Effective shadowing of transition maps for a Hamiltonian saddle-center-center equilibrium

Abstract

We introduce a general methodology to construct globally unstable trajectories near the normally hyperbolic invariant manifold (NHIM) of a saddle-center-center Hamiltonian equilibrium. The proposed approach builds upon the classical geometric mechanism of searching for trajectories near heteroclinic connections between invariant tori of the NHIM. The main ingredient is the Birkhoff normal form near the NHIM, which allows us to control the dynamics of the hyperbolic and center components. We construct transition maps on the NHIM that encode the dynamics along heteroclinic connections. By combining the transition maps with the inner dynamics, given by the restriction of the Hamiltonian flow to the NHIM, we show that these connections can be efficiently (in terms of the travel time) shadowed by true trajectories of the full Hamiltonian system. Finally, the efficiency of this methodology is illustrated in the case of the NHIM associated to the libration point L₁ of the Spatial Restricted 3-Body Problem (SR3BP) for the Sun-Earth system.

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BibTeX

@misc{anderson2026effective,
  title = {{Effective shadowing of transition maps for a Hamiltonian saddle-center-center equilibrium}},
  author = {Rodney L. Anderson and Amadeu Delshams and Marian Gidea and Pablo Roldan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16404},
  eprint = {2610.16404},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16404},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Effective shadowing of transition maps for a Hamiltonian saddle-center-center equilibrium
AU  - Rodney L. Anderson
AU  - Amadeu Delshams
AU  - Marian Gidea
AU  - Pablo Roldan
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16404
UR  - https://arxiv.org/abs/2610.16404
AB  - We introduce a general methodology to construct globally unstable trajectories near the normally hyperbolic invariant manifold (NHIM) of a saddle-center-center Hamiltonian equilibrium. The proposed approach builds upon the classical geometric mechanism of searching for trajectories near heteroclinic connections between invariant tori of the NHIM. The main ingredient is the Birkhoff normal form near the NHIM, which allows us to control the dynamics of the hyperbolic and center components. We construct transition maps on the NHIM that encode the dynamics along heteroclinic connections. By combining the transition maps with the inner dynamics, given by the restriction of the Hamiltonian flow to the NHIM, we show that these connections can be efficiently (in terms of the travel time) shadowed by true trajectories of the full Hamiltonian system. Finally, the efficiency of this methodology is illustrated in the case of the NHIM associated to the libration point L_1 of the Spatial Restricted 3-Body Problem (SR3BP) for the Sun-Earth system.
ER  -

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