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 ∈ ( 0 , 1 ] as the parameter. This reduces the linear stability analysis to the computation of two eigenvalues of a 3 × 3 matrix for each m ∈ ( 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 .