Jitter in Piecewise-Smooth Dynamical Systems with Intersecting Discontinuity Surfaces

Differential equations that switch between different modes of behavior across a surface of discontinuity are used to model, for example, electronic switches, mechanical contact, predator-prey preference changes, and genetic or cellular regulation. Switching in such systems is unlikely to occur precisely at the ideal discontinuity surface, but instead can involve various spatiotemporal delays or noise. If a system switches between more than two modes, across a boundary formed by the intersection of discontinuity surfaces, then its motion along that intersection becomes highly sensitive to such nonidealities. If switching across the surfaces is affected by hysteresis, time delay, or discretization, then motion along the intersection can be affected by erratic variations that we characterize as "jitter". Introducing noise, or smoothing out the discontinuity, instead leads to steady motion along the intersection well described by the so-called canopy extension of Filippov's sliding concept (which applies when the discontinuity surface is a simple hypersurface). We illustrate the results with numerical experiments and an example from power electronics, providing explanations for the phenomenon as far as they are known.