Propulsion by actin polymerization is widely used in cell motility. Here, we investigate a model of the brush range of an actin gel close to a propelled object, describing the force generation and the dynamics of the propagation velocity. We find transitions between stable steady states and relaxation oscillations when the attachment rate of actin filaments to the obstacle is varied. The oscillations set in at small values of the attachment rate via a homoclinic bifurcation. A second transition from a stable steady state to relaxation oscillations, found for higher values of the attachment rate, occurs via a supercritical Hopf bifurcation. The behavior of the model near the second transition is similar that of a system undergoing a canard explosion. Consequently, we observe excitable dynamics also. The model further exhibits bistability between stationary states or stationary states and limit cycles. Therefore, the brush of actin filament ends appears to have a much richer dynamics than was assumed until now.