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

TL;DR

This paper investigates fold phenomena in bifurcations of the figure-eight choreography in the equal-mass three-body problem. It applies a two-dimensional Lyapunov-Schmidt reduced action with three-fold symmetry to derive a finite-order description of the fold, yielding the fold condition $\kappa_0 = \frac{3 A_3^2}{8 A_4}$ and cusp behavior $\Delta S_{\pm}(\kappa)$. Four numerical examples under Lennard-Jones-type and homogeneous potentials illustrate the method, providing expressions for $A_3$ and $A_4$ from time integrals of $L$. The results indicate that folds occur when the symmetry of the Lagrangian is preserved by the bifurcation parameter and that the phenomenon is robust, extending to general few-body problems with similar symmetry.

Abstract

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.

Figures (3)

• Figure 1: $\Delta S_{\pm}(\kappa)$ for $A_3'=0.518$ and $A_4^{(0)}=-8.40$.
• Figure 2: $\Delta S(r_1,r_2)$ for $A_3=0.518, A_4=-8.40$. (a) $\kappa=\kappa_0=-0.012$. (b) $\kappa=0.9\kappa_0$. (c) $\kappa=0$. (d) $\kappa=-0.9\kappa_0$. In (a)--(d), white points represent bifurcation solutions and black fold solutions. Mountain or pond at the center represents solution $q$.
• Figure 3: (a) $\Delta S_\pm(\kappa(T))$ for bifurcation at $T=16.878$ from $\alpha_+$ under LJ potential (\ref{['LJ']}). $\kappa(T)=0.336 - 0.0199 T$. (b) $\Delta S_\pm(\kappa(T))$ for bifurcation at $T=14.836$ from $\alpha_-$ under LJ potential (\ref{['LJ']}). $\kappa(T)=-4.59 + 0.309 T$. (c) $\Delta S_\pm(\kappa(T))$ for bifurcation at $T=17.235$ from $C_y$ bifurcated at $T=14.132$ from $\alpha_+$ under LJ potential (\ref{['LJ']}). $\kappa(T)=-0.671 + 0.0389 T$. (d) $\Delta S_\pm(\kappa(a))$ for bifurcation at $a=0.9966$ from figure-eight choreography under the homogeneous potential (\ref{['homo']}). $\kappa(a)=-0.504 + 0.506 a$. In (a)--(d), solid curves are calculated by numerical exact bifurcation solutions and dashed by (\ref{['Sk']}) using coefficients $(A_3, A_4)$ in table \ref{['tab:res']} and the first order $\kappa(T)$ or $\kappa(a)$ around $\kappa=0$ above.