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 AB Example: Lorenz attractor

Subdivision Algorithm for computations of attractors Dellnitz, Hohmann 1.Subdivision step 2.Selection step

1. SELECTION STEP

2. SUBDIVISION STEP A

1.Subdivision step 2.Selection step In the Subdivision Algorithm we combine these two steps

Global Attractor A Let be a compact subset. We define the global attractor relative 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 do not 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 well with Lohner Algorithm Taylor models with rotation without rotation Still too big, because we cannot integrate too long

Box dimension

Possible problems: 01 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 effect of Taylor methods

Also

wrapping effect we are still not “completely close” to attractor condition not fulfilled Subdivision step Dimension Method II Method III