From the Sharkovskii theorem to periodic orbits for the R\"ossler system

TL;DR

This paper extends Sharkovskii's theorem to certain high-dimensional maps near 1D maps, demonstrating the existence of various periodic orbits, including for the R"ossler system, with proofs supported by computer assistance.

Contribution

It generalizes Sharkovskii's theorem to N-dimensional maps close to 1D, and applies it to the R"ossler system for specific parameters.

Findings

Existence of multiple periodic orbits in high-dimensional maps near 1D maps.

Application of the extended theorem to the R"ossler system.

Computer-assisted proofs validate the theoretical results.

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 $m\triangleright n$ 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.