1. Set Oriented Numerical Methods for the
Approximation of Transport Phenomena
Michael Dellnitz
Institute for Industrial Mathematics and PaSCo
University of Paderborn

2. Participating Institutions / Credits
California Institute of Technology (J. E. Marsden)
University of New South Wales (G. Froyland, M. England)
Technical University Dresden (K. Padberg)
Paderborn group, in particular Mirko Hessel-von Molo, Markus Post, Katrin Witting,
Marcel Schwalb
Research supported by the
Deutsche Forschungsgemeinschaft and by the EU under the project Astronet

3. Contents Introduction to Set Oriented Methods
Computation of Invariant Manifolds
Transport Phenomena
Control
Nonautonomous Systems

4. Introduction

5. Simulation of Chua‘s Circuit

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

7. Relative Global Attractors

8. The Basic Subdivision Scheme
Selection Set

9. Illustration of the Set Oriented Approach
Global Attractor for the Lorenz System

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

11. Global Attractor in Chua‘s Circuit

12. Global Attractor in Chua‘s Circuit
Simulation
Subdivision

13. Invariant Manifolds

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 of a Fixed Point of the Henon Map
Continuation 3
Continuation 2
Continuation 1
Subdivision
Initialization

18. Design of Spacecraft Trajectories: GENESIS

19. GENESIS Trajectory
Problem: construct an energetically efficient
trajectory to a Halo-orbit and back to Earth.
Solution: find appropriate trajectories along the
invariant manifolds of the Halo-orbit.

20. How to Use 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

23. Unstable Manifold of the Halo Orbit

24. Combination with Control
ESA Corner Stone Mission Darwin

25. Combination with Control (D., Junge, Post, Thiere 2006)
Problem statement
Find a low-thrust trajectory for a spacecraft from one planet to a second one
minimize fuel consumption
respect upper bound for thrust of spacecraft
Example: transfer from Earth to Venus by a maximum thrust of 800 mN ?

26. Combination with Control
Typically two steps for trajectory design:
Determine a good initial guess trajectory
global approach and cleverness
simple model (3BP)
Use a local solver to improve trajectory
more realistic model (solar system dynamics)

27. The Model: CR3BP and its Lagrange Points (Circular Restricted 3 Body Problem)
Differential equation in rotating coordinates for a spacecraft
two celestial bodies (primaries) exert gravitational forces on the spacecraft
no gravitational forces exerted by the spacecraft
circular motion of the primaries
planar motion of all three bodies L4 L3 L1 L2 L5

28. obtaining trajectories by patching PCR3BP
...and again: use Invariant Manifolds
obtaining trajectories by patching PCR3BP
Koon, W. S., M. W. Lo, J. E. Marsden, and S. D. Ross [1999-], e.g.:
Constructing a Low Energy Transfer between Jovian Moons
Heteroclinic connections between periodic orbits and...
Shoot the Moon

29. Illustration of the Idea
(Video created by Caltech / JPL)

30. Reachable sets and keeping track of best control
Numerical Treatment
Reachable sets and keeping track of best control

31. Reachable sets and keeping track of best control
Numerical Treatment
Reachable sets and keeping track of best control
u: control
: time
S: start set
software package GAIO:
tree as data structure to store unions of box-shaped sets

32. Example: Earth-Venus Transfer
intersection of the two
reachable sets
red: from Earth, blue: to Venus, yellow: intersection in 4d phase space

33. Statistics and Transport

34. Discretization of the Problem1 2 3 4 5 6 7 8
Transfer Operator Approach

35. Invariant Measure for Chua‘s Circuit
Computation by GAIO; visualization with GRAPE

36. Typical Spectrum of the Markov Chain
Invariant measure
„Almost invariant set“
Magnitude of eigenvalue of related to almost invariance.

37. Almost Invariant Sets in Chua‘s Circuit
Illustration of the sign structure of the
eigenvector for the second largest eigenvalue.
Computation by GAIO; visualization with GRAPE

38. Energy Barrier in the CR3BPL4 L3 L1 L2 L5

39. Spectrum for Jupiter Determine the second largest real positive
eigenvalue:

40. Transport for Jupiter
Eigenvalue:
Eigenvalue:

41. Invariant Measure in the 4BP for Mars / Jupiter
(1024 Boxes)

42. Invariant Measure in the 4BP for Mars / Jupiter
(4096 Boxes)

43. Invariant Measure in the 4BP for Mars / Jupiter
(16384 Boxes)

44. Invariant Measure in the 4BP for Mars / Jupiter
(16384 Boxes)

45. 4BP for Jupiter / Saturn

46. An Alternative: Using the Underlying Graph
(Froyland-D. 2003, D.-Preis et al )
Boxes are vertices
Coarse dynamics represented by edges
Use graph theoretic algorithms in
combination with the multilevel structure

47. Hilda and Quasi-Hilda Asteroids
(D., Junge, Lo, Marsden, Padberg, Preis, Ross, Thiere 2005)
group of minor objects between Mars and Jupiter
owe their longevity to the invariance of a 3:2 resonance with Jupiter
resonance island surrounded by chaotic orbits
Quasi-Hilda objects: Oterma and Gehrels 3

48. Reduction of the System
The 4-dim system is
reduced to a 2-dim
system (energy
restriction and
Poincaré section).

49. Poincaré Section for Fixed Energy Level
Hilda-Asteroids
Quasi-Hilda-Asteroids

50. Set Oriented Approach
Covering of the relevant region with 2-dim boxes

51. Almost Invariant Set and Crosser Line

52. Transport Computations

53. Identifying the Location of Planets

54. Nonautonomous Systems

55. Ocean Dynamics
Goal: Identification of the Lagrangian Coherent Structures (LCS), since they govern transport and mixing
Key contributions to LCS:
Marsden, Dabiri, Shadden, Wiggins, Jones, Lekien, Haller

56. Invariant Manifolds vs. Expansion
Manifolds separate almost invariant sets
Stable manifolds of hyperbolic objects are repelling
Growth of a small perturbation can be measured by expansion rates (finite-time Lyapunov exponents)
Expansion rates are expected to be large along stable manifolds
A curve segment initialized transverse to the stable manifold of a hyperbolic fixed point is stretched along the unstable manifold.

57. Approximation of the direct expansion rate:
Implementation
Set-wise (direct) expansion rate of a box B:
Approximation of the direct expansion rate: B

58. Monterey Bay (2D)
Approximation of LCS in Monterey Bay Integration of drifter trajectories with MANGEN over 144 hours and computation of direct expansion rates. In close cooperation with Caltech.

59. LCS / AIS for a Toy Example

60. Perron-Frobenius Discretization - Time Dependent
(Froyland et al. 2007)
Analyze the temporal evolution of
almost invariant sets by varying t

61. LCS in the Antarctic Circumpolar Current
(Froyland et al. 2007)

62. LCS in the Antarctic Circumpolar Current

63. LCS in the Antarctic Circumpolar Current

64. LCS in the Antarctic Circumpolar Current
Movie made by Marcel Schwalb, Paderborn