New aspects in the unfolding of the nilpotent singularity of codimension three

This paper is concerned with three-dimensional vector fields and more speci fically with the study of dynamics in unfoldings of the nilpotent singulari ty of codimension three. The ultimate goal is to understand the dynamics an d bifurcations in the unfolding of the singularity. However, it is clear fr om the literature that the bifurcation diagram is very complicated and a co mplete study is far beyond the current possibilities, not only from a theor etical point of view but also from a numerical point of view, despite recen t developments of computational methods for dynamical systems, Since all co mplicated dynamical behaviour is known to be of small amplitude, shrinking to the singularity for parameter values tending to the bifurcation paramete r, the aim in this paper is especially to focus on a different aspect that might be interesting in the study of global bifurcation problems in the pre sence of such a nilpotent singularity of codimension three. The notion is i ntroduced of-traffic regulator' and the specific sets called the 'inset' an d 'outset', which give a new framework for studying a transition map in a c ylindrical neighbourhood of the singularity that contains all the non-trivi al dynamics that can bifurcate from the singularity, focusing on the domain s on which the transition map is defined, A list is also given of open prob lems which are believed to be helpful for future investigation of the bifur cations from the nilpotent triple zero singularity in R-3.