The evolution of the stable and unstable manifold of an equilibrium point
type: Item
instance of: scholarly article
title: The evolution of the stable and unstable manifold of an equilibrium point
zbMATH Open document ID: 0936.70016
DOI: 10.1023/A:1008387507657
author: Kenneth R. Meyer
published in: Celestial Mechanics and Dynamical Astronomy
publication date: 2000-05-18T00:00:00Z
review text: The author deals with a Hamiltonian system with two degrees of freedom depending on a parameter \(\nu\) which has an equilibrium point at the origin. Assuming that the eigenvalues of the linearized system are complex for \(\nu<0\) and purely imaginary for \(\nu>0\), the author shows that there exist two-dimensional stable and two-dimensional unstable manifolds for \(\nu<0\) which are gone if \(\nu>0\). The evolution of these submanifolds as \(\nu \to 0\) depends on a certain quantity \(\eta\). In the case \(\eta<0\), the submanifolds detach from the equilibrium as single invariant manifolds, in the case \(\eta>0\) they shrink to the equilibrium and disappear. The proof uses the Sokols'skij normal form of the Hamiltonian \(H=x_2y_1- x_1y_2+\delta {1\over 2} (x^2_1+x^2_2) +\nu\delta {1\over 2}(y^2_1+ y^2_2)+\eta \delta[ {1\over 2}(y^2_1 +y^2_2)]^2\), and a symplectic change to polar coordinates.
Mathematics Subject Classification ID: 70H14
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 37N05
zbMATH DE Number: 1287753
zbMATH Keywords: Sokol'skij normal form of Hamiltonian
zbMATH Keywords: Hamiltonian system with two degrees of freedom
zbMATH Keywords: parameter
zbMATH Keywords: eigenvalues
zbMATH Keywords: linearized system
zbMATH Keywords: polar coordinates
reviewed by: Jan Chrastina
MaRDI profile type: Publication
full work available at URL: a:1008387507657
OpenAlex ID: W2121753428
description: scientific article; zbMATH DE number 1287753