Asteroids and comets are the objects of study for a number of recent scientific missions. It is of interest to find orbits that will neither escape the asteroid nor impact the surface. It is also desirable to find orbits that are periodic, to simplify mission design, and also polar, so that the entire surface including the polar regions can be viewed. The dynamics about such bodies are affected by numerous perturbations, making closed-form analytic solutions to the equations of motion difficult to find, if not impossible. Semi-analytic and numerical methods can sometimes be useful and are employed in this work. In this effort, the asteroid is modeled as a rotating triaxial ellipsoid. Four asteroids were chosen for study. Lagrange's Planetary Equations are used to generate plots of changes of periapsis distance over a single orbit, for several sizes of orbits about each asteroid. Resonance, the condition in which the orbit period is commensurate or nearly commensurate with the asteroid's rotational period, and its effects on polar orbits are the focus of this study. These orbits are used as inputs into a differential corrections algorithm in an attempt to produce periodic orbits. Finally, a correlation is made between the success of the differential corrections program and the qualitative nature of the corresponding periapsis plot.