Hopf bifurcation in a generalized Goodwin model with delay

Abstract

In the theory of dynamical systems, delay differential equations have an important place. While in a non-delayed dynamical system the rate of change of state variables depends instantaneously on the state variables, in delayed dynamical systems this functional dependence can be with a time delay. In real life problems, this may occur, for example, when the signals transmitted to the processor of a physical system that collects and evaluates signals from different points in space are transmitted with a time difference due to the path difference. Methods and simulation tools are available in the literature for analysing the stability of a dynamic system formulated without delay, either locally at equilibrium points or globally. The "stable" and "unstable" conditions that we encounter in stability analysis can be target conditions according to the physical model under investigation. For example, in a dynamic system that approximately models the vibrations of a structural element vibrating under the effect of an earthquake, it is desired that the vibrations evolve to zero equilibrium point over time and that the zero equilibrium point is stable. In a mechanical system which is desired to generate energy with its vibrations, it will be the target condition that the vibrations are not damped. Stability analysis is performed to determine the parameter conditions that will give the stable and unstable conditions of the equilibrium points. However, if the dynamical system modelling the relevant physical system actually has a delayed time dynamics, the system may actually be unstable in a parameter set that is predicted as a stable equilibrium point by the non-delayed analysis. Therefore, the analysis of the relevant dynamics needs to be carried out in the formulation of the theory of delayed dynamical systems. Goodwin's model is one of the well-known dynamical systems in macroeconomics which formulates the mechanism between the employment ratio and the wage share in a closed economy. The model is formulated under the assumptions of steady technical progress and steady growth in technical force. Only two factors of production are considered: labour and capital. Working class consume all their wages, whereas all profits are invested by the capital holders. A constant capital-output ratio is assumed, and the relation between the inflation rate and unemployment rate is determined by a linearized Phillips curve. There is an argument in the literature that the functional dependence of the Phillips curve, which expresses the relationship between the inflation rate and the unemployment rate, depends on the time delay. There are only a few publications that consider this dependence with a delay and dynamically analyse modified versions of the Goodwin model. The Goodwin model, which is essentially a mathematical economics analogue of the predator-prey system of population dynamics, despite its simplicity, explains to some extent the periodic behaviour of state variables observed at certain time intervals.