Time-Varying Shadows of Quasi-Periodic Motion Across Sections of the Flow Within Nearly Time-Periodic Three-Body Dynamics

Quasi-periodic behavior underlying gravitational three-body dynamics may supply naturally bounded reference trajectories to control spacecraft motion. To insert or maintain a spacecraft into bounded motion that resembles quasi-periodic behavior, guidance algorithms may require target position and velocity values, potentially at a fixed epoch. Within lower fidelity models that are autonomous, Poincaré maps are a particularly effective tool to identify target conditions for nearly quasi-periodic motion; however, challenges remain in transitioning the application of Poincaré maps to higher fidelity models that are nearly time-periodic. In fact, Poincaré maps are often visualized in lower dimensional spaces for inspection; such visualizations do not often supply a static and comprehensive description for the underlying dynamical structures, when dynamics are nearly time-periodic. In this work, we explore a framework to facilitate the interpretation of Poincaré map patterns associated with epoch-dependent solutions and states that are projected to lower dimensional position and velocity spaces. The introduction of chaos indicators may reveal regions of the projection that produce bounded motion as a function of the given epoch and/or initial state perturbation. Within precisely time-periodic systems, these regions may be the image, or shadow, of underlying torus manifolds. In the Poisson sense, shadows of quasi-periodic motion may be considered regions of stability within the lower dimensional space. Within the projection space, chaos indicators may capture time variations of a reference shadow or reveal special perturbation patterns. The framework is presented in two case studies: the first study demonstrates application within binary asteroid dynamics that are precisely time-periodic; the second study demonstrates application within nearly time-periodic dynamics (i.e., an ephemeris representation of the Earth-Moon system), when solutions resemble quasi-periodic motion only over finite time intervals.