Semianalytical Computation of Heteroclinic Connections Between Center Manifolds with the Parameterization Method
Miquel Barcelona, Àlex Haro, Josep–Maria Mondelo
Abstract
This paper presents a methodology for the computation of whole sets of heteroclinic connections between isoenergetic slices of center manifolds of center x center 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 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.