Linear Stability Analysis of Symmetric Periodic Simultaneous Binary Collision Orbits in the Planar Pairwise Symmetric Four-Body Problem

We apply the symmetry reduction method of Roberts to numerically analyze the linear stability of a one-parameter family of symmetric periodic orbits with regularizable simultaneous binary collisions in the planar pairwise symmetric four-body problem with a mass \(m\in(0,1]\) as the parameter. This reduces the linear stability analysis to the computation of two eigenvalues of a \(3\times 3\) matrix for each \(m\in(0,1]\) obtained from numerical integration of the linearized regularized equations along only the first one-eighth of each regularized periodic orbit. The results are that the family of symmetric periodic orbits with regularizable simultaneous binary collisions changes its linear stability type several times as \(m\) varies over \((0,1]\), with linear instability for \(m\) close or equal to 0.01, and linear stability for \(m\) close or equal to 1.