Lateral vibrations of stabilized bottomhole assemblies of drillstrings often exhibit quite complicated patterns. Previous work has taken into account nonlinearities due to fluid damping, wall contact, eccentric collars, and initial curvature of the collar. This paper addresses the effects of these forces, along with linear and parametric excitation of the lateral motion due to an applied dynamic weight on bit (WOB). Four state-space nonlinear ordinary differential equations in two-dimensional space are used to describe the model. A time variant zero-mean axial WOB is considered and described as a stationary random process. Random and chaotic whirling characteristics of drillstrings drillstrings are investigated using the minimum phase-volume deconvolution technique. The reported study serves to point out that even in ordinary engineering practice problems, rather complex phenomena can appear, and should thus be taken into consideration in the design and operation stages.