Approximation of Attractors Using the Subdivision Algorithm
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Approximation of Attractors Using the Subdivision Algorithm
Dr. Stefan SiegmundPeter Taraba
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What is an attractor?
Attractor is a set A, which is
Invariant under the dynamics
attraction
AB
Example: Lorenz attractor
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Subdivision Algorithm for computations of attractors
Dellnitz, Hohmann
1. Subdivision step2. Selection step
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1. SELECTION STEP
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2. SUBDIVISION STEP
A
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1. Subdivision step2. Selection step
In the Subdivision Algorithm we combine these two steps
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Global Attractor A
Let be a compact subset. We define the global attractorrelative to by
In general
p
q
p,q – hyperbolic fixed points& heteroclinic connection
Q
is 1-time map
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We can miss some boxes
That’s why use of interval arithmetics (basic operations,Lohner algorithm, Taylor models) will ensure that we donot miss any box
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Example – Lorenz attractor
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Interval analysis
Discrete maps work also with basic interval operations
Lohner algorithm
More complex continuous diff. eq.(Lorenz …) does not work wellwith Lohner Algorithm
Taylor models
with rotationwithout rotation
Still too big, becausewe cannot integratetoo long
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Box dimension
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Possible problems:
0 1
We have to take map
or in continuous time enlarge
There exist such such that we get only those boxes, which contain A
hyperbolic
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Disadvantage of this limit is that it converges slowly
Method I
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This approximation is usually better (converges faster)
Method II
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Why should we use Taylor models?
1. we will not miss any boxes, we will get rigorous covering of relative attractors
2. there is a hope we can get closer covering of attractor
3. we will get better approximation of dimension
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2. there is a hope we can get closer covering of attractor
Memory limitations
Computation time limitation
we can not continue in subdivision
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3. we will get better approximation of dimension
Wrapping effectof Taylor methods
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Also
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wrappingeffect
we are stillnot “completelyclose” to attractor
condition not fulfilled
Subdivision step
Dimension
Method II
Method III