How often do simple dynamical processes have infinitely many coexisting sinks?
type: Item
instance of: scholarly article
title: How often do simple dynamical processes have infinitely many coexisting sinks?
zbMATH Open document ID: 0602.58036
DOI: 10.1007/BF01463400
author: James A. Yorke
author: Laura Tedeschini-Lalli
published in: Communications in Mathematical Physics
publication date: 1986-01-01T00:00:00Z
review text: This work concerns the nature of chaotic dynamical processes. Sheldon Newhouse wrote on dynamical processes (depending on a parameter \(\mu)\) \(x_{n+1}=T(x_ n;\mu)\), where x is in the plane, such as might arise when studying Poincaré return maps for autonomous differential equations in \({\mathbb{R}}^ 3\). He proved that if the system is chaotic there will very often be existing parameter values for which there are infinitely many periodic attractors coexisting in a bounded region of the plane, and that such parameter values \(\mu\) would be dense in some interval. The fact that infinitely many coexisting sinks can occur brings into question the very nature of the foundations of chaotic dynamical processes. We prove, for an apparently typical situation, that Newhouse's construction yields only a set of parameter values \(\mu\) of measure zero.
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 3971830
zbMATH Keywords: chaotic dynamical processes
zbMATH Keywords: periodic attractors
zbMATH Keywords: Newhouse's construction
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01463400
OpenAlex ID: W1985863580
description: scientific article; zbMATH DE number 3971830