  Discontinuous time derivatives are used to model threshold-dependent
switching in such diverse applications as dry friction, electronic control, and
biological growth. In a continuous flow, a discon- tinuous derivative can
generate multiple outcomes from a single initial state. Here we show that well
defined solution sets exist for flows that become multi-valued due to grazing a
discontinuity. Loss of determinism is used to quantify dynamics in the limit of
infinite sensitivity to initial conditions, then applied to the dynamics of a
superconducting resonator and a negatively damped oscillator.
