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

In the figure-eight choreography in the classical three-body problem, both side bifurcation solutions sometimes fold at one side of the bifurcation point with cusp of action. Three numerical examples of such fold for figure-eight choreography under the Lennard-Jones-type potential and one under the homogeneous potential are introduced. Up to the forth order of representation variable of the Lyapunov-Schmidt reduced action in two dimension with three-fold symmetry, the fold is analyzed.

💡 Research Summary

The paper investigates a subtle bifurcation phenomenon occurring in the classic equal‑mass three‑body problem’s figure‑eight choreography. While the figure‑eight orbit is a highly symmetric periodic solution, certain parameter variations (the period T for Lennard‑Jones‑type potentials or the exponent a for homogeneous potentials) cause an equivariant bifurcation that partially breaks the symmetry. The authors focus on the “three‑fold‑type” bifurcation, where the isotropy subgroup of the Hessian’s degenerate eigenspace is isomorphic to the cyclic group C₃ or the dihedral group D₃.

Using Lyapunov‑Schmidt (LS) reduction, the infinite‑dimensional action functional is projected onto the two‑dimensional critical eigenspace spanned by orthonormal eigenfunctions ϕ₁, ϕ₂. Imposing the three‑fold symmetry reduces the reduced action S(r,θ) to the polar form

 S(r,θ)=S(q)+κ r²+ (A₃/3!) r³ sin 3θ + (A₄/4!) r⁴+⋯,

where κ is the critical eigenvalue crossing zero, and A₃, A₄ are higher‑order coefficients derived from the original Lagrangian. Assuming A₃·A₄=0, the variational equations yield two families of nontrivial solutions:

1. r₋(κ) ≈