Bifurcations of Two Symmetric Families of Dziobek Configurations

Abstract

In this work, we investigate bifurcations of Dziobek configurations in four and five-body problems, considering the exponent of the potential function of each system to be negative and less than minus one. The aim of this study is to find new central configurations. Initially, we investigate the bifurcations of the equilateral triangular configuration with bodies of unit mass at its vertices and a body with a mass of arbitrary value at its center. Using the Liapunov- Schmidt reduction method and the Equivariant Branching Theorem, we find three families of central configurations that bifurcate from the degenerate centered triangular configuration. In the Newtonian case, we performed a complete analysis of the solutions found and also found three families of central configurations with the same behavior as well as in (MEYER; SCHMIDT, 1987). Next, we study the bifurcations of a Dziobek configuration of the five-body problem in space. More precisely, we consider the regular tetrahedral configuration with bodies of unit mass at the vertices and a body of arbitrary mass at the center. Firstly, we analyze what happens in a neighborhood of the degenerate configuration by varying three of the vertex masses in the same fashion. Next, we vary two of the vertex masses equally. We use the Liapunov-Schmidt reduction method, the equivariance of the equations that describe the problem and Taylor’s formula to obtain new central configurations. In the first case, we found four new symmetrical families that arise from the degenerate configuration and, in the second, we found three new symmetrical families.