Stability Switches and Hopf Bifurcation in a Kaleckian Model of Business Cycle

Abstract

This paper considers a Kaleckian type model of business cycle based on a nonlinear delay differential equation, whose associated characteristic equation is a transcendental equation with delay dependent coefficients. Using the conventional analysis introduced by Beretta and Kuang (2002), we show that the unique equilibrium can be destabilized through a Hopf bifurcation and stability switches may occur. Then some properties of Hopf bifurcation such as direction, stability, and period are determined by the normal form theory and the center manifold theorem.

1. Introduction

The fact that transformation of inputs into outputs does not occur instantaneously but exhibits a gestation lag was first approached formally and systematically by Kalecki [1], who introduced a time delay between the moment when an investment decision is made and the delivering of the finished real investment. In his analysis, Kalecki showed that a gestation period can cause cycles in the economy. The mathematical description of Kalecki's model of business cycle leads to the following differential equation with delay, which describes the change in net investment: 𝐾̇(𝑡)=𝑎𝜏𝐾(𝑡)−𝑎𝜏+𝑚𝐾(𝑡−𝜏), (1) where 𝑡 is time, 𝐾 represents the net level of investment, 𝜏 is the delay between taking the decision to invest and carrying it out, and 𝑚 is the degree of effect of the capital stock on profit rates. The term 𝑎 is the percentage of profits reinvested, usually ranging from 0.8 to 0.95. Henceforth, we assume 𝑎<1. Delay differential equations have been widely employed for economic modeling in the last twenty years. In particular, it has been shown by many authors that the introduction of a delay has the advantage of producing more complex dynamics. Among others, we recall Asea and Zak [2], Bianca et al. [3, 4], Bianca and Guerrini [5], Szydłowski and Krawiec [6], Matsumoto and Szidarovszky [7], Szydłowski [8, 9], and Zak [10]. Their results indicate that the time delay is an important control parameter in economic processes and can often cause an otherwise stable system to oscillate.

Recently, Casal and Dibeh [11] made the linear model in (1) nonlinear by assuming that the negative effect of the capital stock, which is represented in (1) by 𝑚𝐾(𝑡−𝜏), increases disproportionally as the capital stock builds up. Specifically, they considered the nonlinear effect of 𝐾 on profitability and, hence, on investment to be represented by the function 𝐺(𝐾(𝑡),𝐾(𝑡−𝜏))=𝑚𝐾(𝑡−𝜏)+𝑛𝐾(𝑡)+𝜀𝐾(𝑡)³, (2) where 𝑛 is the degree of responsiveness of investment to the level of capital stock, and 𝜀 is a small parameter. Under assumption (2), the economy happens to be described by the following delay differential equation: 𝐾̇(𝑡)=𝑎𝜏𝐾(𝑡)−𝑎−𝑛𝜏+𝑚𝐾(𝑡−𝜏)−𝜀𝐾(𝑡)³. (3) Notice that when 𝑛=𝜀=0, (3) reduces to the Kalecki's model (1).

Casal and Dibeh [11] solved (3) by using approximation methods developed by Kryloff and Bogoliuboff [12] and applied to delay differential equations by Clark and Sebasta [13]. Moreover, Deeba et al. [14] worked out (3) using the Adomian decomposition algorithm. Limit cycle solutions, taking positive, zero, and negative values, were found for a wide range of the model's parameters. However, we remark that the dynamic analysis of (3) is still far from being complete.

The purpose of this paper is to perform a thorough analysis on the stability switches and Hopf bifurcation of the nonlinear model (3) when the time delay is taken as the bifurcation parameter. First of all, the linear stability about the trivial equilibrium is investigated by considering the characteristic transcendental equation associated with the linearized Kaleckian model. Unfortunately, the coefficients of this equation (and not only the transcendental term in it) depend on the delay, which makes the stability analysis rather complicated from the mathematical standpoint. However, a method that combines graphical information with analytical work has been developed by Beretta and Kuang [15] in order to determine stability switches for delay equations, which do not lend themselves to classical methods because of the aforementioned difficulties. Such an approach allows us to show that stability switches and Hopf bifurcations may occur as the delay increases. In addition, using the normal form reduction and the center manifold theory, suitable formulas for determining the direction of Hopf bifurcation and the stability of bifurcation periodic solution are obtained. Finally, some numerical simulations are performed to illustrate our qualitative results.

2. Local Stability and Existence of Hopf Bifurcation

Equation (3) is a delay differential equation with delay dependent coefficients. To investigate the local asymptotic stability of its unique steady state 𝐾_∗=0, we consider the linear part of (3) at this equilibrium. This gives 𝐾̇(𝑡)=𝑎𝜏𝐾(𝑡)−𝑎−𝑛𝜏+𝑚𝐾(𝑡−𝜏). (4) The corresponding characteristic equation, obtained by substituting 𝐾(𝑡)=𝑒^𝜆𝑡 into (4), is 𝑃()𝑎𝜆,𝜏=𝜆−𝜏+𝑎−𝑛𝜏+𝑚𝑒^−𝜆𝜏=0. (5) Equation (5) is a transcendental equation having, in general, an infinite number of complex roots. It is well known that the equilibrium of (4) is locally asymptotically stable if each of the characteristic roots of (5) has negative real part. Thus, the marginal stability is determined by 𝜆=0 and 𝜆=𝑖𝜔(𝜔>0). We start our analysis with the case of no delay.

Lemma 1. The equilibrium point is locally asymptotically stable when the delay is very small.

Proof. An application of L’Hopital’s rule gives lim_𝜏→0+𝑃(𝜆,𝜏)=𝜆+𝑛+lim_𝜏→0+𝑎−1+𝑒^−𝜆𝜏𝜏+𝑚lim_𝜏→0+𝑒^−𝜆𝜏=0⇒𝜆=−𝑚+𝑛1−𝑎<0. (6) Thus, the statement holds.

Let 𝜏>0. As 𝜏 varies, stability switches may occur in correspondence with any critical value of 𝜏 at which a root of (5) transitions from having negative to having positive or null real part. If this occurs, then there must be a critical value of 𝜏 such that the characteristic equation has a purely imaginary root. Notice that the case 𝜆=0 does not need to be considered since it is not a root for (5). In fact, 𝑃(0,𝜏)=𝑛+𝑚≠0. We begin by looking for a purely imaginary root of (5). Let 𝜆=𝑖𝜔, with 𝜔>0, be a solution of (5). Then 𝑃()𝑎𝑖𝜔,𝜏=𝑖𝜔−𝜏+𝑎−𝑛𝜏+𝑚𝑒^{−𝑖𝜔𝜏}=0. (7) Breaking the polynomial up into its real and imaginary parts, and writing the exponential in (7) in terms of trigonometric functions, we get 𝑎𝜔=𝜏𝑎+𝑚sin𝜔𝜏,𝜏𝑎−𝑛=𝜏+𝑚cos𝜔𝜏. (8) Squaring each equation in (8) and summing the results yields 𝐹(𝜔,𝜏)≡𝜔²−𝑎𝜏+𝑚²+𝑎𝜏−𝑛²=0. (9)

Lemma 2. Let ()(𝐼=0,+∞,𝑖𝑓𝑛≤𝑚,0,𝜏),𝑖𝑓𝑛>𝑚, (10) where 𝜏=2𝑎.𝑛−𝑚 (11) For 𝜏∈𝐼, (9) has a unique positive root given by 𝜔(𝜏)=(𝑚+𝑛)𝑚−𝑛+2𝑎𝜏. (12)

Proof. If 𝐹(𝜔,𝜏)=0 has positive roots, then one has 𝑎𝜏+𝑚²−𝑎𝜏−𝑛²=(𝑚+𝑛)𝑚−𝑛+2𝑎𝜏>0⇒𝑚−𝑛+2𝑎𝜏>0, (13) completing the proof.

We have shown that, for 𝜏∈𝐼, there exists an 𝜔=𝜔(𝜏)>0, given by (12), such that 𝐹(𝜔,𝜏)=0. Let 𝜃(𝜏)∈(0,𝜋) be defined for 𝜏∈𝐼 by (𝜃(𝜏=𝜔(𝜏)sin))()(𝜃(𝜏=()𝑎/𝜏+𝑚,cos))𝑎/𝜏−𝑛().𝑎/𝜏+𝑚 (14) Then, 𝜔(𝜏)𝜏=𝜃(𝜏)+2𝑗𝜋, 𝑗∈ℕ₀=ℕ∪{0}. Following Beretta and Kuang [15], we introduce the maps 𝜏_𝑗∶𝐼→ℝ⁺₀, 𝜏_𝑗(𝜏)=𝜃(𝜏)+2𝑗𝜋𝜔(𝜏),𝑗∈ℕ₀,𝜏∈𝐼. (15) The expression of 𝜏_𝑗(𝜏) can be obtained from (14) as follows: 𝜏_𝑗(𝜏)=1𝜔(𝜏)tan⁻¹𝜔(𝜏)()𝑎𝑎/𝜏−𝑛+2𝑗𝜋,if𝜏≤𝑛,1𝜔(𝜏)tan⁻¹𝜔(𝜏)()+𝜋𝑎𝑎/𝜏−𝑛2𝑗+1,if𝜏>𝑛. (16)

Remark 3. When 𝑛>𝑚, 𝜏=2𝑎/(𝑛−𝑚)>𝑎/𝑛, so that (16) leads to the cases 𝜏<𝑎/𝑛 and 𝑎/𝑛<𝜏<𝜏.

As a result, 𝑖𝜔_∗, with 𝜔_∗=𝜔(𝜏_∗)>0, is a purely imaginary root of (7) if and only if 𝜏_∗ is a zero of the function 𝑆_𝑗(𝜏), for some 𝑗∈ℕ₀, where 𝑆_𝑗(𝜏) is defined by 𝑆_𝑗(𝜏)=𝜏−𝜏_𝑗(𝜏),𝑗∈ℕ₀,𝜏∈𝐼. (17)

Proposition 4. Assume that the equation 𝑆_𝑗(𝜏)=0 has a positive root 𝜏_∗∈𝐼 for some 𝑗∈ℕ₀. Then there exists a pair of simple purely imaginary roots 𝜆=±𝑖𝜔(𝜏_∗) of (5) at 𝜏=𝜏_∗, crossing the imaginary axis from left to right if Γ(𝜏_∗)>0 and from right to left if Γ(𝜏_∗)<0, where Γ𝜏_∗𝑑()=signRe𝜆𝑑𝜏_{𝜆=𝑖𝜔∗}=sign𝑑𝑆_𝑗(𝜏)𝑑𝜏_𝜏=𝜏∗. (18)

Proof. We start proving that 𝜆=𝑖𝜔_∗ is a simple root for (5). If 𝜆=𝑖𝜔_∗ has repeated roots, then 𝑑𝑃(𝑖𝜔_∗,𝜏_∗)/𝑑𝜆=0 gives rise to the contradiction 𝜔_∗=0. Differentiating (5) with respect to 𝜏 we find that 𝑑𝜆=−𝑎𝑑𝜏𝜏²−𝑎𝜏²𝜆−(𝑎/𝜏)+𝑛()𝑎𝑎/𝜏+𝑚−𝜆𝜏−𝜏𝑎+𝑛1+𝜏𝜆−𝜏+𝑛⁻¹. (19) We remark that 𝑑()signRe𝜆𝑑𝜏_{𝜆=𝑖𝜔∗}=signRe𝑑𝜆𝑑𝜏⁻¹_{𝜆=𝑖𝜔∗}. (20) From (12) we have 𝜔(𝜏)𝜔^′(𝜏)()𝑎=−𝑚+𝑛𝜏², (21) so that (19) yields Re𝑑𝜆𝑑𝜏⁻¹_{𝜆=𝑖𝜔∗}=Re𝑎/𝜏_∗−𝑛−𝜏_∗𝑎/𝜏_∗+𝑚²+𝑖𝜔_∗−𝜔_∗𝜔^′_∗+𝑖𝜔_∗−𝑎/𝜏²_∗+𝑎/𝜏_∗+𝑚²=−𝜔_∗𝜔^′_∗𝑎𝜏_∗−𝑛−𝜏_∗𝑎𝜏_∗+𝑚²+𝜔²_∗−𝑎𝜏²_∗+𝑎𝜏_∗+𝑚²×𝜔_∗𝜔^′_∗²+𝜔²_∗−𝑎𝜏²_∗+𝑎𝜏_∗+𝑚²²⁻¹. (22) Next, differentiating (16) and (17) with respect to 𝜏 gives 𝑑𝑆_𝑗𝜏_∗𝑑𝜏=1−𝑑𝜏_𝑗𝜏_∗𝑎𝑑𝜏=1−𝜏²_∗𝜔²_∗−−𝑎𝜏_∗+𝑎−𝑛𝜏_∗+𝑚²𝜏_∗×𝜔_∗𝜔^′_∗𝑎𝜏_∗+𝑚²𝜔²_∗⁻¹=𝑎𝜏_∗+𝑚²𝜔²_∗−𝑎𝜏²_∗𝜔²_∗+−𝑎𝜏_∗+𝑎−𝑛𝜏_∗+𝑚²𝜏_∗𝜔_∗𝜔^′_∗×𝑎𝜏_∗+𝑚²𝜔²_∗⁻¹. (23) A comparison of (22) and (23) leads to the conclusion.

According to the previous result, the knowledge of the geometric shape of the functions 𝑆_𝑗(𝜏), 𝜏∈𝐼, the location of their zeros 𝜏_∗, and the sign of Γ(𝜏_∗) allow us to determine at which delay values stability switching of the equilibrium eventually occur.

The following properties of 𝑆_𝑗(𝜏) can be verified using (17):(1)𝑆_𝑗(𝜏)>𝑆_𝑗+1(𝜏), 𝑗∈ℕ₀. Hence, if 𝑆₀ has no zeros in 𝐼, nor does 𝑆_𝑗 for all 𝑗>0.(2) Let 𝑛>𝑚. Then 𝐼=(0,𝜏). As 𝜏→𝜏⁻, 𝜔(𝜏)→0, while sin(𝜗(𝜏))→0 and cos(𝜗(𝜏))→−1. As a consequence, 𝜗(𝜏)→𝜋 and 𝑆_𝑗(𝜏)→−∞ as 𝜏→𝜏⁻.(3) Let 𝑛≤𝑚. Then 𝐼=(0,+∞). As 𝜏→+∞, 𝜔(𝜏)→√𝑚²−𝑛²≥0, while 𝜏_𝑗(𝜏) tends to a positive number. Hence, 𝑆_𝑗(𝜏)→+∞ as 𝜏→+∞.(4) Let 𝜏∈𝐼. As 𝜏→0⁺,𝜔(𝜏)→+∞ yielding 𝑆_𝑗(𝜏)→0 for 𝑗∈ℕ₀.(5) Let 𝜏∈𝐼 be small, with 𝜏≤𝑎/𝑛. An application of L’Hopital's rule implies 𝜔(𝜏)/(𝑎/𝜏−𝑛)→0 as 𝜏→0⁺. Hence, a Taylor series expansion gives 𝜏₀(𝜏)=1/(𝑎/𝜏−𝑛). From this we derive that 𝑑𝑆₀(𝜏)/𝑑𝜏<0 if 𝜏≤𝑎/𝑛<2𝑎/𝑛.

According to the above analysis, we state the following conclusions for (3).

Theorem 5. (1) Let 𝐼=(0,𝜏) (case 𝑛>𝑚). If the function 𝑆₀(𝜏) has no zeros in 𝐼, then the equilibrium 𝐾_∗=0 is locally asymptotically stable for all 𝜏≥0.
(2) Let 𝐼=(0,+∞) (case 𝑛≤𝑚). The function 𝑆_𝑗(𝜏), 𝑗∈ℕ₀, has a simple zero in 𝐼.
(3) If the function 𝑆_𝑗(𝜏) has a finite number of simple zeros in 𝐼, for some fixed 𝑗∈ℕ₀, then they must be even if 𝐼=(0,𝜏), say 𝜏₁<𝜏₂<⋯<𝜏_2𝑟, and odd if 𝐼=(0,+∞), say 𝜏₁<𝜏₂<⋯<𝜏_2𝑟+1. In both cases, the equilibrium 𝐾_∗=0 is locally asymptotically stable if 𝜏∈(𝜏_2𝑘,𝜏_2𝑘+1)(𝑘=0,1,2,…,𝑟;𝜏₀=0) and unstable otherwise. Furthermore, a Hopf bifurcation occurs at 𝜏₁,𝜏₂,…,𝜏_2𝑟+1.

3. Properties of Hopf Bifurcation

In the previous section, we have obtained the conditions under which a family of periodic solutions bifurcates from the equilibrium 𝐾_∗=0 at the critical value 𝜏_∗. In this section, we wills study the direction, stability, and period of the Hopf bifurcation. Following the ideas of Hassard et al. [16], we derive the explicit formulae for determining the properties of the Hopf bifurcation at the critical value of 𝜏_∗ by using the normal form and the center manifold theory. Throughout this section, we always assume that (3) undergoes Hopf bifurcation at the equilibrium 𝐾_∗=0 for 𝜏=𝜏_∗, and then ±𝑖𝜔(𝜏_∗) are the corresponding purely imaginary roots of the characteristic equation at the equilibrium 𝐾_∗=0. For notational convenience, let 𝜏=𝜏_∗+𝜇,𝜇∈ℝ. Then 𝜇=0 is the Hopf bifurcation point for (3). Define by 𝐶=𝐶([−𝜏_∗,0],ℝ) the Banach space of continuous real-valued functions that map [−𝜏_∗,0] into ℝ. Then (3) can be written as a functional differential equation in the phase space 𝐶. Let 𝜑∈𝐶 and define the linear operator: 𝐿_𝜇𝜑=𝑎𝜏𝜑(0)−𝑎−𝑛𝜏+𝑚𝜑(−𝜏), (24) and the nonlinear operator: 𝑓𝜇,𝜑=−𝜀𝜑(0)³. (25) There is a bounded variation function 𝜂(𝜗,𝜇) with 𝜗∈[−𝜏_∗,0] such that 𝐿_𝜇𝜑=⁰_−𝜏∗𝑑𝜂𝜗,𝜇𝜑(𝜗). (26) In fact, we can choose 𝑑𝜂(𝜗,𝜇)=[(𝑎/𝜏−𝑛)𝛿(𝜗)−(𝑎/𝜏+𝑚)𝛿(𝜗+𝜏)]𝑑𝜗, where 𝛿(𝜗) is the Dirac delta function. Define operators 𝐴𝜇(𝜗)𝜑=𝑑𝜑𝑑𝜗,𝜗∈−𝜏_∗,0_0−𝜏∗,𝜑(𝑠)𝑅𝜇𝑑𝜂𝑠,𝜇,𝜗=0,𝜑=0,𝜗∈−𝜏_∗,𝑓,0𝜇,𝜑,𝜗=0. (27) Then (3) can be rewritten as follows: 𝑢̇_𝑡=𝐴𝜇𝑢_𝑡+𝑅𝜇𝑢_𝑡, (28) where 𝑢_𝑡(𝜗)=𝑢(𝑡+𝜗) for 𝜗∈[−𝜏_∗,0]. Let 𝐶=𝐶([0,𝜏_∗],ℝ). For 𝜓∈𝐶, define the operator 𝐴^∗ as 𝐴^∗𝜇𝜓(𝑠)=−𝑑𝜓(𝑠)𝑑𝑠,𝑠∈0,𝜏_∗,_0−𝜏∗𝑑𝜂𝑟,𝜇𝜓(−𝑟),𝑠=0. (29) Next, define the bilinear inner product of 𝜑∈𝐶 and 𝜓∈𝐶 as =𝜓,𝜑𝜓(0)𝜑(0)−⁰_{𝜗=−𝜏∗}^𝜗_𝜉=0𝜓(𝜉−𝜗)𝑑𝜂(𝜗,0)𝜑(𝜉)𝑑𝜉, (30) where the overbar denotes complex conjugation. Then 𝐴(0) and 𝐴^∗(0) are adjoint operators, whose eigenvalues are 𝑖𝜔(𝜏_∗) and −𝑖𝜔(𝜏_∗), respectively. A direct computation shows that the eigenvectors associated with them are 𝑞(𝜗)=𝑒^{𝑖𝜔(𝜏∗)𝜗},𝜗∈−𝜏_∗,𝑞,0_∗(𝑠)=𝐵𝑒^{𝑖𝜔(𝜏∗)𝑠},𝑠∈0,𝜏_∗, (31) respectively, where 1𝐵=1−((𝑎/𝜏)+𝑚)𝜏_∗𝑒^{𝑖𝜔(𝜏∗)𝜏∗}. (32) Notice that ⟨𝑞^∗,𝑞⟩=0. In addition, this value of 𝐵 guarantees that ⟨𝑞^∗,𝑞⟩=1. In the remainder of this section, following the theory and the same notations as in Hassard et al. [16], we compute the coordinates to describe the center manifold 𝒞 at 𝜇=0. Let 𝑢_𝑡 be the solution of (3) when 𝜇=0. Define 𝑧(𝑡)=𝑞^∗,𝑢_𝑡,𝑊(𝑡,𝜗)=𝑢_𝑡(𝜗)−2Re𝑧(𝑡)𝑞(𝜗). (33) On the center manifold 𝒞, we have 𝑊(𝑡,𝜗)=𝑊(𝑧(𝑡),𝑧(𝑡),𝜗), where 𝑊(𝑧,𝑧,𝜗)=𝑊₂₀(𝜗)𝑧²2+𝑊₁₁(𝜗)𝑧𝑧+𝑊₀₂(𝜗)𝑧²2+⋯. (34) In the above relations 𝑧 and 𝑧 are local coordinates for the center manifold 𝒞 in the direction of 𝑞^∗ and 𝑞^∗. From (27) for solutions 𝑢_𝑡∈𝒞, since 𝜇=0, we have 𝑧̇(𝑡)=𝑞^∗,𝑢̇_𝑡=𝑞^∗,𝐴𝜇𝑢_𝑡+𝑞^∗,𝑅𝜇𝑢_𝑡=𝑖𝜔𝜏_∗𝑧+𝑞^∗(0)𝑓₀(𝑧,𝑧), (35) with 𝑓₀(𝑧,𝑧)=𝑓0,𝑊(𝑧,𝑧,0)+2Re𝑧(𝑡)𝑞(0). (36) Equation (35) can be rewritten as 𝑧̇(𝑡)=𝑖𝜔𝜏_∗𝑧(𝑡)+𝑔(𝑧,𝑧), (37) where 𝑔(𝑧,𝑧)=𝑞^∗(0)𝑓0,𝑊(𝑧,𝑧,0)+2Re𝑧(𝑡)𝑞(0)=𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧6+⋯. (38) Hence, (35) and (38) yield 𝑊̇=𝑢̇_𝑡−𝑧𝑞̇−𝑧̇𝑞=𝐴(0)𝑊−2Re𝑞^∗(0)𝑓₀𝑞(𝜗),𝜗∈−𝜏_∗,𝐴(0),0𝑊−2Re𝑞^∗(0)𝑓₀𝑞(0)+𝑓₀(,𝜗=0,=𝐴𝑊+𝐻𝑧,𝑧,𝜗), (39) where 𝐻(𝑧,𝑧,𝜗)=𝐻₂₀(𝜗)𝑧²2+𝐻₁₁(𝜗)𝑧𝑧+𝐻₀₂(𝜗)𝑧²2+⋯. (40) Noticing that 𝑢_𝑡(𝜗)=𝑊(𝑧,𝑧,𝜗)+𝑧𝑞(𝜗)+𝑧𝑞(𝜗), we get 𝑢_𝑡(0)=𝑊(𝑧,𝑧,0)+𝑧+𝑧. Thus, 𝑓₀(𝑧,𝑧)=𝑓0,𝑊(𝑧,)𝑧(𝑡)𝑞(0[𝑊(𝑧,0+2Re)=−𝜀𝑧,𝑧,0)+𝑧+𝑧]³=(−3𝜀)𝑧²𝑧+(−3𝜀)𝑧𝑧²+⋯. (41) On the other hand, we have 𝑔(𝑧,𝑧)=𝑞^∗(0)𝑓₀(𝑧,𝑧)=𝑞^∗(0)𝑓0,𝑢_𝑡=𝐵−𝜀𝑢_𝑡(0)³=𝐵(−𝜀)𝑧²𝑧+(−𝜀)𝑧𝑧²+⋯=𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧6+⋯. (42) Comparing these coefficients we find 𝑔₂₀=𝑔₁₁=𝑔₀₂𝑔=0,₂₁=−6𝜀𝐵=−6𝜀1−((𝑎/𝜏)+𝑚)𝜏_∗𝑒^{−𝑖𝜔(𝜏∗)𝜏∗}. (43) Based on the above analysis, each 𝑔_𝑖𝑗 is computed and we can calculate the following quantities: 𝑐₁(0)=𝑖2𝜔𝜏_∗𝑔₁₁𝑔₂₀−2𝑔₁₁²−𝑔₀₂²3+𝑔₂₁2,𝜇₂𝑐=−Re₁(0)Re𝜆^′𝜏_∗,𝛽₂=2Re𝑐₁(0,𝑇₂)𝑐=−𝐼𝑚₁(0)+𝜇₂𝐼𝑚𝜆^′𝜏_∗𝜔𝜏_∗, (44) which determine the properties of bifurcating periodic solutions. Specifically, 𝜇₂,𝛽₂, and 𝑇₂ determine the direction, stability, and period of the corresponding Hopf bifurcation, respectively (see Hassard et al. [16]). A direct computation shows 𝑐₁(0)=𝑔₂₁2=−3𝜀1−𝑎/𝜏_∗+𝑚𝜏_∗cos𝜔𝜏_∗𝜏_∗𝑚²𝜏²_∗+2(𝑎𝑚+𝑛)𝜏_∗+(1−𝑎)²+𝑖3𝜀𝑎/𝜏_∗+𝑚𝜏_∗cos𝜔𝜏_∗𝜏_∗𝑚²𝜏²_∗+2(𝑎𝑚+𝑛)𝜏_∗+(1−𝑎)², (45) so that using (8) we get 𝛽₂=−6𝜀1−𝑎+𝑛𝜏_∗𝑚²𝜏²_∗+2(𝑎𝑚+𝑛)𝜏_∗+(1−𝑎)²<0. (46)

Theorem 6. (1) The direction of the Hopf bifurcation of (3) at the trivial equilibrium 𝐾_∗=0 when 𝜏=𝜏_∗ is subcritical (resp., supercritical) if 𝜇₂<0 (resp., 𝜇₂>0); that is, there exists a bifurcating periodic solution for 𝜏<𝜏_∗ (resp. 𝜏>𝜏_∗) in the sufficiently small 𝜏_∗-neighborhood.
(2) The bifurcating periodic solution is locally asymptotically stable since 𝛽₂<0.
(3) The period of the bifurcating periodic solution decreases (resp., increases) with respect to 𝜏 if𝑇₂<0 (resp., 𝑇₂>0).

4. Numerical Simulations

In this section, we present numerical results to verify the analytical predictions obtained in the previous sections. We set two sets of parameters (𝑎, 𝑚, 𝑛, and 𝜀).(i)𝑎=0.95, 𝑚=0.1, 𝑛=0.15, 𝜀=0.01, and 𝜏=38.

𝑆₀(𝜏) has two positive roots in this case (see Figure 1(a)), say 𝜏₁≈4.5 and 𝜏₂≈11.8, while 𝑆₁(𝜏) is strictly negative (see Figure 1(b)). Hence, all 𝑆_𝑗(𝜏) functions with 𝑗≥1 have no real roots. The equilibrium is asymptotically stable when 𝜏∈[0,𝜏₁) (see Figure 2(a), where 𝜏=3) and 𝜏∈(𝜏₂,𝜏) (see Figure 2(b), where 𝜏=14). When 𝜏∈(𝜏₁,𝜏₂), the equilibrium is unstable (see Figure 3, where 𝜏=4.6). At 𝜏₁ and 𝜏₂, Hopf bifurcation occurs. Furthermore, we know that 𝛽₂<0 and Γ(𝜏_∗)>0. Therefore, the Hopf bifurcation at the equilibrium is supercritical and the bifurcating periodic solutions are orbitally asymptotically stable (see Figure 3, where 𝜏=4.6).(ii)𝑎=0.95, 𝑚=0.15, 𝑛=0.1, and 𝜀=0.01.

(a)

(b)

Figure 1 

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(b)

Figure 2 

Figure 3 

𝑆₀(𝜏) does intersect once with the 𝜏-axis at 𝜏₁≈0.76 (see Figure 4(a)) and 𝑆₁(𝜏) is strictly negative (see Figure 4(b)). Hence, the equilibrium is asymptotically stable for 𝜏∈[0,𝜏₁) (see Figure 5(a), where 𝜏=0.7), and sustained oscillations occur when 𝜏∈(𝜏₁,+∞) due to Hopf bifurcations (see Figure 5(b), where 𝜏=0.77). Also in this case, having 𝛽₂<0 and Γ(𝜏_∗)>0, the bifurcation is supercritical and the bifurcating periodic solution is orbitally asymptotically stable (see Figure 5(b), where 𝜏=0.77).

(a)

(b)

Figure 4 

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Figure 5