Spatial periodic orbits in the equilateral circular restricted four-body problem: computer-assisted proofs of existence

  • Jaime Burgos-GarcíaEmail author
  • Jean-Philippe Lessard
  • J. D. Mireles James
Original Article


We use validated numerical methods to prove the existence of spatial periodic orbits in the equilateral restricted four-body problem. We study each of the vertical Lyapunov families (up to symmetry) in the triple Copenhagen problem, as well as some halo and axial families bifurcating from planar Lyapunov families. We consider the system with both equal and non-equal masses. Our method is constructive and non-perturbative, being based on a posteriori analysis of a certain nonlinear operator equation in the neighborhood of a suitable approximate solution. The approximation is via piecewise Chebyshev series with coefficients in a Banach space of rapidly decaying sequences. As by-product of the proof, we obtain useful quantitative information about the location and regularity of the solution.


Gravitational four-body problem Spatial periodic orbits Chebyshev spectral methods Computer-assisted existence proofs 



The authors offer their thanks to the two anonymous referees who read the submitted version of the manuscript. The final published version is greatly improved thanks to their insightful comments and questions. The first author was supported by PRODEP grant UACOAH-PTC-416, and the third author was partially supported by NSF grants DMS-1813501 and DMS-1700154 and by the Alfred P. Sloan Foundation Grant G-2016-7320. The authors would like to thank J.B. van den Berg for many helpful conversations in the early stages if this work.


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Authors and Affiliations

  1. 1.Facultad de Ciencias Físico MatemáticasUniversidad Autónoma de CoahuilaSaltilloMéxico
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  3. 3.Department of Mathematical SciencesFlorida Atlantic UniversityBoca RatonUSA

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