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

Abstract

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.