Homoclinic bifurcations in the unfolding of the nilpotent singularity of codimension 4 in R4

Abstract

The rich variety of homoclinic phenomena exhibited by the limit family of any generic unfolding of a fourdimensional nilpotent singularity of codimension-four is discussed. Specifically, numerical techniques based on the Taylor integrator and the expansion of the invariant manifolds were designed for this family. A partial bifurcation diagram which includes, besides a suggestive catalogue of local bifurcations of equilibria, folds and period doublings of periodic orbits is also given. These results are certainly the first steps towards a much more ambitious goal: to achieve a general understanding of these codimension-four unfoldings.