almost invariant sets and transport in the solar system
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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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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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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
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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Applied Mathematics
Realization of the Subdivision Step
},,1:{),( nircyRyrcR iiin for
Boxes are indeed boxes
Subdivision by bisection
Data structure
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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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Applied Mathematics
Global Attractor in Chua´s Circuit
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Subdivision
Simulation
Global Attractor in Chua´s Circuit
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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
University of Paderborn
Applied Mathematics
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
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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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
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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Applied Mathematics
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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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
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