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

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Abstract

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.

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Notes

  1. currently renamed Artemis-I

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Acknowledgements

This work was completed at Tsinghua University with the support of the 2015 Chinese National Postdoctoral International Exchange Program and the National Natural Science Fund for Distinguished Young Scholars of China (No. 11525208). The authors also would like to thank the anonymous reviewers and Dr. Chappaz for their useful and detailed suggestions.

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  1. Davide Guzzetti

    Present address: Department of Aerospace Engineering, Auburn University, 211 Davis Hall, Auburn, AL, 36849, USA

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  1. School of Aerospace Engineering, Tsinghua University, Qinghuayuan 1, Beijing, 100084, China

    Davide Guzzetti & Hexi Baoyin

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Guzzetti, D., Baoyin, H. Time-Varying Shadows of Quasi-Periodic Motion Across Sections of the Flow Within Nearly Time-Periodic Three-Body Dynamics. J Astronaut Sci 68, 855–890 (2021). https://doi.org/10.1007/s40295-021-00284-x

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