Bifurcations of Central Configurations in the Four-Body Problem with some equal masses

TL;DR

This paper investigates how central configurations in the four-body problem change as some masses become equal, identifying bifurcation points through numerical and rigorous methods, and classifying these bifurcations.

Contribution

It combines numerical continuation with rigorous bifurcation analysis to classify bifurcations in four-body configurations with equal masses.

Findings

Identified mass parameters where solution counts change

Proved existence of bifurcations rigorously

Classified types of bifurcations in the system

Abstract

We study the bifurcations of central configurations of the Newtonian four-body problem when some of the masses are equal. First, we continue numerically the solutions for the equal mass case, and we find values of the mass parameter at which the number of solutions changes. Then, using the Krawczyk method and some result of equivariant bifurcation theory, we rigorously prove the existence of such bifurcations and classify them.