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Rosette central configurations, degenerate central configurations and bifurcations. (English) Zbl 1175.70013
Summary: 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 $\epsilon = m_2/m_1$. We show that, as $\mu$ varies, if $n > 3$, there is a degenerate central configuration and a bifurcation for every $\epsilon > 0$, while if $n = 3$ there is a bifurcation only for some values of $\epsilon$.

70F10$n$-body problems
Full Text: DOI arXiv
[4] Hampton, M. and Moeckel, R.: 2006, ’Finiteness of relative equilibria of the four-body problem’, Inv. Math., to appear. · Zbl 1083.70012
[5] · Zbl 0328.57012 · doi:10.1090/S0002-9904-1975-13794-3
[6] · Zbl 0332.58008 · doi:10.2307/1970964
[7] · Zbl 1138.70008 · doi:10.1007/s10569-004-1991-2
[8] · Zbl 0203.26102 · doi:10.1007/BF01389805
[9] · doi:10.1007/BFb0068618