Session II.1 - Computational Dynamics
Poster
Attracting period 3 implies all natural periods for multidimensional maps
Anna Gierzkiewicz
Jagiellonian University in Krakow, Poland - This email address is being protected from spambots. You need JavaScript enabled to view it.
We present the method from [ A. G., P. Zgliczynski, J Differ Equ, 314 (2022),733--751] 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 Roessler 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, etc. We also expect that this method works for DDEs. The proofs are computer-assisted with the use of CAPD library for C++.
Joint work with Piotr Zgliczynski (Jagiellonian University, Poland).