University of Paderborn

Applied Mathematics

Michael DellnitzDepartment of Mathematics

University of Paderborn

Almost Invariant Sets andTransport in the Solar System

University of Paderborn

Applied Mathematics

Overview

almost invariant sets

invariant measures

global attractorsinvariant manifolds

invariant sets(mission design; zero finding)

statistics(molecular dynamics;transport problems)

set orientednumerical

methods

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Applied Mathematics

Simulation of Chua´s Circuit

yz

zyxy

xm

xmyx

)3

( 310

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Numerical Strategy

1. Approximation of the invariant set A

2. Approximation of the dynamical behavior on A

A

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The Multilevel Approachfor the Lorenz System

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Relative Global Attractors

nnjj RRfxfx :),(1

define compact For nRQ

0

)(

j

jQ QfA

Relative Global Attractor

.QAQ inside sets invariant the all contains

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Applied Mathematics

The Subdivision Algorithm

. covering boxes

of collection a be Let

Q

k

A

C 1

Subdivision

).diam( )diam( and

that such Construct

1ˆ

ˆ

ˆ

1

kkCBCB

k

CCBB

C

kk

Selection Set

BBfCBCBC kkkˆ)(ˆˆ:ˆ 1 s.t.

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Applied Mathematics

Example: Hénon Map

parameters and babxyaxyxf ),,1(),( 2

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distance Hausdorff the

Then Define

),(

.

h

BQkCB

k

0),(lim kQ

kQAh

A Convergence Result

Proposition [D.-Hohmann 1997]:

Remark:

Results on the speed of convergence can be obtained if possesses a hyperbolic structure.QA

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Applied Mathematics

Realization of the Subdivision Step

},,1:{),( nircyRyrcR iiin for

Boxes are indeed boxes

Subdivision by bisection

Data structure

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Realization of the Selection Step

Standard choice of test points:

• For low dimensions: equidistant distribution on edges of boxes.• For higher dimensions: stochastic distribution inside the boxes.

.,)(1 jiBBf ji whethercheck to have We

Use test points:

? points test all for ji ByByf )(

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Global Attractor in Chua´s Circuit

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Subdivision

Simulation

Global Attractor in Chua´s Circuit

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Invariant Manifolds

. i.e. , of point fixed a be Let ppffp )(

jpxfxpW

jpxfxpWju

js

for

for

)(:)(

)(:)(

Stable and unstable manifold of p

)( pW s

)( pW u

p

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Example: Pendulum

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Computing Local Invariant Manifolds

Idea:

)( pWA

pN

ulocN

of odneighborho small a for

Let p be a hyperbolic fixed point

N

p )( pW uAN

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Applied MathematicsInitializationSubdivisionContinuation 1Continuation 2Continuation 3

Covering of an Unstable Manifold for a Fixed Point of the Hénon Map

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Applied Mathematics

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.

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GENESIS Trajectory

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Invariant Manifolds

Stable manifold

Unstable manifold

Halo orbit

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Earth

Halo orbit

Unstable Manifoldof the Halo Orbit

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Unstable Manifoldof the Halo Orbit

Flight along the manifold

Computation with GAIO, University of Paderborn

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)(

))((

)(1

l

lkkl

kld

Bm

BBfmp

pP

with chain Markov eApproximat

Invariant Measures:Discretization of the Problem

Galerkin approximation using the functions

dihiBi ,,1

dBB ,,1 coveringBox

)stochastic measure; Lebesgue ( dPm

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Applied Mathematics

Invariant Measure for Chua´s Circuit

Computation by GAIO; visualization with GRAPE

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Applied Mathematics

Invariant Measurefor the Lorenz System

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Typical Spectrum of the Markov Chain

Invariant measure

„Almost invariant set“

We consider the simplest situation...

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

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At the Other End

This map has norelevant eigenvalue

except for theeigenvalue 1

(using a result fromBaladi 1995).

Let‘s pick amap between

the two extremes

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A Map with a Nontrivialrelevant Eigenvalue

This map has arelevant eigenvalue

of modulus less than one.

Essential spectrumof continuous problem

(Keller ´84)

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Corresponding Eigenfunctions

Eigenfunction forthe eigenvalue 1

Eigenfunction forthe eigenvalue < 1

positive on (0,0.5) andnegative on (0.5,1)

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Almost Invariant Sets

and if measure the to respect

withinvariant almost- is :Definition

).())((

0)(

1 AAAf

A

XA

.

1

MP

PP d

for

that such operator transfer

the of eigenvalue an be letNow

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Almost Invariance and Eigenvalues

. to respect withinvariant almost- is if

Then withset a be Let

A

AXA

2

1

.5.0)(

Proposition:

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Example

796.02/)592.01(]5.0,0[ and A

Second eigenfunction of the 1D-map:

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Almost Invariant Setsin Chua´s Circuit

Computation by GAIO; Visualization with GRAPE

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Applied Mathematics

Transport in the Solar System(Computations by Hessel, 2002)

Idea: Concatenate the CR3BPs for

• Neptune• Uranus• Saturn• Jupiter• Mars

and compute the probabilities for transitionsthrough the planet regions.

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Spectrum for Jupiter

Detemine the secondlargest real positiveeigenvalue:

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Transport for Jupiter

Eigenvalue: 0.9998

Eigenvalue: 0.9982

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Transport for Neptune

Eigenvalue: 0.999947

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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

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Using the Underlying Graph(Froyland-D. 2001, D.-Preis 2001)

Boxes are verticesCoarse dynamics represented by edges

Use graph theoretic algorithms incombination with the multilevel structure

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Using Graph Partitioningfor Jupiter (Preis 2001–)

Green – green: 0.9997Red – red: 0.9997

Yellow – yellow: 0.8733

Green – yellow: 0.065Red – yellow: 0.062

T: approx. 58 days

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4BP for Jupiter / Saturn

Invariant measure

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Applied Mathematics

4BP for Jupiter / Saturn

Almost invariant sets

University of Paderborn

Applied Mathematics

4BP for Saturn / Uranus

Almost invariant sets

University of Paderborn

Applied Mathematics

Contact

http://www.upb.de/math/~agdellnitz

Papers and software at