Bifurcation sequences in the secular 3D planetary 3-Body problem: a geometric approach

We implement the geometric method proposed in [14, 17, 21] to analytically predict the sequence of bifurcations leading to a change of stability and/or the appearance of new periodic orbits in the secular 3D planetary three body problem. Stemming from the analysis in [1], we examine various normal form models as regards the extent to which they lead to a phase space dynamics qualitatively similar as that in the complete system. For fixed total angular momentum, the phase space in Hopf variables is the 3D sphere, and the complete sequence of bifurcations of new periodic orbits can be recovered through formulas yielding the tangencies or degenerate intersections between the sphere and the surfaces of a constant second integral of motion of the normal form flow. In particular, we find the critical values of the second integral giving rise to pitchfork and saddle-node bifurcations of new periodic orbits in the system. This analysis renders possible to predict the most important structural changes in the phase space, as well as the emergence of new possible stable periodic planetary orbital configurations which can take place as the mutual inclination between the two planets is allowed to increase.