Consistent four-body invariant-manifold framework for designing Earth–Moon to Sun–Earth low-energy heteroclinics

We present a unified computational framework for constructing heteroclinic connections between the Earth–Moon and Sun–Earth systems within the non-autonomous consistent quasi-bicircular problem. The approach relies on expressing hyperbolic invariant manifolds through multivariate Fourier–Taylor parameterizations and performing a consistent change of variables that embeds both dynamical environments into a common synodical frame. We quantify the accuracy of each transformation, introduce a multi-branch roadmap for propagating Earth–Moon manifold legs to a Sun–Earth Poincaré section, and employ mean curves together with generalized Poincaré sections to compute accurate initial guesses. These are refined via a boundary-value formulation. As an application, we obtain direct and short winding heteroclinic transfers between the Earth–Moon L ₂ and Sun–Earth L ₂ regions. The framework provides a coherent and effective methodology for analyzing low-cost transport between coupled subsystems and for exploring complex dynamical pathways in multi-body models.