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

Abstract

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.

Global Existence of Periodic Solutions in a Delayed Kaldor-Kalecki Model

“…is clearly continuous in (λ, τ ) ∈ C × R + . This justifies hypothesis (A3) in [11, p. 4814], for the considered system (12). A stationary solution (E * , τ ) is called a center if det(∆ (E * ,τ ) (im 2π ω0 )) = 0 for some positive integer m. A center (E * , τ ) is said to be isolated if it is the only center in some neighborhood of (E * , τ ).…”

“…z is a p-periodic solution of system (12) , and let l(E * , τ n , 2π ω0 ) denote the connected component of (E * , τ n , 2π ω0 ) in Σ, where ω 0 and τ n are defined respectively in (7) and (8). Proof.…”

“…Under hypotheses (H1) and (H2) of Theorem 1, the periodic solutions of system(12) are uniformly bounded.…”

“…In this paper we use a method on the global existence of periodic solutions given by Wu [11] to study the following delayed Kaldor-Kalecki model of business cycle (see [5,12,14,15]):…”

“…where Y is the income, K is the capital stock, α is the adjustment coefficient in the goods market, δ is the depreciation rate of capital stock, I is the investment function, S is the saving and τ is the time delay needed for new capital to be installed (see [1]). In [12,2008], we investigated the local Hopf bifurcation, that is proven to exist as the delay cross some critical value τ 0 . In [13,2009], we established an explicit algorithm for determining the direction of Hopf bifurcation and the stability or instability of the bifurcating branch of periodic solutions using the methods presented by Diekmann et al in [16,1995].…”