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Studying network of symmetric periodic orbit families of the Hill problem via symplectic invariants

Cengiz Aydin, Alexander Batkhin

TL;DR

The paper addresses how symmetric periodic orbit families in the spatial Hill 3BP interconnect at bifurcation points. It introduces and leverages symplectic invariants, notably the Conley–Zehnder index and local Floer homology, to construct bifurcation graphs that reveal the network structure among natural families such as $g$, $f$, planar Lyapunov, vertical Lyapunov, halo, and $\mathcal B_0^{\pm}$. By combining KS regularization, symmetry analysis, and index theory, the authors identify explicit and implicit connections across multiple covering orders (e.g., $n$-th covers, $n+1$-th covers, etc.) and demonstrate how spatial dynamics and regularized coordinates enable a richer network than planar analyses alone. The results provide a systematic framework to predict and interpret bifurcation pathways, with potential implications for trajectory design and celestial mechanics where such orbit families serve as organizing structures.

Abstract

In the framework of the spatial circular Hill three-body problem we illustrate the application of symplectic invariants to analyze the network structure of symmetric periodic orbit families. The extensive collection of families within this problem constitutes a complex network, fundamentally comprising the so-called basic families of periodic solutions, including the orbits of the satellite $g$, $f$, the libration (Lyapunov) $a,c$, and collision $\mathcal B_0$ families. Since the Conley-Zehnder index leads to a grading on the local Floer homology and its Euler characteristics, a bifurcation invariant, the computation of those indices facilitates the construction of well-organized bifurcation graphs depicting the interconnectedness among families of periodic solutions. The critical importance of the symmetries of periodic solutions in comprehending the interaction among these families is demonstrated.

Studying network of symmetric periodic orbit families of the Hill problem via symplectic invariants

TL;DR

The paper addresses how symmetric periodic orbit families in the spatial Hill 3BP interconnect at bifurcation points. It introduces and leverages symplectic invariants, notably the Conley–Zehnder index and local Floer homology, to construct bifurcation graphs that reveal the network structure among natural families such as , , planar Lyapunov, vertical Lyapunov, halo, and . By combining KS regularization, symmetry analysis, and index theory, the authors identify explicit and implicit connections across multiple covering orders (e.g., -th covers, -th covers, etc.) and demonstrate how spatial dynamics and regularized coordinates enable a richer network than planar analyses alone. The results provide a systematic framework to predict and interpret bifurcation pathways, with potential implications for trajectory design and celestial mechanics where such orbit families serve as organizing structures.

Abstract

In the framework of the spatial circular Hill three-body problem we illustrate the application of symplectic invariants to analyze the network structure of symmetric periodic orbit families. The extensive collection of families within this problem constitutes a complex network, fundamentally comprising the so-called basic families of periodic solutions, including the orbits of the satellite , , the libration (Lyapunov) , and collision families. Since the Conley-Zehnder index leads to a grading on the local Floer homology and its Euler characteristics, a bifurcation invariant, the computation of those indices facilitates the construction of well-organized bifurcation graphs depicting the interconnectedness among families of periodic solutions. The critical importance of the symmetries of periodic solutions in comprehending the interaction among these families is demonstrated.

Paper Structure

This paper contains 29 sections, 3 theorems, 46 equations, 26 figures, 19 tables.

Table of Contents

  1. Introduction
  2. SCR3BP and Hill 3BP
  3. Equations of motion
  4. Linear symmetries and Lagrange points
  5. Kustaanheimo--Stiefel regularization of Hill 3BP
  6. Regularization of equations of motion
  7. Lidov's method of computing critical solutions of family $\mathcal{B}_0$
  8. Bifurcations of periodic orbit families with double symmetry
  9. Out-of-plane branching from planar doubly symmetric critical orbits
  10. Branching from vertical collision symmetric critical orbits
  11. Interaction of Conley--Zehnder index with bifurcation points
  12. Conley--Zehnder indices of natural periodic orbits families
  13. Results
  14. Families $g$, $g'$ and $f$
  15. Planar and vertical Lyapunov, halo orbits and $\mathcal{B}_0^{\pm}$
  16. ...and 14 more sections

Key Result

Proposition 1

Let a doubly symmetric planar periodic solution with period $T$ have the stability index $S$ equal to $\cos(2\pi p/q)$, where $p, q$ are coprime integrals.

Figures (26)

  • Figure 1: Right: Lagrange points in the CR3BP; left: Lagrange points in the Hill 3BP.
  • Figure 2: The index jump.
  • Figure 3: Left top: Bifurcation graph related to $g$ and $g'$. Plots on the top shows $g$-orbits. The gray dashed $g$-orbit at $\Gamma = 4.49999$ is where symmetry-breaking bifurcation of $g'$ happens; some $g'$-orbits are plotted in the second row in green with symmetric ones. Middle right shows $f$-orbits in purple and middle left their planar and vertical stability indices. Diagrams at the bottom show vertical stability indices associated to families $g$ (left) and $g'$ (right).
  • Figure 4: Top middle and left: Plots of planar Lyapunov orbits in gray (red dot corresponds to $L_2$); below is the corresponding vertical stability diagram. First gray dashed orbit at $\Gamma = 4.005312$ corresponds to $a^{(1,1)}$ and second gray dashed orbit at $\Gamma = 1.228063$ corresponds to $a^{(1,2)}$. Right: Plots of vertical Lyapunov orbits in green, from top to bottom; second dashed orbit is degenerate.
  • Figure 5: Bridge between planar and vertical Lyapunov orbits. Right: Critical planar Lyapunov orbit $a^{(1,2)}$ (gray) at $\Gamma = 1.228063$ and critical vertical Lyapunov orbit (green) at $\Gamma = 0.512430$. Left: Some orbits in blue associated to the bridge.
  • ...and 21 more figures

Theorems & Definitions (7)

  • Definition 2.1
  • Proposition : BatkhinPCS2020
  • Conjecture : On symmetries of spatial solutions interacting with $\mathcal{B}_0$ family
  • Lemma 4.1
  • proof
  • Corollary 4.2
  • Remark 4.3