Bifurcations of periodic orbits in the 3D secular planetary 3-Body problem: an approach through an integrable Hamiltonian system

We analyze, through a geometric description, the sequence of bifurcations of periodic orbits in a Hamiltonian model derived from the normalization of the secular 3D planetary three body problem. Stemming from the results in (Mastroianni & Efthymiopoulos 2023) we analyze the phase space of the corresponding integrable approximation. In particular, we propose a normal form leading to an integrable Hamiltonian whose sequence of bifurcations is qualitatively the same as that in the complete system. Using as representation of the phase space the 3D-sphere in the Hopf variables space, we geometrically analyze phase-space dynamics through the sequence of bifurcations leading to the appearance of fixed points of the secular Hamiltonian flow, i.e., periodic orbits in the complete system. Moreover, through a semi-analytical method, we find the critical values of the second integral giving rise to pitchfork and saddle-node bifurcations characterising the dynamics.