In this paper, the authors present a discrete system model to study the coupled axial–torsional dynamics of a drill string. The model is developed taking into account state-dependent time delay and nonlinearities due to dry friction and loss of contact. Simulations are carried out by using a 32-segment model with 128 states. Bit bounce is observed through time histories of axial vibrations, while stick-slip phenomenon is noted in the torsion response. The normal strain contours of this spatial–temporal system demonstrate the existence of strain wave propagation along the drill string. The shear strain wave exhibits features of wave nodes and wave loops along the drill string, which indicate that the torsional motion has the properties of a standing wave. When the penetration rate is varied, qualitative changes are observed in the system response. The observed behavior includes chaotic and hyperchaotic dynamics. Stability analysis reveals a stable region for the degenerate one-segment model. This stable region becomes infinitesimally small, as the resolution of spatial discretization is increased. This finding suggests that drill-string motions have a high likelihood of being self-exited in practical drilling operations.