kilin> TBP> Bifurcation

Periodic solutions bifurcated from figure-eight choreographies of the equal mass three-body problem

Hiroshi FUKUDA

(2019.01.09)

Preprint arXiv:1901.00115


The following movies are periodic solutions bifurcated from the figure-eight choreographic solutions to the equation of motion \[ \frac{d^2 q_i}{dt^2} = - \frac{\partial U}{\partial q_i}, \; i = 0,1,2. \] Periods \( T \)'s of the movies are relatively correct. See arXiv:1901.00115 on initial conditions.

Homogeneous potential

\[ U = \sum_{i < j}u_a(r_{i j}), \; u_a(r)=-\frac{1}{r^a}, \; r_{ij}=|q_i-q_j| \]

\( D_{x y} \) at \( a=0.9766 \) bifurcated from \( a=0.9966 \)

, Δt=T/12/, \( T=1 \)

\( D_{x y} \) at \( a=1.0000 \) bifurcated from \( a=0.9966 \)

, Δt=T/12/, \( T=1 \)

\( D_{x y} \) at \( a=1.0166 \) bifurcated from \( a=0.9966 \)

, Δt=T/12/, \( T=1 \)

\( D_x \) at \( a=1.3425 \) bifurcated from \( a=1.3424 \)

, Δt=T/12/, \( T=1 \)

\( D_2 \) at \( a=1.3425 \) bifurcated from \( a=1.3424 \)

, Δt=T/12/, \( T=1 \)

The Lennard-Jones-type potential

\[ U = -\sum_{i < j}u^{LJ}(r_{i j}), \; u^{LJ}(r)=-\frac{1}{r^6}+\frac{1}{r^{12}}, \; r_{ij}=|q_i-q_j| \]

\( D_{x y} \) bifurcated from \( \alpha_+ \) at the left side of \( T=16.878 \)

, Δt=T/12/, \( T=20 \)

\( D_{x y} \) bifurcated from \( \alpha_+ \) at the right side of \( T=16.878 \)

, Δt=T/12/, \( T=20 \)

\( D_{x y} \) bifurcated from \( \alpha_- \) at the left side of \( T=14.836 \)

, Δt=T/12/, \( T=20 \)

\( D_{x y} \) bifurcated from \( \alpha_- \) at the right side of \( T=14.836 \)

, Δt=T/12/, \( T=20 \)

\( D_{x} \) bifurcated from \( \alpha_+ \) at \( T=16.111 \)

, Δt=T/12/, \( T=20 \)

\( D_{2} \) bifurcated from \( \alpha_+ \) at \( T=16.111 \)

, Δt=T/12/, \( T=20 \)

\( D_{x} \) bifurcated from \( \alpha_- \) at \( T=14.861 \)

, Δt=T/12/, \( T=20 \)

\( D_{2} \) bifurcated from \( \alpha_- \) at \( T=14.861 \)

, Δt=T/12/, \( T=20 \)

\( C_{x} \) bifurcated from \( \alpha_- \) at \( T=14.595 \)

, Δt=T/12/, \( T=20 \)

\( C_{2} \) bifurcated from \( \alpha_+ \) at \( T=18.615 \)

, Δt=T/12/, \( T=20 \)

\( C_{y} \) bifurcated from \( \alpha_+ \) at \( T=17.132 \)

, Δt=T/12/, \( T=20 \)