One-parameter families of mappings of the interval with negative Schwarzian derivative, in which monotonicity of bifurcations breaks down
type: Item
instance of: scholarly article
title: One-parameter families of mappings of the interval with negative Schwarzian derivative, in which monotonicity of bifurcations breaks down
zbMATH Open document ID: 0697.58028
DOI: 10.1007/BF01060393
published in: Ukrainian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: The paper gives examples of \(C^ 3\)-convex unimodal (i.e. having a unique extremum at a point c) functions f: \(I\to I\), where \(I=[0,1]\), \(f(0)=f(1)=0\), \(f(c)=1\) with the Schwarzian \(Sf(x)=f'''(x)/f'(x)- ((3/2)(f''(x)/f'(x))^ 2<0\) for \(x\in I\setminus \{c\}\) such that, for the family \(\{f_{\lambda}\}_{\lambda \in (0,1]}\), where \(f_{\lambda}=\lambda \cdot f\), a monotonicity of bifurcations of cycles is violated. This means that there is \(\lambda >\lambda_ f(n)=\inf \{\lambda (0,1]: f_{\lambda}\) has a cycle of the period \(n\}\) such that \(f_{\lambda}\) has no cycle of period n. Some consequences of the fact are discussed.
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 26A18
zbMATH DE Number: 4143091
zbMATH Keywords: one-dimensional dynamical systems
zbMATH Keywords: periodic points
zbMATH Keywords: bifurcation of periodic points
author: Sergiĭ Kolyada
reviewed by: Wojciech Kryszewski
MaRDI profile type: Publication
cites work: Sharkovskii's order for the appearance of superstable cycles in one-parameter families of simple real maps: An elementary proof
cites work: Q3668545
cites work: Shift-maximal sequences in function iteration: Existence, uniqueness and multiplicity
description: scientific article; zbMATH DE number 4143091