Local Hopf Bifurcation and Stability of Limit Cycle in a Delayed Kaldor-kalecki Model

We consider a delayed Kaldor-Kalecki business cycle model. We first consider the existence of local Hopf bifurcation, and we establish an explicit algorithm for determining the direction of the Hopf bifurcation and the stability or instability of the bifurcating branch of periodic solutions using the methods presented by O. Diekmann et al. in [1]. In the end, we conclude with an application.


Introduction and mathematical models
In a recent paper [2], we formulate a delayed Kaldor-Kalecki business cycle model by introducing the Kalecki's time delay [3] in the Kaldor model [4] 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. The dynamics are studied in terms of local stability and of the description of the Hopf bifurcation, that is proven to exists as the delay (taken as a parameter of bifurcation) cross some critical value.
In this paper, we reconsider the model (1) and we establish an explicit algorithm for determining the direction of the Hopf bifurcation and the stability or instability of the bifurcating branch of periodic solutions using the methods presented by O. Diekmann et al. in [1].
The first model in this optic is proposed by Kalecki in [3,1935]. 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 [5,1936] and Kalecki in [6,1937], Kaldor in [4,1940] 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 [7][8][9] for more information).
Based on the Kaldor model of business cycle and the Kalecki's idea on time delay, Krawiec and Szydłowski in [10,1999] 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 [10,11,2000], Krawiec and Szydblowski 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 [13,2005], they investigate the stability of limit cycle. Zhang and Wei [14,2004], 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. Additionally we establish an explicit algorithm for determining the direction of the Hopf bifurcation and the stability or instability of the bifurcating branch of periodic solutions using the methods presented by O. Diekmann et al. in [1]. In the end, we give a numerical illustrations.

Steady state
In the following proposition, we give a sufficient conditions for the existence and uniqueness of positive equilibrium E * of the system (4).

Local stability and local Hopf bifurcation 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 (7) is where The local stability of the steady state E * is a result of the localization of the roots of the characteristic equation (8). 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 (8) reads as hence, according to the Hurwitz criterion, we have the following lemma.
. We now return to the study of equation (8) with τ > 0.
According to the Hopf bifurcation theorem [15], we establish sufficient conditions for the local existence of periodic solutions.

Direction of Hopf bifurcation
In this section we use a formula on the direction of the Hopf bifurcation given by Diekman in [1] to formulate an explicit algorithm about the direction and the stability of the bifurcating branch of periodic solutions of (4).
Normalizing the delay τ by scaling t → t τ and effecting the change U (t) = Y (τ t) and V (t) = K(τ t), the system (4) is transformed into By the translation Z(t) = (U, V ) − (Y * , K * ), system (12) is written as a functional differential equation in C := C([−1, 0], R 2 ), where L(τ ) : C → R 2 the linear operator and h : C × R → R 2 the nonlinear part of (13) are given respectively by: Using the Riesz representation theorem (see [15]), we obtain where δ(.) denotes the Dirac function. Let A(τ ) denotes the generator of semigroup generated by the linear part of (13) and A = A(τ 0 ).

Then,
for ϕ = (ϕ 1 , ϕ 2 ) ∈ C. From Theorem 1, a Hopf bifurcation occurs at the critical value τ = τ 0 . By the Taylor expansion of the time delay function τ (ε) near the critical value τ 0 , we have The sign of τ 2 determines either the bifurcation is supercritical (if τ 2 > 0) and periodic orbits exist for τ > τ 0 , or it is subcritical (if τ 2 < 0) and periodic orbits exist for τ < τ 0 . The term τ 2 may be calculated (see [1]) using the formula, where M 0 is the characteristic matrix of the linear part of (13), where D i 1 h, i = 2, 3, denotes the i − th derivative of h with respect to ϕ, P (θ) denotes the eigenvector of A, P (θ) denotes it conjugate eigenvector and p, q are defined later.
To study the direction of Hopf bifurcation, one needs to calculate the second and third derivatives of nonlinear part of (13) with respect to ϕ, and Then and As iω 0 is a solution of (8) at τ = τ 0 , then iω 0 is an eigenvalue of A and there exist a corresponding eigenvector of the form P (θ) = pe iω0θ where p = (p 1 , p 2 ) ∈ C 2 , satisfy the equations: Then one may assume and calculate So, from (22) and (23), we have and Now, consider A * , a conjugate operator of A, A * : C([0, 1], R 2 ) → R 2 , defined by, Let Q(s) = qe iω0s be the eigenvector for A * associated to the eigenvalue iω 0 , q = (q 1 , q 2 ) T . One needs to choose q such that the inner product (see [15]), If we take q 2 = 0, then q 1 = 1 and from (27), we have From the expression of M 0 in (19), we have and From (25), (26), (30), (31), we deduce, and where Now, from (19) we have and from the proof of Theorem 1, we have Consequently we deduce the following result: where Re(c) is given by (34).
Note that, Theorem 4 provides an explicit algorithm for detecting the direction and stability of Hopf bifurcation.
By the previous proposition, we have if τ < 2.9929, then system (4) have a stable equilibrium point E * . Fig. 1 shows that behavior of system (4) is stable for τ = 2. If we increase the value of τ, then we find a stable periodic solution occurs at τ 0 = 2.9929 and E * becomes unstable for τ > 2.9929. Fig.2 show that is E * unstable for τ > 2.9929.   (4) is unstable when τ = 3.