# 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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