  This article examines dynamical systems on a class of K3 surfaces in
$\mathbb{P}^{2} \times \mathbb{P}^{2}$ with an infinite automorphism group. In
particular, this article develops an algorithm to find $\mathbb{Q}$-rational
periodic points using information modulo $p$ for various primes $p$. The
algorithm is applied to exhibit K3 surfaces with $\mathbb{Q}$-rational periodic
points of primitive period $1,...,16$. A portion of the algorithm is then used
to determine the Riemann zeta function modulo 3 of a particular K3 surface and
find a family of K3 surfaces with Picard number two.
