Almost Invariant Sets and Transport in the Solar System

University of Paderborn

Applied Mathematics

Michael DellnitzDepartment of Mathematics


Almost Invariant Sets andTransport in the Solar System


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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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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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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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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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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


Applied MathematicsInitializationSubdivisionContinuation 1Continuation 2Continuation 3

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


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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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Invariant Measure for Chua´s Circuit

Computation by GAIO; visualization with GRAPE


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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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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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4BP for Jupiter / Saturn

Almost invariant sets


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4BP for Saturn / Uranus

Almost invariant sets


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Contact

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

Papers and software at

Almost Invariant Sets and Transport in the Solar System

Documents

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