Poincar\'e-Bendixson Theorem for Hybrid Systems

TL;DR

This paper extends the Poincaré-Bendixson theorem to two-dimensional hybrid dynamical systems and introduces a method to compute the Poincaré return map derivative, aiding stability analysis.

Contribution

It provides the first Poincaré-Bendixson theorem for 2D hybrid systems and a technique to compute the return map derivative for stability assessment.

Findings

Proves a Poincaré-Bendixson theorem for 2D hybrid systems

Develops a method to compute the derivative of the Poincaré return map

Establishes a Poincaré-Bendixson theorem for certain 1D hybrid systems

Abstract

The Poincar\'e-Bendixson theorem plays an important role in the study of the qualitative behavior of dynamical systems on the plane; it describes the structure of limit sets in such systems. We prove a version of the Poincar\'e-Bendixson Theorem for two dimensional hybrid dynamical systems and describe a method for computing the derivative of the Poincar\'e return map, a useful object for the stability analysis of hybrid systems. We also prove a Poincar\'e-Bendixson Theorem for a class of one dimensional hybrid dynamical systems.