1. Almost Invariant Sets and Transport in the Solar System
Michael Dellnitz
Department of Mathematics
University of Paderborn

2. (mission design; zero finding)
Overview
invariant sets
(mission design; zero finding)
GAIO
invariant
manifolds
global attractors
statistics
(molecular dynamics;
transport problems)
set oriented
numerical methods
almost
invariant sets
invariant
measures

3. Simulation of Chua´s Circuit

4. Numerical Strategy A
1. Approximation of the invariant set A
2. Approximation of the dynamical
behavior on A

5. The Multilevel Approach for the Lorenz System

6. Relative Global Attractors

7. The Subdivision Algorithm
Selection Set

8. Example: Hénon Map

9. A Convergence Result Proposition [D.-Hohmann 1997]: Remark:
Results on the speed of convergence can be obtained if possesses a hyperbolic structure.

10. Realization of the Subdivision Step
Boxes are indeed boxes
Subdivision by bisection
Data structure

11. Realization of the Selection Step
Use test points:
Standard choice of test points:
For low dimensions: equidistant distribution on edges of boxes.
For higher dimensions: stochastic distribution inside the boxes.

12. Global Attractor in Chua´s Circuit

13. Global Attractor in Chua´s Circuit
Simulation
Subdivision

14. Stable and unstable manifold of p
Invariant Manifolds
Stable and unstable manifold of p

15. Example: Pendulum

16. Computing Local Invariant Manifolds
Let p be a hyperbolic fixed point
Idea: AN p

17. Covering of an Unstable Manifold for a Fixed Point of the Hénon Map
Continuation 3
Continuation 2
Continuation 1
Subdivision
Initialization

18. Discussion
The algorithm is in principle applicable to manifolds of arbitrary dimension.
The numerical effort essentially depends on the dimension of the invariant manifold (and not on the dimension of state space).
The algorithm works for general invariant sets.

19. GENESIS Trajectory

20. Invariant Manifolds
Unstable manifold
Stable manifold
Halo orbit

21. Unstable Manifold of the Halo Orbit
Earth
Halo orbit

22. Unstable Manifold of the Halo Orbit
Flight along the manifold
Computation with GAIO, University of Paderborn

23. Invariant Measures: Discretization of the Problem
Galerkin approximation using the functions

24. Invariant Measure for Chua´s Circuit
Computation by GAIO; visualization with GRAPE

25. Invariant Measure for the Lorenz System

26. Typical Spectrum of the Markov Chain
Invariant measure
„Almost invariant set“
We consider the simplest situation...

27. Analyzing Maps with Isolated Eigenvalues (D.-Froyland-Sertl 2000)

28. At the Other End This map has no relevant eigenvalue except for the
(using a result from
Baladi 1995).
Let‘s pick a
map between
the two extremes

29. A Map with a Nontrivial relevant Eigenvalue
This map has a
relevant eigenvalue
of modulus less than one.
Essential spectrum
of continuous problem
(Keller ´84)

30. Corresponding Eigenfunctions
Eigenfunction for
the eigenvalue 1
Eigenfunction for
the eigenvalue < 1
positive on (0,0.5) and
negative on (0.5,1)

31. Almost Invariant Sets

32. Almost Invariance and Eigenvalues
Proposition:

33. Example
Second eigenfunction of the 1D-map:

34. Almost Invariant Sets in Chua´s Circuit
Computation by GAIO; Visualization with GRAPE

35. Transport in the Solar System (Computations by Hessel, 2002)
Idea: Concatenate the CR3BPs for
Neptune
Uranus
Saturn
Jupiter
Mars
and compute the probabilities for transitions
through the planet regions.

36. Spectrum for Jupiter Detemine the second largest real positive
eigenvalue:

37. Transport for Jupiter
Eigenvalue:
Eigenvalue:

38. Transport for Neptune
Eigenvalue:

39. Quantitative Results For the Jacobian constant C = 3.004 we obtain for
the probability to pass each planet within ten years:
Neptune:
Uranus:
Saturn: 0.011
Jupiter: 0.074

40. Using the Underlying Graph (Froyland-D. 2001, D.-Preis 2001)
Boxes are vertices
Coarse dynamics represented by edges
Use graph theoretic algorithms in
combination with the multilevel structure

41. Using Graph Partitioning for Jupiter (Preis 2001–)
Green – green:
Red – red:
Yellow – yellow:
Green – yellow: 0.065
Red – yellow: 0.062
T: approx. 58 days

42. 4BP for Jupiter / Saturn
Invariant measure

43. 4BP for Jupiter / Saturn
Almost invariant sets

44. 4BP for Saturn / Uranus
Almost invariant sets

45. Contact
Papers and software at