Event--Selected Vector Field Discontinuities Yield Piecewise--Differentiable Flows

摘要

We study a class of discontinuous vector fields brought to our attention by multilegged 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, it is known that the vector field yields a well--defined flow that is Lipschitz continuous. We extend these results by showing this flow is piecewise--differentiable, so that 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 Poincaré maps for periodic orbits), show that 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 first-- and second--order phase oscillator models abstracted from the legged locomotion application domain that motivated our interest in this class of models.