Rosette Central Configurations, Degenerate central configurations and bifurcations

In this paper we find a class of new degenerate central configurations and bifurcations in the Newtonian $n$-body problem. In particular we analyze the Rosette central configurations, namely a coplanar configuration where $n$ particles of mass $m_1$ lie at the vertices of a regular $n$-gon, $n$ particles of mass $m_2$ lie at the vertices of another $n$-gon concentric with the first, but rotated of an angle $\pi \mathrm{/}n$, and an additional particle of mass $m_0$ lies at the center of mass of the system. This system admits two mass parameters $\mu =m_0\mathrm{/}{m}_1$ and $\text{ep}=m_2\mathrm{/}{m}_1$. We show that, as $\mu$ varies, if $n>3$, there is a degenerate central configuration and a bifurcation for every $\text{ep}>0$, while if $n=3$ there is a bifurcations only for some values of $ϵ$.