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

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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