Magnetic track brakes (MTB) are additional braking systems for railway vehicles and are utilised at emergency and low adhesion conditions. Operations at low velocities can cause harmful self-excited vibrations in the mechanical structure. While the onset of self-excited vibrations was studied by linear stability analysis in literature, nonlinear stability analysis and the post-critical behaviour of the coupled electromagnetic-mechanical system have not yet been addressed. In this work, continuation methods are applied to a minimal model of the MTB to study occurring bifurcations and the dynamic behaviour after the loss of stability. Super-critical Hopf bifurcations are identified, and a second co-existing branch of stable periodic orbits with stick-slip transitions is found.