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Heteroclinic Connections between Periodic Orbits in Planar Restricted Circular Three Body Problem - A Computer Assisted Proof

D. Wilczak, P. Zgliczynski

TL;DR

The paper addresses proving the existence of heteroclinic and homoclinic connections between Lyapunov orbits in the planar restricted circular three-body problem at the Oterma parameters, and it establishes a four-symbol symbolic dynamics built on these connections. It combines topological covering relations with $C^1$-level hyperbolicity via computer-assisted proofs, leveraging system symmetries to reduce computational cost. The main contributions are rigorous proofs of two-way heteroclinic connections between $L_1^*$ and $L_2^*$, transversal homoclinic connections in both interior and exterior regions, and a corresponding four-symbol dynamical coding, together with a practical computational framework (Lohner-based) for ODE-based proofs. These results provide a rigorous topological mechanism for rapid heliocentric transitions observed in comets like Oterma and demonstrate a scalable method for validating complex dynamical structures in celestial mechanics.

Abstract

The restricted circular three-body problem is considered for the following parameter values $C=3.03$, $μ=0.0009537$ - the values for {\em Oterma} comet in the Sun-Jupiter system. We present a computer assisted proof of an existence of homo- and heteroclinic cycle between two Lyapunov orbits and an existence of symbolic dynamics on four symbols built on this cycle.

Heteroclinic Connections between Periodic Orbits in Planar Restricted Circular Three Body Problem - A Computer Assisted Proof

TL;DR

The paper addresses proving the existence of heteroclinic and homoclinic connections between Lyapunov orbits in the planar restricted circular three-body problem at the Oterma parameters, and it establishes a four-symbol symbolic dynamics built on these connections. It combines topological covering relations with -level hyperbolicity via computer-assisted proofs, leveraging system symmetries to reduce computational cost. The main contributions are rigorous proofs of two-way heteroclinic connections between and , transversal homoclinic connections in both interior and exterior regions, and a corresponding four-symbol dynamical coding, together with a practical computational framework (Lohner-based) for ODE-based proofs. These results provide a rigorous topological mechanism for rapid heliocentric transitions observed in comets like Oterma and demonstrate a scalable method for validating complex dynamical structures in celestial mechanics.

Abstract

The restricted circular three-body problem is considered for the following parameter values , - the values for {\em Oterma} comet in the Sun-Jupiter system. We present a computer assisted proof of an existence of homo- and heteroclinic cycle between two Lyapunov orbits and an existence of symbolic dynamics on four symbols built on this cycle.

Paper Structure

This paper contains 27 sections, 28 theorems, 117 equations, 12 figures, 9 tables.

Table of Contents

  1. Introduction and statement of results
  2. Properties of PCR3BP: Poincaré maps and symmetries
  3. Symmetry properties of PCR3BP
  4. Topological tools
  5. h-sets
  6. Covering relations
  7. Covering relation on the plane with one nominally expanding direction ($u=1$)
  8. Representation of the h-sets
  9. Action of $R$ on h-sets
  10. $C^1$ tools
  11. General theorems
  12. How to prove an existence of an heteroclinic orbit, fuzzy sets.
  13. The Lyapunov orbits.
  14. Existence of Lyapunov orbits.
  15. Hyperbolicity in the neighborhood of Lyapunov orbits.
  16. ...and 12 more sections

Key Result

Theorem 1.1

For PCR3BP with $C=3.03$, $\mu=0.0009537$ there exist two periodic solutions in the Jupiter region, $L_1^*$ and $L_2^*$ , called Lyapunov orbits, and there exists heteroclinic connections between them, in both directions. Moreover for both orbits $L^*_1$ and $L_2^*$ there exists a homoclinic orbit i

Figures (12)

  • Figure 1: Hills region for PCR3BP with $C=3.03$, $\mu=0.0009537$ from KLMR.
  • Figure 2: An example of h-set on the plane.
  • Figure 3: An example of an $f-$covering relation: $N_{0} \Longrightarrow N_{0},N_{1}$
  • Figure 4: An example of an $R$-symmetric h-set
  • Figure 5: The Lyapunov orbits and the location of $x_i^*$.
  • ...and 7 more figures

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Theorem 3.6
  • Definition 3.7
  • Remark 3.8
  • ...and 36 more