Attracting period 3 implies all natural periods for multidimensional maps

We present the method from \cite{Gierz:1} for finding a wide variety of periodic orbits for multidimensional maps with an attracting $n$-periodic orbit. The set of periods is induced by the Sharkovskii ordering `$\triangleleft$' of natural numbers: \[ 3\triangleleft 5 \triangleleft 7 \triangleleft \cdots \triangleleft 2\cdot 3 \triangleleft 2 \cdot 5 \triangleleft \cdots \triangleleft 2^2\cdot 3 \triangleleft 2^2 \cdot 5 \triangleleft \dots \triangleleft 2^k \triangleleft 2^{k-1} \triangleleft \cdots \triangleleft 2^2 \triangleleft 2 \triangleleft 1\text{.} \]
As an example, we prove the existence of $n$-periodic orbits for all $n\in\mathbb{N}$ in the R\"ossler system with a $3$-periodic orbit, the existence of $n$-periodic orbits for all $n\in\mathbb{N}\setminus\{3\}$ in a similar system with a $5$-periodic orbit, {\it etc.} We also expect that this method works for DDEs (joint work in progress with R. Szczelina). The proofs are computer-assisted with the use of CAPD library for C++.
\begin{thebibliography}{99} \bibitem{Gierz:1} \textsc{A. Gierzkiewicz, P. Zgliczy\'nski}, From the Sharkovskii theorem to periodic orbits for the R\"ossler system, \textit{J. Differential Equations}, \textbf{314}(2022), 733--751. \end{thebibliography}