  A two-fold is a singular point on the discontinuity surface of a
piecewise-smooth vector field, at which the vector field is tangent to the
discontinuity surface on both sides. If an orbit passes through an invisible
two-fold (also known as a Teixeira singularity) before settling to regular
periodic motion, then the phase of that motion cannot be determined from
initial conditions, and in the presence of small noise the asymptotic phase of
a large number of sample solutions is highly random. In this paper we show how
the probability distribution of the asymptotic phase depends on the global
nonlinear dynamics. We also show how the phase of a smooth oscillator can be
randomised by applying a simple discontinuous control law that generates an
invisible two-fold. We propose that such a control law can be used to
desynchronise a collection of oscillators, and that this manner of phase
randomisation is fast compared to existing methods (which use fixed points as
phase singularities) because there is no slowing of the dynamics near a
two-fold.
