  We extend Sharkovskii's theorem to the cases of $N$-dimensional maps which
are close to 1D maps, with an attracting $n$-periodic orbit. We prove that,
with relatively weak topological assumptions, there exist also $m$-periodic
orbits for all $m\triangleright n$ in Sharkovskii's order, in the nearby.
  We also show, as an example of application, how to obtain such a result for
the R\"ossler system with an attracting periodic orbit, for four sets of
parameter values. The proofs are computer-assisted.
