Bifurcation analysis of figure-eight choreography in the three-body problem based on crystallographic point groups

2024/06/11 by Hiroshi Fukuda, H. Ozaki, Fukuda, Hiroshi +1 · 1 citation
Engineering · Physics and Astronomy · #Astro and Planetary Science #FOS: Physical sciences #Mathematical Physics (math-ph) #Nuclear physics research studies #Spacecraft Dynamics and Control

paper · pdf · doi:10.48550/arxiv.2406.07717

openalex publication_date 2024/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The bifurcation of figure-eight choreography is analyzed by its symmetry group based on the variational principle of the action. The irreducible representations determine the symmetry and the dimension of the Lyapunov-Schmidt reduced action, which yields four types of bifurcations in the sequence of the bifurcation cascade. Type 1 bifurcation, represented by trivial representation, bifurcates two solutions. Type 2, by non-trivial one-dimensional representation, bifurcates two congruent solutions. Type 3 and 4, by two-dimensional irreducible representations, bifurcate two sets of three and six congruent solutions, respectively. We analyze numerical bifurcation solutions previously published and four new ones: non-symmetric choreographic solution of type 2, non-planar solution of type 2, y-axis symmetric solution of type 3, and non-symmetric solution of type 4.

Cited by

• Fold of a bifurcation solution from the figure-eight choreography in the three body problem

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