Stability and bifurcation analysis for the Kaldor–Kalecki model with a discrete delay and a distributed delay

• Propose a Kaldor–Kalecki model with both discrete and distributed delays.
• Discuss the effect of the distributed delay on system dynamics.
• Analyze Hopf bifurcation by multiple scales.

In this paper, a Kaldor–Kalecki model of business cycle with both discrete and distributed delays is considered. With the corresponding characteristic equation analyzed, the local stability of the positive equilibrium is investigated. It is found that there exist Hopf bifurcations when the discrete time delay passes a sequence of critical values. By applying the method of multiple scales, the explicit formulae which determine the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are derived. Finally, numerical simulations are carried out to illustrate our main results.