On a multivalued version of the Sharkovskii theorem and its application to differential inclusions. II
type: Item
instance of: scholarly article
title: On a multivalued version of the Sharkovskii theorem and its application to differential inclusions. II
zbMATH Open document ID: 1083.34010
author: Jan Andres
author: Libor Jüttner
author: Karel Pastor
published in: Set-Valued Analysis
publication date: 2005-06-15T00:00:00Z
review text: This work is dedicated to the extension of the Sharkovskii theorem for \(M\)-maps and its applications to differential equations and inclusions. One of the main results of this paper is the following theorem: Let an \(M\)-map \(\varphi:\mathbb R\to \mathbb R\) have an \(n\)-orbit. Then \(\varphi\) has also a \(k\)-orbit, for every \(k < n\), with the exception of at most three orbits. A multi-valued map \(\varphi:\mathbb R\to \mathbb R\) is called an \(M\)-map if it is upper-semicontinuous and its nonempty sets of values are either single points or compact intervals. An \(n\)-orbit associated to an \(M\)-map \(\varphi\) is a sequence \(\{ x_1,\dots,x_n\}\) of points \(x_j\in {\mathbb{R}}\) such that \(x_{j+1}\in \varphi(x_j)\), \(j=1,2,\dots ,n\), where \(x_{n+1}=x_1\) and the orbit cannot be formed by going \(p\)-times arround a shorter \(m\)-orbit, where \(mp=n\). The order relation \(k < n\) is in the sense of Sharkovskii. These results complete results given in the paper of the first author, J. Fišer and the second author [Set-Valued Anal. 10, 1--14 (2002; Zbl 1082.37048)], and it is proved that they are sharp by constructing counter-examples.
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 58C06
zbMATH DE Number: 2176207
zbMATH Keywords: Sharkovskii's theorem
zbMATH Keywords: multi-valued version
zbMATH Keywords: periodic orbits
zbMATH Keywords: application to differential equations and inclusions
reviewed by: Adriana Buică
MaRDI profile type: Publication
full work available at URL: s11228-004-8200-z
OpenAlex ID: W2546687151
cites work: Q3374434
cites work: On a multivalued version of the Sharkovskii theorem and its application to differential inclusions
cites work: Period three plays a negative role in a multivalued version of Sharkovskii's theorem
cites work: Q3994508
cites work: Q3836181
cites work: Period Three Implies Chaos
cites work: Multivalued mappings and Filippov's operation
cites work: Q4752342
cites work: Q5751242
DOI: 10.1007/S11228-004-8200-Z
Recommended article: Q5400546
Recommended article: A multivalued version of Sharkovskiĭ's theorem holds with at most two exceptions
Recommended article: On an extension of Sarkovskii's order
Recommended article: Period three plays a negative role in a multivalued version of Sharkovskii's theorem
Recommended article: From the Sharkovskii theorem to periodic orbits for the Rössler system
Recommended article: A Simple Proof of Sharkovsky's Theorem Revisited
Recommended article: Rotation numbers, twists and a Sharkovskii–Misiurewicz-type ordering for patterns on the interval
Recommended article: The minimal number of periodic orbits of periods guaranteed in Sharkovskii's theorem
Recommended article: Q2998687
Recommended article: Q3169571
description: scientific article; zbMATH DE number 2176207