^′(𝜏⁰₁)=0.0618. It follows from Theorem 6 that (𝑝^∗,0) is asymptotically stable as far as 𝜏⁰₁∈[0,0.2046) and system (82) undergoes a Hopf bifurcation when 𝜏⁰₁=0.2046. Furthermore, using Lemma 7, after simple computation, we have 𝛼=0.0249+0.8772𝑖,𝛽=0.4512−0.7308𝑖,𝐷=0.0029+0.5189𝑖. (83) Then we can obtain 𝑔₂₀=−0.0256−0.0033𝑖,𝑔₁₁𝑔=0.0001−0.0108𝑖,₀₂=0.0256−0.0030𝑖,𝑔₂₁=0.0005+0.0020𝑖. (84) It is easy to obtain 𝐶₁(0)=−0.00061286−0.00029893𝑖,𝜇₂𝛽=0.0099>0,₂=−0.0012<0. (85) Therefore, the Hopf bifurcation of system (82) is supercritical, and the bifurcation periodic solutions are stable. This conclusions can be verified by the numerical simulation in Figures 5, 6, 7, and 8.

Figure 1 

𝜏1=0, 𝜏2=0.68, (𝑝∗,0) is asymptotically stable.

Figure 2 

𝜏1=0, 𝜏2=0.68, (𝑝∗,0) is asymptotically stable.

Figure 3 

𝜏1=0, 𝜏2=0.7273, periodic solution bifurcates from (𝑝∗,0).

Figure 4 

𝜏1=0, 𝜏2=0.7237, (𝑝∗,0), periodic solution bifurcates from (𝑝∗,0).

Figure 5 

𝜏1=0.15, 𝜏2=0.68, (𝑝∗,0) is asymptotically stable.

Figure 6 

𝜏1=0.15, 𝜏2=0.68, (𝑝∗,0) is asymptotically stable.

Figure 7 

𝜏1=0.2046, 𝜏2=0.68, periodic solution bifurcates from (𝑝∗,0).

Figure 8 

𝜏1=0.2046, 𝜏2=0.68, (𝑝∗,0), periodic solution bifurcates from (𝑝∗,0).

6. Conclusion

The delay in production process has been considered in many economic models. However, few of them considered the delay of consumer consumption. In this paper, we first establish a class of economic models with two delays; the delay in production process and consumer consumption are both considered, and then the dynamics of this system have been investigated. Specifically, stability of equilibrium point and the existence of Hopf bifurcation, in great detail, are studied. We have derived some sufficient conditions to ensure that all the characteristic roots have negative real parts. We also show that a Hopf Bifurcation will occur once some parameters pass through critical values; that is, a family of periodic orbits bifurcates from the positive equilibrium point. At last, the direction of Hopf bifurcation and the stability of the bifurcating periodic orbits are discussed by applying the normal form theory and the center manifold theorem. Simulations show that the theoretically predicted values are in excellent agreement with the numerically observed behavior. Our results extend and complement some earlier publications.

Acknowledgments

The authors are extremely grateful to the editor and the anonymous reviewers for their constructive and valuable comments, which have contributed much to the improvement of this paper. This work was jointly supported by the National Natural Science Foundation of China under Grants nos. 11101053, 70921001, and 71171024, the Key Project of Chinese Ministry of Education under Grant no. 211118, the Excellent Youth Foundation of Educational Committee of Hunan Provincial no. 10B002, and the Scientific Research Funds of Hunan Provincial Science and Technology Department of China.