Motivated by a stochastic differential equation describing the dynamics of interfaces, we study the bifurcation behaviour of a more general class of such equations. These equations are characterized by a two-dimensional phase space (describing the position of the interface and an internal degree of freedom). The noise accounts for thermal fluctuations of such systems. The models considered show a saddle-node bifurcation and have furthermore homoclinic orbits, i.e. orbits leaving an unstable fixed point and returning to it. Such systems display intermittent behaviour. The presence of noise combined with the topology of the phase space leads to unexpected behaviour as a function of the bifurcation parameter, i.e. of the driving force of the system. We explain this behaviour using saddle-point methods and considering global topological aspects of the problem. This then explains the non-monotonic force–velocity dependence of certain driven interfaces