Invariant manifolds of a non-autonomous quasi-bicircular problem computed via the parameterization method

Abstract

The parameterization method (pm) has been used to compute high-order parameterizations of invariant manifolds of vector fields at fixed points. This paper extends such approach to invariant manifolds of periodically-perturbed vector fields about a periodic orbit with the same frequency, with a direct application on the libration points of the Sun-Earth-Moon system. The Sun-Earth-Moon environment is modeled by the so-called Quasi-Bicircular Model (qbcp), which is a coherent restricted four-body model that describes the motion of a spacecraft under the simultaneous gravitational influences of the Earth, the Moon, and the Sun. The pm is adapted to account for the explicit time-dependency of the corresponding vector field. This new procedure yields high-order periodic semi-analytical approximations of the center manifolds about the libration points L 1,2 of the periodically-perturbed Sun-(Earth+Moon) and Earth-Moon systems. These approximations are then used to initialize the computation of Poincaré maps, which allow to get a qualitative description of the non-autonomous dynamics near the equilibrium points. It is shown that, with this new approach, the semi-analytical description of the center manifolds in a coherent four-body environment is valid in a neighborhood significant enough to be used in practice. In particular, the well-known Halo orbit bifurcation is recovered in all cases.

Using spectral submanifolds for optimal mode selection in nonlinear model reduction

“…These conditions and the existence and uniqueness results for non-autonomous SSMs are stated in theorem 4 of Haller & Ponsioen [11], which is deduced from the abstract results on whiskers of invariant tori by Haro & de la Llave [15] (theorem 4.1). We refer the reader to [16][17][18] for further applications of the parametrization method to invariant manifolds attached to periodic orbits.…”

Geometry of transit orbits in the periodically-perturbed restricted three-body problem

“…Additional perturbations will lead to additional bifurcations in the topology of the Lagrange point dynamical replacement (see Figure 1). For instance, quasi-periodic Lagrange manifolds in systems with two or more perturbations of incommensurate period will generate hyperbolic structures controlling transit (Gómez et al, 2003;Bihan et al, 2017;.…”

Global analysis of direct transfers from Lunar orbits to Sun-Earth libration point regimes

“…The main transfer examples include, Sun-Earth libration point missions [1,7], low-energy capture into Mars halo orbit [24,31], and transfers between collinear libration point orbits [4,12,14]. However, when the gravitational force-field created by any of the Sun, Earth, and Moon cannot be ignored, the complexity increases [19,20]. A simple way to deal with a four-body regime is to uncouple the model into two uncoupled three-body problems, i.e., the Sun-Earth-spacecraft and the Earth-Moon-spacecraft systems.…”

“…Earth-to-Moon transfers in the RTBP have been studied in [9,25]. Transfers between libration points in a four-body coherent model have been studied in [19,20].…”