Approximation of Attractors Using
the Subdivision Algorithm
Dr. Stefan Siegmund
Peter Taraba
What is an attractor?
Attractor is a set A, which is
Invariant under the dynamics
attraction
B
A
Example: Lorenz attractor
Dellnitz, Hohmann
Subdivision Algorithm for computations of attractors
1. Subdivision step
2. Selection step
1. SELECTION STEP
2. SUBDIVISION STEP
A
In the Subdivision Algorithm we combine these two steps
1. Subdivision step
2. Selection step
Global Attractor A
Let
relative to
be a compact subset. We define the global attractor
by
is 1-time map
In general
p,q – hyperbolic fixed points
& heteroclinic connection
q
p
Q
We can miss some boxes
That’s why use of interval arithmetics (basic operations,
Lohner algorithm, Taylor models) will ensure that we do
not miss any box
Example – Lorenz attractor
Interval analysis
Discrete maps work also with basic interval operations
Lohner algorithm
with rotation
without rotation
Still too big, because
we cannot integrate
too long
More complex continuous diff. eq.
(Lorenz …) does not work well
with Lohner Algorithm
Taylor models
Box dimension
Possible problems:
0
1
We have to take map
or in continuous time enlarge
hyperbolic
There exist such
such that we get only those boxes, which contain A
Method I
Disadvantage of this limit is that it converges slowly
Method II
This approximation is usually better (converges faster)
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
2. there is a hope we can get closer covering of attractor
Memory limitations
Computation time limitation
we can not continue in subdivision
3. we will get better approximation of dimension
Wrapping effect
of Taylor methods
Also
wrapping
effect
Dimension
Method III
we are still
not “completely
close” to attractor
condition not fulfilled
Method II
Subdivision step