  We provide conditions to guarantee the occurrence of Shilnikov bifurcations
in analytic unfoldings of some Hopf-Zero singularities through a beyond all
order phenomenon: the exponentially small breakdown of invariant manifolds
which coincide at any order of the normal form procedure. The conditions are
computable and satisfied by generic singularities and generic unfoldings.
  The existence of Shilnikov bifurcations in the $\mathcal{C}^r$ case was
already argued by Guckenheimer in the $80$'s. About the same time, endowing the
space of $\mathcal{C}^\infty$ unfoldings with a convenient topology,
persistence and density of the Shilnikov phenomenon was proved by Broer and
Vegter in 1984. However, since the proof involves the use of flat
perturbations, this approach is not valid in the analytic context. What is
more, none of the mentioned approaches provide a computable criteria to decide
whether a given unfolding exhibits Shilnikov bifurcations or not.
  This work ends the whole discussion by showing that, under generic and
checkable hypothesis, any analytic unfolding of a Hopf-Zero singularity within
the appropriate class contains Shilnikov homoclinic orbits, and as a
consequence chaotic dynamics.
