A Note on the Triple-Zero Linear Degeneracy: Normal Forms, Dynamical and Bifurcation Behaviors of an Unfolding

This paper is devoted to the analysis of bifurcations in a three-parameter unfolding of a linear degeneracy corresponding to a triple-zero eigenvalue. We carry out the study of codimension-two local bifurcations of equilibria (Takens–Bogdanov and Hopf-zero) and show that they are nondegenerate. This allows to put in evidence the presence of several kinds of bifurcations of periodic orbits (secondary Hopf,…) and of global phenomena (homoclinic, heteroclinic). The results obtained are applied in the study of the Rössler equation.