  For a continuous flow on a compact metric space, the aim of this paper is to
prove a Conley-type decomposition of the strong chain recurrent set. We first
discuss in details the main properties of strong chain recurrent sets. We then
introduce the notion of strongly stable set as an invariant set which is the
intersection of the $\omega$-limits of a specific family of nested and
definitively invariant neighborhoods of itself. This notion strengthens the one
of stable set; moreover, any attractor results strongly stable. We then show
that strongly stable sets play the role of attractors in the decomposition of
the strong chain recurrent set; indeed, we prove that the strong chain
recurrent set coincides with the intersection of all strongly stable sets and
their complementaries.
