From the Sharkovskii theorem to periodic orbits for the Rössler system

Abstract

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▹n in Sharkovskii's order, in the nearby.We also show, as an example of application, how to obtain such a result for the Rössler system with an attracting periodic orbit, for four sets of parameter values. The proofs are computer-assisted.