  We study a class of discontinuous vector fields brought to our attention by
multi-legged animal locomotion. Such vector fields arise not only in
biomechanics, but also in robotics, neuroscience, and electrical engineering,
to name a few domains of application. Under the conditions that (i) the vector
field's discontinuities are locally confined to a finite number of smooth
submanifolds and (ii) the vector field is transverse to these surfaces in an
appropriate sense, we show that the vector field yields a well-defined flow
that is Lipschitz continuous and piecewise-differentiable. This implies that
although the flow is not classically differentiable, nevertheless it admits a
first-order approximation (known as a Bouligand derivative) that is
piecewise-linear and continuous at every point. We exploit this first-order
approximation to infer existence of piecewise-differentiable impact maps
(including Poincar\'{e} maps for periodic orbits), show the flow is locally
conjugate (via a piecewise-differentiable homeomorphism) to a flowbox, and
assess the effect of perturbations (both infinitesimal and non-infinitesimal)
on the flow. We use these results to give a sufficient condition for the
exponential stability of a periodic orbit passing through a point of multiply
intersecting events, and apply the theory in illustrative examples to
demonstrate synchronization in abstract first- and second-order phase
oscillator models.
