Restricted Three-Body Problem Fixed-Point Identification via High-Order Transfer Maps

Periodic orbits (POs) play a central role in the circular restricted three-body problem (CRTBP). This paper presents a framework for identifying single- and multirevolution POs by searching for fixed points of selected Poincaré maps of the CRTBP. Unlike conventional continuation approaches that follow known solution branches from seed orbits, the proposed method reformulates the multirevolution PO search as a fixed-point identification problem over chains of single-revolution high-order transfer maps (HOTMs). Within a differential algebra (DA) framework, polynomial HOTMs are constructed for both planar and spatial CRTBP, while the Jacobi constant is used to reduce the number of independent variables. To preserve map accuracy across the search domain, an automatic domain splitting (ADS) strategy is employed, combined with feasibility estimation to prune subdomains and reduce computational burden. A two-stage HOTM-based polynomial optimization procedure is then introduced: the first stage identifies combinable subdomain sequences, and the second refines them into fixed-point solutions. The method is applied to the Earth–moon CRTBP, where planar POs with up to nine revolutions and spatial POs with up to four revolutions are identified. The results recover known families, including distant retrograde and Lyapunov orbits, and also reveal additional higher-period candidate solutions with hybrid dynamic characteristics.