# Morse index and bifurcation for figure-eight choreographies of the equal mass three-body problem

## TL;DR

This paper investigates how the Morse index influences bifurcations in figure-eight choreographies of the equal mass three-body problem under specific potentials, revealing that Morse index changes precisely indicate bifurcation points and the number of bifurcated solutions.

## Contribution

It demonstrates that Morse index changes are both necessary and sufficient for bifurcation in these choreographies and provides numerical evidence linking Morse index variations to bifurcation occurrences.

## Key findings

- Morse index changes occur at bifurcation points.
- All bifurcating solutions are approximated by variational functions.
- Number of bifurcated solutions equals the change in Morse index.

## Abstract

We report on the Morse index and periodic solutions bifurcating from the figure-eight choreography for the equal mass three-body problem under homogeneous potential $-1/r^a$ for $a \ge 0$, and under Lennard-Jones (LJ) type potential $1/r^{12}-1/r^6$, where $r$ is a distance between bodies. It is shown that the Morse index changes at a bifurcation point and all solutions bifurcating are approximated by variational functions responsible for the change of the Morse index. Inversely we observed %numerically bifurcation occurs at every point where the Morse index changes for the figure-eight choreography under $-1/r^a$, and for $\alpha$ solution under LJ type potential, where $\alpha$ solution is a figure-eight choreography tending to that under $-1/r^6$ for infinitely large period. Thus, to our numerical studies, change of the Morse index is not only necessary but also sufficient condition for bifurcation for these choreographies. Further we observed that the change of the Morse index is equal to the number of bifurcated solutions regarding solutions with congruent orbits as the same solution.