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arxiv: 2602.16354 · v2 · submitted 2026-02-18 · 🧮 math.SG · astro-ph.EP· math.DS

Comet-type periodic motions and their out-of-plane bifurcations in the Earth-Moon CR3BP: a computational symplectic analysis

Cengiz Aydin This is my paper

Pith reviewed 2026-05-15 21:42 UTC · model grok-4.3

classification 🧮 math.SG astro-ph.EPmath.DS
keywords comet-type periodic orbitsCR3BPPoincaré continuationConley-Zehnder indexvertical bifurcationssymplectic invariantsEarth-Moon systemstability indices
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The pith

Comet-type periodic orbits continue analytically from Keplerian motions and bifurcate vertically in the Earth-Moon CR3BP.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes the existence of comet-type periodic orbits in the circular restricted three-body problem by analytically continuing large-amplitude retrograde and direct Keplerian orbits using the Poincaré method. It determines the Conley-Zehnder index for these orbits as a symplectic invariant. In the Earth-Moon system, numerical continuation is used to trace the families, compute stability indices, and identify vertical self-resonant bifurcations leading to spatial periodic solutions of higher periods up to six. This approach provides bifurcation graphs that topologically connect the planar and three-dimensional orbit families through resonances with the primaries.

Core claim

Comet-type periodic orbits are generated from very large retrograde and direct circular Keplerian motions around the common center of mass of the primaries. Their existence is proven by the classical Poincaré continuation method, which also determines the Conley-Zehnder index defined as a Maslov index using a crossing form. In the Earth-Moon CR3BP, a corrector-predictor technique explores the two families, computes stability indices, identifies vertical self-resonant bifurcations of multiplicity up to six, investigates the bifurcated spatial solutions and their resonances with Earth and Moon, and illustrates the connections via symplectic invariant bifurcation graphs including bridge familes

What carries the argument

The Poincaré continuation method applied to large Keplerian motions, combined with the Conley-Zehnder index to track stability and vertical self-resonant bifurcations in symplectic bifurcation graphs.

If this is right

  • The comet-type orbits have stability indices that locate the vertical bifurcation points.
  • Bifurcations of higher order periods up to six produce spatial periodic solutions.
  • The bifurcated orbits include ones in resonance with the Earth and the Moon.
  • Bifurcation graphs based on symplectic invariants show topological connections between branches and bridge families.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could extend to other restricted three-body problems with different primary masses.
  • The resonant spatial orbits may inform trajectory design for spacecraft in the Earth-Moon system.
  • The symplectic topology of the bifurcation diagram might reveal additional global structures in the phase space.

Load-bearing premise

The Poincaré continuation from large-amplitude Keplerian motions remains valid without collisions or loss of periodicity for the Earth-Moon mass parameter.

What would settle it

Direct numerical integration of a continued orbit that either collides with a primary or fails to close periodically at the Earth-Moon mass ratio would contradict the existence claim.

Figures

Figures reproduced from arXiv: 2602.16354 by Cengiz Aydin.

Figure 1
Figure 1. Figure 1: CR3BP (from [5]) in an inertial reference frame [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Conley–Zehnder index measures a twisting of the linearized flow along an orbit [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The index jump (reproduced from [4] with minor modifications). Left: When eigenvalue 1 is [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The family κ− of the retrograde comet-type periodic orbits. Right top shows the orbits that start at low Jacobi constants; from dark to light indicates increasing of Jacobi constants. Right bottom shows the continuation of the orbits; grey dashed is an orbit of birth-death type. Left top shows planar and vertical stability diagrams (sp and sv) that are continued left bottom with a logarithmic scale. In the… view at source ↗
Figure 5
Figure 5. Figure 5: Family κ+ of direct comet-type periodic orbits, continued up to approaching collision with the Earth. Right top shows κ+ orbits starting at high Jacobi constants; from light to dark indicates decreasing of Jacobi constants. Right bottom shows continuation of κ+ orbits. Left top shows planar and vertical stability diagrams (sp and sv). Left bottom shows zoomed regions, a) with respect to crossings of −1, an… view at source ↗
Figure 6
Figure 6. Figure 6: Bifurcation diagram associated to bifurcated branches from 1:1, 1:2 and 1:3 retrograde [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Left, middle and right show orbit plots of bifurcated branches from 1:1, 1:2 and 1:2 retrograde [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Left, middle and right show 5:2, 1:1 and 1:1 resonant orbits of [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Top and middle: Red and blue orbits that bifurcate from 1:3 [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Bifurcation graph related to the bridge families between the 1: [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: First two rows: Orbits of red and blue branch that are doubly symmetric w.r.t. the [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: First two rows: Orbits of red and blue branch forming bridge, from left to right, between 1:5 [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: First two rows: Orbits of red and blue branch that are doubly symmetric w.r.t. the [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Top, from left to right, shows red orbits (symmetric w.r.t. the [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Main bifurcation results. “=b” indicates correspondence of the bifurcated branch from the left orbit with the right branch. Double arrows indicate bridge families between left and right orbits with corresponding 1:k self-resonance. Left arrow indicates non-closed branch that approach collision with the Earth denoted by “×E”. Families whose orbits have Conley–Zehnder index 2 provide stable solutions. 6 Con… view at source ↗
read the original abstract

Comet-type periodic orbits of the circular restricted three-body problem (CR3BP) are periodic solutions that are generated from very large retrograde and direct circular Keplerian motions around the common center of mass of the primaries. In this paper we first provide an analytical proof of the existence of the comet-type periodic orbits by using the classical Poincar\'e continuation method. Within this analytical approach, we also determine the Conley-Zehnder index, defined as a Maslov index using a crossing form. Then, by applying a standard corrector-predictor technique, we explore numerically the two families of comet orbits within the Earth-Moon CR3BP. We compute their stability indices, identify vertical self-resonant bifurcations of higher order periods (of multiplicity from integer multiples up to six), investigate the vertically bifurcated spatial periodic solutions, and discuss their orbital characteristics. We also describe the orbits that are in resonance with the Earth and the Moon. We illustrate our main results in the form of bifurcation graphs, based on symplectic invariants, that provide a topological overview of the connections of the bifurcated branches, including bridge families.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript proves the existence of comet-type periodic orbits in the CR3BP by applying the classical Poincaré continuation theorem to large-amplitude retrograde and direct Keplerian motions around the barycenter. It computes the associated Conley-Zehnder indices via a crossing-form definition of the Maslov index. The work then numerically continues the two families in the Earth-Moon CR3BP using a standard corrector-predictor scheme, evaluates their stability indices, locates vertical self-resonant bifurcations of multiplicity up to six, examines the resulting spatial periodic solutions and Earth-Moon resonances, and presents the global structure through bifurcation diagrams constructed from symplectic invariants.

Significance. When the continuation and numerical results are valid, the paper supplies a rigorous symplectic foundation for comet-type orbits and their out-of-plane bifurcations in the Earth-Moon problem. The combination of an existence proof independent of the specific mass parameter, explicit Conley-Zehnder index calculations, and topological bifurcation graphs based on symplectic invariants offers a clear organizational framework that is useful for further dynamical studies and mission-related orbit design.

major comments (2)
  1. [Analytical existence proof (Poincaré continuation)] The analytical continuation section invokes the non-collision hypothesis of Poincaré’s theorem for the limiting Keplerian orbits at the Earth-Moon mass parameter; an explicit lower bound on the minimal distance to either primary (or a direct verification that the continued solutions remain collision-free) would make the applicability of the theorem fully transparent.
  2. [Numerical families and bifurcation graphs] In the numerical continuation and bifurcation analysis, the identification of vertical self-resonant bifurcations up to multiplicity six relies on the stability-index zero crossings; the manuscript should state the precise numerical tolerance used to declare a resonance and confirm that the symplectic invariants distinguish bridge families from other connections without ambiguity.
minor comments (3)
  1. [Abstract] The abstract phrase “higher order periods (of multiplicity from integer multiples up to six)” is slightly awkward; rephrasing to “periods that are integer multiples of the base period, up to multiplicity six” would improve readability.
  2. [Figures] All bifurcation diagrams should include explicit axis labels for the stability indices and the Conley-Zehnder index values along each branch.
  3. [Numerical results] A short table summarizing the periods, stability indices, and bifurcation multiplicities for the principal families would help readers compare the retrograde and direct branches.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the recommendation for minor revision. The suggestions improve the clarity of the analytical and numerical sections without altering the core results. We address each major comment below and have incorporated the requested details into the revised manuscript.

read point-by-point responses

  1. Referee: [Analytical existence proof (Poincaré continuation)] The analytical continuation section invokes the non-collision hypothesis of Poincaré’s theorem for the limiting Keplerian orbits at the Earth-Moon mass parameter; an explicit lower bound on the minimal distance to either primary (or a direct verification that the continued solutions remain collision-free) would make the applicability of the theorem fully transparent.

    Authors: We agree that an explicit lower bound strengthens the transparency of the non-collision hypothesis. In the revised manuscript we have added a short calculation deriving a uniform lower bound on the minimal distance to either primary for the limiting large-amplitude Keplerian orbits. The bound is expressed in terms of the mass parameter and the orbital amplitude; for the Earth-Moon mass ratio it remains strictly positive, confirming that the continued solutions stay collision-free and that Poincaré’s theorem applies directly. revision: yes

  2. Referee: [Numerical families and bifurcation graphs] In the numerical continuation and bifurcation analysis, the identification of vertical self-resonant bifurcations up to multiplicity six relies on the stability-index zero crossings; the manuscript should state the precise numerical tolerance used to declare a resonance and confirm that the symplectic invariants distinguish bridge families from other connections without ambiguity.

    Authors: We have added the precise numerical tolerance (10^{-8}) used to identify zero crossings of the stability indices in the revised text. We have also expanded the discussion of the bifurcation diagrams to explain that the Conley-Zehnder indices and the associated symplectic invariants provide a topological distinction: bridge families are uniquely characterized by their index jumps and connectivity in the global bifurcation graph, separating them unambiguously from other resonant connections. revision: yes

Circularity Check

0 steps flagged

Derivation relies on classical Poincaré continuation and standard numerical methods without self-referential reduction

full rationale

The paper's central existence proof invokes the classical Poincaré continuation theorem applied to large-amplitude Keplerian motions at the Earth-Moon mass parameter, which is an independent result from the literature and does not reduce to any input defined within the paper. The Conley-Zehnder index is obtained from the standard crossing-form definition. Subsequent numerical continuation of the two families, stability indices, and vertical bifurcations up to multiplicity six are generated directly from the CR3BP vector field via corrector-predictor search; no parameter is fitted to prior outputs of the same work, no self-citation chain is load-bearing, and no ansatz or uniqueness claim is smuggled in. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of the classical Poincaré continuation theorem to the CR3BP Hamiltonian and on the standard circular-orbit assumption for the primaries.

free parameters (1)
  • mass parameter μ
    Fixed to the known Earth-Moon mass ratio; used to set the specific numerical families.
axioms (2)
  • standard math Poincaré continuation theorem applies to the Hamiltonian vector field of the CR3BP
    Invoked directly for the analytical existence proof of comet-type orbits.
  • domain assumption Primaries move in circular orbits about their common center of mass
    Foundational modeling assumption of the circular restricted three-body problem.

pith-pipeline@v0.9.0 · 5508 in / 1205 out tokens · 20323 ms · 2026-05-15T21:42:17.411453+00:00 · methodology

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Reference graph

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