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

TL;DR

This work analyzes bifurcations of the figure-eight choreography in the equal-mass three-body problem by exploiting the symmetry group $D_{6h}$ and an action-based variational framework. It derives four bifurcation types via the Lyapunov–Schmidt reduction of the action and classifies bifurcating branches according to irreducible representations of the symmetry group, providing explicit reduced-action forms for 1D and 2D cases. Numerical results for Lennard–Jones and homogeneous potentials yield new non-symmetric, non-planar, and plane-symmetric solutions, illustrating the theory's predictive power. The approach offers a general equivariant bifurcation toolkit for symmetric periodic solutions in few-body systems and can be extended to symmetry-changing regimes and inverse bifurcations.

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.

Figures (7)

• Figure 1: Diagram for cascade of bifurcations. Shapes with $n$-fold symmetry show groups $D_{n(h)}$ or $C_{n(h)}$. The symbol with the dark edge is $D_{n(h)}$, the light edge $C_{n(h)}$, the rounded edge $n$ accompanying $h$, and the grayed face non-planar. $n \ge 3$ is choreographic. Lines connecting symbols show fold $n$ of representation $D_{n}$ or $C_{n}$. The solid line is $n=2$, dot-dashed 3, and dashed 6, respectively. Factor ${\cal C}^x$ in generators is omitted.
• Figure 2: Orbit at $T=45$ for non-symmetric choreography $q^{(b)^2}$ with $C_6\{\mu{\cal C}\}C$ bifurcated at $T^{(b)^2}=19.020$ from $q^{(b)}$ with $D_{6}\{\mu{\cal C},{\cal S}\}C_i$ in table \ref{['LJ18615']}. Initial conditions: $(x_1,y_1) = (1.2013307,0.85317488)$, $(x_2,y_2) = (-1.8559322,-0.32791656)$, $(\dot{x}_1,\dot{y}_1) = (-0.30882003,-0.13297343)$ and $(\dot{x}_2,\dot{y}_2) = (-0.041060813,0.14612712)$.
• Figure 3: Orbits at $T=30$ for $y$-axis symmetric solution $q^{(b)^2}$ with $D_2\{{\cal M}, \mu\}D_y$fukuda2023 and their action values, bifurcated at $T^{(b)^2}=17.235$ from $q^{(b)}$ with $C_{6h}\{{\cal CM},\mu\}C_y$. (a) $q^{(b)^2}$ from the left side of bifurcation point, (b) from right side, and (c) action value $S(q^{(b)^2})-S(q^{(b)})$. Initial conditions: (a) $(x_1, y_1) = (0, 0.10529629)$, $(x_2, y_2) = (1.5421890, 0.45426224)$, $(\dot{x}_1, \dot{y}_1) = (0.30563750, -0.37778377)$ and $(\dot{x}_2, \dot{y}_2) = (-0.17440457, 0.19744777)$. (b) $(x_1, y_1) = (0, 0.0014326920)$, $(x_2, y_2) = (1.6469521, 0.47710996)$, $(\dot{x}_1, \dot{y}_1) = (0.23166197, -0.38278558)$ and $(\dot{x}_2, \dot{y}_2) = (-0.12592133, 0.18861645)$.
• Figure 4: $S(q^{(b)}+\phi r)-S(q^{(b)})$, $r=(r_1,r_2)$ at $T=17.259$ for $y$-axis symmetric solution $q^{(b)^2}$ with $D_2\{{\cal M}, \mu\}D_y$fukuda2023 bifurcated at $T^{(b)^2}=17.235$ from $q^{(b)}$ with $C_{6h}\{{\cal CM},\mu\} C_y$.
• Figure 5: Orbits at $T=20$ for non-symmetric solution $q^{(b)^2}$ with $C_2\{\mu\}D$ and their action values, bifurcated at $T^{(b)^2}=17.785$ from the $q^{(b)}$ with $C_{6h}\{{\cal CM},\mu\}C_y$. (a) $q^{(b)^2}$ with higher action value, (b) with lower action value, and (c) their action values $S(q^{(b)^2})-S(q^{(b)})$. Initial conditions: (a) $(x_1, y_1) = (0, 0.045)$, $(x_2, y_2) = (1.4396475, 0.5347018)$, $(\dot{x}_1,\dot{y}_1)=(0.34271348, 0.30563750)$ and $(\dot{x}_2,\dot{y}_2)=(-0.17598165, 0.25858920)$. (b) $(x_1, y_1) = (0, 0.045)$, $(x_2,y_2)=(1.4327086, 0.40489872)$, $(\dot{x}_1,\dot{y}_1)=(0.35818585, -0.52798612)$ and $(\dot{x}_2,\dot{y}_2)=(-0.12592133, 0.18861645)$.

Theorems & Definitions (3)

• Theorem 1
• Corollary 1
• Corollary 2