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

2026/01/31 by Hiroshi Fukuda, Hiroshi Ozaki
Physics and Astronomy · Mathematics · #math-ph #math.MP

paper · pdf

arxiv created 2026/08/04 · arxiv updated 2026/08/07

Abstract

In the figure-eight choreography in the classical three-body problem, both-side bifurcation solutions sometimes fold on one side of the bifurcation point with cusp of action. Three numerical examples of a such fold for figure-eight choreography under the Lennard-Jones-type potential and one under the homogeneous potential are introduced. Up to the fourth order of representation variable of the Lyapunov-Schmidt reduced action in two dimensions with three-fold symmetry, the fold is analyzed. It is shown that the bifurcation solutions fold under a condition given by the third and fourth expansion coefficients.

Citations

• Bifurcation analysis of figure-eight choreography in the three-body problem based on crystallographic point groups
• Variational principle of action and group theory for bifurcation of figure-eight solutions
• Morse index and bifurcation for figure-eight choreographies of the equal mass three-body problem
• Figure-eight choreographies of the equal mass three-body problem with Lennard-Jones-type potentials
• A remarkable periodic solution of the three-body problem in the case of equal masses
• A Remarkable Periodic Solution of the Three-Body Problem in the Case of Equal Masses
• Braids in classical dynamics