Linear stability of double–double orbits in the parallelogram four-body problem

We study the linear stability of a two-parameter family of periodic orbits in the parallelogram four-body problem. This family was numerically found and named as double–double orbits by Vanderbei. A demonstration of such an orbit is shown. By introducing new transformations and applying Roberts' symmetry reduction method, the linear stability can be simplified to the calculation of two eigenvalues. A global picture of linear stability of this set with respect to the two parameters is given for the first time. Actually, most of the double–double orbits are numerically proved to be linearly stable.