In this paper we find a class of new degenerate central configurations andbifurcations in the Newtonian $n$-body problem. In particular we analyze theRosette central configurations, namely a coplanar configuration where $n$particles of mass $m_1$ lie at the vertices of a regular $n$-gon, $n$ particlesof mass $m_2$ lie at the vertices of another $n$-gon concentric with the first,but rotated of an angle $\pi/n$, and an additional particle of mass $m_0$ liesat the center of mass of the system. This system admits two mass parameters$\mu=m_0/m_1$ and $\ep=m_2/m_1$. We show that, as $\mu$ varies, if $n> 3$,there is a degenerate central configuration and a bifurcation for every$\ep>0$, while if $n=3$ there is a bifurcations only for some values of$\epsilon$.
展开▼
机译：在本文中，我们在牛顿$ n $体问题中发现了一类新的简并的中心构型和分支。特别是，我们分析了Rosette中心配置，即共面配置，其中$ n $质量$ m_1 $的粒子位于规则的$ n $ -gon的顶点，$ n $质量$ m_2 $的粒子位于另一$$的顶点n $ -gon与第一个同心，但旋转角度为$ \ pi / n $，并且质量为$ m_0 $的附加粒子位于系统的质量中心。该系统允许两个质量参数$ \ mu = m_0 / m_1 $和$ \ ep = m_2 / m_1 $。我们显示，随着$ \ mu $的变化，如果$ n> 3 $，则存在退化的中央配置，并且每个$ \ ep> 0 $都有分叉，而如果$ n = 3 $，则仅对某些分叉\ epsilon $的值。
展开▼
