A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: The cusp of order 4

Abstract

In this paper we give an example of a family of polynomial vector fields with three limit cycles appearing simultaneously on a Hopf bifurcation (H) of order 3 and vanishing simultaneously in a homoclinic loop bifurcation (HL) of order 3. The region with three limit cycles is a topological 3-simplex. The system is a generalization of Bogdanov's system. At the same time we give the bifurcation diagram for the universal unfolding of the cusp of order 4. This bifurcation diagram is a cone. It contains a cone on the bifurcation diagram with the three limit cycles inside a 3-simplex region, plus saddle-node and cusp bifurcations of lower order.