  Mathematical models involving switches --- in the form of differential
equations with discontinuities --- can accomodate real-world non-idealities
through perturbations by hysteresis, time-delay, discretization, and noise.
These are used to model the processes associated with switching in electronic
control, mechanical contact, predator-prey preferences, and genetic or cellular
regulation. The effect of such perturbations on rapid switching dynamics about
a single switch are somewhat benign: in the limit that the size of the
perturbation goes to zero the dynamics is given by Filippov's sliding solution.
When multiple switches are involved, however, perturbations can have
complicated effects, as shown in this paper. In the zero-perturbation limit,
hysteresis, time-delay, and discretization cause erratic variation or `jitter'
for stable sliding motion, whilst noise generates a relatively regular sliding
solution similar to the canopy solution (an extension of Filippov's solution to
multiple switches). We illustrate the results with a model of a switched power
circuit, and showcase a variety of complex phenomena that perturbations can
generate, including chaotic dynamics, exit selection, and coexisting sliding
solutions.
