Hopf Bifurcation Analysis in a Delayed Kaldor-kalecki Model of Business Cycle

In this paper, we analyze the model of business cycle with time delay set forth by A. Krawiec and M. Szydłowski [1]. Our goal in this model is to introduce the time delay into capital stock and gross product in capital accumulation equation. The dynamics are studied in terms of local stability and of the description of the Hopf bifurcation, that is proven to exist as the delay (taken as a parameter of bifurcation) cross some critical value. Additionally we conclude with an application.


Introduction and mathematical models
Great attention has been paid to equations with delay, which have significant economical and biological background (see for example [2][3][4][5][6][7][8][9]). In most application of delay differential equations in investment processes, the need of incorporation of a time delay is often the result of the time interval required between investment decision and installation of investment capital [10,11].In general, delay differential equations exhibit much more complicated dynamics than ordinary differential equations since time delay could cause a stable equilibrium to become unstable and cause the system to fluctuate.
In this paper, we consider the Kaldor-Kalecki model of business cycle with time delay as follows: where Y is the gross product, K is the capital stock, α is the adjustment coefficient in the goods market, δ is the depreciation rate of capital stock, I(Y, K) is the investment function, S(Y, K) is the saving and τ is the time delay needed for new capital to be installed.
Clearly, introducing time delay into capital stock and gross product in capital accumulation equation is more reasonable, because the change in the capital stock is due to the past investment decisions (see [12, p. 103]).
The first model in this optic is proposed by Kalecki (in 1935 [10]).The main characteristic feature of his model is the distinction between investment decisions and implementation, i.e. there is a time delay after which capital equipment is available for production.
Besides the influence of Keynes (in 1936 [13]) and Kalecki (in 1937 [14]), Kaldor (in 1940 [15]) presented a nonlinear model of business cycle by an ordinary differential equations as follows: In this model the nonlinearity of investment and saving function leads to limit cycle solution (see also [16][17][18] for more information).
Based on the Kaldor model of business cycle and the Kalecki's idea on time delay, Krawiec and Szydłowski (in 1999, [1]) proposed the following Kaldor-Kalecki model of business cycle: The fundamental characteristics of this model is the nonlinearity of investment function and the inclusion of time delay into the gross product in capital accumulation equation.
In ( [1] and [6], 2000), Krawiec and Szydłowski investigated the stability and Hopf bifurcation of a positive equilibrium E * of system (3) in the special case of small time delay.In ( [12], 2001), they showed that for a small time delay parameter the Kaldor-Kalecki model assumes the form of the Lienard equation.In ( [19], 2005), they investigate the stability of limit cycle.Zhang and Wei ( [9], 2004) investigated local and global existence of Hopf bifurcation for (3).
In this work, the dynamics of the system (1) are studied in terms of local stability and of the description of the Hopf bifurcation, that is proven to exist as the delay (taken as a parameter of bifurcation) cross some critical value.A numerical illustrations is given to compare our results and the ones (3) of Krawiec-Szydłowski model [1].

Steady state
In the following proposition, we give a sufficient conditions for the existence and uniqueness of positive equilibrium E * of the system (4).
Proposition 1. Suppose that (ii) I(0) > 0; Then there exists a unique equilibrium E * = (Y * , K * ) of system (4), where Y * is the positive solution of and K * is determined by Proof.(Y, K) is a steady-state of (4) if Let us assume that Y > 0 and K > 0 satisfy (7).Then and In view of hypotheses (i), (ii) and (iii) of Proposition 1, it's clear that equation ( 9) has a unique solution Y * > 0. This concludes the proof.

Local stability analysis
Let y = Y − Y * and k = K − K * .Then by linearizing system ( 4) around (Y * , K * ) we have The characteristic equation associated to system (10) is where and The local stability of the steady state E * is a result of the localization of the roots of the characteristic equation (11).In order to investigate the local stability of the steady state, we begin by considering the case without delay τ = 0.In this case the characteristic equation ( 11) reads as hence, according to the Hurwitz criterion, we have the following lemma.

Theorem 2. Assume that
. Then E * is locally asymptotically stable for all τ ≥ 0.
Proof.From Lemma 1, (H3) implies that the characteristic equation ( 11) has all roots with negative real parts for τ = 0 and no purely imaginary roots for τ > 0. Thus, E * is locally asymptotically stable for all τ ≥ 0.

Hopf bifurcation occurrence
According to the Hopf bifurcation theorem [25], we establish sufficient conditions for the local existence of periodic solutions.Theorem 3.Under hypotheses (H1) and (H2) of Theorem 1, a Hopf bifurcation of periodic solutions of system (4) occurs at E * when τ = τ 0 .
Proof.For the proof of this theorem we apply the Hopf bifurcation theorem introduced in [25].From Lemma 1, the characteristic equation ( 11) has a pair of imaginary roots ±iω 0 at τ = τ 0 .In the first, lets show that iω 0 is simple: Consider the branch of characteristic roots λ(τ ) = ν(τ )+iω(τ ), of equation ( 11) bifurcating from iω 0 at τ = τ 0 .By derivation of (11) with respect to the delay τ, we obtain If we suppose, by contradiction, that iω 0 is not simple, the right hand side of (18) gives αγ + iω 0 = 0, and leads a contradiction with the fact that α and γ are positive.

Effect of additional delay
Let's compare the principal results of systems (3) (see Krawiec-Szydłowski model in [1]) and ( 4) by a numerical illustration.Consider the following Kaldor-type investment function: .
The following numerical simulations are given for system (4) for τ = 2, and τ = 3 and for system (3) for τ = 3.As τ 0 < τ c , we think that it's more interesting to introduce the delay τ into both gross product and capital stock (see also [12, p. 103]).

Effect of changing parameters
Now, let's show how the critical delay τ 0 and the period of oscillations P 0 change as the model parameters move.
In Fig. 4, we construct the family of curves τ 0 (α, β, γ, δ) assuming that three of parameters α, β, γ and δ are fixed.For values of γ which are less than a critical value γ c = 0.004, the condition of existence of equilibrium is violated (see Fig. 4(a)).For values of β (resp.δ) which are less (resp.greater) than a critical value β c = 0.07 (resp.δ c = 0.28), the system will not exhibit a Hopf bifurcation (see Fig.