  In this paper we study the existence of heteroclinic cycles in generic
unfoldings of nilpotent singularities. Namely we prove that any nilpotent
singularity of codimension four in $\mathbb{R}^4$ unfolds generically a
bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic
orbits to equilibrium points with two pairs of complex eigenvalues. We also
prove that any nilpotent singularity of codimension three in $\mathbb{R}^3$
unfolds generically a bifurcation curve of heteroclinic cycles between two
saddle-focus equilibrium points with different stability indexes. Under generic
assumptions these cycles imply the existence of homoclinic bifurcations.
Homoclinic orbits to equilibrium points with complex eigenvalues are the
simplest configurations which can explain the existence of complex dynamics as,
for instance, strange attractors. The proof of the arising of these dynamics
from a singularity is a very useful tool, particularly for applications.
