Structure change of stable and unstable manifolds in two-dimensional maps: Period-doubling bifurcation
type: Item
instance of: scholarly article
title: Structure change of stable and unstable manifolds in two-dimensional maps: Period-doubling bifurcation
zbMATH Open document ID: 0771.58036
DOI: 10.1016/0960-0779(92)90002-5
author: Yoshihiro Yamaguchi
author: Kiyotaka Tanikawa
published in: Chaos, Solitons and Fractals
publication date: 1992-09-27T00:00:00Z
review text: The authors' summary: ``In dissipative orientations preserving two- dimensional maps, the tangled structure of stable and unstable manifolds of saddles born in the sequence of the period-doubling bifurcation is investigated using a geometrical method. Two theorems are obtained. The first theorem makes clear the existence of persistent transverse intersection between the stable and unstable manifolds over a large parameter interval. Using this theorem, the accumulation relations of the unstable manifolds are obtained. The second theorem specifies the type of the first asymptotic heteroclinic tangency, and gives the order relation of several critical values at which the first asymptotic heteroclinic tangency and the first homoclinic tangency occur. Numerical evidence supporting the theoretical result is also given.''.
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 64954
zbMATH Keywords: stable manifolds
zbMATH Keywords: unstable manifolds
zbMATH Keywords: period-doubling bifurcation
reviewed by: J. Kolomý
MaRDI profile type: Publication
full work available at URL: 0960-0779(92)90002-5
OpenAlex ID: W2050740405
cites work: Ergodic theory of chaos and strange attractors
cites work: The dynamics of the Hénon map
cites work: A two-dimensional mapping with a strange attractor
cites work: Q3696187
cites work: Q3769642
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: A theorem on the first heteroclinic tangency in two-dimensional maps. Orientation-preserving cases
cites work: New type of heteroclinic tangency in two-dimensional maps
cites work: On the first tangency of stable and unstable manifolds in C r planar diffeomorphisms
cites work: On Morse-Smale dynamical systems
cites work: Basin boundary metamorphoses: Changes in accessible boundary orbits
description: scientific article; zbMATH DE number 64954