The region exterior to the Moon—yet still inside Earth’s effective gravitational sphere of influence—hosts a dense and expansive network of exterior mean-motion resonances (MMRs) that organize low-energy pathways for transport between the exterior and interior realms. In the Poincaré map at osculating orbit apogee, computed using the planar CR3BP, stable resonant orbits appear as stable islands (often asymmetric) interleaved with unstable periodic orbits in the chaotic sea whose stable/unstable manifolds generate heteroclinic connections and low-energy inter-resonance transitions. Such manifold-guided pathways are foundational to ballistic capture, multi-swingby strategies, and “superhighway” style transfers that exploit lunar gravity. In addition, classic Earth–Moon transfer studies emphasize that the Sun’s perturbation can be an enabling ingredient for low-energy access to the lunar region and beyond (Koon et al. 2022; McCarthy et al. 2020), not merely an error term.
This study computes the region of influence for prominent lunar MMRs of the form 1:n, specifically the 1:2, 1:3 and 1:4 resonances, using the PCR3BP, following existing methods (Koon et al. 2022). Using an apogee Poincaré map, we delineate MMR regions in the plane of semi-major axis and synodic longitude of perigee (relative to the Earth–Moon line). Our Poincaré maps reveal the 1:2, 1:3 and 1:4 resonance “islands”, existing as two stable asymmetric libration zones and a weak unstable symmetric libration zone, all embedded within a larger strong unstable symmetric resonance zone. After computation of the unstable resonant periodic orbits by continuation methods, the corresponding stable and unstable manifolds are visualized, forming the separatrix whose extent quantifies the region of influence of symmetric unstable resonant orbits.
We identify a variety of weak and strong heteroclinic connections linking interior resonant orbits (2:1 and 3:1) to the 1:3 exterior resonant orbit by leveraging the tube dynamics of L1–L2 Lyapunov orbits, thereby demonstrating the existence of low-energy transfer pathways between interior and exterior regions. This comprehensive analysis, incorporating the full PCR3BP model, accurately delineates the true region of influence of exterior resonances of the form 1:n. These findings inform potential low-energy routes for spacecraft trajectories from the interior to exterior Earth–Moon realm—pathways that may, in turn, facilitate trajectories beyond the Earth–Moon vicinity to nearby three-body regimes, such as the Sun–Earth libration points.
However, many lunar exterior resonant orbits exhibit large excursions away from the Earth–Moon line, making them especially sensitive to solar perturbations in addition to effects associated with the Moon’s elliptic motion (Park & Howell, 2024; Gómez et al., 2002). In these regimes the dynamics are no longer well-modeled as autonomous: periodic solar forcing transforms fixed-point organization into phase-dependent invariant structures of a stroboscopic Poincaré map. From a mission-design standpoint, this raises a concrete “survival” question: which 1:n resonant structures persist, and under what phasing/epoch conditions, when the Sun is included? In the forced setting, stable island chains deform into invariant structures of the stroboscopic map, while unstable resonant objects may persist as normally hyperbolic structures with time-dependent manifolds; generically, fixed points also bifurcate into quasi-periodic tori (Fitzgerald, 2022).
To connect the PCR3BP catalog to higher-fidelity force models, we adopt a continuation-and-bridging strategy that leverages one-frequency intermediate models and consistent time scalings (Gómez et al., 2002; Park & Howell, 2024), supported by dominance diagnostics that quantify where acceleration due to lunar eccentricity or solar gravity dominates the local acceleration budget. We investigate “survival of exterior resonances” by investigating those members of the sidereal resonant family that are also synodic resonant, so that a repeatable Earth-Moon-Sun geometry is maintained (Gupta, 2024).
Taken together, the global apogee-based Poincaré map provides a phase-space global atlas of exterior resonances, and tracking its deformation under solar forcing identifies which resonance islands, separatrices, and manifold corridors remain coherent as repeatable (stroboscopic) structures—thereby directly informing robust, epoch-aware low-energy transfer design within cislunar space and to neighboring three-body gateways such as the Sun–Earth libration region.