Electronic Journal of Qualitative Theory of Differential Equations Spec. Ed. I, 2009 No. 27, 1–20; http://www.math.u-szeged.hu/ejqtde/

BIFURCATION ANALYSIS OF A KALDOR-KALECKI MODEL OF BUSINESS CYCLE WITH TIME DELAY

Liancheng Wang ^{1} and Xiaoqin P. Wu ^{2}

1 Department of Mathematics & Statistics Kennesaw State University Kennesaw, GA 30144, USA e-mail: lwang5@kennesaw.edu 2 Department of Mathematics, Computer & Information Sciences Mississippi Valley State University Itta Bena, MS 38941, USA e-mail: xpaul wu@yahoo.com

Honoring the Career of John Graef on the Occasion of His Sixty-Seventh Birthday

Abstract In this paper, we investigate a Kaldor-Kalecki model of business cycle with delay in both the gross product and the capital stock. Stability analysis for the equilibrium point is carried out. We show that Hopf bifurcation occurs and periodic solutions emerge as the delay crosses some critical values. By deriving the normal forms for the system, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established. Examples are presented to confirm our results.

Key words and phrases: Kaldor-Kalecki model of business cycle, Hopf bifurcation, periodic solutions, stability. AMS (MOS) subject classifications: 34K18

1 Introduction

In this paper, we study the Kaldor-Kalecki model of business cycle with delay of the following form:

\{ \begin{matrix}\frac{dY (t)}{dt} = α[I(Y (t), K(t)) − S(Y (t), K(t))], \\ \frac{dK(t)}{dt} = I(Y (t − τ), K(t − τ)) − qK(t), \end{matrix}  (1)

where Y is the gross product, K is the capital stock, α > 0 is the adjustment coefficient in the goods market, q ∈ (0, 1) is the depreciation rate of capital stock, I(Y, K) and S(Y, K) are investment and saving functions, and τ ≥ 0 is a time lag representing delay for the investment due to the past investment decision.

A business model in this line was first proposed by Kalecki [11], in which the idea of a delay of the implementation of a business decision was introduced. Later on, Kalecki [12] and Kaldor [10] proposed and studied business models using ordinary differential equations and nonlinear investment and saving functions. They showed that periodic solutions exist under the assumption of nonlinearity. Similar models were also analyzed by several authors and the existence of limit cycles were established due to the nonlinearity, see [4, 7, 23]. Krawiec and Szydlowski [14, 15, 16] combined the two basic models of Kaldor’s and Kalecki’s and proposed the following Kaldor-Kalecki model of business cycle:

\{ \begin{matrix}\frac{dY (t)}{dt} = α[I(Y (t), K(t)) − S(Y (t), K(t))], \\ \frac{dK(t)}{dt} = I(Y (t − τ), K(t)) − qK(t). \end{matrix}

This model has been studied intensively since its introduction, see [17, 19, 20, 21, 22, 24]. It is argued that a more reasonable model should include delays in both the gross product and capital stock, because the change in the capital stock is also caused by the past investment decisions [17]. Adding a delay to capital stock K leads to System (1).

As in [14], also see [1, 2, 22], using the following saving and investment functions S and I, respectively,

S(Y, K) = γY, I(Y, K) = I(Y) − βK

where β > 0 and γ ∈ (0, 1) are constants, System (1) becomes the following system:

\{ \begin{matrix}\frac{dY (t)}{dt} = α[I(Y (t)) − βK(t) − γY (t)], \\ \frac{dK(t)}{dt} = I(Y (t − τ)) − βK(t − τ) − qK(t). \end{matrix}  (2)

Kaddar and Talibi Alaoui [9] studied System (2). They gave a condition for the characteristic equation of the linearized system to have a pair of purely imaginary roots and showed that the Hopf bifurcation may occur as the delay τ passes some critical values. However, they did not give the stability of the periodic solution and the direction of the Hopf bifurcation.

In this paper, we first give a more detailed discussion of the distribution of the eigenvalues of the linearized system of (2). So local stability of the equilibrium point is established. Conditions are found under which the Hopf bifurcation occurs and periodic solutions emerge as the delay crosses some critical values. By deriving the normal forms for System (2) using the normal form theory developed by Faria and Magalhães [5, 6], the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established. Finally, some examples are presented to illustrate our theoretical results.

2 Distribution of Eigenvalues

Throughout the rest of this paper, we assume that I(s) is a nonlinear function, C^{3}, and that System (2) has an isolated equilibrium point (Y^{∗}, K^{∗}). Let I^{∗} = I(Y^{∗}), u_{1} = Y − Y^{∗}, u_{2} = K − K^{∗}, and i(s) = I(s + Y^{∗}) − I^{∗}. Then System (2) can be transformed into

\{ \begin{matrix}\frac{du_{1}(t)}{dt} = α[i(u_{1}(t)) − βu_{2}(t) − γu_{1}(t)], \\ \frac{du_{2}(t)}{dt} = i(u_{1}(t − τ)) − βu_{2}(t − τ) − qu_{2}(t). \end{matrix}  (3)

Let the Taylor expansion of i at 0 be

i(u) = ku + i^{(2)}u^{2} + i^{(3)}u^{3} + O(|u|^{4})

where

k = i′(0) = I′(Y^{∗}), i^{(2)} = \frac{1}{2}i′′(0) = \frac{1}{2}I′′(Y^{∗}), i^{(3)} = \frac{1}{3!}i′′′(0) = \frac{1}{3!}I′′′(Y^{∗}).

The linear part of System (3) at (0, 0) is

\{ \begin{matrix}\frac{du_{1}(t)}{dt} = α[(k − γ)u_{1}(t) − βu_{2}(t)], \\ \frac{du_{2}(t)}{dt} = ku_{1}(t − τ) − βu_{2}(t − τ) − qu_{2}(t), \end{matrix}  (4)

and its corresponding characteristic equation is

λ^{2} + [q − α(k − γ)]λ − αq(k − γ) + (βλ + αβγ)e^{−λτ} = 0.  (5)

For τ = 0, Equation (5) becomes

λ^{2} + [q + β − α(k − γ)]λ − αq(k − γ) + αβγ = 0.  (6)

Define

k_{1} = \frac{βγ}{q} + γ, k_{2} = \frac{q + β}{α} + γ,

and for the rest of the paper, we always assume k_{1} ≤ k_{2}. For the case that k_{1} > k_{2}, the discussion can be carried out similarly.

Theorem 2.1. Let τ = 0. If k < k_{1}, all roots of Equation (6) have negative real parts, and hence (Y^{∗}, K^{∗}) is asymptotically stable. If k > k_{1}, Equation (6) has a positive root and a negative root, and hence (Y^{∗}, K^{∗}) is unstable.

Now assume τ > 0. Let ωi (ω > 0) be a purely imaginary root of Equation (5). After plugging it into Equation (5) and separating the real and imaginary parts, we have

ω^{2} + αq(k − γ) = αβγ cos(ωτ) + βω sin(ωτ),
[q − α(k − γ)]ω = αβγ sin(ωτ) − βω cos(ωτ). (7)

Adding squares of two equations yields

ω^{4} + [q^{2} − β^{2} + α^{2}(k − γ)^{2}]ω^{2} + α^{2}q^{2}(k − γ)^{2} − α^{2}β^{2}γ^{2} = 0.  (8)

Let

A = q^{2} − β^{2} + α^{2}(k − γ)^{2},
B = α^{2}q^{2}(k − γ)^{2} − α^{2}β^{2}γ^{2}.

If A ≥ 0 and B ≥ 0, Equation (8) has no positive roots. If B < 0, Equation (8) has a unique positive root

ω_{+} = \sqrt{\frac{−A + \sqrt{A^{2} − 4B}}{2}} .

If A < 0, B > 0, and A^{2} − 4B > 0, Equation (8) has two positive roots

ω_{±} = \sqrt{\frac{−A ± \sqrt{A^{2} − 4B}}{2}} .

Solving Equation (7) for sin(ωτ) and cos(ωτ) yields

sin(ωτ) = \frac{ω^{3} + [αqk − α^{2}γ(k − γ)]ω}{α^{2}βγ^{2} + βω^{2}} ,
cos(ωτ) = \frac{α^{2}qγ(k − γ) + (αk − q)ω^{2}}{α^{2}βγ^{2} + βω^{2}} .

Define

l_{1}^{±} = \frac{ω_{±}^{3} + [αqk − α^{2}γ(k − γ)]ω_{±}}{α^{2}βγ^{2} + βω_{±}^{2}} ,
l_{2}^{±} = \frac{α^{2}qγ(k − γ) + (αk − q)ω_{±}^{2}}{α^{2}βγ^{2} + βω_{±}^{2}} .

We, thus, have the following result.

Lemma 2.1. Let ω_{±} and l_{i}^{±} (i = 1, 2) be defined above.

(i) If B < 0, then there exists a sequence of positive numbers {τ_{j}^{+}}_{j=0}^{∞} such that τ_{0}^{+} < τ_{1}^{+} < τ_{2}^{+} < ⋯ < τ_{j}^{+} < ⋯, and Equation (5) has a pair of purely imaginary roots ±iω_{+} when τ = τ_{j}^{+}.

(ii) If A < 0, B > 0, and A^{2} − 4B > 0, then there exist two sequences of positive numbers {τ_{j}^{+}}_{j=0}^{∞} and {τ_{j}^{−}}_{j=0}^{∞} such that τ_{0}^{+} < τ_{1}^{+} < τ_{2}^{+} < ⋯ < τ_{j}^{+} < ⋯, τ_{0}^{−} < τ_{1}^{−} < τ_{2}^{−} < ⋯ < τ_{j}^{−} < ⋯, and Equation (5) has a pair of purely imaginary roots ±iω_{±} when τ = τ_{j}^{±}.

Here τ_{j}^{±} (j = 0, 1, 2, ⋯) are defined below

τ_{j}^{±} = \frac{1}{ω_{±}} \{ \begin{matrix}arccos l_{2}^{±} + 2jπ, & if l_{1}^{±} > 0, \\ 2π − arccos l_{2}^{±} + 2jπ, & if l_{1}^{±} < 0. \end{matrix}

Define λ(τ) = σ(τ) + iω(τ) to be the root of Equation (5) such that σ(τ_{j}^{±}) = 0 and ω(τ_{j}^{±}) = ω_{±}, respectively.

Lemma 2.2. Let σ(τ) and τ_{j}^{±} be defined above. Then

σ′(τ_{j}^{+}) > 0, σ′(τ_{j}^{−}) < 0.

Proof. Differentiate Equation (5) with respect to τ yields

(\frac{dλ}{dτ})^{−1} = \frac{[2λ + q − α(k − γ)]e^{λτ} + β}{λβ(λ + αγ)} − \frac{τ}{λ}

and a calculation gives

Re (\frac{dλ}{dτ})^{−1} _{τ=τ_{j}^{±}} = \frac{2ω_{±}^{2} + α^{2}(k − γ)^{2} + q^{2} − β^{2}}{β^{2}(α^{2}γ^{2} + ω_{±}^{2})} = \frac{2ω_{±}^{2} + A}{β^{2}(α^{2}γ^{2} + ω_{±}^{2})}

which gives

Re (\frac{dλ}{dτ})^{−1} _{τ=τ_{j}^{±}} = \frac{±\sqrt{A^{2} − 4B}}{β^{2}(α^{2}γ^{2} + ω_{±}^{2})},

completing the proof.

To discuss the distribution of the roots of Equation (5), we will need the following lemma due to Ruan and Wei [18].

Lemma 2.3. Consider the exponential polynomial

P (λ, e^{−λτ}) = p(λ) + q(λ)e^{−λτ}

where p, q are real polynomials such that deg(q) < deg(p) and τ ≥ 0. As τ varies, the total number of zeros of P (λ, e^{−λτ}) on the open right half-plane can change only if a zero appears on or crosses the imaginary axis.

Now we turn our attention to the relationship between A, B and our system parameters. We look at the following two cases.

Case I. β ≤ q. In this case, A ≥ 0.

1. B ≥ 0 ⟺ k ≥ k_{1} or k ≤ −βγ/q + γ;

2. B < 0 ⟺ |k − γ| < βγ/q.

Case II. β > q. In this case,

1. A ≥ 0, B ≥ 0 ⟺ k ≥ max\{\sqrt{β^{2} − q^{2}}/α + γ, k_{1}\} or k ≤ min\{−\sqrt{β^{2} − q^{2}}/α + γ, −βγ/q + γ};

2. B < 0 ⟺ |k − γ| < βγ/q;

3. A < 0, B > 0 ⟺ βγ/q < |k − γ| < \sqrt{β^{2} − q^{2}}/α.

The discussions above, Theorem 2.1 and Lemmas 2.1, 2.2 and 2.3 imply the following Lemma 2.4.

Lemma 2.4. Assume β ≤ q. Let τ_{j}^{+} be defined in Lemma 2.1. Then we have

(i) if B ≥ 0, then all roots of Equation (5) have negative real parts when k < −βγ/q + γ and Equation (5) has roots with negative real parts and roots with positive real parts when k > k_{1};

(ii) if B < 0, or |k − γ| < βγ/q, all roots of Equation (5) have negative real parts for all τ ∈ [0, τ_{0}^{+}); Equation (5) has a pair of purely imaginary roots ±iω_{+} and all other roots have negative real parts when τ = τ_{0}^{+}; it has 2(j+1) roots with positive real parts and all other roots have negative real parts when τ ∈ (τ_{j}^{+}, τ_{j+1}^{+}), j = 0, 1, 2, · · · .

Lemma 2.5. Assume β > q. Let τ_{j}^{±} be defined in Lemma 2.1. Then we have

(i) if A ≥ 0, B ≥ 0, then all roots of Equation (5) have negative real parts when k < min\{−\sqrt{β^{2} − q^{2}}/α+γ, −βγ/q+γ\}, and Equation (5) has roots with negative real parts and roots with positive real parts when k > max\{\sqrt{β^{2} − q^{2}}/α + γ, k_{1}\};

(ii) if B < 0, or if |k − γ| < βγ/q, all roots of Equation (5) have negative real parts for all τ ∈ [0, τ_{0}^{+}); Equation (5) has a pair of purely imaginary roots ±iω_{+} and all other roots have negative real parts when τ = τ_{0}^{+}; it has 2(j+1) roots with positive real parts and all other roots have negative real parts when τ ∈ (τ_{j}^{+}, τ_{j+1}^{+}), j = 0, 1, 2, · · · .

(iii) if A < 0, B > 0 and A^{2}−4B > 0, then we have βγ/q < |k−γ| < \sqrt{β^{2} − q^{2}}/α and A^{2} − 4B > 0. Assume that A^{2} − 4B > 0. If −\sqrt{β^{2} − q^{2}}/α + γ < k < −βγ/q + γ, all roots of Equation (5) have negative real parts for all τ ∈ [0, τ_{0}^{+}), Equation (5) has roots with positive real parts when τ ∈ (τ_{0}^{+}, τ_{m}^{−}) where m is the smallest positive integer such that τ_{m}^{−} > τ_{0}^{+}, it has a pair of purely imaginary roots ±iω_{+} and all other roots have negative real parts when τ = τ_{0}^{+}. if τ_{m}^{−} < τ_{1}^{+}, Equation (5) has two roots with positive real parts and all other roots have negative real parts when τ ∈ (τ_{0}^{+}, τ_{m}^{−}) and all roots of Equation (5) have negative real parts when τ ∈ (τ_{m}^{−}, τ_{1}^{+}). If k_{1} < k < \sqrt{β^{2} − q^{2}}/α + γ, Equation (5) has roots with negative real parts and roots with positive real parts.

The following Hopf bifurcation theorems follow immediately. Theorem 2.2. Assume β ≤ q. Let τ_{j}^{+} be defined in Lemma 2.1. Then we have

(i) the equilibrium point (Y^{∗}, K^{∗}) is asymptotically stable for all τ ≥ 0 when k < −βγ/q + γ and it is unstable for all τ ≥ 0 when k > k_{1};

(ii) the equilibrium point (Y^{∗}, K^{∗}) is asymptotically stable for all τ ∈ [0, τ_{0}^{+}) and unstable for all τ > τ_{0}^{+} when |k − γ| < βγ/q. System (2) undergoes a Hopf bifurcation at (Y^{∗}, K^{∗}) when τ = τ_{j}^{+} for j = 0, 1, 2, · · · .

Theorem 2.3. Assume β > q. Let τ_{j}^{±} be defined in Lemma 2.1. Then we have

(i) the equilibrium point (Y^{∗}, K^{∗}) is asymptotically stable for all τ ≥ 0 when k < min\{−\sqrt{β^{2} − q^{2}}/α + γ, −βγ/q + γ\}, and unstable for all τ ≥ 0 when k > max\{\sqrt{β^{2} − q^{2}}/α + γ, k_{1}\};

(ii) the equilibrium point (Y^{∗}, K^{∗}) is asymptotically stable for all τ ∈ [0, τ_{0}^{+}) and unstable for all τ > τ_{0}^{+} when |k − γ| < βγ/q. System (2) undergoes a Hopf bifurcation at (Y^{∗}, K^{∗}) when τ = τ_{j}^{+} for j = 0, 1, 2, · · · .

(iii) Assume A^{2} − 4B > 0. The equilibrium point (Y^{∗}, K^{∗}) is asymptotically stable for all τ ∈ [0, τ_{0}^{+}) and unstable when τ ∈ (τ_{0}^{+}, τ_{m}^{−}) where m is defined in Lemma 2.5 when −\sqrt{β^{2} − q^{2}}/α + γ < k < −βγ/q + γ. System (2) undergoes a Hopf bifurcation at (Y^{∗}, K^{∗}) when τ = τ_{j}^{+} for j = 0, 1, 2, · · · . When k_{1} < k < \sqrt{β^{2} − q^{2}}/α+γ, the equilibrium point (Y^{∗}, K^{∗}) is unstable. System (2) undergoes a Hopf bifurcation at (Y^{∗}, K^{∗}) when τ = τ_{j}^{±} for j = 0, 1, 2, · · · .

3 Direction and Stability of Hopf Bifurcation

From Section 2, we know that at (Y^{∗}, K^{∗}) the characteristic equation of linearized System (2) has a pair of purely imaginary roots ±iω_{±} if τ = τ_{j}^{±} for each j under some conditions. Under these conditions, as the delay τ passes the critical values τ_{j}^{±}, Hopf bifurcation occurs and periodic solutions emerge. In this section, by deriving a normal form for System (2) using a normal form theory developed by Faria and Magalhães [5, 6], we study the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions.

We first normalize the delay in System (2) by rescaling t → t/τ to get the following system

\{ \begin{matrix}\frac{du_{1}(t)}{dt} & = & ατ[(k − γ)u_{1}(t) − βu_{2}(t) + i^{(2)}u_{1}^{2}(t) + i^{(3)}u_{1}^{3}(t)] + O(|u_{1}|^{4}), \\ \frac{du_{2}(t)}{dt} & = & τ[ku_{1}(t − 1) − βu_{2}(t − 1) − qu_{2}(t) + i^{(2)}u_{1}^{2}(t − 1) \\ & & +i^{(3)}u_{1}^{3}(t − 1)] + O(|u_{1}|^{4}). \end{matrix}  (9)

8 L. Wang & X. Wu Let τ_{c} = τ_{j}^{±} and τ = τ_{c} + μ. Then μ is the bifurcation parameter for System (9) and System (9) becomes

\{ \begin{matrix}\frac{du_{1}(t)}{dt} & = & α(τ_{c} + μ)[(k − γ)u_{1}(t) − βτu_{2}(t) + i^{(2)}u_{1}^{2}(t) + i^{(3)}u_{1}^{3}(t)] \\ & & +O(|u_{1}|^{4}), \\ \frac{du_{2}(t)}{dt} & = & (τ_{c} + μ)[ku_{1}(t − 1) − βu_{2}(t − 1) − qu_{2}(t) + i^{(2)}u_{1}^{2}(t − 1) \\ & & +i^{(3)}u_{1}^{3}(t − 1)] + O(|u_{1}|^{4}). \end{matrix}  (10)

The linearization of System (10) at (0, 0) is

\{ \begin{matrix}\frac{du_{1}(t)}{dt} = ατ_{c}[(k − γ)u_{1}(0) − βu_{2}(0)], \\ \frac{du_{2}(t)}{dt} = τ_{c}[ku_{1}(−1) − βu_{2}(−1) − qu_{2}(0)]. \end{matrix}  (11)

Let

η(θ) = Aδ(θ) + Bδ(θ + 1)

where

A = τ_{c} (\begin{matrix}α(k − γ) & −αβ \\ 0 & −q \end{matrix}) , B = τ_{c} (\begin{matrix}0 & 0 \\ k & −β \end{matrix}) .

Let C = C([−1, 0], ℂ^{2}) and define a linear operator L on C as follows:

Lφ = ∫_{−1}^{0} dη(θ)φ(θ), ∀φ ∈ C.

Then System (10) can be transformed into

Ẋ(t) = LX_{t} + F (X_{t}, μ),

where X = (u_{1}, u_{2})^{T} , X_{t} = X(t + θ), θ ∈ [−1, 0], and F (X_{t}, μ) = (F^{1}, F^{2})^{T} where

F^{1} = α[(k − γ)μu_{1}(0) − βμu_{2}(0) + τ_{c}i^{(2)}u_{1}^{2}(0) + τ_{c}i^{(3)}u_{1}^{3}(0)] + h.o.t.,
F^{2} = kμu_{1}(−1) − βμu_{2}(−1) − qμu_{2}(0) + τ_{c}i^{(2)}u_{1}^{3}(−1) + τ_{c}i^{(3)}u_{1}^{2}(−1) + h.o.t.,

where “h.o.t” represents high order terms. Write the Taylor expansion of F as

F (φ, μ) = \frac{1}{2}F_{2}(φ, μ) + \frac{1}{3!}F_{3}(φ, μ) + h.o.t..

Take the enlarged space of C

BC = {φ : [−1, 0] → ℂ^{2} : φ is continuous on [−1, 0), ∃ lim_{θ→0^{−}} φ(θ) ∈ ℂ^{2}}. Then the elements of BC can be expressed as ψ = φ + X_{0}ν, φ ∈ C and

X_{0}(θ) = \{ \begin{matrix}0, −1 ≤ θ < 0, \\ I, θ = 0, \end{matrix}

EJQTDE Spec. Ed. I, 2009 No. 27

where I is the identity matrix on C and the norm of BC is |φ + X_{0}ν| = |φ| + |ν|_{ℂ^{2}}. Let C^{1} = C^{1}([−1, 0], ℂ^{2}). Then the infinitesimal generator A : C^{1} → BC associated with L is given by

Aφ = φ̇ + X_{0}[Lφ − φ̇(0)] = \{ \begin{matrix}φ̇, & −1 ≤ θ < 0, \\ Aφ(0) + Bφ(−1), & θ = 0, \end{matrix}

and its adjoint

A^{∗}ψ = \{ \begin{matrix}−ψ̇, & 0 < s ≤ 1, \\ ψ(0)A + ψ(1)B, & s = 0, \end{matrix} for ∀ψ ∈ C^{1∗},

where C^{1∗} = C^{1}([0, 1], ℂ^{2∗}). Let C′ = C([0, 1], ℂ^{2∗}) and for φ ∈ C and ψ ∈ C′, define a bilinear inner product between C and C′ by

⟨ψ, φ⟩ = ψ(0)φ(0) − ∫_{−1}^{0} ∫_{0}^{θ} ψ(ξ − θ)dη(θ)φ(ξ)dξ
= ψ(0)φ(0) + ∫_{−1}^{0} ψ(ξ + 1)Bφ(ξ)dξ.

From Section 2, we know that ±iτ_{c}ω_{0} are eigenvalues of A and A^{∗}, where ω_{0} = ω_{+} or ω_{−}. Now we compute eigenvectors of A associated with iτ_{c}ω_{0} and eigenvectors of A^{∗} associated with −iτ_{c}ω_{0}. Let q(θ) = (ρ, k)^{T} e^{iτ_{c}ω_{0}θ} be an eigenvector of A associated with iτ_{c}ω_{0}. Then Aq(θ) = iτ_{c}ω_{0}q(θ). It follows from the definition of A that

(\begin{matrix}−α(k − γ)τ_{c} + iτ_{c}ω_{0} & αβτ_{c} \\ −kτ_{c}e^{−iτ_{c}ω_{0}} & βτ_{c}e^{−iτ_{c}ω_{0}} + qτ_{c} + iτ_{c}ω_{0} \end{matrix}) q(0) = 0.

We can obviously choose q(θ) = (ρ, k)^{T} e^{iτ_{c}ω_{0}θ} where ρ = β + (q + iω_{0})e^{iτ_{c}ω_{0}}.

Similarly, we can find an eigenvector p(s) of A^{∗} associated with −iτ_{c}ω_{0}

p(s) = \frac{1}{D}(σ, αβ)e^{iτ_{c}ω_{0}s}, where σ = −βe^{iτ_{c}ω_{0}} − q + iω_{0}

with D being a constant to be determined such that ⟨p̄(s), q(θ)⟩ = 1. In fact, since

⟨p̄(s), q(θ)⟩ = \frac{1}{D̄}[kαβ(1 + (ρ − β)e^{−iτ_{c}ω_{0}}) + ρσ̄]

we have D = kαβ(1 + (ρ̄ − β)e^{iτ_{c}ω_{0}}) + ρ̄σ. Let P be spanned by q, q̄ and P^{∗} by p, p̄. Then C can be decomposed as

C = P ⊕ Q where Q = {φ ∈ C : ⟨ψ, φ⟩ = 0, ∀ψ ∈ P^{∗}}.

Let Q^{1} = Q ∩ C^{1}. Let Φ(θ) = (q(θ), q̄(θ)) and Ψ(s) = \binom{p̄(s)}{p(s)}. Then Φ̇ = ΦJ and Ψ̇ = −JΨ where J = diag(iτ_{c}ω_{0}, −iτ_{c}ω_{0}). Define the projection π : BC → P by

π(φ + X_{0}ν) = Φ[(Ψ, φ) + Ψ(0)ν].

Let u = Φx + y, namely

u_{1}(θ) = e^{iτ_{c}ω_{0}θ}ρx_{1} + e^{−iτ_{c}ω_{0}θ}ρ̄x_{2} + y_{1}(θ),
u_{2}(θ) = e^{iτ_{c}ω_{0}θ}kx_{1} + e^{−iτ_{c}ω_{0}θ}kx_{2} + y_{2}(θ).

Then System (10) can be decomposed as

\{ \begin{matrix}ẋ = Jx + Ψ(0)F (Φx + y, μ), \\ ẏ = A_{Q^{1}}y + (I − π)X_{0}F (Φx + y, μ). \end{matrix}

This can be rewritten as

\{ \begin{matrix}ẋ = Jx + \frac{1}{2}f_{2}^{1}(x, y, μ) + \frac{1}{3!}f_{3}^{1}(x, y, μ) + h.o.t., \\ ẏ = A_{Q^{1}}y + \frac{1}{2}f_{2}^{2}(x, y, μ) + \frac{1}{3!}f_{3}^{2}(x, y, μ) + h.o.t., \end{matrix}  (12)

where

f_{j}^{1}(x, y, μ) = Ψ(0)F_{j}(Φx + y, μ), f_{j}^{2}(x, y, μ) = (I − π)X_{0}F_{j}(Φx + y, μ).

According to the normal form theory due to Faria and Magalhães [5, 6, 8], on the center manifold, System (12) can be transformed as the following normal form:

ẋ = Jx + \frac{1}{2}g_{2}^{1}(x, 0, μ) + \frac{1}{3!}g_{3}^{1}(x, 0, μ) + h.o.t.

where g_{j}^{1}(x, 0, μ) is a homogeneous polynomial of degree j in (x, μ). Let Y be a normed space and j, p ∈ ℕ. Let

V_{j}^{p}(Y) = {∑_{|q|=j} c_{q}x^{q} : q ∈ ℕ_{0}^{q}, c_{q} ∈ Y }

with norm | ∑_{|q|=j} c_{q}x^{q}| = ∑_{|q|=j} |c_{q}|_{Y}. Define M_{j} to be the operator in V_{j}^{4}(ℂ^{2} × ker π) with the range in the same space by

M_{j}(p, h) = (M_{j}^{1}p, M_{j}^{2}h),

where (M_{j}^{1}p)(x, μ) = [J, p(·, μ)](x) = D_{x}p(x, μ)Jx − Jp(x, μ). It is easy to check that V_{j}^{3}(ℂ^{2}) = Im(M_{j}^{1}) ⊕ Ker(M_{j}^{1}) and

Ker(M_{j}^{1}) = {μ^{l}x^{q}e_{k} : (q, λ̄) = λ_{k}, k = 1, 2, q ∈ ℕ_{0}^{2}, |(q, l)| = j}.

Kaldor-Kalecki Model of Business Cycle 11 Hence

ker(M_{2}^{1}) = Span \{\binom{μx_{1}}{0}, \binom{0}{μx_{2}}\} ,

ker(M_{3}^{1}) = Span \{\binom{x_{1}^{2}x_{2}}{0}, \binom{μ^{2}x_{1}}{0}, \binom{0}{x_{1}x_{2}^{2}}, \binom{0}{μ^{2}x_{2}}\} .

Define

f̃_{3}^{1}(x, 0, μ) = f_{3}^{1}(x, 0, μ) + \frac{3}{2}[(D_{x}f_{2}^{1})(x, 0, μ)U_{2}^{1}(x, μ) + (D_{y}f_{2}^{1})(x, 0, μ)U_{2}^{2}(x, μ)]

where

U_{2}^{1}(x, μ)|_{μ=0} = (M_{2}^{1})^{−1}Proj_{Im(M_{2}^{1})}f_{2}^{1}(x, 0, 0) = (M_{2}^{1})^{−1}f_{2}^{1}(x, 0, 0)

and U_{2}^{2}(x, μ) is determined by

(M_{2}^{2}U_{2}^{2})(x, μ) = f_{2}^{2}(x, 0, μ).

Then

g_{2}^{1}(x, 0, μ) = Proj_{ker(M_{2}^{1})}f_{2}^{1}(x, 0, μ), g_{3}^{1}(x, 0, μ) = Proj_{ker(M_{3}^{1})}f̃_{3}^{1}(x, 0, μ).

Let us compute g_{2}^{1}(x, 0, μ) first. Since

\frac{1}{2}f_{2}^{1}(x, 0, μ) = (\begin{matrix}a_{1}μx_{1} + a_{2}μx_{2} + a_{20}x_{1}^{2} + a_{11}x_{1}x_{2} + a_{02}x_{2}^{2} \\ ā_{2}μx_{1} + ā_{1}μx_{2} + ā_{02}x_{1}^{2} + ā_{11}x_{1}x_{2} + ā_{20}x_{2}^{2} \end{matrix}) ,

where

a_{1} = −\frac{α}{D̄}[kβ(q + (β − ρ)e^{−iτ_{c}ω_{0}}) + σ̄(k(β − ρ) + γρ)],
a_{2} = −\frac{α}{D̄}[kβ(q + σ̄ + βe^{iτ_{c}ω_{0}}) − ρ̄(kβe^{iτ_{c}ω_{0}} + (k − γ)σ̄)],
a_{20} = \frac{αρ^{2}τ_{c}}{D̄} i^{(2)}(e^{−2iτ_{c}ω_{0}}β + σ̄),  (13)
a_{11} = \frac{2α|ρ|^{2}τ_{c}}{D̄} i^{(2)}(β + σ̄),
a_{02} = \frac{αρ̄^{2}τ_{c}}{D̄} i^{(2)}(e^{2iτ_{c}ω_{0}}β + σ̄),

then

\frac{1}{2}g_{2}^{1}(x, 0, μ) = \frac{1}{2}Proj_{ker(M_{2}^{1})}f_{2}^{1}(x, 0, μ) = (\begin{matrix}a_{1}μx_{1} \\ ā_{1}μx_{2} \end{matrix}) .

EJQTDE Spec. Ed. I, 2009 No. 27

Next we compute \frac{1}{3!}g_{3}^{1}(x, 0, μ) = \frac{1}{3!}Proj_{ker(M_{3}^{1})}f̃_{3}^{1}(x, 0, μ). Since the term O(μ^{2}|x|) is irrelevant to determine the generic Hopf bifurcation, we have

\frac{1}{3!}g_{3}^{1}(x, 0, μ) = \frac{1}{3!}Proj_{ker(M_{3}^{1})}f̃_{3}^{1}(x, 0, μ) = \frac{1}{3!}Proj_{S}f̃_{3}^{1}(x, 0, 0) + O(μ^{2}|x|)
= \frac{1}{3!}Proj_{S}f_{3}^{1}(x, 0, 0) + \frac{1}{4}Proj_{S}[(D_{x}f_{2}^{1})(x, 0, 0)U_{2}^{1}(x, 0)
+ (D_{y}f_{2}^{1})(x, 0, 0)U_{2}^{2}(x, 0)] + O(μ^{2}|x|).

where

S = Span \{\binom{x_{1}^{2}x_{2}}{0}, \binom{0}{x_{1}x_{2}^{2}}\} .

Step 1. Compute \frac{1}{3!}Proj_{ker(M_{3}^{1})}f_{3}^{1}(x, 0, 0). Since

\frac{1}{3!}f_{3}^{1}(x, 0, 0) = \binom{a_{30}x_{1}^{3} + a_{21}x_{1}^{2}x_{2} + a_{12}x_{1}x_{2}^{2} + a_{03}x_{2}^{3}}{ā_{03}x_{1}^{3} + ā_{12}x_{1}^{2}x_{2} + ā_{21}x_{1}x_{2}^{2} + ā_{30}x_{2}^{3}}

where

a_{30} = \frac{αρ^{3}τ_{c}}{D̄} [i^{(3)}(βe^{−3iτ_{c}ω_{0}} + σ̄)],
a_{21} = \frac{3α|ρ|^{2}ρτ_{c}}{D̄} [i^{(3)}(βe^{−iτ_{c}ω_{0}} + σ̄)],
a_{12} = \frac{3α|ρ|^{2}ρ̄τ_{c}}{D̄} [i^{(3)}(βe^{iτ_{c}ω_{0}} + σ̄)],
a_{03} = \frac{αρ̄^{3}τ_{c}}{D̄} [i^{(3)}(βe^{3iτ_{c}ω_{0}} + σ̄)],

we have

\frac{1}{3!}Proj_{ker(M_{3}^{1})}f_{3}^{1}(x, 0, 0) = \binom{a_{21}x_{1}^{2}x_{2}}{ā_{21}x_{1}x_{2}^{2}}.

Step 2. Compute \frac{1}{2}Proj_{S}[D_{x}f_{2}^{1}(x, 0, 0)U_{2}^{1}(x, 0)]. The elements of the canonical basis of V_{2}^{2}(ℂ^{2}) are

\binom{x_{1}^{2}}{0}, \binom{x_{1}x_{2}}{0}, \binom{x_{2}^{2}}{0}, \binom{μx_{1}}{0}, \binom{μx_{2}}{0}, \binom{μ^{2}}{0},

\binom{0}{x_{1}^{2}}, \binom{0}{x_{1}x_{2}}, \binom{0}{x_{2}^{2}}, \binom{0}{μx_{1}}, \binom{0}{μx_{2}}, \binom{0}{μ^{2}},

whose images under \frac{1}{iω_{0}} M_{2}^{1} are, respectively

\binom{x_{1}^{2}}{0}, −\binom{x_{1}x_{2}}{0}, −3\binom{x_{2}^{2}}{0}, \binom{0}{0}, −2\binom{μx_{2}}{0}, \binom{μ^{2}}{0},

3\binom{0}{x_{1}^{2}}, \binom{0}{x_{1}x_{2}}, −\binom{0}{x_{2}^{2}}, 2\binom{0}{μx_{1}}, \binom{0}{0}, \binom{0}{μ^{2}}.

Hence

U_{2}^{1}(x, 0) = \frac{1}{iω_{0}} \binom{a_{20}x_{1}^{2} − a_{11}x_{1}x_{2} − \frac{1}{3}a_{02}x_{2}^{2}}{\frac{1}{3} ā_{02}x_{1}^{2} + ā_{11}x_{1}x_{2} − ā_{20}x_{2}^{2}},

and

\frac{1}{2}Proj_{S}[D_{x}f_{2}^{1}(x, 0, 0)U_{2}^{1}(x, 0)] = \binom{C_{1}x_{1}^{2}x_{2}}{C̄_{1}x_{1}x_{2}^{2}}

where

C_{1} = \frac{2}{iω_{0}} (2|a_{02}|^{2} − 3a_{20}a_{11} + 3|a_{11}|^{2})
= −\frac{iα^{2}|ρ|^{2}ρτ_{c}^{2}(i^{(2)})^{2}}{12D̄|D|^{2}} [6D̄ρ̄|β + σ|^{2} − 3Dρ(β + σ̄)(βe^{−2iτ_{c}ω_{0}} + σ̄)
+ D̄ρ̄|βe^{−2iτ_{c}ω_{0}} + σ|^{2}].

Step 3. Compute \frac{1}{2}Proj_{S}[(D_{y}f_{2}^{1})(x, 0, 0)U_{2}^{2}(x, 0)], where U_{2}^{2}(x, 0) is a second-order homogeneous polynomial in (μ, x_{1}, x_{2}) with coefficients in Q^{1}. Let

h(x)(θ) = U_{2}^{2}(x, 0) = h_{20}(θ)x_{1}^{2} + h_{11}(θ)x_{1}x_{2} + h_{02}(θ)x_{2}^{2}.

The coefficients h_{jk} = (h_{jk}^{1}, h_{jk}^{2})^{T} are determined by M_{2}^{2}h(x) = f_{2}^{2}(x, 0, 0) or

D_{x}h(x)Bx − A_{Q^{1}}(h(x)) = (I − π)X_{0}F_{2}(Φx, 0)

which is equivalent to

ḣ(x) − D_{x}h(x)Bx = ΦΨ(0)F_{2}(Φx, 0),
ḣ(x)(0) − Lh(x) = F_{2}(Φx, 0),

where ḣ denotes the derivative of h(x)(θ) with respect to θ. Note that

F_{2}(Φx, 0) = A_{20}x_{1}^{2} + A_{11}x_{1}x_{2} + A_{02}x_{2}^{2}

where

A_{20} = (2i^{(2)}αρ^{2}τ_{c}, 2i^{(2)}ρ^{2}τ_{c}e^{−2iτ_{c}ω_{0}})^{T} ,
A_{11} = (4i^{(2)}α|ρ|^{2}τ_{c}, 2i^{(2)}α|ρ|^{2}τ_{c})^{T} ,
A_{02} = (2i^{(2)}αρ̄^{2}τ_{c}, 2i^{(2)}ρ̄^{2}τ_{c}e^{2iτ_{c}ω_{0}})^{T} .

Comparing the coefficients of x_{1}^{2}, x_{1}x_{2}, x_{2}^{2} of these equations, it is not hard to verify that h̄_{02} = h_{20}, h̄_{11} = h_{11} and that h_{20}, h_{11} satisfy the following equations

\{ \begin{matrix}ḣ_{20} − 2iτ_{c}ω_{0}h_{20} = ΦΨ(0)A_{20}, \\ ḣ_{20}(0) − Lh_{20} = A_{20}, \end{matrix}  (14)

14 L. Wang & X. Wu and

\{ \begin{matrix}ḣ_{11} = ΦΨ(0)A_{11}, \\ ḣ_{11}(0) − Lh_{11} = A_{11}. \end{matrix}  (15)

Noting that f_{2}^{1}(x, 0, 0) = ΨF_{2}(Φx, 0), we deduce

\frac{1}{2}(D_{y}f_{2}^{1})h(x, 0, 0) = \binom{\frac{ατ_{c}i^{(2)}}{2D̄} [β(ρe^{−iτ_{c}ω_{0}}x_{1} + ρ̄e^{iτ_{c}ω_{o}}x_{2})h^{1}(−1) + σ̄(ρx_{1} + ρ̄x_{2})h^{1}(0)]}{\frac{ατ_{c}i^{(2)}}{2D} [β(ρe^{−iτ_{c}ω_{0}}x_{1} + ρ̄e^{iτ_{c}ω_{o}}x_{2})h^{1}(−1) + σ̄(ρx_{1} + ρ̄x_{2})h^{1}(0)]}

where

h^{1}(−1) = h_{20}^{1}(−1)x_{1}^{2} + h_{11}^{1}(−1)x_{1}x_{2} + h_{02}^{1}(−1)x_{2},
h^{1}(0) = h_{20}^{1}(0)x_{1}^{2} + h_{11}^{1}(0)x_{1}x_{2} + h_{02}^{1}(0)x_{2}^{2}.

and hence

\frac{1}{2}Proj_{S}[(D_{y}f_{2}^{1})h](x, 0, 0) = \binom{C_{2}x_{1}^{2}x_{2}}{C̄_{2}x_{1}x_{2}^{2}},

where

C_{2} = \frac{ατ_{c}i^{(2)}}{D̄} [e^{−iω_{0}τ_{0}}βρh_{11}^{1}(−1) + ρσ̄h_{11}^{1}(0) + e^{iω_{0}τ_{0}}βρ̄h_{20}^{1}(−1) + ρ̄σ̄h_{20}^{1}(0)].

Here h_{20}, h_{11} are determined by System (14) and System (15). After long but basic calculations, we obtain

h_{20}^{1}(0) = (2i^{(2)}αρ^{2}e^{−3τ_{c}ω_{0}}(D̄(−ie^{2iτ_{c}ω_{0}}(2De^{3iτ_{c}ω_{0}}σ)ω(−iq + 2ω_{0}) +kαβ(β + e^{2iτ_{c}ω_{0}}σ)((−1 + e^{2iτ_{c}ω_{0}})β − 2ie^{iτ_{c}ω_{0}}ω_{0})) +e^{iτ_{c}ω_{0}}(β + e^{2iτ_{c}ω_{0}}σ)(−ie^{iτ_{c}ω_{0}}kαβ + ie^{3iτ_{c}ω_{0}}kαβ +2βω_{0} + 2e^{2iτ_{c}ω_{0}}(q + 2iω_{0})ω_{0})ρ̄) + D(2e^{iτ_{c}ω_{0}}ρ(β + e^{2iτ_{c}ω_{0}}(q + 2iω_{0}))ω_{0} −ikαβ((−1 + e^{2iτ_{c}ω_{0}})β + ρ − 2e^{2iτ_{c}ω_{0}}ρ − 2ie^{3iτ_{c}ω_{0}}ω_{0}))(β + e^{iτ_{c}ω_{0}} σ̄)) /(ω_{0}D(−iβ + e^{2iτ_{c}ω_{0}}(−iq + 2ω_{0})) − α(βγ − ie^{2iτ_{c}ω_{0}}(k − γ)(−iq + 2ω_{0}))D̄), h_{20}^{1}(−1) =

\frac{2i^{(2)}αρ^{2}e^{−5τ_{c}ω_{0}}}{ω_{0}|D|^{2}} [i(−1 + e^{2iτ_{c}ω_{0}})(e^{2iτ_{c}ω_{0}}(β + e^{2iτ_{c}ω_{0}}σ)D̄ρ̄ + Dρ(β + e^{2iτ_{c}ω_{0}} σ̄))

+(D̄(−ie^{2iτ_{c}ω_{0}}(2De^{2iτ_{c}ω_{0}}ω_{0}(−iq + 2ω) + kαβ(β + e^{2iτ_{c}ω_{0}}σ)((−1 + e^{iτ_{c}ω_{0}})β −2ie^{2iτ_{c}ω_{0}}ω_{0})) + e^{iτ_{c}ω_{0}}(β + e^{2iτ_{c}ω_{0}})(−ie^{iτ_{c}ω_{0}}kαβ + ie^{3iτ_{c}ω_{0}}kαβ + 2βω_{0} +2e^{2iτ_{c}ω_{0}}(q + 2iω_{0})ω_{0})ρ̄) + D(2e^{2iτ_{c}ω_{0}}ρ(β + e^{2iτ_{c}ω_{0}}(q + 2iω_{0}))ω_{0} −ikαβ((−1 + e^{2iτ_{c}ω_{0}})β + ρ − e^{2iτ_{c}ω_{0}} − 2ie^{3iτ_{c}ω_{0}}))(β + e^{2iτ_{c}ω_{0}} σ̄))] /(2ω_{0}(−iβ + e^{2iτ_{c}ω_{0}}(−iq + 2ω)) − α(βγ − ie^{2iτ_{c}ω_{0}}(k − γ)(−iq + 2ω_{0})))

EJQTDE Spec. Ed. I, 2009 No. 27

and

h_{11}^{1}(0) =

\frac{4i^{(2)}|ρ|^{2}e^{−iτ_{c}ω_{0}}}{(βγ + (−k + γ)q)|D|^{2}} [e^{iτ_{c}ω_{0}} D̄(−Dq + kαβ(β + σ)(−1 + e^{iτ_{c}ω_{0}}βτ_{c})

+(β + σ)(−q + β(−1 + e^{iτ_{c}ω_{0}}kατ_{c}))ρ̄) − D(e^{iτ_{c}ω_{0}}(kαβ − (β + q)ρ) +kαβ(−β + ρ)τ_{c})(β + σ̄)],

h_{11}^{1}(−1)

\frac{4i^{(2)}|ρ|^{2}e^{−iτ_{c}ω_{0}}}{|D|^{2}} [−e^{−2iτ_{c}ω_{0}}ατ_{c}((β + σ)D̄ρ̄ + De^{−2iτ_{c}ω_{0}}ρ(β + σ̄))

−\frac{1}{βγ + (−k + γ)q} (D̄e^{iτ_{c}ω_{0}}(−Dq + kαβ(β + σ)(−1 + e^{iτ_{c}ω_{0}}βτ_{c})

−(β + σ)(−q + β(−1 + e^{iτ_{c}ω_{0}}kατ_{c}))ρ̄) − D(e^{iτ_{c}ω_{0}}(kαβ − (β + q)ρ) +kαβ(−β + ρ)τ_{c})(β + σ̄))].

Collecting the results above, we obtain

\frac{1}{3!}g_{3}^{1}(x, 0, μ) = \binom{b_{21}x_{1}^{2}x_{2}}{b̄_{21}x_{1}x_{2}^{2}} + O(μ^{2}|x|),

where b_{21} = a_{21} + \frac{1}{2}(C_{1} + C_{2}). Therefore, System (10) can be transformed into the following normal form:

\{ \begin{matrix}ẋ_{1} = iτ_{c}ω_{0}x_{1} + a_{1}μx_{1} + b_{21}x_{1}^{2}x_{2} + h.o.t., \\ ẋ_{2} = −iτ_{c}ω_{0}x_{2} + ā_{1}μx_{2} + b̄_{21}x_{1}x_{2}^{2} + h.o.t., \end{matrix}  (16)

where a_{1} is given in (13). Let x_{1} = w_{1}+iw_{2}, x_{2} = w_{1}−iw_{2} and w_{1} = r cos ξ, w_{2} = r sin ξ. Then (16) can be further written as

\{ \begin{matrix}ṙ = aμr + br^{3} + h.o.t., \\ ξ̇ = τ_{c}ω_{0} + h.o.t., \end{matrix}

where a = Re[a_{1}] and b = Re[b_{21}]. Hence the first Lyapunov coefficient is l_{1}(μ) = b + O(μ), see [3, 13].

Theorem 3.1. Let a and b be given above.

(i) The bifurcating periodic solution is stable if b < 0, and unstable if b > 0;

(ii) The Hopf bifurcation is supercritical if ab < 0, and subcritical if ab > 0.

Remark. The coefficient a is given by

a = Re[a_{1}] =

\frac{k^{2}α^{2}(q^{2} + ω_{0}^{2})}{β^{2}(α^{2}γ^{2} + ω_{0}^{2})|D|^{2}} [−αω_{0}^{2}(β^{2} − q^{2} − ω_{0}^{2})(β^{2}γ + (k − γ)(q^{2} + ω_{0}^{2}))

+α^{3}(−β^{4}γ^{3} + β^{2}γ(k^{2} − 3kγ + 2γ^{2})(q^{2} + ω_{0}^{2}) + (k − γ)^{3}(q^{2} + ω_{0}^{2})^{2} +ω_{0}^{4}(β^{2}q − (q^{2} + ω_{0}^{2})(q − ω_{0}^{2})) − α^{4}(k − γ)^{2}((k − γ)^{2}q(1 + qτ_{c})(q^{2} + ω_{0}^{2}) −β^{2}γ^{2}(q + q^{2}τ_{c} + τ_{c}ω_{0}^{2})) + α^{2}ω_{0}^{2}(2kγ(−β^{2}q + 2q^{3} + q^{4}τ_{c} + 2qω_{0}^{2} − τ_{c}ω_{0}^{4}) +k^{2}(β^{2}q − 2q^{3} − q^{4}τ_{c} − 2qω^{2} + τ_{c}ω_{0}^{4}) + γ^{2}(−(q^{2} + ω_{0}^{2})(2q + q^{2}τ_{c} − τ_{c}ω_{0}^{2}) +β^{2}(2q + q^{2}τ_{c} + τ_{c}ω_{0}^{2}))))].

Although the explicit algorithm is derived to compute b, it is difficult to determine the sign of b for general α, β, γ, k, q. But if i^{(2)} = 0, it is easy to see C_{1} = C_{2} = 0 and hence b can be simply expressed as

b = −\frac{3i^{(3)}}{|D|^{2}} (β^{2} + q^{2} + ω_{0}^{2} + 2βq cos(τ_{c}ω_{0}) − 2βω sin(τ_{c}ω_{0}))(−kαβ^{2}q + 2β^{2}q
+2β^{2}q^{2} + q^{4} − kαβ^{2}q^{2}τ_{c} + 2β^{2}ω_{0}^{2} + 2q^{2}ω^{2} − kα^{2}β^{2}τ_{c}ω_{0}^{2} + ω_{0}^{4}
+β(β^{2}q + 3q(q^{2} + ω_{0}^{2}) − kα(q^{2} + q^{3}τ_{c} − ω_{0}^{2} + qτ_{c}ω_{0}^{2})) cos(τ_{c}ω_{0})
+β^{2}(q^{2} − ω_{0}^{2}) cos(2τ_{c}ω_{0}) − β^{3}ω_{0} sin(τ_{c}ω_{0}) + 2kαβq sin(τ_{c}ω_{0})
−3βq^{2}ω_{0} sin(τ_{c}ω_{0})
+kαβq^{2}τ_{c}ω_{0} sin(τ_{c}ω_{0}) − 3βω_{0}^{3} sin(τ_{c}ω_{0}) − 2β^{3}qω_{0} sin(2τ_{c}ω_{0})).

4 Numerical Simulations

In this section, we give some examples to illustrate the theoretical results obtained in the previous sections.

Example 1. Let α = 1, β = 0.8, γ = 0.5625, q = 0.9 and

I(s) = tanh(0.5s).

Then (0, 0) is an equilibrium point of System (2), k = 0.5, i^{(2)} = 0, i^{(3)} = −0.041667. Hence ω_{+} = 0.6066, and τ_{0}^{+} = 3.1382. Take τ = 2.5. According to Theorem 2.2 (ii), the trivial equilibrium point (0, 0) is asymptotically stable, (Figure 1).

Example 2. Let α = 1, β = 0.8, γ = 0.5625, q = 0.9 and

I(s) = tanh(0.5s).

Kaldor-Kalecki Model of Business Cycle 17 Figure 1: The equilibrium point (0, 0) is asymptotically stable when τ < τ_{0}^{+}. Figure 2: The stable periodic orbit generated by Hopf bifurcation when β < q. Figure 3: The stable periodic orbit generated by Hopf bifurcation when β > q.

EJQTDE Spec. Ed. I, 2009 No. 27

Then k = 0.5, i^{(2)} = 0, i^{(3)} = −0.041667 and hence ω_{+} = 0.6066, τ_{0}^{+} = 3.1382. Take τ_{c} = τ_{0}^{+}, μ = 0.001. After using the algorithm in Section 3, we have

a = 2.0772, b = −0.0362,

and hence the bifurcating periodic solution is stable and the Hopf bifurcation is supercritical (Figure 2).

Example 3. Let α = 0.1, β = 0.9, γ = 0.5625, q = 0.5 and

I(s) = tanh(0.9s).

Then k = 0.9, i^{(2)} = 0, i^{(3)} = −0.243 and hence ω_{+} = 0.7503, τ_{0}^{+} = 2.7185. Take τ_{c} = τ_{0}^{+}, μ = 0.001. After using the algorithm in Section 3, we have

a = 1.2365, b = −0.0002,

and hence the bifurcating periodic solution is stable and the Hopf bifurcation is supercritical (Figure 3).

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