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

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.