Some Enumeration Problems on Central Configurations at the Bifurcation Points

Abstract

In “Counting central configurations at the bifurcation points,” we proposed an algorithm to rigorously count central configurations in some cases that involve one parameter. Here, we improve our algorithm to consider three harder cases: the planar $(3+1)$ -body problem with two equal masses; the planar 4-body problem with two pairs of equal masses which have an axis of symmetry containing one pair of them; the spatial 5-body problem with three equal masses at the vertices of an equilateral triangle and two equal masses on the line passing through the center of the triangle and being perpendicular to the plane containing it. While all three problems have been studied in two parameter cases, numerical observations suggest new results at some points on the bifurcation curves. Applying the improved version of our algorithm, we count at those bifurcation points. As a result, for the $(3+1)$ -body problem, we identify three points on the bifurcation curve where there are 8 central configurations, which adds to the known results of $8,9,10$ ones. For our 4-body case, at the bifurcation points, there are 3 concave central configurations, which adds to the known results of $2,4$ ones. For our 5-body case, at the bifurcation point, there is 1 concave central configuration, which adds to the known results of $0,2$ ones.