A unified approach to explain contrary effects of hysteresis and smoothing in nonsmooth systems

Piecewise smooth dynamical systems make use of discontinuities to model switching between regions of smooth evolution. This introduces an ambiguity in prescribing dynamics at the discontinuity: should it be given by a limiting value on one side or other of the discontinuity, or a member of some set containing those values? One way to remove the ambiguity is to regularize the discontinuity, the most common being either to smooth out the discontinuity, or to introduce a hysteresis between switching in one direction or the other across the discontinuity. Here we show that the two can in general lead to qualitatively different dynamical outcomes. We then define a higher dimensional model with both smoothing and hysteresis, and study the competing limits in which hysteretic or smoothing effect dominate the behaviour, only the former of which correspond to Filippov's standard `sliding modes'.