Kathrin Padberg

Transport in Dynamical Astronomy and Multibody Problems

We combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resonance regions as well as transport rates and bottlenecks between regions in dynamical systems. This methodology can be applied to a variety of multibody problems, including those in molecular modeling, chemical reaction rates and dynamical astronomy. In this paper we focus on problems in dynamical astronomy to illustrate the power of the combination of these different numerical tools and their applicability. In particular, we compute transport rates between two resonance regions for the three-body system consisting of the Sun, Jupiter and a third body (such as an asteroid). These resonance regions are appropriate for certain comets and asteroids.

Lagrangian structures and transport in turbulent magnetized plasmas

Using direct numerical simulations, it is established that under realistic conditions, the turbulent transport in magnetized fusion plasmas tends to be in a regime for which the cross-field dynamics of tracers can be described neither by Eulerian nor by random walk approaches. Instead, the concept of Lagrangian coherent structures turns out to be a useful tool for studying such systems. Here, networks of repelling and attracting material lines—acting as barriers for transport—become important. The latter are analogues of the stable and unstable manifolds in the static case. This opens up new possibilities for interpreting and analyzing turbulent transport in magnetized plasmas.

Set Oriented Approximation of Invariant Manifolds: Review of Concepts for Astrodynamical Problems

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 ...

Continuous and Discrete Concepts for Detecting Transport Barriers in the Planar Circular Restricted Three Body Problem

In the last two decades, the mathematical analysis of material transport has received considerable interest in many scientific fields, in particular in astrodynamics. In this contribution we will focus on the numerical detection and approximation of transport barriers in the solar system. For this we consider and combine several techniques for the mathematical treatment of transport processes—using both continuous concepts from dynamical systems theory and discrete ideas from graph theory. For the demonstration of our results we consider the planar circular restricted three body problem with Sun and Jupiter as primaries, a simple model for describing the motion of asteroids in the solar system.