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Computer Assisted Proofs of the Existence of High-Period Fixed Points


Abstract

We describe an efficient method to rigorously prove the existence and attractiveness of high period periodic points through verified numerical computations. The proof consists of two parts: first the existence and then the attractiveness. To prove the existence, we only require a non-verified, numerical approximation of the derivative of the map at the fixed point. In the second part of the proof we require knowledge of the exact derivative. The use of Taylor Models in COSY INFINITY to carry out the calculations very successfully controls the dependency problem commonly encountered in verified numerics.

In comparison, we also implemented the same proof using traditional interval arithmetic. This approach is more complicated, as it requires calculations to be carried out in higher precision than the standard double precision. Also, interval methods suffer significantly from the dependency problem, which requires splitting of the domain into smaller, more tractable pieces.

We then apply both algorithms to prove the existence of a period 15 point in a Henon map very close to the standard parameters. Using high precision intervals, we obtain a very tight enclosure of the periodic point with a precision of up to 70 decimal digits. By examining the Jacobian of the 15th iterate of the map, we lastly establish uniqueness and attractiveness of this periodic point.


A. Wittig, M. Berz, S. Newhouse, Proceedings of the Fields Institute, in print


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