In the current paper we suggest a new robust algorithm to search for cycles of arbitrary length in non-linear autonomous discrete dynamical systems. With the help of the computer we were able to find (unstable) cycles for several basic maps of nonlinear science: Hénon, Holmes cubic, Ikeda, Lozi, Elhaj-Sprott. The theoretical part of the paper is based on properties of a new family of extremal polynomials that contains the Fejér and Suffridge polynomials. The associated combination of geometric complex analysis and discrete dynamics seems to be a new phenomenon, both theoretical and practical. A novelty of this paper is in the discovery of a close connection between two seemingly disconnected fields: extremal polynomials and cycles in dynamical systems.
|Number of pages||24|
|Journal||New York Journal of Mathematics|
|State||Published - 2019|
Bibliographical noteFunding Information:
We are thankful to an anonymous referee for carefully reading the paper and finding several typos. M.T. was supported in part by the NSF grant DMS–1636435.
© 2019, University at Albany. All rights reserved.
- Chaos control
- Discrete dynamical systems
- Extremal polynomials
ASJC Scopus subject areas
- Mathematics (all)