Semi-analytic construction of global transfers between quasi-periodic orbits in the spatial R3BP (2501.02485v2)

Abstract: Consider the spatial restricted three-body problem, as a model for the motion of a spacecraft relative to the Sun-Earth system. We focus on the dynamics near the equilibrium point $L_1$, located between the Sun and the Earth. We show that we can transfer the spacecraft from a quasi-periodic orbit that is nearly planar relative to the ecliptic to a quasi-periodic orbit that has large vertical amplitude, at zero energy cost. (In fact, the final orbit has the maximum vertical amplitude that can be obtained through the particular mechanism that we consider. Moreover, the transfer can be made through any prescribed sequence of quasi-periodic orbits in between). Our transfer mechanism is based on selecting trajectories homoclinic to a normally hyperbolic invariant manifold (NHIM) near $L_1$, and then gluing them together. We present a theoretical result establishing the existence of such transfer orbits, and we verify numerically its applicability to our model. We provide several explicit constructions of such transfers, and also develop an algorithm to design trajectories that achieve the shortest transfer time for this particular mechanism. The change in the vertical amplitude along a homoclinic trajectory can be described via the scattering map. We develop a new tool, the `Standard Scattering Map' (SSM), which is a series representation of the exact scattering map. We use the SSM to obtain a complete description of the dynamics along homoclinic trajectories. The SSM can be used in many other situations, from Arnold diffusion problems to transport phenomena in applications.