The research in this thesis focuses on the study of chaotic transport in phase space applied to the analysis of ionization in two different atomic systems. The first system consists of an atom exposed to periodic alternating positive and negative external electric field pulses, and the second system consists of an atom placed in constant external electric and magnetic fields. The main goal of this thesis is to develop a connection between the general mathematical constructs of a dynamical system, and an atomic experiment. As such, this research acts as a bridge between the areas of applied math and experimental physics--Abstract.