Computing quasi-periodic trajectories of the RTBP in O(N log N) operations
- Paper number
IAC-17,C1,9,3,x40488
- Year
2017
- Abstract
Quasi-periodic (QP) trajectories play a fundamental role in many models in Astrodynamics, and provide many additional trajectory options in mission design beyond periodic motion [1,2]. They can be obtained locally by semi-analytical methods and globally by numerical ones. A numerical method that has progressively gained attention in the Astrodynamics community [1,2] is imposing invariance of a curve in the torus through the time-$T$ flow (stroboscopic map), where $T$ is one of the periods of the torus. Using a Fourier series approximation for the curve with $N$ Fourier modes leads to a system of non-linear equations whose solution usually requires $O(N^3)$ operations. This is a computational bottleneck, especially for tori of more than two dimensions. There is a new approach to the theoretical and computational study of invariant manifolds in dynamical systems known as the Parameterization Method [3,4,5]. In the computational side of this approach, differential correction is done in a functional framework, after the application of a transformation that makes use of geometrical properties that enable the computation of each Newton correction in a sequence of steps that are diagonal in either time or frequency domain. The computational benefit comes from the fact that the FFT allows to switch between these domains in $O(N\log N)$ operations. In this paper, some theory and algorithms from [5] will be extended to enable the simultaneous computation of families of partially hyperbolic invariant tori together with the linear approximation of their stable and unstable manifolds. As a test, some of the families of tori of the RTBP (Lissajous, halo, quasi-halo) will be computed. \bigskip \noindent{\bf References} \medskip \noindent[1] D.~Guzzetti, N.~Bosanac, A.~Haapala, K.~C. Howell, and D.~C. Folta. Rapid trajectory design in the earth--moon ephemeris system via an interactive catalog of periodic and quasi-periodic orbits. {\em Acta Astronautica}, 126:439--455, 2016. \medskip \noindent[2] N.~Baresi and D.~J. Scheeres. Quasi-periodic invariant tori of time-periodic dynamical systems: applications to small body exploration. Paper IAC-16,C1,7,4,x32824, 67th International Astronautical Congress, 26-30 September 2016, Guadalajara, Mexico. \medskip \noindent[3] R.~de~la Llave, A.~Gonz{\'a}lez, {\`A}.~Jorba, and J.~Villanueva. K{AM} theory without action-angle variables. {\em Nonlinearity}, 18(2):855--895, 2005. \medskip \noindent[4] {\`A}.~Haro and R.~de~la Llave. A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: numerical algorithms. {\em Discrete Contin. Dyn. Syst. Ser. B}, 6(6):1261--1300, 2006. \medskip \noindent[5] A.~Haro, M.~Canadell, J.-L. Figueras, A.~Luque, and J.~M. Mondelo. {\em The parameterization method for invariant manifolds: from rigorous results to effective computations}. Applied Mathematical Sciences. Springer, 2016.- Abstract document
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