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

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 Lya...

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Bibliographic Details
Published in:Celestial mechanics and dynamical astronomy 2019, Vol.131 (1), p.1-36, Article 2
Main Authors: Burgos-García, Jaime, Lessard, Jean-Philippe, James, J. D. Mireles
Format: Article
Language:English
Subjects:
Aerospace Technology and Astronautics
Astrophysics and Astroparticles
Banach spaces
Bifurcations
Chebyshev approximation
Classical Mechanics
Dynamical Systems and Ergodic Theory
Four body problem
Geophysics/Geodesy
Numerical methods
Orbits
Original Article
Physics
Physics and Astronomy
Sequences
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Summary: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.
ISSN:0923-2958
1572-9478
DOI:10.1007/s10569-018-9879-8