Approximation of Attractors Using the Subdivision Algorithm

Dr. Stefan SiegmundPeter Taraba

What is an attractor?

Attractor is a set A, which is

Invariant under the dynamics

attraction

AB

Example: Lorenz attractor

Subdivision Algorithm for computations of attractors

Dellnitz, Hohmann

1. Subdivision step2. Selection step

1. SELECTION STEP

2. SUBDIVISION STEP

A

1. Subdivision step2. Selection step

In the Subdivision Algorithm we combine these two steps

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

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

Example – Lorenz attractor

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

Box dimension

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

Disadvantage of this limit is that it converges slowly

Method I

This approximation is usually better (converges faster)

Method II

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 effectof Taylor methods

Also

wrappingeffect

we are stillnot “completelyclose” to attractor

condition not fulfilled

Subdivision step

Dimension

Method II

Method III