The Evolution of the Stable and Unstable Manifold of an Equilibrium Point
Author: Meyer, Kenneth
Source: Celestial Mechanics and Dynamical Astronomy, Volume 70, Number 3, March 1998 , pp. 159-165(7)
Publisher: Springer
Abstract:
We consider the evolution of the stable and unstable manifolds of an equilibrium point of a Hamiltonian system of two degrees of freedom which depends on a parameter, ν. The eigenvalues of the linearized system are complex for ν < 0 and purely imaginary for ν > 0. Thus for ν < 0 the equilibrium has a two-dimensional stable manifold and a two-dimensional unstable manifold, but for ν > 0 these stable and unstable manifolds are gone. We study the system defined by the truncated generic normal form in this situation.One of two things happens depending on the sign of a certain quantity in the normal form expansion. In one case the two families detach as a single invariant manifold and recedes from the equilibrium as ν tends away from 0 through positive values. In the other case the stable and unstable manifold are globally connected for ν < 0 and the whole structure of these manifolds shrinks to the equilibrium as ν → 0 and disappears.These considerations have interesting implications about Strömgren's conjecture in celestial mechanics and the blue sky catastrophe of Devaney.Keywords: stable manifold; bifurcation; restricted three-body problem; Strömgren's conjecture
Document Type: Research article
DOI: http://dx.doi.org/10.1023/A:1008387507657
Affiliations: 1: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio, 45221-0025, U.S.A., Email: ken.meyer@uc.edu
Publication date: 1998-03-01
- In this: publication
- By this: publisher
- In this Subject: Astronomy
- By this author: Meyer, Kenneth

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