Stable and unstable manifolds asymptotic to the outermost KAM curve
type: Item
instance of: scholarly article
title: Stable and unstable manifolds asymptotic to the outermost KAM curve
zbMATH Open document ID: 0777.58022
DOI: 10.1016/0022-247X(92)90313-3
author: Kiyotaka Tanikawa
author: Yoshihiro Yamaguchi
published in: Journal of Mathematical Analysis and Applications
publication date: 1992-09-27T00:00:00Z
review text: The authors' summary: ``A one-parameter family of area-preserving piecewise linear maps is considered. Behavior of the stable and unstable manifolds starting from a hyperbolic fixed point is investigated in relation to the outermost KAM (Kolmogorov-Arnold-Moser) curve of a neighboring elliptic fixed point. It is proved that, at least for certain specific values of the bifurcation parameter, each of the stable and unstable manifolds has a sequence of points that tends to the outermost KAM curve of the stable fixed point. As a corollary, homoclinic points also accumulate to the outermost KAM curve. Though simple, these seem to be the first rigorous examples of such an asymptotic behavior of stable and unstable manifolds. It is found that stable and unstable manifolds form a polygon for a sequence of parameter values even if the dynamical system given by the map is not integrable''. Reviewer's comment: This paper seems to be an important contribution to the theory of dynamical systems induced by linear maps.
Mathematics Subject Classification ID: 37C75
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 62830
zbMATH Keywords: stable manifolds
zbMATH Keywords: unstable manifolds
zbMATH Keywords: Kolmogorov-Arnold-Moser curve
zbMATH Keywords: homoclinic point
zbMATH Keywords: hyperbolic fixed points
zbMATH Keywords: elliptic fixed points
reviewed by: Andrzej Pelczar
MaRDI profile type: Publication
cites work: SMALL DENOMINATORS AND PROBLEMS OF STABILITY OF MOTION IN CLASSICAL AND CELESTIAL MECHANICS
cites work: Q3294369
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description: scientific article; zbMATH DE number 62830