Sharkovsky orderings of higher order difference equations
type: Item
instance of: scholarly article
title: Sharkovsky orderings of higher order difference equations
zbMATH Open document ID: 1060.39010
author: Uwe an der Heiden
author: Mann-Lin Liang
published in: Discrete and Continuous Dynamical Systems
publication date: 2004-10-27T00:00:00Z
review text: The authors invistigate the periodic nature of the \(k\)-th order nonlinear difference equation \(x_{n}=f(x_{n-k}).\) They introduce a new definition of the \(k\)-th Sharkovsky ordering, which involve a special case the Sharkovsky ordering with \(k=1\) and extending the Sharkovsky Ordering Theorem of the nonlinear difference equation \(x_{n}=f(x_{n-1}).\) They prove that the set of periods of a periodic solutions \(\{x_{n}\}\) of \(x_{n}=f(x_{n-k})\) is given by the \(k\)-th ordering set \(S_{k}(p),\) where \(p\) is a cycle of minimal period of \(x_{n}=f(x_{n-1})\). Some applications are given for the number of attracting cycles of \(x_{n}=f(x_{n-k})\).
reviewed by: S. H. Saker
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 37F20
zbMATH DE Number: 2110776
zbMATH Keywords: nonlinear difference equation
zbMATH Keywords: periodic solutions
zbMATH Keywords: Sharkovsky ordering
zbMATH Keywords: attracting cycles
MaRDI profile type: Publication
full work available at URL: dcds.2004.11.599
OpenAlex ID: W2030861188
DOI: 10.3934/DCDS.2004.11.599
description: scientific article; zbMATH DE number 2110776