Periodic orbits as centers of stability in the secular 3D planetary three body problem

In a previous paper, we have developed an analytical model of the secular 3D planetary problem by expanding the perturbation function up to the twelfth order in the eccentricities and the inclinations. Although the expansion is limited the model is able to describe with accuracy most of the observed systems of exoplanets. With the help of this model we were able to describe the geometry of the phase space of a typical system. The kernel of this description is a series of surfaces of section showing the chaotic and the regular domains of the phase space. We have observed in this previous paper that a family of unstable periodic orbits is responsible for the chaoticity, while we have hinted that the islands of stability are organized around stable periodic orbits. In this contribution we compute the main families of periodic orbits of the problem and show that indeed they are responsible for sculpting the phase space.