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

TL;DR

The paper develops a geometric framework based on Hopf variables to predict sequences of bifurcations of periodic orbits in the 3D secular planetary three-body problem, focusing on transitions from apsidal corotation resonance states as mutual inclination grows. By reducing the problem to a sphere of fixed radius $\sigma_0$ and analyzing the intersection with constant-energy surfaces $\mathcal{C}_{\sigma_0,\mathcal{E}}$, the authors classify bifurcations into CPI (tangencies) and CPII (degenerate intersections), providing analytic criteria and, in low-order truncations, explicit bifurcation values. They compare three integrable models—$\mathcal{H}_{int}$, the secular normal form $\mathcal{H}_{int}^{(1)}$, and the octupole-based $\tilde{\mathcal{H}}_{int}^{(N_P=3,N_{bk}=4)}$—and show that higher-order normal forms better reproduce the full system's bifurcation sequence, with the octupole case yielding fully tractable analytic expressions for critical values. The method robustly captures the emergence of new stable orbital configurations as angular momentum and inclination vary, offering a practical analytical tool for understanding the architecture of exoplanetary systems. This approach provides a bridge between near-integrable models and complex, non-integrable dynamics, with direct relevance to interpreting observed mutually inclined planetary systems and their long-term stability.

Abstract

We implement the geometric method proposed in ([9], [3], [16]) 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 [17], 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.

Figures (14)

• Figure 1: Poincaré surfaces of section $\mathcal{PS}_{\mathcal{H}_{sec}}(\mathcal{E};\mathrm{AMD})$ in the plane $(X_2, Y_2)$ with $\mathrm{AMD}$ fixed and different values of energy. The surfaces of section are computed by a numerical integration of trajectories in the Hamiltonian truncated at multipolar degree $N_{P}=5$, order $N_{bk}=12$ in the eccentricities, and energies (from left to right) $\mathcal{E}= -6.77\cdot 10^{-5},-2.53\cdot 10^{-5}, -1.92\cdot 10^{-5}, {-1.73\cdot 10^{-5},} -1.17\cdot 10^{-5}$,$-2.61\cdot 10^{-6}$.
• Figure 2: The permissible domain in $\mathcal{E}$ and $\sigma_0$ consistent with a fixed value of the system's $\mathrm{AMD}$. The numerical curves shown above refer to the example of the integrable model $\mathcal{Z}=\mathcal{H}_{int}$. Fixing a value of $\sigma_0$, the two limiting curves yield the corresponding limiting energies $\mathcal{E}_L(\sigma_0)$, $\mathcal{E}_R(\sigma_0)$. Alternatively, fixing a value of the energy we obtain the limits $\sigma_{0,min}(\mathcal{E}),\sigma_{0,max}(\mathcal{E})$. The above limits can be computed analytically as explained in maseft2023 (computing the tangencies between the sphere of radius $\sigma_0$ described in \ref{['sphere']} and the constant energy surface reported in \ref{['energy.curve']}). In summary, there exist two critical values of the energy $\mathcal{E}_{min}$, $\mathcal{E}_{2,3}$ such that, for any value of the energy in the interval $\mathcal{E}_{min}<\mathcal{E}<\mathcal{E}_{2,3}$, $\sigma_0$ is limited from below by a minimum value $\sigma_{0,min}(\mathcal{E})$, while the limit from above $\sigma_{0,max}$ is posed only by the globally maximum value $\sigma_{0,max}=\sigma_{0,\mathrm{AMD}}=\mathrm{AMD}$. The limiting values $\mathcal{E}=\mathcal{E}_{min}$ and $\mathcal{E}=\mathcal{E}_{2,3}$ correspond to the co-planar ACR orbits A (anti-aligned) and B (aligned) respectively. On the other hand, for energies larger than $\mathcal{E}_{2,3}$ there are no possible co-planar orbits, and $\sigma_{0,max}$ as well becomes a decreasing function of $\mathcal{E}$. Overall, as we move from left to right or top to bottom, we obtain orbits of higher mutual inclination. For the Hamiltonian $\mathcal{H}_{int}$ (see text for parameter values) we compute the values $\mathcal{E}_{min}=-1.18\cdot 10^{-4}$ and $\mathcal{E}_{2,3}=-6.77\cdot 10^{-5}$ and $\sigma_{0_{max}}=\mathrm{AMD}=0.0162044$.
• Figure 3: Top frame: Poincaré surfaces of section $\mathcal{PS}_{\mathcal{H}_{int}}(\mathcal{E};\mathrm{AMD})$ for the integrable Hamiltonian model $\mathcal{H}_{int}$ at the values of the energies (from top to bottom, left to right) $\mathcal{E}= -6.77\cdot 10^{-5},-1.8\cdot 10^{-5},-1.74\cdot 10^{-5},-1.7\cdot 10^{-5}, -1.67\cdot 10^{-5}, -1.6\cdot 10^{-5}$,$-1.43\cdot 10^{-5}, -4.05\cdot 10^{-7}$. Bottom frame: Phase portraits by contour plots $\mathcal{P}_{\mathcal{H}_{int}}(\sigma_0;\mathrm{AMD})$ with decreasing values of $\sigma_0$ (from top to bottom, left to right) $\sigma_0= 1.62\cdot 10^{-2}, 6.15\cdot 10^{-3}, 5.93\cdot 10^{-3}, 5.85\cdot 10^{-3}, 5.78\cdot 10^{-3}, 5.62\cdot 10^{-3}, 5.304\cdot 10^{-3}, 1.74\cdot 10^{-4}$. The two alternative representations of the phase portraits, corresponding to $\mathcal{P_Z}(\mathcal{E})$ (top) or $\mathcal{P_Z}(\sigma_0)$ (bottom), for $\mathcal{Z}=\mathcal{H}_{int}$, yield an equivalent bifurcation sequence.
• Figure 4: An example of inner tangency between $\mathcal{S}_{\sigma_0}$ and $\mathcal{C}_{\sigma_0,\, \mathcal{E}^{(P_1)}}$ at the point $P_1$ (left), and outer tangency at the point $P_2$ (right). The example is taken from the analysis of the integrable model $\mathcal{Z}=\mathcal{H}_{int}$ (see second panel of the second row of Fig. \ref{['Fig.Poin_Hopf_INTEG']}, corresponding to the phase portrait $\mathcal{P_Z}(\sigma_0)$ with ${\sigma_0}= 6.15\cdot 10^{-3}$).
• Figure 5: Graphical representation of the critical points of second kind $F_1$ and $F_2$ in the elliptic case (see text). The example is taken from the choice of integrable model $\mathcal{Z}=\mathcal{H}_{int}$, and it corresponds to the phase portrait $\mathcal{P_Z}(\sigma_0)$ with $\sigma_0=5.93\cdot 10^{-3}$ (the third panel in the top row of Fig. \ref{['Fig.Poin_Hopf_INTEG']}). The top left panel shows the intersection of the sphere $\mathcal{S}_{\sigma_0}$ with an energy surface $\mathcal{C}_{\sigma_0,\, \mathcal{E}}$ for $\mathcal{E}\approx\mathcal{E}^{(F)}$ and the right top panel for $\mathcal{E}=\mathcal{E}^{(F)}$. The bottom panels show the corresponding projections to the plane $\sigma_2=0$ (intersection of the curves $\mathcal{C}_{\sigma_0,\, \mathcal{E}}^{(\sigma_2=0)}$ with the circles $\mathcal{S}_{\sigma_0}^{(\sigma_2=0)}$). We have $\mathcal{E}=\mathcal{E}_1^{(F)}=-1.7362\cdot 10^{-5}$ and $\mathcal{E}^{(F)}=-1.7366\cdot 10^{-5}$, in which the corresponding surfaces are a cylinder ( top left panel) reducing to a straight line ( top right panel). The curve $\mathcal{C}_{\mathcal{Z}}(\sigma_0, \mathcal{E}_{1}^{(F)})$ in the corresponding phase portraits $\mathcal{P_Z}(\sigma_0)$ (see third panel of first row of Fig. \ref{['Fig.Poin_Hopf_INTEG']}) is plotted in green.

Theorems & Definitions (5)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Proposition 3.4
• proof