Celestial Mechanics and Dynamical Astronomy

, Volume 114, Issue 1–2, pp 77–106 | Cite as

Boundary-value problem formulations for computing invariant manifolds and connecting orbits in the circular restricted three body problem

  • R. C. Calleja
  • E. J. Doedel
  • A. R. Humphries
  • A. Lemus-Rodríguez
  • E. B. Oldeman
Original Article

Abstract

We demonstrate the remarkable effectiveness of boundary value formulations coupled to numerical continuation for the computation of stable and unstable manifolds in systems of ordinary differential equations. Specifically, we consider the circular restricted three-body problem (CR3BP), which models the motion of a satellite in an Earth–Moon-like system. The CR3BP has many well-known families of periodic orbits, such as the planar Lyapunov orbits and the non-planar vertical and halo orbits. We compute the unstable manifolds of selected vertical and halo orbits, which in several cases leads to the detection of heteroclinic connections from such a periodic orbit to invariant tori. Subsequent continuation of these connecting orbits with a suitable end point condition and allowing the energy level to vary leads to the further detection of apparent homoclinic connections from the base periodic orbit to itself, or the detection of heteroclinic connections from the base periodic orbit to other periodic orbits. Some of these connecting orbits are of potential interest in space mission design.

Keywords

Restricted three-body problem Boundary value problems Invariant manifolds Connecting orbits Numerical continuation Stable and unstable manifolds 

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • R. C. Calleja
    • 1
  • E. J. Doedel
    • 2
  • A. R. Humphries
    • 1
  • A. Lemus-Rodríguez
    • 3
  • E. B. Oldeman
    • 2
  1. 1.Mathematics and StatisticsMcGill UniversityMontrealCanada
  2. 2.Computer ScienceConcordia UniversityMontrealCanada
  3. 3.Mathematics and StatisticsConcordia UniversityMontrealCanada

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