A degenerate Hartman theorem
type: Item
instance of: scholarly article
title: A degenerate Hartman theorem
zbMATH Open document ID: 0885.58084
DOI: 10.1007/BF02761038
author: Thomas J. Morrisey
published in: Israel Journal of Mathematics
publication date: 1998-04-26T00:00:00Z
review text: The importance of the Hartman theorem in the study of dynamical systems and particulary for the three-body problem is well-known. The author considers the case of a real analytic diffeomorphism, \(f:\mathbb{R}^2\to \mathbb{R}^2\), with \(q\) as a non-hyperbolic fixed point and \(\Delta f(q)= \text{Id}\). The author also announces several ideas as a basis for further study. He considers a vector bundle with base space a 2-manifold with fiber dimension 2, and he constructs the vector cone by fixing an element of the base space and by choosing a particular subset within the fiber over this element. He proves that this subset is closed under addition and scalar multiplication and he studies an orbit for the element of the base space and its associated cone field. It is also shown that the tangent space of the stable manifold of the equilibrium point acts on by the derivative of the diffeomorphism, and it is observed how this derivative influences the vector cone. A coordinate system with domain the base space is obtained by applying these ideas to the inverse diffeomorphism. Briefly, imposing nontrivial sufficient conditions on a 2-dimensional diffeomorphism ensures that the diffeomorphism is topologically conjugate to a decoupled map.
Mathematics Subject Classification ID: 37N99
Mathematics Subject Classification ID: 70F07
zbMATH DE Number: 952562
zbMATH Keywords: non-hyperbolic equilibrium point
zbMATH Keywords: tangent space
zbMATH Keywords: vector cone
zbMATH Keywords: cone field
zbMATH Keywords: Hartman theorem
zbMATH Keywords: three-body problem
zbMATH Keywords: real analytic diffeomorphism
MaRDI profile type: Publication
cites work: A stable manifold theorem for degenerate fixed points with applications to celestial mechanics
cites work: Q3952851
cites work: Homoclinic orbits and oscillation for the planar three-body problem
cites work: Q4692840
full work available at URL: https://doi.org/10.1007/bf02761038
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description: scientific article; zbMATH DE number 952562