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Data-Driven Prediction of Chaotic Transition in Periapsis Poincaré Maps

Shanshan Pan, Taiki Urashi, Mai Bando, Yasuhiro Yoshimura, Hongru Chen, Toshiya Hanada

Abstract

Chaotic trajectories in multi-body dynamical systems play a crucial role in designing low-energy trajectories in astrodynamics. However, predicting these trajectories is inherently difficult, as small errors in initial conditions can grow exponentially, making long-term predictions unreliable. This study introduces a novel methodology using Dynamic Mode Decomposition (DMD) to predict chaotic transitions in the periapsis Poincaré map of the circular restricted three-body problem (CRTBP). Unlike standard DMD approaches that model continuous equations of motion, the proposed method approximates deformations in a low-dimensional Poincaré map, enabling trajectory prediction and revealing transition structures. Two approaches are developed: the Local Deformation Map-based DMD (LDMD) and the Global Deformation Map-based DMD (GDMD). LDMD constructs discrete maps to track local deformations of periapsis sets, while GDMD captures global deformations using widely distributed data. A key advantage of this framework is that it approximates nonlinear chaotic transport using a linear operator, which enables fast prediction of periapsis evolution via matrix powers and direct access to geometric structures. To validate the proposed method, the deformation map is applied to design ballistic transfer trajectories to the Moon using a targeting strategy, demonstrating its practical relevance in astrodynamics. This work highlights the potential of data-driven modeling to bridge chaotic dynamics with systematic trajectory design.

Data-Driven Prediction of Chaotic Transition in Periapsis Poincaré Maps

Abstract

Chaotic trajectories in multi-body dynamical systems play a crucial role in designing low-energy trajectories in astrodynamics. However, predicting these trajectories is inherently difficult, as small errors in initial conditions can grow exponentially, making long-term predictions unreliable. This study introduces a novel methodology using Dynamic Mode Decomposition (DMD) to predict chaotic transitions in the periapsis Poincaré map of the circular restricted three-body problem (CRTBP). Unlike standard DMD approaches that model continuous equations of motion, the proposed method approximates deformations in a low-dimensional Poincaré map, enabling trajectory prediction and revealing transition structures. Two approaches are developed: the Local Deformation Map-based DMD (LDMD) and the Global Deformation Map-based DMD (GDMD). LDMD constructs discrete maps to track local deformations of periapsis sets, while GDMD captures global deformations using widely distributed data. A key advantage of this framework is that it approximates nonlinear chaotic transport using a linear operator, which enables fast prediction of periapsis evolution via matrix powers and direct access to geometric structures. To validate the proposed method, the deformation map is applied to design ballistic transfer trajectories to the Moon using a targeting strategy, demonstrating its practical relevance in astrodynamics. This work highlights the potential of data-driven modeling to bridge chaotic dynamics with systematic trajectory design.
Paper Structure (16 sections, 22 equations, 26 figures, 3 tables)

This paper contains 16 sections, 22 equations, 26 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Data-Driven Modeling Procedure for Chaotic Trajectories
  3. The Planar Circular Restricted Three-Body Problem
  4. Periodic Orbits and Invariant Manifolds
  5. Periapsis Poincaré Map
  6. Dynamic Mode Decomposition
  7. Numerical Simulations
  8. Local Deformation Map
  9. Data Recovery by LDMD
  10. Prediction of Periapsis Not in the Data
  11. Analysis of FTLE Field
  12. Global Deformation Map
  13. Data Recovery by GDMD
  14. Discussion
  15. Application of discrete map to spacecraft trajectory design
  16. ...and 1 more sections

Figures (26)

  • Figure 1: Lyapunov orbit near $L_1$, showing stable manifolds (magenta) and an example trajectory (blue) passing through the manifold.
  • Figure 2: Example resonant orbit with a 3:1 resonance ratio.
  • Figure 4: The PPM for Jacobi constant $C=3.172602661563305$.
  • Figure 5: Schematic representation of the discrete map $A$, illustrating the deformation of the periapsis set.
  • Figure 6: Time evolution of periapsis sets obtained from numerical integration.
  • ...and 21 more figures