Bifurcations in an economic growth model with a distributed time delay transformed to ODE
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ORIGINAL PAPER
Bifurcations in an economic growth model with a distributed time delay transformed to ODE Luca Guerrini · Adam Krawiec
· Marek Szydłowski
Received: 14 February 2020 / Accepted: 16 July 2020 © The Author(s) 2020
Abstract In this paper, we consider a model of economic growth with a distributed time-delay investment function, where the time-delay parameter is a mean time delay of the gamma distribution. Using the linear chain trick technique, we transform the delay differential equation system into an equivalent one of ordinary differential equations (ODEs). Since we are dealing with weak and strong kernels, our system will be reduced to a three- and four-dimensional ODE system, respectively. The occurrence of Hopf bifurcation is investigated with respect to the following two parameters: time-delay parameter and rate of growth parameter. Sufficient criteria on the existence and stability of a limit cycle solution through the Hopf bifurcation are presented in case of time-delay parameter. Numerical studies with the Dana and Malgrange investment
function show the emergence of two Hopf bifurcations with respect to the rate growth parameter. In this case, we have been able to detect the existence of stable long-period cycles in the economy. According to the time-delay and adjustment speed parameters, the range of admissible values of the rate of growth parameter breaks down into three intervals. First, we have stable focus, then the limit cycle and finally again the stable solution with two Hopf bifurcations. Such behavior appears for some middle interval of the admissible range of values of the rate of growth parameter.
L. Guerrini Department of Management, Polytechnic University of Marche, Piazza Martelli 8, 60121 Ancona, AN, Italy e-mail: [email protected]
1 Introduction
A. Krawiec (B) Institute of Economics, Finance and Management, Jagiellonian University, Łojasiewicza 4, 30-348 Kraków, Poland e-mail: [email protected] M. Szydłowski Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Kraków, Poland M. Szydłowski Mark Kac Complex Systems Research Centre, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland e-mail: [email protected]
Keywords Kaldor–Kalecki growth model · Distributed time delay · Bifurcation analysis · Hopf bifurcation · Linear chain trick
In economics, many processes depend on past events, so it is natural to use time-delay differential equations to model economic phenomena. Two main areas of applications are business cycle and economic growth theories. In recent decades, the analysis of the effect of investment delay has been the focus of extensive examination as a tool for endogenous cycles to explain business cycles and growth cycles. Differential equations with time delay (discrete or distributed) and their mathematical methods have been seen to be the most adequate tools to model the business cycle and growth in an economy where the investment delay plays a cru-
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cial role [1–5], as well as in physics,
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