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University of Paderborn
Applied Mathematics
Michael DellnitzDepartment of Mathematics
University of Paderborn
Almost Invariant Sets andTransport in the Solar System
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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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University of Paderborn
Applied Mathematics
Simulation of Chua´s Circuit
yz
zyxy
xm
xmyx
)3
( 310
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University of Paderborn
Applied Mathematics
Numerical Strategy
1. Approximation of the invariant set A
2. Approximation of the dynamical behavior on A
A
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University of Paderborn
Applied Mathematics
The Multilevel Approachfor the Lorenz System
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University of Paderborn
Applied Mathematics
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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University of Paderborn
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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University of Paderborn
Applied Mathematics
Example: Hénon Map
parameters and babxyaxyxf ),,1(),( 2
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University of Paderborn
Applied Mathematics
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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University of Paderborn
Applied Mathematics
Realization of the Subdivision Step
},,1:{),( nircyRyrcR iiin for
Boxes are indeed boxes
Subdivision by bisection
Data structure
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University of Paderborn
Applied Mathematics
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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University of Paderborn
Applied Mathematics
Global Attractor in Chua´s Circuit
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University of Paderborn
Applied Mathematics
Subdivision
Simulation
Global Attractor in Chua´s Circuit
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University of Paderborn
Applied Mathematics
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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University of Paderborn
Applied Mathematics
Example: Pendulum
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University of Paderborn
Applied Mathematics
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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University of Paderborn
Applied MathematicsInitializationSubdivisionContinuation 1Continuation 2Continuation 3
Covering of an Unstable Manifold for a Fixed Point of the Hénon Map
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University of Paderborn
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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University of Paderborn
Applied Mathematics
GENESIS Trajectory
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University of Paderborn
Applied Mathematics
Invariant Manifolds
Stable manifold
Unstable manifold
Halo orbit
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University of Paderborn
Applied Mathematics
Earth
Halo orbit
Unstable Manifoldof the Halo Orbit
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University of Paderborn
Applied Mathematics
Unstable Manifoldof the Halo Orbit
Flight along the manifold
Computation with GAIO, University of Paderborn
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University of Paderborn
Applied Mathematics
)(
))((
)(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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University of Paderborn
Applied Mathematics
Invariant Measure for Chua´s Circuit
Computation by GAIO; visualization with GRAPE
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University of Paderborn
Applied Mathematics
Invariant Measurefor the Lorenz System
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University of Paderborn
Applied Mathematics
Typical Spectrum of the Markov Chain
Invariant measure
„Almost invariant set“
We consider the simplest situation...
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University of Paderborn
Applied Mathematics
Analyzing Maps with IsolatedEigenvalues (D.-Froyland-Sertl 2000)
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University of Paderborn
Applied Mathematics
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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University of Paderborn
Applied Mathematics
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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University of Paderborn
Applied Mathematics
Corresponding Eigenfunctions
Eigenfunction forthe eigenvalue 1
Eigenfunction forthe eigenvalue < 1
positive on (0,0.5) andnegative on (0.5,1)
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University of Paderborn
Applied Mathematics
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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University of Paderborn
Applied Mathematics
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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University of Paderborn
Applied Mathematics
Example
796.02/)592.01(]5.0,0[ and A
Second eigenfunction of the 1D-map:
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University of Paderborn
Applied Mathematics
Almost Invariant Setsin Chua´s Circuit
Computation by GAIO; Visualization with GRAPE
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University of Paderborn
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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University of Paderborn
Applied Mathematics
Spectrum for Jupiter
Detemine the secondlargest real positiveeigenvalue:
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University of Paderborn
Applied Mathematics
Transport for Jupiter
Eigenvalue: 0.9998
Eigenvalue: 0.9982
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University of Paderborn
Applied Mathematics
Transport for Neptune
Eigenvalue: 0.999947
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University of Paderborn
Applied Mathematics
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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University of Paderborn
Applied Mathematics
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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University of Paderborn
Applied Mathematics
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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University of Paderborn
Applied Mathematics
4BP for Jupiter / Saturn
Invariant measure
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University of Paderborn
Applied Mathematics
4BP for Jupiter / Saturn
Almost invariant sets
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University of Paderborn
Applied Mathematics
4BP for Saturn / Uranus
Almost invariant sets
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University of Paderborn
Applied Mathematics
Contact
http://www.upb.de/math/~agdellnitz
Papers and software at