Bifurcation diagram of a cubic three-parameter autonomous system

Abstract

In this paper, we study the cubic three-parameter autonomous planar system ẋ1 = k1 + k2x1 - x31 - x2, ẋ2 = k3x1 - x2 where k2, k3 > 0. Our goal is to obtain a bifurcation diagram; i.e., to divide the parameter space into regions within which the system has topologically equivalent phase portraits and to describe how these portraits are transformed at the bifurcation boundaries. Results may be applied to the macroeconomical model IS-LM with Kaldor's assumptions. In this model existence of a stable limit cycles has already been studied (Andronov-Hopf bifurcation). We present the whole bifurcation diagram and among others, we prove existence of more difficult bifurcations and existence of unstable cycles.