Dynamics of a delayed business cycle model with general investment function

• A delayed business cycle model is formulated and rigorously analyzed.
• Well-posedness of the model and local stability of the economic equilibrium are determined.
• Direction and stability of the Hopf bifurcation are investigated.
• Global existence of bifurcating periodic solutions is established.
• Numerical simulations are presented to illustrate our theoretical results.

The aim of this paper is to study the dynamics of a delayed business cycle model with general investment function. The model describes the interaction of the gross product and capital stock. Furthermore, the delay represents the time between the decision of investment and implementation. Firstly, we show that the model is well posed by proving the global existence and boundedness of solutions. Secondly, we determine the economic equilibrium of the model. By analyzing the characteristic equation, we investigate the stability of the economic equilibrium and the local existence of Hopf bifurcation. Also, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theory. Moreover, the global existence of bifurcating periodic solutions is established by using the global Hopf bifurcation theory. Finally, our theoretical results are illustrated with some numerical simulations.