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

TL;DR

This paper introduces a semi-analytical method for computing heteroclinic connections between center manifolds in Hamiltonian systems, combining parameterization, novel meshing, and manifold matching techniques.

Contribution

It develops a new approach that uncouples center and hyperbolic parts, explicitly represents slices of manifolds, and avoids numerical integration, enhancing the computation of heteroclinic connections.

Findings

Successfully computed heteroclinic connections in the Earth-Moon system.

Applied method to identify connections at multiple energy levels.

Facilitated potential space mission trajectory design.

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.