Event-Selected Vector Field Discontinuities Yield Piecewise-Differentiable Flows

TL;DR

This paper establishes conditions under which discontinuous vector fields, inspired by multi-legged animal locomotion, produce well-defined, piecewise-differentiable flows, enabling analysis of stability and impact maps in complex dynamical systems.

Contribution

It introduces a framework for analyzing flows of discontinuous vector fields with finite smooth discontinuity surfaces, including their derivatives and stability properties.

Findings

Flow is Lipschitz continuous and piecewise-differentiable under specified conditions.

Existence of piecewise-differentiable impact and Poincaré maps is demonstrated.

Sufficient conditions for exponential stability of periodic orbits are provided.

Abstract

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.