  We consider diffeomorphisms $f$ with heteroclinic cycles associated to
saddles $P$ and $Q$ of different indices. We say that a cycle of this type can
be stabilized if there are diffeomorphisms close to $f$ with a robust cycle
associated to hyperbolic sets containing the continuations of $P$ and $Q$. We
focus on the case where the indices of these two saddles differ by one. We
prove that, excluding one particular case (so-called twisted cycles that
additionally satisfy some geometrical restrictions), all such cycles can be
stabilized.
