Set oriented approximation of invariant manifolds: Review of concepts for astrodynamical problems

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During the last decade set oriented methods have been developed for the approximation and analysis of complicated dynamical behavior. These techniques do not only allow the computation of invariant sets such as attractors or invariant manifolds. Also statistical quantities of the dynamics such as invariant measures, transition probabilities, or (finite-time) Lyapunov exponents, can be efficiently approximated. All these techniques have natural applications in the numerical treatment of problems in astrodynamics. In this contribution we will give an overview of the set oriented numerical methods and how they are successfully used for the solution of astrodynamical tasks. For the demonstration of our results we consider the (planar) circular restricted three body problem. In particular, we approximate invariant manifolds of periodic orbits about the L1 and L2 equilibrium points and show an extension to the application of a continuous control force. Moreover, we demonstrate that expansion rates (finite-time Lyapunov exponents), which so far have mainly been applied in fluid dynamics, can provide useful information on the qualitative behavior of trajectories in the context of astrodynamics. The set oriented numerical methods and their application to astrodynamical problems discussed in this contribution serve as further important steps towards understanding the pathways of comets or asteroids and the design of energy-efficient trajectories for spacecraft.

Original languageEnglish
Title of host publicationNew Trends in Astrodynamics and Applications III : Princeton, New Jersey, 16-18 August 2006
EditorsEdward Belbruno
Number of pages10
Place of PublicationPrinceton, New Jersey
PublisherAmerican Institute of Physics Inc.
Publication date07.02.2007
ISBN (Print)0-7354-0389-9, 978-0-7354-0389-5
Publication statusPublished - 07.02.2007
Externally publishedYes
EventInternational Conference on New Trends in Astrodynamics and Applications - American Institute of Physics, University princeton, Princeton, United States
Duration: 16.08.200618.08.2006
Conference number: 3

    Research areas

  • Finite-time Lyapunov exponents, Hamiltonian systems, Invariant manifolds, Reachable sets, Set oriented numerics, Space mission design, Three body problem
  • Mathematics