Codimension-2 bifurcations of the Kaldor model of business cycle

In this paper, complete analysis is presented to study codimension-2 bifurcations for the nonlinear Kaldor model of business cycle. Sufficient conditions are given for the model to demonstrate Bautin and Bogdanov–Takens (BT) bifurcations. By computing the first and second Lyapunov coefficients and performing nonlinear transformation, the normal forms are derived to obtain the bifurcation diagrams such as Hopf, homoclinic and double limit cycle bifurcations. Some examples are given to confirm the theoretical results.

Research highlights
► The conditions are given such that the characteristic equation may have purely imaginary roots and double zero roots.
► Purely imaginary roots lead us to study Hopf and Bautin bifurcations and to calculate the first and second Lyapunov coefficients.
► Double zero roots lead us to study Bogdanov–Takens (BT) bifurcation.
► Bifurcation diagrams for Bautin and BT bifurcations are obtained by using the normal form theory.