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

Abstract

This paper is concerned with three-dimensional vector fields and more specifically with the study of dynamics in unfoldings of the nilpotent singularity of codimension three. The ultimate goal is to understand the dynamics and bifurcations in the unfolding of the singularity. However, it is clear from the literature that the bifurcation diagram is very complicated and a complete study is far beyond the current possibilities, not only from a theoretical point of view but also from a numerical point of view, despite recent developments of computational methods for dynamical systems. Since all complicated dynamical behaviour is known to be of small amplitude, shrinking to the singularity for parameter values tending to the bifurcation parameter, 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 presence of such a nilpotent singularity of codimension three. The notion is introduced of 'traffic regulator' and the specific sets called the 'inset' and 'outset', which give a new framework for studying a transition map in a cylindrical neighbourhood of the singularity that contains all the non-trivial dynamics that can bifurcate from the singularity, focusing on the domains on which the transition map is defined. A list is also given of open problems which are believed to be helpful for future investigation of the bifurcations from the nilpotent triple zero singularity in 3.