Dynamics of the delayed business cycle model with expectation and general saving function

Abstract

In this paper, we study a class of business cycle models related to the relationship between the gross product and the capital stock. The model is extended by adding general terms of the investment and the saving functions. In addition, a time delay which is represented an expectation and time lag for investment is also added into the model. The aim for this work is to find the sufficient conditions in which the model’s solution is periodic. We study the existence of unique positive equilibrium. Next, the linearization method is used to analyze local behavior of the model. In addition, by considering time delay as a bifurcation parameter, we investigate sufficient conditions for the existence of a Hopf bifurcation of the model. Our results are applied to represent conditions for an occurrence of business cycles, which can happen in many business situations. Finally, some numerical simulations are illustrated to support our theoretical results.