Shil'nikov configurations in any generic unfolding of the nilpotent singularity of codimension three on R3

We are interested in the dynamics arising in generic unfoldings of the nilpotent singularity of codimension three. Since Shil'nikov-type homoclinic orbits make up the simplest scenario where very complicated dynamics for three-dimensional vector fields can be shown, the aim of this paper is to prove analytically their existence in any of such unfoldings.