Rosette Central Configurations, Degenerate central configurations and bifurcations

TL;DR

This paper investigates new degenerate central configurations and bifurcations in the Newtonian n-body problem, focusing on Rosette configurations with specific mass arrangements and analyzing how these configurations change with varying mass parameters.

Contribution

It introduces a class of degenerate central configurations called Rosette configurations and analyzes their bifurcations depending on the number of particles and mass ratios.

Findings

Degenerate configurations occur for n > 3 as mass parameters vary.

Bifurcations are present for all positive epsilon when n > 3.

For n=3, bifurcations occur only at specific epsilon values.

Abstract

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 $\pi/n$, and an additional particle of mass $m_0$ lies at the center of mass of the system. This system admits two mass parameters $\mu=m_0/m_1$ 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 $\epsilon$.