Identifying Fixed Points in the Three-Body Problem Using a High-Order Transfer Map

Abstract: Periodic orbits (POs) play a central role in the circular restricted three-body problem (CRTBP). This paper introduces a method to search for POs by identifying single- and multiple-revolution fixed points in chosen Poincare maps that describe the CRTBP dynamics, with a theoretical capability to detect all fixed points across arbitrary revolution counts exhaustively.First, high-order transfer maps (HOTMs), represented as polynomials, are constructed within the differential algebra (DA) framework for both planar and spatial CRTBP to map states between successive Poincare section crossings, with the Jacobi constant used to reduce the number of independent variables. Next, an automatic domain splitting (ADS) strategy is employed to generate subdomains, preserving HOTM accuracy, with an integrated feasibility estimation to reduce ADS's computation burden.Then, a two-stage HOTM-based polynomial optimization framework is introduced, first identifying combinable subdomain sequences and then refining the fixed point solutions. Finally, the method is applied to the Earth-Moon CRTBP, identifying POs up to nine revolutions in the planar case and four in the spatial case. Known families such as distant retrograde orbits (DROs) and Lyapunov orbits are recovered, along with a previously undocumented family that exhibits a hybrid character between DROs and Lyapunov orbits.