Searching for cycles in non-linear autonomous discrete dynamical systems

D. Dmitrishin, A. Stokolos, M. Tohaneanu

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)603-626
Number of pages24
JournalNew York Journal of Mathematics
Volume25
StatePublished - 2019

Bibliographical note

Funding 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.

Publisher Copyright:
© 2019, University at Albany. All rights reserved.

Keywords

  • Chaos control
  • Discrete dynamical systems
  • Extremal polynomials

ASJC Scopus subject areas

  • Mathematics (all)

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