Semi-analytical methods in orbital dynamics: leveraging computational advances for improved dynamical analysis

We present the semi-analytical parameterization method and Floquet-based procedures to study different periodic orbits and their associated invariant manifolds in non-autonomous Hamiltonian systems. The approach includes the construction of a coordinate map and the associated reduced vector field, both expressed in the form of Fourier-Taylor expansions. The computational load and numerical sensitivity are benchmarked with respect to the manifold dimension and expansion orders. Focusing on the quasi-bicircular problem, the invariant manifolds in the neighborhood of dynamical equivalents of SEL_1,2, EML_1,2,3,4, the resonant 2:1 Halo orbit, as well as the 3:1 NRHO, are parameterized, and the accuracy and convergence are analysed based on posteriori error estimations.