Semianalytical Computation of Heteroclinic Connections Between Center Manifolds with the Parameterization Method

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 98-126, March 2024.
Abstract. This paper presents a methodology for the computation of whole sets of heteroclinic connections between isoenergetic slices of center manifolds of center [math] center [math] 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 isoenergetic slices of the center manifolds, using a novel strategy that avoids numerical integration of the reduced differential equations and makes an explicit three-dimensional 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 Poincaré section. The methodology is applied to obtain the whole set of isoenergetic heteroclinic connections from the center manifold of [math] to the center manifold of [math] in the Earth-Moon circular, spatial restricted three-body problem, for nine increasing energy levels that reach the appearance of halo orbits in both [math] and [math]. Some comments are made on possible applications to space mission design.