Hybrid Method for Computing Resonant Families Traversing Total Bifurcations in the PCR3BP

Continuation methods are used in numerous applications that focus on generating families of periodic orbits with common topology and characteristics, rather than achieving single disjoint trajectories. However, they work by sequentially tracking parameters associated with the system as they vary, thus creating a solution path. This study proposes a nonsequential hybrid method to compute families of resonant periodic orbits traversing total bifurcations of types 1 and 2 in the planar circular restricted three-body problem, including members with arbitrary proximity to the secondary that preserve both the intrinsic resonance of the family and its topology. A thorough description of the development is given, which combines the theory of generating orbits with an analytical asymptotic approximation of resonant orbits. Unification is achieved numerically by incorporating a two-stage differential correction scheme that modulates the minimum distances to the secondary of the closest approaching orbits. Families are constructed orbit-by-orbit simultaneously. The operation of the hybrid method is illustrated with the exterior 5:9 and 6:7 resonant families in the Jupiter–Europa system. The mechanism of type 1 bifurcations is further explored to analyze the impact of family approximation geometries to the singularity on the nonlinear physical behavior in the nearest orbits.