Data-Driven Prediction of Chaotic Transition in Periapsis Poincar\'e 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 Poincar\'e 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 Poincar\'e 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.