Differential algebra methods applied to continuous abacus generation and bifurcation detection: application to periodic families of the Earth-Moon system

The return of human space missions to the Moon puts the Earth-Moon system (EMS) at the center of attention. Hence, studying the periodic solutions to the circular restricted three-body problem (CR3BP) is crucial to ease transfer computations, find new solutions, or to better understand these orbits. This work proposes a novel continuation method of periodic families using differential algebra (DA) mapping. We exploit DA with automatic control of the truncation error to represent each family of periodic orbits as a set 2D Taylor polynomial maps. These maps guarantee the access to any point of the family without any numerical propagation, providing a continuous abacus. When applied to the halo family at L-1 and L-2, the planar Lyapunov at L-1 and L-2, the distant retrograde orbit (DRO) family, and the butterfly family, we show that the DA-based 2D mapping is asymptotically more efficient than point-wise methods by at least two orders of magnitude, with controlled accuracy. To assist the computation of family of periodic orbits, we propose a novel DA-based automatic bifurcation detection algorithm that enables the continuous mapping of the family's bifurcation criteria. A bifurcation study on the halo L-2 shows identical results as point-wise methods while highlighting two undocumented bifurcations.