Sil′nikov Bifurcations in Generic 4-Unfoldings of a Codimension-4 Singularity

The main aim of this paper is to prove analytically the existence of homoclinic orbits of focus-saddle type in generic unfoldings of a codimension-4 singularity whose 2-jet in normal form is given by: Moreover, if the parameters are defined on open sets then these homoclinic orbits satisfy the hypothesis of Sil′nikov′s theorem. In this way, we are able to give a method of obtaining families of quadratic vector fields on 3 which display strange attractors.