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.