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Approximation of Attractors Using the Subdivision Algorithm

Approximation of Attractors Using the Subdivision Algorithm. Dr. Stefan Siegmund Peter Taraba. B. A. What is an attractor?. Attractor is a set A , which is. Invariant under the dynamics. attraction. Example: Lorenz attractor. Dellnitz, Hohmann.

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Approximation of Attractors Using the Subdivision Algorithm

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  1. Approximation of Attractors Using the Subdivision Algorithm Dr. Stefan Siegmund Peter Taraba

  2. B A What is an attractor? Attractor is a set A, which is Invariant under the dynamics attraction Example: Lorenz attractor

  3. Dellnitz, Hohmann Subdivision Algorithm for computations of attractors • Subdivision step • Selection step

  4. 1. SELECTION STEP

  5. A 2. SUBDIVISION STEP

  6. In the Subdivision Algorithm we combine these two steps • Subdivision step • Selection step

  7. q p Global Attractor A Let be a compact subset. We define the global attractor relative to by is 1-time map In general p,q – hyperbolic fixed points & heteroclinic connection Q

  8. 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

  9. Example – Lorenz attractor

  10. 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

  11. Box dimension

  12. Possible problems: 0 1 We have to take map or in continuous time enlarge hyperbolic such that we get only those boxes, which contain A There exist such

  13. Method I Disadvantage of this limit is that it converges slowly

  14. Method II This approximation is usually better (converges faster)

  15. 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

  16. 2. there is a hope we can get closer covering of attractor Memory limitations Computation time limitation we can not continue in subdivision

  17. Wrapping effect of Taylor methods 3. we will get better approximation of dimension

  18. Also

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

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