Jumping stable manifolds for dissipative maps of the plane
type: Item
instance of: scholarly article
title: Jumping stable manifolds for dissipative maps of the plane
zbMATH Open document ID: 0686.58008
DOI: 10.1016/0167-2789(89)90097-3
author: Stephen M. Hammel
author: Christopher K. R. T. Jones
published in: Physica D
publication date: 1989-01-01T00:00:00Z
review text: An essential assumption of the authors is that basin boundaries arise `typically' from stable manifolds of saddle points. Other separatrices are ignored. The `discontinuity' argument is inspired by low accuracy numerical computations. A numerical study of a particular \({\mathbb{C}}\to {\mathbb{C}}\) map, due to Ikeda, is used as confirming evidence. For \({\mathbb{C}}\to {\mathbb{C}}\) maps it was already known to Julia (70 years ago) that basin boundaries can have an extremely complicated structure, not necessarily related to saddle-point manifolds. Other non-`typical' cases can be found in books, for example [in the reviewer and \textit{C. Mira}: Recurrences and discrete dynamical systems (Lect. Notes Math. 809, 1980; Zbl 0449.58003)].
Mathematics Subject Classification ID: 58D15
Mathematics Subject Classification ID: 65Q05
zbMATH DE Number: 4124640
zbMATH Keywords: maps
zbMATH Keywords: basin boundaries
zbMATH Keywords: stable manifolds of saddle points
Wikidata QID: Q57927232
reviewed by: Igor Gumowski
MaRDI profile type: Publication
full work available at URL: 0167-2789(89)90097-3
OpenAlex ID: W2049710227
cites work: Basin boundary metamorphoses: Changes in accessible boundary orbits
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Differentiable dynamical systems
description: scientific article; zbMATH DE number 4124640