Application of machine learning to discover dynamical structures in 4D Poincaré maps in the circular restricted three-body problem

Abstract

The spatial Circular Restricted Three-Body Problem (CR3BP) produces complex orbital dynamics that are difficult to classify at scale using traditional, manual Poincaré map analysis. A Poincaré map simplifies continuous orbital motion by recording the points where a trajectory intersects a chosen surface in phase space, converting a continuous path into a discrete set of crossings that reveal underlying topological structure. These crossings make it easier to characterize trajectory behavior, but interpreting them visually becomes impractical for large datasets and as Poincaré maps evolve from two into four dimensions. To address this challenge, an unsupervised clustering pipeline for discovering dynamical structures in four-dimensional (4D) Poincaré maps is introduced. Each trajectory’s Poincaré crossings are transformed into a compact feature representation capturing geometric dispersion, statistical, and frequency-based characteristics, enabling density-based clustering without prior labels. The method is evaluated across multiple Jacobi constants to assess robustness, and expanded to investigate chaining dynamical structures to find similar structures across different Jacobi constants. The results demonstrate that feature-based clustering can recover meaningful dynamical structures in high-dimensional trajectory data.