  In this paper, we explore the three-dimensional chaotic set near a homoclinic
cycle to a hyperbolic bifocus at which the vector field has negative
divergence. If the invariant manifolds of the bifocus satisfy a non-degeneracy
condition, a sequence of hyperbolic suspended horseshoes arises near the cycle,
with one expanding and two contracting directions. We extend previous results
on the field and we show that the first return map to a given cross section may
be approximated by a map exhibiting heteroclinic tangencies associated to two
periodic orbits. When the cycle is broken, under an additional hypothesis about
the coexistence of two heteroclinically related periodic points (one without
dominated splitting into one-dimensional sub-bundles), the heteroclinic
tangencies can be slightly modified in order to satisfy Tatjer's conditions for
a generalized tangency of codimension two. This configuration may be seen as
the organizing center, by which one can obtain Bogdanov-Takens bifurcations and
therefore, strange attractors, infinitely many sinks and non-trivial
contracting wandering domains.
