Linear stability analysis of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem

Abstract

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.