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

Abstract

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