Hard-to-reach oil and gas reservoirs are nowadays accessed by directional drilling techniques, which use a rotary steerable system to drill complex curved boreholes. This paper aims at providing understanding of the complex behavior of directional drilling systems by developing a model for the borehole evolution and providing a dynamic analysis of the resulting model. The planar evolution of the borehole path is modeled in the form of a delay complementarity system, which accounts for undergauged stabilizers and a saturation of the bit orientation with respect to the borehole orientation. These are essential nonlinearities from a practical point of view. The pursued dynamic analysis reveals that these systems induce steady-state oscillations in the borehole path, which are related to the planar equivalent of the highly detrimental borehole spiraling observed in practice. The model and dynamic analysis provide essential insights and can serve in the further development of control techniques to track borehole paths while mitigating borehole spiraling.