Local Hopf Birurcation and Stability of Limit Cycle in a Delayed Kaldor-Kalecki Model

A Kaddar,H Talibi Alaoui

doi:10.15388/na.2009.14.3.14499

Abstract

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.

Highlights

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
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
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

Summary

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: dY dt. 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: dK (2). In this model the nonlinearity of investment and saving function leads to limit cycle solution (see [7,8,9] for more information). 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.

Steady state and stability analysis

Steady state

Local stability and local Hopf bifurcation analysis

Direction of Hopf bifurcation

Application