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

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.


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

L. Wang & X. Wu
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: 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: 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.

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 Let the Taylor expansion of i at 0 be where The linear part of System (3) at (0, 0) is and its corresponding characteristic equation is For τ = 0, Equation (5) becomes Define 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.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 Adding squares of two equations yields Let If A ≥ 0 and B ≥ 0, Equation ( 8) has no positive roots.If B < 0, Equation ( 8) has a unique positive root If A < 0, B > 0, and A 2 − 4B > 0, Equation ( 8) has two positive roots Solving Equation (7) for sin(ωτ ) and cos(ωτ ) yields We, thus, have the following result.
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.

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 EJQTDE Spec.Ed.I, 2009 No. 27

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 The linearization of System ( 10) at (0, 0) is where ) and define a linear operator L on C as follows: Then System (10) can be transformed into where where "h.o.t" represents high order terms.Write the Taylor expansion of F as Then the elements of BC can be expressed as ψ = ϕ + X 0 ν, ϕ ∈ C and where I is the identity matrix on C and the norm of BC is |ϕ Then the infinitesimal generator A : C 1 → BC associated with L is given by and its adjoint where ) and for ϕ ∈ C and ψ ∈ C ′ , define a bilinear inner product between C and C ′ by 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 iτcω 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 βτ c e −iτcω 0 + qτ c + iτ c ω 0 q(0) = 0.
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

Figure 3 :
Figure 3: The stable periodic orbit generated by Hopf bifurcation when β > q.