1 / 45

Almost Invariant Sets and Transport in the Solar System

Almost Invariant Sets and Transport in the Solar System. Michael Dellnitz Department of Mathematics University of Paderborn. Overview. invariant sets (mission design; zero finding). statistics (molecular dynamics; transport problems). GAIO. invariant manifolds. global attractors.

keith
Download Presentation

Almost Invariant Sets and Transport in the Solar System

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. Overview invariant sets (mission design; zero finding) statistics (molecular dynamics; transport problems) GAIO invariant manifolds global attractors set oriented numericalmethods 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 Approachfor the Lorenz System

  6. Relative Global Attractors Relative Global Attractor

  7. The Subdivision Algorithm Subdivision Selection Set

  8. Example: Hénon Map

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

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

  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. Invariant Manifolds Stable and unstable manifold of p

  15. Example: Pendulum

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

  17. Covering of an Unstable Manifold for a Fixed Point of the Hénon Map Continuation 1 Continuation 2 Continuation 3 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 Manifoldof the Halo Orbit Earth Halo orbit

  22. Unstable Manifoldof 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 Measurefor the Lorenz System

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

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

  28. At the Other End This map has no relevant eigenvalue except for the eigenvalue 1 (using a result from Baladi 1995). Let‘s pick a map between the two extremes

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

  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 Setsin 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: 0.9982 Eigenvalue: 0.9998

  38. Transport for Neptune Eigenvalue: 0.999947

  39. Quantitative Results For the Jacobian constant C = 3.004 we obtain for the probability to pass each planet within ten years: • Neptune: 0.0002 • Uranus: 0.0003 • 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 Partitioningfor Jupiter (Preis 2001–) Green – green: 0.9997 Red – red: 0.9997 Yellow – yellow: 0.8733 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 http://www.upb.de/math/~agdellnitz

More Related