Topological Decoupling Near Planar Parabolic Orbits

Clark Robinson*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


In two different three body problems, oscillatory orbits have been shown to exist for the three-body problem in Celestial Mechanics: Sitnikov, Alekseev, Moser, and McGehee considered a spatial problem which had one degree of freedom; Easton, McGehee, and Xia considered a planar problem which had at least three degrees of freedom. Both situations involve analyzing the motion as one particle with mass m3 goes to infinity while the other two masses stay bounded in elliptic motion. Motion with m3 at infinity corresponds to a periodic orbit in the first problem and the Hopf flow on S3 in the second problem, both of which are normally degenerately hyperbolic. The proof of the existence of oscillatory orbits uses stable and unstable manifolds for these degenerate cases. In order to get the symbolic dynamics which shows the existence of oscillation, the orbits which go near infinity need to be controlled for an unbounded length of time. In this paper, we prove that the flow near infinity for the Easton–McGehee example with three degrees of freedom is topologically equivalent to a product flow, i.e., a Grobman–Hartman type theorem in the degenerate situation.

Original languageEnglish (US)
Pages (from-to)337-351
Number of pages15
JournalQualitative Theory of Dynamical Systems
Issue number2
StatePublished - Oct 1 2015


  • Celestial mechanics
  • Parabolic orbits
  • Three-body problem

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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