"We present an algorithm for a rigorous, global periodic point finder.
This algorithm allows us to rigorously identify small
enclosures of fixed and periodic points of a sufficiently smooth map
$f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ in a given region
$K\subset\mathbb{R}^n$. If the derivative of $f$ is known, we can
furthermore verify uniqueness of the fixed or periodic points in each
enclosure.
We then proceed to present an implementation of this algorithm in
Taylor Model arithmetic using COSY INFINITY. The application of this
implementation to the locally hyperbolic Hénon map demonstrates the
power of the
Taylor Model approach. Taylor Models combine the speed of numerical
methods with the low overestimation of symbolic computation, thus
avoiding the problems introduced by other verified numerical methods
such as interval arithmetic. This allows us to compute small enclosures
of all periodic points with period 13 in the attractor of the Hénon map.
