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

(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

Simulation of Chua´s Circuit

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

The Multilevel Approach for the Lorenz System

Relative Global Attractors

The Subdivision Algorithm Selection Set

Example: Hénon Map

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

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

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.

Global Attractor in Chua´s Circuit

Global Attractor in Chua´s Circuit Simulation Subdivision

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

Example: Pendulum

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

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

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.

GENESIS Trajectory

Invariant Manifolds Unstable manifold Stable manifold Halo orbit

Unstable Manifold of the Halo Orbit Earth Halo orbit

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

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

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

Invariant Measure for the Lorenz System

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

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

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

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)

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

Almost Invariant Sets

Almost Invariance and Eigenvalues Proposition:

Example Second eigenfunction of the 1D-map:

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

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.

Spectrum for Jupiter Detemine the second largest real positive eigenvalue:

Transport for Jupiter Eigenvalue: 0.9982 Eigenvalue: 0.9998

Transport for Neptune Eigenvalue: 0.999947

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

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

Using Graph Partitioning for 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

4BP for Jupiter / Saturn Invariant measure

4BP for Jupiter / Saturn Almost invariant sets

4BP for Saturn / Uranus Almost invariant sets

Contact Papers and software at http://www.upb.de/math/~agdellnitz