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

TL;DR

This paper analyzes fold bifurcation solutions in the figure-eight choreography of the three-body problem, using numerical examples and Lyapunov-Schmidt reduction to understand the conditions for solution folding.

Contribution

It introduces numerical examples of fold bifurcations in the figure-eight choreography under various potentials and analyzes the fold conditions up to fourth order in Lyapunov-Schmidt reduction.

Findings

Fold bifurcations occur under specific conditions related to expansion coefficients.

Numerical examples demonstrate folds under Lennard-Jones-type and homogeneous potentials.

Analysis up to fourth order reveals the mathematical criteria for solution folding.

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.