• Fold of a bifurcation solution from the figure-eight choreography in the three body problem (2026)

• Bifurcation analysis of figure-eight choreography in the three-body problem based on crystallographic point groups (2025)

• Irreducible Orthogonal Representations of Point Groups for bifurcation analysis of figure eight choreography in the three-body problem using Mathematica (in Japanese) (2024)

• A proposition on similar unitary/orthogonal matrices (in Japanese) (2024) 

• Periodic solutions bifurcated from figure eight choreography: non-planar eight and non-symmetric eight (2023)

• Bifurcation of Simó H solution bifurcated from figure-eight choreographies of the equal mass three-body problem (2019)

• Morse index and bifurcation for figure-eight choreographies of the equal mass three-body problem (2019)

The following movies are periodic solutions bifurcated from the figure-eight choreographic solutions to the equation of motion

d2qi/dt2 = -∂U/∂qi, U=Σi>ju(qi-qj), i = 0,1,2.

Legends

• f8: the figure-eight choreography under homogeneous interaction potential

• α± : the figure-eight choreography under Lennerd-Jones potential

• ↼: two-fold-type bifurcation 

• ←/←: three-fold-type bifurcation with dihedral / cyclic representation group 

• ⇐/⇐: six-fold-type bifurcation  with dihedral / cyclic representation group

• Indent level of movie: the generation of bifurcations from the figure-eight

• L/R: bifurcation from left/right side 

• Hi/Lo: higher/lower action value 

• Light grayed symbol: spatial symmetry of orbits used in preprint2019

• >/< button: stop movie at next/previous T/n (initially n=12) 

D2h{S,M,μ}Dxy ← f8 a=0.9966 

D2{SM,μ}Dx D2{CS,μ}Di ⇐ f8 a=1.3424

D2h{S,M,μ}Dxy ← α+ T=16.878 

D2h{S,M,μ}Dxy ← α- T=14.836 

D2{SM,μ}Dx D2{CS,μ}Di ⇐ α+ T=16.111

D2{SM,μ}Dx D2{CS,μ}Di ⇐ α- T=14.861

D6{μC,SM}Cx ↼ α- T=14.595 

D6{μC,S}C2 ↼ α+ T=18.615 

C6h{CM,μ}Cy ↼ α+ T=17.132