## 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 $$\epsilon$$.

### MSC:

 70F10 $$n$$-body problems
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### References:

 [4] Hampton, M. and Moeckel, R.: 2006, ’Finiteness of relative equilibria of the four-body problem’, Inv. Math., to appear. · Zbl 1083.70012
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