TY - CHAP

T1 - Set oriented approximation of invariant manifolds

T2 - International Conference on New Trends in Astrodynamics and Applications

AU - Dellnitz, Michael

AU - Padberg, Kathrin

AU - Post, Marcus

AU - Thiere, Bianca

N1 - Conference code: 3

PY - 2007/2/7

Y1 - 2007/2/7

N2 - 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.

AB - 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.

KW - Finite-time Lyapunov exponents

KW - Hamiltonian systems

KW - Invariant manifolds

KW - Reachable sets

KW - Set oriented numerics

KW - Space mission design

KW - Three body problem

KW - Mathematics

UR - http://www.scopus.com/inward/record.url?scp=33947403602&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/7214a3d0-10e4-31f2-91fe-cd6cf1f04b26/

U2 - 10.1063/1.2710046

DO - 10.1063/1.2710046

M3 - Article in conference proceedings

AN - SCOPUS:33947403602

SN - 0-7354-0389-9

SN - 978-0-7354-0389-5

T3 - AIP conference proceedings

SP - 90

EP - 99

BT - New Trends in Astrodynamics and Applications III

A2 - Belbruno, Edward

PB - American Institute of Physics Inc.

CY - Princeton, New Jersey

Y2 - 16 August 2006 through 18 August 2006

ER -