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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. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the existence and linear stability of the Broucke periodic orbit in the planar three-body problem. In each period of this orbit, there are two binary collisions (or BC for short) between the outer bodies, while the inner body reaches its minimum or maximum at the time of each BC. A surprising simple existence proof of this orbit is given. The initial condition of this orbit is shown to be a supremum of some well-chosen set. The linear stability is then analyzed by Robertsʼ symmetry reduction method. It is shown that the Broucke periodic orbit with equal masses is linearly stable. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four-body problem. The Hamiltonian of the differential system is regularized by a Levi–Civita type transformation and an appropriate scaling of time. The initial condition of this orbit is shown to be the infimum of some well-chosen set. This existence proof is direct and surprisingly simple. Further, a careful study shows that this orbit has a symmetry group isomorphic to the dihedral group . Then Robertsʼ symmetry reduction method is applied to show the linear stability. It turns out that the rhomboidal periodic orbit in the planar equal mass four-body problem is linearly stable. [Copyright &y& Elsevier]