Fully Numerical Methods for Continuing Families of Quasi-Periodic Invariant Tori in Astrodynamics
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Fully Numerical Methods for Continuing Families of Quasi-Periodic Invariant Tori in Astrodynamics Nicola Baresi1 · Zubin P. Olikara1 · Daniel J. Scheeres1
© American Astronautical Society, part of Springer Nature 2018
Abstract Quasi-periodic invariant tori are of great interest in astrodynamics because of their capability to further expand the design space of satellite missions. However, there is no general consent on what is the best methodology for computing these dynamical structures. This paper compares the performance of four different approaches available in the literature. The first two methods compute invariant tori of flows by solving a system of partial differential equations via either central differences or Fourier techniques. In contrast, the other two strategies calculate invariant curves of maps via shooting algorithms: one using surfaces of section, and one using a stroboscopic map. All of the numerical procedures are tested in the co-rotating frame of the Earth as well as in the planar circular restricted three-body problem. The results of our numerical simulations show which of the reviewed procedures should be preferred for future studies of astrodynamics systems. Keywords Dynamical systems theory · Quasi-periodic invariant tori · Continuation · Partial differential equations · Boundary value problems
Nicola Baresi
[email protected] Zubin P. Olikara [email protected] Daniel J. Scheeres [email protected] 1
Department of Aerospace Engineering Sciences, University of Colorado Boulder, 429 UCB, Boulder, CO 80309, USA
J of Astronaut Sci
Introduction The application of dynamical system theory to astrodynamics has opened the doors to many design options that can significantly expand the capabilities of spacecraft missions. Advantages of satellites around libration point orbits include easy accessibility and constant visibility from Earth as well as cheap transfer opportunities along invariant manifolds [7, 10, 11]. The next step is the computation of quasi-periodic (QP) invariant tori, which can further extend the design space by providing mission designers with new transfer opportunities [21] as well as initial conditions to establish bounded relative motion in non-trivial dynamical environments [3, 6]. For all these reasons, quasi-periodic invariant tori are of great interest to the astrodynamics community, and several strategies have been developed throughout the last decades to compute these objects. Although there exist analytical and semianalytical methods [7, 9, 15, 25], this paper focuses on fully numerical procedures that have been successfully applied for computing quasi-periodic solutions in relevant astrodynamics problems such as the Restricted Three Body Problem (RTBP) [29]. Generally speaking, numerical methods can be divided into two categories: methods that compute invariant tori of flows, and methods that calculate invariant curves of maps (Fig. 1). Both strategies aim at calculating diffeomorphisms u(θ ) : Td → Rn whose images are invariant for the consi
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