Continuation and bifurcations of concave central configurations in the four and five body-problems for homogeneous force laws

The central configurations given by an equilateral triangle and a regular tetrahedron with equal masses at the vertices and a body at the barycenter have been widely studied in [9] and [14] due to the phenomena of bifurcation occurring when the central mass has a determined value m⁎m⁎. We propose a variation of this problem setting the central mass as the critical value m⁎m⁎ and letting a mass at a vertex to be the parameter of bifurcation. In both cases, 2D and 3D, we verify the existence of bifurcation, that is, for a same set of masses we determine two new central configurations. The computation of the bifurcations, as well as their pictures have been performed considering homogeneous force laws with exponent a<−1a<−1.