  We show that all eccentric timelike orbits in Reissner-Nordstr\"{o}m
spacetime can be classified using a taxonomy that draws upon an isomorphism
between periodic orbits and the set of rational numbers. By virtue of the fact
that the rationals are dense, the taxonomy can be used to approximate aperiodic
orbits with periodic orbits. This may help reduce computational overhead for
calculations in gravitational wave astronomy. Our dynamical systems approach
enables us to study orbits for both charged and uncharged particles in spite of
the fact that charged particle orbits around a charged black hole do not admit
a simple one-dimensional effective potential description. Finally, we show that
comparing periodic orbits in the RN and Schwarzschild geometries enables us to
distinguish charged and uncharged spacetimes by looking only at the orbital
dynamics.
