"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.