Semi-analytical computation of heteroclinic connections between center manifolds with the parameterization method

Abstract

This paper presents methodology for the computation of whole sets of heteroclinic connections between iso-energetic slices of center manifolds of center x center x saddle fixed points of autonomous Hamiltonian systems. It involves: (a) computing Taylor expansions of the center-unstable and center-stable manifolds of the departing and arriving fixed points through the parameterization method, using a new style that uncouples the center part from the hyperbolic one, thus making the fibered structure of the manifolds explicit; (b) uniformly meshing iso-energetic slices of the center manifolds, using a novel strategy that avoids numerical integration of the reduced differential equations and makes an explicit 3D representation of these slices as deformed solid ellipsoids; (c) matching the center-stable and center-unstable manifolds of the departing and arriving points in a Poincar\'e section. The methodology is applied to obtain the whole set of iso-energetic heteroclinic connections from the center manifold of L2 to the center manifold of L1 in the Earth-Moon circular, spatial Restricted Three-Body Problem, for nine increasing energy levels that reach the appearance of Halo orbits in both L1 and L2. Some comments are made on possible applications to space mission design.