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

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

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

Introduction

Simulation of Chua‘s Circuit

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

Relative Global Attractors

The Basic Subdivision Scheme Selection Set

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

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

Global Attractor in Chua‘s Circuit

Global Attractor in Chua‘s Circuit Simulation Subdivision

Invariant Manifolds

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

Design of Spacecraft Trajectories: GENESIS

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.

How to Use 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

Unstable Manifold of the Halo Orbit

Combination with Control ESA Corner Stone Mission Darwin

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 ?

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)

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

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

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

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

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

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

Statistics and Transport

Discretization of the Problem 1 2 3 4 5 6 7 8 Transfer Operator Approach

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

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

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

Energy Barrier in the CR3BP L4 L3 L1 L2 L5

Spectrum for Jupiter Determine the second largest real positive eigenvalue:

Transport for Jupiter Eigenvalue: 0.9982 Eigenvalue: 0.9998

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

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

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

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

4BP for Jupiter / Saturn

An Alternative: Using the Underlying Graph (Froyland-D. 2003, D.-Preis et al. 2002-) Boxes are vertices Coarse dynamics represented by edges Use graph theoretic algorithms in combination with the multilevel structure

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

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

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

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

Almost Invariant Set and Crosser Line

Transport Computations

Identifying the Location of Planets

Nonautonomous Systems

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

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.

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

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.

LCS / AIS for a Toy Example

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

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

LCS in the Antarctic Circumpolar Current

LCS in the Antarctic Circumpolar Current

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