Poincar\'{e}-Hopf theorem for hybrid systems

TL;DR

This paper extends the Poincaré-Hopf index theorem to hybrid dynamical systems, accommodating non-smooth guard sets, variable mode dimensions, and multivalued resets, broadening its applicability.

Contribution

It introduces a generalized Poincaré-Hopf theorem tailored for hybrid systems with non-smooth guards, variable dimensions, and multivalued resets, expanding theoretical understanding.

Findings

Generalized Poincaré-Hopf theorem for hybrid systems

Applicable to non-smooth guard sets and variable mode dimensions

Handles arbitrary multivalued reset maps

Abstract

A generalization of the Poincar\'{e}-Hopf index theorem applicable to hybrid dynamical systems is obtained. For the hybrid systems considered, guard sets are not assumed to be smooth; distinct "modes" are not assumed to have constant dimension; and resets are arbitrary multivalued maps (relations).