Three-dimensional periodic oscillations generating from plane periodic ones around the collinear Lagrangian points

The three families of three-dimensional periodic oscillations which include the infinitesimal periodic oscillations about the Lagrangian equilibrium pointsL ₁,L ₂ andL ₃ are computed for the value =0.00095 (Sun-Jupiter case) of the mass parameter. From the first two vertically critical (|a _v |=1) members of the familiesa, b andc, six families of periodic orbits in three dimensions are found to bifurcate. These families are presented here together with their stability characteristics. The orbits of the nine families computed are of all types of symmetryA, B andC. Finally, examples of bifurcations between families of three-dimensional periodic solutions of different type of symmetry are given.