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We show that all eccentric timelike orbits in Reissner-Nordstr\"om 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 Reissner-Nordstr\"om and Schwarzschild geometries enables us to distinguish charged and uncharged spacetimes by looking only at the orbital dynamics.
Misra et al. (Mon,) studied this question.