Talk 3

Title:  Bifurcations of Centered Symmetric Central Configurations

Speaker:  Michelle Gonzaga

Affiliation: Universidade Federal de Pernambuco (UFPE) 

Abstract:  In the N-body problem, the study of bifurcations has proved to be a useful way of obtaining new central configurations from known degenerate ones, as well as answering finiteness and enumeration questions. In this talk, we investigate bifurcations of symmetric classes of central configurations involving four and five bodies, namely the centered equilateral triangle and the centered regular tetrahedron. Results in the current literature have been obtained for a changing central mass and the Newtonian potential, or for a varying vertex mass and homogeneous potentials with exponents less than -1. For the centered equilateral triangle, considering homogeneous potentials with exponents less than −1/5, we classify all symmetric central configurations that arise as the central mass increases through its degenerate value. For the centered regular tetrahedron, we vary two and three of the vertex masses in an analogous manner and classify all new central configurations for exponents less than or equal to −1. We show that all central configurations bifurcating from the degenerate centered regular tetrahedron must be symmetric when two or three vertex masses are varied equally. Finally, we discuss bifurcations of a centered square central configuration and indicate directions that can be explored in the search for new central configurations bifurcating from the degenerate configuration.