Multi-parameter bifurcations of the Kaldor–Kalecki model of business cycles with delay

Abstract

In this paper, we study the following Kaldor–Kalecki model of business cycles describing the interaction of the gross product Y and the capital product K: {dY(t)dt=α[I(Y(t))−βK(t)−γY(t)],dK(t)dt=I(Y(t−τ))−(β+q)K(t). Single and multi-parameter bifurcation analysis of the system is carried out to perform center manifold reduction. We show that this system can exhibit codimension-1 singularity (transcritical and pitchfork bifurcations) and codimenion-2 singularity (nontransversal and symmetric BT bifurcations) under certain conditions on the time delay τ and the nonlinearity of I(Y) by using the normal form method for DDEs due to Faria and Magalhães. In addition, we use the normal forms of the system to predict bifurcation diagrams such as pitchfork bifurcations, homoclinic bifurcations and double limit cycle bifurcations, which are confirmed by numerical simulations.