  Switch-like behaviour in dynamical systems may be modelled by highly
nonlinear functions, such as Hill functions or sigmoid functions, or
alternatively by piecewise-smooth functions, such as step functions. Consistent
modelling requires that piecewise-smooth and smooth dynamical systems have
similar dynamics, but the conditions for such similarity are not well
understood. Here we show that by smoothing out a piecewise-smooth system one
may obtain dynamics that is inconsistent with the accepted wisdom --- so-called
Filippov dynamics --- at a discontinuity, even in the piecewise-smooth limit.
By subjecting the system to white noise, we show that these discrepancies can
be understood in terms of potential wells that allow solutions to dwell at the
discontinuity for long times. Moreover we show that spurious dynamics will
revert to Filippov dynamics, with a small degree of stochasticity, when the
noise magnitude is sufficiently large compared to the order of smoothing. We
apply the results to a model of a dry-friction oscillator, where spurious
dynamics (inconsistent with Filippov's convention or with Coulomb's model of
friction) can account for different coefficients of static and kinetic
friction, but under sufficient noise the system reverts to dynamics consistent
with Filippov's convention (and with Coulomb-like friction).
