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Homoclinic dynamics in a restricted four-body problem: transverse connections for the saddle-focus equilibrium solution set

  • Shane Kepley
  • J. D. Mireles JamesEmail author
Original Article
  • 101 Downloads

Abstract

We describe a method for computing an atlas for the stable or unstable manifold attached to an equilibrium point and implement the method for the saddle-focus libration points of the planar equilateral restricted four-body problem. We employ the method at the maximally symmetric case of equal masses, where we compute atlases for both the stable and unstable manifolds. The resulting atlases are comprised of thousands of individual chart maps, with each chart represented by a two-variable Taylor polynomial. Post-processing the atlas data yields approximate intersections of the invariant manifolds, which we refine via a shooting method for an appropriate two-point boundary value problem. Finally, we apply numerical continuation to some of the BVP problems. This breaks the symmetries and leads to connecting orbits for some nonequal values of the primary masses.

Keywords

Gravitational 4-body problem Invariant manifolds High-order Taylor methods Automatic differentiation Numerical continuation 

Mathematics Subject Classification

70K44 34C45 70F15 

Notes

Acknowledgements

The authors would like to sincerely thank two anonymous referees who carefully read the submitted version of the manuscript. Their suggestions greatly improved the final version. The second author was partially supported by NSF Grant DMS-1813501. Both authors were partially supported by NSF Grant DMS-1700154 and by the Alfred P. Sloan Foundation Grant G-2016-7320.

Compliance with ethical standards

Conflict of interest

The authors of this manuscript certify that they have no affiliations with or involvement in any organization or entity with any financial interest (such as honoraria, educational grants, participation in speaker’s bureaus, membership, employment, consultancies, stock ownership, or other equity interest, and expert testimony or patent-licensing arrangements), or nonfinancial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA
  2. 2.Department of Mathematical SciencesFlorida Atlantic UniversityBoca RatonUSA

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