On quadratic symplectic mappings
type: Item
instance of: scholarly article
title: On quadratic symplectic mappings
zbMATH Open document ID: 0806.58024
DOI: 10.1007/BF02572331
published in: Mathematische Zeitschrift
publication date: 1994-12-21T00:00:00Z
full work available at URL: https://eudml.org/doc/174660
review text: In this paper the algebraic form of quadratic symplectic mappings in \(\mathbb{R}^{2n}\) with the standard symplectic structure is investigated. For the planar case this leads to the familiar Hénon maps. For \(n \geq 2\) it is shown that any quadratic map can be written as product of a shear map to be multiplied from the right and left by affine symplectic maps. As a consequence it is shown that the inverse map of a quadratic symplectic map is also quadratic. In the generic case normal forms of such quadratic maps are derived, where special attention is given to the 4-dimensional case. Also generalized symplectic maps are considered and the question of strange attractors for such maps is raised. In an aside the stability of elliptic fixed points for area-preserving quadratic mappings in the plane is discussed where Feldman's theorem in the theory of transcendental numbers is used.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 37G05
zbMATH DE Number: 704927
zbMATH Keywords: quadratic symplectic mappings
zbMATH Keywords: Henon maps
zbMATH Keywords: inverse map
zbMATH Keywords: normal forms
zbMATH Keywords: strange attractors
author: Jürgen Moser
MaRDI profile type: Publication
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OpenAlex ID: W1991100653
description: scientific article; zbMATH DE number 704927