Morse index and bifurcation for figure-eight choreographies of the equal mass three-body problem
Morse index and bifurcation for figure-eight choreographies of the equal mass three-body problem
复制标题
DOI：
10.1088/1751-8121/ab1270
复制
发表时间：
2019-01
期刊：
影响因子：
--
通讯作者：
H. Fukuda;T. Fujiwara;H. Ozaki
中科院分区：
文献类型：
--
作者：
H. Fukuda;T. Fujiwara;H. Ozaki
本文研究了等质量三体问题在齐次势−1/ra和Lennard-Jones（LJ）型势下的莫尔斯指数和周期解，其中r是物体之间的距离.证明了莫尔斯指标在分歧点处发生变化，所有分歧解都可用引起莫尔斯指标变化的变分函数来近似.我们立刻观察到，在莫尔斯指数发生变化的每一点上，对于-1/ra下的8字形编排，以及对于LJ型势下的解，其中解是一个8字形编排，在无穷大的周期内趋向于-1/r6下的编排。因此，对于我们的数值研究，莫尔斯指数的变化不仅是必要的，而且是一个充分条件的分歧，这些编排。此外，我们观察到，莫尔斯指数的变化等于分歧的解决方案的数量，解决方案的全等轨道为同一个解决方案。
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/ra for , and under Lennard-Jones (LJ) type potential , 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 bifurcation occurs at every point where the Morse index changes for the figure-eight choreography under −1/ra, and for solution under the LJ type potential, where solution is a figure-eight choreography tending to that under −1/r6 for an infinitely large period. Thus, to our numerical studies, change of the Morse index is not only necessary but also a 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.