Abstract Equations of motion are derived to describe the hunting mode of a railway axle running at constant velocity along straight track. It is assumed that the wheel and rail-head profiles take some arbitrary shape. This shape gives rise to non-linearities in the equations. The equations are first linearized, and approximate expressions derived for the frequency of the oscillation and conditions of stability. Asymptotic stability for all initial conditions of the non-linear system is then considered in the manner of Aiserman, and the equations are examined for stable limit-cycles by applying the first approximations of Kryloff and Bogoliuboff. It is shown that, when running at low velocities, the axle will execute limit-cycle oscillations even though the wheel's flanges do not contact the rails. Small increases in velocity, however, quickly result in flange contact.