Parameterization method applied to astrodynamics and celestial mechanics

Abstract

This research project aims to develop analytical and numerical techniques based on the Parameterization Method to compute high-dimensional hyperbolic invariant sets in Hamiltonian systems with applications in Astrodynamics and Celestial Mechanics. These dynamical structures constitute an essential part of the skeleton of the dynamics in the phase space and are fundamental to compute transfer and parking orbits in Preliminary Project of Modern Space Missions in the Solar System. Specifically, we aim to develop and apply efficient techniques to compute two-dimensional hyperbolic tori belonging to the central manifolds of the collinear points of the Spacial Circular Restricted Three-Body Problem and other center-center-saddle type equilibria of the model. Additionally, we will apply this technique to calculate the respective stable and unstable invariant manifolds. In collaboration with one of the pioneers of the method, we will have the novel opportunity to extend the parameterization method to a class of dynamical systems not treated yet. The efficiency demonstrated by these tools in other mathematical models allows the systematization of computing cantorians biparametric families of solutions, as well as, to address open issues such as the dynamical understanding of the limits of the central manifolds. Besides these hyperbolic structures, the method further allows the estimation of the errors involved. The results of our project will allow the future globalization of these invariant sets in the phase space, in order to calculate homoclinic and heteroclinic connections that are essential to design transfer trajectories in the Solar System. In the future, we may also extend the application of the method to other dynamic systems of interest. (AU)